Fatigue Theory

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The nCode Book of Fat igue Theory

Durability Management Overview. .......................................................................................5Section 1. Introduction to Fatigue........................................................................................ 8

1.0 Background. ................................................................................................................82.0 The History of Fatigue. .............................................................................................103.0 High Cycle Fatigue vs Low Cycle Fatigue. .............................................................124.0 Computerised Fatigue Analysis. .............................................................................13

Section 2 - The S-N Approach ............................................................................................ 141.0 Introduction. ..............................................................................................................142.0 Stress Cycles. ............................................................................................................143.0 The S-N Curve ...........................................................................................................164.0 The Influence of Mean Stress. .................................................................................215.0 Factors Influencing Fatigue Life. .............................................................................25

Section 3 - The Local Strain Approach ..............................................................................341.0 Introduction. ..............................................................................................................342.0 The Microscopic Aspects of Fatigue Failure.......................................................... 343.0 Accounting for Plasticity. .........................................................................................364.0 Example 1. Calculating Cyclic Stress-Strain Response. .......................................505.0 Strain-Life Characteristics. ......................................................................................526.0 Example 2. Transition Life and Shot Peening. .......................................................567.0 Determination of Cyclic Fatigue Properties. ..........................................................578.0 The Effect of Mean Stress. .......................................................................................599.0 Factors Influencing Fatigue Life. .............................................................................61

Section 4 - Multiaxial Considerations ................................................................................621.0 Introduction. ..............................................................................................................622.0 The Multiaxial Stress-Strain State. ..........................................................................623.0 Equivalent Stress-Strain Approaches..................................................................... 664.0 Critical Plane Approaches........................................................................................ 72

Section 5 - The Statistical Nature of Fatigue..................................................................... 741.0 Background. ..............................................................................................................742.0 Representation of Fatigue Data on a Statistical Basis. .........................................753.0 The Statistical Distribution Function. .....................................................................754.0 Probability of Failure at a Finite Life. ...................................................................... 755.0 Probability of Failure for Infinite Life. ..................................................................... 786.0 Handling Statistics Under Random Loading Conditions. .....................................787.0 The Absolute Accuracy of Fatigue Life Estimation. ..............................................83

Section 6- Fracture Mechanics Based Fatigue Crack Growth Analysis......................... 841.0 Introduction ............................................................................................................... 842.0 Complexing Effects ...................................................................................................873.0 Background to Fatigue Crack Growth Analysis..................................................... 88

Section 7 - Fatigue Modeling And Analysis Within The FATIMAS System ................. 1051.0 Introduction ............................................................................................................. 1052.0 An Overview of Fatigue Analysis Methodologies ................................................ 1053.0 Materials Data for Fatigue Modelling ....................................................................1144.0 Overview of the Fatigue Analysis Software System ............................................1255.0 Fatigue Analysis Case Studies .............................................................................. 155

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Durabil i ty Management Overview.

Engineers are faced with many decisions during the product development process. The time between conceptual design and the finished production item has to be minimised. This is combined with the need to save weight, select optimum materials and economise on production processes whilst satisfying operational demands placed on the product. Accurate measurement, data acquisition and analysis, and testing are key factors in the process of calculating product performance. Most products must survive in a variable loading environment and the predominant failure mechanism under these conditions is fatigue. Exploiting fatigue knowledge and the use of computer based analysis techniques at an early stage in the design process can dramatically reduce the development period. The designer has the opportunity to estimate the effects of changing component shape, material and even vibration modes on durability performance. Durability prediction can be integrated with design and thus optimise performance prior to undertaking expensive durability testing.

Product performance is dictated by the loads experienced in service, the distribution of stresses and strains in the product due to these loads and the behaviour of the product materials under these conditions. Combining this information in computer models enables rapid evaluation of component durability, Figure 1. The power of this approach is the speed at which the effects on durability of changes in material, shape and loads can be assessed. Hence, expensive prototype, laboratory and service trials can be minimised and the design from the "drawing board", the modern CAD system, will be much closer to the final production version.

Figure 1

Once analytical optimisation of the component has been completed, a prototype is manufactured and tested to evaluate durability and performance, Figure 2. The tests can take many forms, ranging from operation in service or simulated service on rigs, through accelerated testing on proving grounds or

Service Loads

Stress Analysis

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Fatigue Analysis Product Life

Properties

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test machines to constant amplitude testing. The evaluation can vary from a single component, through component systems, to complete vehicles or structures. It is clear that the further removed testing becomes from direct service, the more analytical input is necessary. Testing is usually essential at some stage of product development but cost, response time and limited flexibility make it unattractive at an early stage where a range of design options require consideration.

Figure 2

To achieve these demands and remain competitive, manufacturers must respond by adopting advanced techniques for design, materials' selection, durability and performance prediction and evaluation.

The power of implementing these techniques is best realised through :

- interactive computer aided design tools to create productdesigns, analyse behaviour and assess fatigue life

- the use of databases to rapidly retrieve data such as serviceloads, material properties, component test results andproduction/manufacturing hardware performance

- integrating design, manufacture and production engineeringfrom the initial concept to final production;

- complementary product testing and evaluation to provide

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crucial feedback to refine predictive methods and establishconfidence in final product performance.

Ideally, the process represents a continuous, organised and documented evaluation of all aspects of product design/development to ensure total acceptability for all criteria from customer needs to production. Simultaneous engineering utilising advanced CAE tools provides the opportunity to undertake rapid evaluations of options and changes early in the process to avoid the need for expensive changes close to and after start up of production.

Effective integration involves multi-disciplinary teams and technology tools which are straightforward to use and encourage communication. The capability to model components and systems in the computer and analyse behaviour under simulated loading provides the means to rapidly assess a wide range of design options before an expensive prototype is built. A large percentage, some 75%, of the final product cost is defined at the design stage and this must, therefore, be the best place for cost improvements and planning of production methods. The key areas of design and evaluation expertise are not particularly novel. However, in industry to date :

- the capabilities often tend to be dispersed and difficult tointegrate

- the focus of product teams must be the product and technologyoften lags behind

- the latest software/hardware tools are often not understoodor available

- the software/hardware tools are not used to maximum benefit

- the knowledge and application of databases is limited.

It is now recognised that by overcoming these limitations and implementing the speed and power of next-generation software technology, manufacturers can exploit the promise of simultaneous engineering, and bring innovative products to market more quickly, at less cost with better quality and performance. This quality in engineering development offers a vital competitive advantage.

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Section 1. Introduction to Fatigue.

1.0 Background.

Static or quasi static loading is rarely observed in modern engineering components or structures. For this reason, designers must address themselves to the implications of repeated load, fluctuating loads, and rapidly applied loads. By far the majority of component designs involve parts subjected to fluctuating or cyclic loads. Such loading induces fluctuating or cyclic stresses that often result in failure by fatigue. Indeed, it is often said that 95% of all structural failures occur through a fatigue mechanism.LK

It is worth noting at the outset that the term fatigue, coined more than a hundred years ago, may not be the best choice of terminology, since many aspects of the phenomenon are distinctly different from the biological counterpart. For example, it is next to impossible to detect any progressive changes in material behaviour during the fatigue process and, therefore, failures often occur without warning. Also, periods of rest, with the fatigue stress removed, do not lead to any measurable healing or recovery. Thus the damage done during the fatigue process is cumulative, and generally unrecoverable. From this stand-point, the German term Betriebsfestikeit, (operational strength) is a better descriptor of the phenomenon. However, since Betriebsfestikeit, has 17 characters and fatigue only 7 we shall continue to use the term fatigue!

Fatigue, although a complex subject, has not been neglected by the research community. Estimates indicate that if one wished to keep up with the literature by reading a paper each working day, one would fall behind by more than a year for each year of reading. Furthermore, attempting to catch up with the backlog would be virtually impossible. Yet the designer or test analyst is increasingly challenged by the demands of higher performance, lower weight, and longer life, and all this at a reasonable cost and in as short a time as possible ! These apparently conflicting demands can only be overcome through a consideration of the problems associated with fatigue resistant designs. Up until recently, these problems were summarised as :

- life calculations are usually less accurate then strengthcalculations. Order of magnitude errors in life estimates arenot unusual

- fatigue properties cannot be accurately deduced from othermechanical properties, they need to be measured directly

- full-scale prototype testing is usually necessary to assurean acceptable life

- laboratory results of different but otherwise "identical" tests may differ widely, requiring statistical interpretation

- material and designs must often be selected to provide slowcrack growth and, if possible, detection of cracks beforethey become dangerous.

- "fail-safe" design concepts must often be implemented in orderto achieve acceptable reliability. That is, even if astructural element fails, the structure must remain intactand remain able to support the loads in the short term.

Modern advances, have to some extent, mitigated these problems. For example, these days it is

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usual to consider life estimates, either calculated or measured, to be within a factor of two rather then ten. Furthermore, computerised analysis of thousands of laboratory data sets do point to acceptable empirical correlations between monotonic tensile data and fatigue parameters.

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2.0 The History of Fatigue.

For centuries it has been known that wood or metal can be made to break by repeatedly bending it back and forth with a large amplitude. However, it came as something of a surprise, not to say shock, when it was discovered that repeated stressing can produce fracture even when the stress amplitude is apparently well within the elastic range of the material. The first fatigue investigations seem to have been reported by a German mining engineer, W. A. S. Albert, who in 1829 performed some repeated loading tests on iron chain. Some of the earliest fatigue failures in service occurred in the axles of stage coaches. When railway systems began to develop rapidly in the middle of the nineteenth century, fatigue failures of railway axles became a widespread problem that began to draw attention to cyclic loading effects. This was the first time that many similar components had been subjected to millions of cycles at stress levels well below the monotonic tensile yield stress. As is often the case with unexplained service failures, attempts were made to reproduce the failures in the laboratory. Between 1852 and 1870 the German railway engineer August Wöhler set up and conducted the first systematic fatigue investigation; from this point of view he may be regarded as the grandfather of modern fatigue thinking. He conducted tests on full-scale railway axles and also on small-scale bending, torsion and axial cyclic loading for several different materials. Some of Wöhler's data, shown in Figure 3, are for Krupp axle steel and are plotted, in terms of nominal stress vs cycles to failure, on what has become very well known as the S-N diagram. Each curve on such a diagram is still referred to as a Wöhler line.

Figure 3 S-N data reported by Wöhler.Note, 1 centner = 50 kg, 1 zoll = 1 inch, 1 centner / zoll2 ~ 0.75 MPa

At about the same time, other engineers began to concern themselves with problems of failures associated with fluctuating loads in bridges, marine equipment and power generation machines. By

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1900 over 80 papers had been published on the subject of fatigue failures. During the first part of the twentieth century more effort was placed on understanding the mechanisms of the fatigue process rather than just observing its results. This activity finally led, in the late fifties and early sixties, to the development of the two approaches, one based on linear elastic fracture mechanics, LEFM, to explain how cracks propagate, and the so-called Coffin-Manson local strain methodology to explain crack initiation. Most recently, Miller and his colleagues at Sheffield University have been working on ways of finding a unified theory of metal fatigue, based on crack growth on a microscopic, macroscopic and structural level.

From this vast wealth of knowledge one thing becomes clear, modern designers and engineers will not create more fatigue resistant components by indulging in more experimentation, although the need for more research is ever present. From a practical point of view, a more profitable approach is the implementation and efficient use of the knowledge which is available today !

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3.0 High Cycle Fatigue vs Low Cycle Fatigue.

Over the years, fatigue failure investigations have led to the observation that the fatigue process actually embraces two domains of cyclic stressing or straining that are distinctly different in character, and in each of which failure occurs by apparently different physical mechanisms. One domain of cyclic loading is that for which significant plastic strain occurs during at least some of the loading cycles. This domain involves some large cycles, relatively short lives and is usually referred to as low-cycle fatigue. The other domain of cyclic loading is that for which the stress or strain cycles are largely confined to the elastic range. This domain is associated with low loads and long lives and is commonly referred to as high-cycle fatigue. Low cycle fatigue is typically associated with fatigue lives between about 10 to 100,000 cycles and high cycle fatigue to lives greater than 100,000 cycles.

Later, it will be explained how to distinguish more precisely between these two domains, for now suffice to say that fatigue solutions, that is remedies for extending fatigue life, are different in each domain. In the high cycle domain, measures such as shot-peening (Kugelschuss in German) or other surface hardening treatments, or the use of higher tensile materials are beneficial; whereas in the low cycle domain, where ductility and resistance to plastic flow are important, they are inappropriate.

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4.0 Computerised Fatigue Analysis.

From the above discussion it should be clear by now that, prior to contemplating a fatigue analysis, several pieces of information must be to hand. Firstly, a description of the cyclic loading environment, secondly, a characterisation of the geometry of the component in question and lastly, details of the cyclic properties of the material from which the component is to be, or was, manufactured. Figure 4 provides a simple block diagram of the process.

Figure 4 Inputs Required for a Fatigue Analysis

A main objective is to visit each of the three main areas of input, which are: loading environment, geometry, and materials' data, and in turn explore their contents so that we are better able to understand the process as a whole. In the light of the need to compete, produce better products more efficiently, more quickly and at lower cost, the challenge is how best to create and implement a durability process which will allow these requirements to be fulfilled.

Loading

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Computer Analysis Life

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Section 2 - The S-N Approach

1.0 Introduction.

It has been recognised since 1830 that a metal subjected to a repetitive or fluctuating load will fail at a stress level lower than that required to cause fracture on a single application of the load. The nominal stress method was the first approach developed to try to understand this failure process and is still widely used in applications where the applied stress is nominally within the elastic range of the material and the number of cycles to failure is large. From this point of view, the nominal stress approach, is best suited to that area of the fatigue process known as high cycle fatigue. The nominal stress method does not work well in the low cycle region where the applied strains have a significant plastic component. In this region a strain based methodology must be used.

2.0 Stress Cycles.

Before looking in more detail at the nominal stress procedure it is worth considering the general or typical types of cyclic stresses which contribute to the fatigue process, such as those below. ..

Figure 5 Typical fatigue stress cycles, (a) fully reversed (b) offset, (c) random.

Figure 5(a) illustrates a fully reversed stress cycle with a sinusoidal form. This is an idealised loading condition typical of that found in rotating shafts operating at constant speed without overloads. For this kind of stress cycle, the maximum and minimum stresses are of equal magnitude but opposite sign. Usually tensile stress is considered to be positive and compressive stress negative. Figure 5(b) illustrates the more general situation where the maximum and minimum stresses are not equal, in this case they are both tensile, and so define an offset for the cyclic loading. Figure 5(c) illustrates a more complex, random loading pattern which is more representative of the cyclic stresses found in real structures.

From the above it is clear that a fluctuating stress cycle can be considered to be made up of two components, a static or steady state stress Sm, and an alternating or variable stress amplitude Sa. It

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is also often necessary to consider the stress range,Sr, which is the algebraic difference between the maximum and minimum stress in a cycle.

sr = smax - smin

The stress amplitude,sa, then is one half the stress range.

sa = sr / 2 = (smax - smin ) / 2

The mean stress, is the algebraic mean of the maximum and minimum stress in the cycle.

sm = (smax + smin ) / 2

Two ratios are often defined for the representation of mean stress, the stress or R ratio, and the amplitude ratio A.

R = smin / smax

A = sa / sm = (1-R) / (1+R)

The following table illustrates some R values for common loading conditions.

R ratio Loading Condition

R > 1 Both Smax and Smin are negative. Negative mean stress.

R = 1 Static loading.

0 < R < 1 Both Smax and Smin are positive. Positive mean stress, |Smax| > |Smin|.

R = 0 Zero to tension loading, Smin = 0

R = -1 Fully reversed loading,|Smax| = |Smin| zero mean stress.

R < 0 |Smax| < |Smin| , Smax approaching zero.

R infinite Smax equal to zero.

Table 1: R ratio for some common loading conditions.

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3.0 The S-N Curve

Between 1852 and 1870 the German railway engineer August Wöhler set up and conducted the first systematic fatigue investigation. Wöhler conducted cyclic tests on full-scale railway axles and also on small-scale bending, torsion and push-pull specimens of several different materials. Some of Wöhler's data, shown in Figure 3, are for Krupp axle steel and are plotted, in terms of nominal stress vs cycles to failure, on what has become known as the S-N diagram. Typically, the S-N relationship is determined for a specific value of Sm, R or A. Note that in dealing with the nominal stress approach, the convention is that nominal stress is usually referred to as S and localised stress by the Greek counterparts eg. σ.

3.1 Procedure for determining the S-N Curve

Most determinations of fatigue properties have been made in completely reversed bending,i.e. R = -1, by means of the so-called rotating bend test. One example is the R. R. Moore test, which uses four point loading to apply a constant moment to a rotating (1750 rpm) cylindrical hour-glass-shaped specimen. Specimens, which are typically between 6 to 8 mm in diameter in the test section, are usually polished to a mirror finish prior to testing.

Figure 6 The R.R Moore fatigue testing machine.

The stress level at the surface of the specimen is calculated using the elastic beam equation, even if the resulting value exceeds the yield strength of the material.

S = Mc / I

where :

S is the nominal stress acting normal to the cross-sectionM is the bending momentc is the distance of the surface from the neutral axisI is the moment of inertia

For the circular section of the R. R. Moore specimen the beam equation reduces to :

S = 32 M / p d3

bearings bearingsSpecimen

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where :

d is the diameter of the specimen.

The usual laboratory procedure for determining an S-N curve is to test the first specimen at a high stress, about two thirds of the static tensile stress of the material, where failure is expected in a fairly small number of cycles. The test stress is decreased for each succeeding specimen until one or two specimens do not fail before at least 107 cycles. For materials which exhibit it, the highest stress at which no failure occurs, a runout, is taken to be the fatigue limit. For situations where an infinite life design requires a probability of survival to be associated with it, more complex testing and analysis procedures, such as the Probit and staircase methods, have been developed to determine the mean and variance of the fatigue limit . For materials which do not exhibit a fatigue limit, tests are usually terminated between 107 and 108 cycles, and the concept of an endurance limit at either 107 or 108 cycles defined.

The S-N curve is usually determined through the use of about 15 specimens. However, it is generally found that results are accompanied by a large amount of scatter and some form of statistical analysis should be applied, see the section on the Statistical Nature of Fatigue for more details.

S-N data is nearly always presented in the form of a log-log plot of alternating stress, amplitude Sa or range Sr, versus cycles to failure, with the actual Wöhler line representing the mean of the data. Certain materials, eg. steels, display a fatigue limit, Se, which represents an alternating stress level below which the material has an infinite life. For most engineering purposes, infinite is taken to be 1 million cycles. Great care must be exercised when designing on the basis of a fatigue limit, since it has a nasty habit of disappearing due to periodic overloads, corrosion, and elevated temperature.

Figure 7 Idealised form of the S-N curve.

When plotted on log-log scales, the relationship between alternating stress, S, and number of cycles to failure, N can be described by a straight line, Figure 7. The slope of the line, b, (after Basquin, a prominent worker who first proposed the law) can be derived from the following:

b = - (logS - logSo) / (logNo - logN)

logNo - logN = -1/b log(S/So)

logN = logNo+ 1/b log(S/So)

N No

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N = No (S/So)1/b

Sometimes, for convenience, the term 1/b is replaced by the letter k,

N = No (S/So)k

The above equation says that if we know the Basquin slope, b, and any other co-ordinate pair, (No,So) then for a given stress amplitude S, the number of cycles can be calculated directly. Typically No is taken to be 106 cycles and the corresponding stress amplitude is taken to be an endurance limit, usually denoted as Se or S6, so that the above equation may be rewritten as :

N = (S/Se)k x 106

3.2 Example 1. A Simple Life Estimation.

For a material with an endurance limit of 250 MPa, and a Basquin slope, b, of -0.1, calculate number of cycles to failure at a stress amplitude of 300 MPa. Under these conditions,

N = (300 / 250)-10 x 106 = 161,000 cycles

3.3 Limits of the S-N Curve.

As mentioned above, the S-N approach is applicable to situations where cyclic loading is essentially elastic.

Figure 8 Typical S-N curves for ferrous and non-ferrous metals.

This means that the S-N curve should be confined on the life axis to numbers greater than about 10,000 cycles in order to ensure no significant plasticity is occurring.

Indeed great care must be taken in using the above S-N equations in situations where lives less than 10,000 cycles are being estimated. Figure 8 shows typical S-N curves for both ferrous and non-ferrous metals. The points to note in Figure 8 are the limits of the logN axis, the presence of a fatigue limit for the mild steel and the absence of a fatigue limit for the aluminium alloy. Because both materials represented in Figure 8 have relatively low yield stresses, the life axis is confined to begin at 105 cycles at which point the alternating stress is about 350 and 300 MPa respectively for the two alloys.

3.4 Tensile Properties and the S-N Curve.

Through many years of experience, particularly with steels, empirical relationships between fatigue and tensile properties have been developed. These relationships, are not soundly based in science,

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however, they remain useful tools for engineers for assessing fatigue performance. When the S-N curves for a number of different steels of varying strengths are plotted as the ratio of endurance limit, i.e., the stress amplitude at 106 cycles, S6, to ultimate tensile strength, Su, all the curves tend to all fall onto a single curve which implies that :

S6 = Se ~ 0.5 Su for Su < 1400 MPa

and

S6 = Se ~ 700 MPa for Su > 1400 MPa

Figure 9 Generalised S-N curve for wrought steels.

In addition to this, the stress at 103 cycles,S3, can be approximated by 0.9 Su and so, utilising these approximations, a generalised S-N curve, can be generated for wrought steels, see Figure 9.

Methods of representing the S-N curve in the range 1 to 103 cycles have been developed but they must be treated with extreme caution. They usually use some percentage of the ultimate strength, Su, or true fracture stress, sf, as a measure of the stress amplitude at either 1 or 1/4 cycles. The main difficulty with employing this approach is that the deduced S-N curves are extremely flat in the low cycle region, and this makes estimates of life particularly inaccurate. The reason for this apparent flatness is the large plastic strain which results from the high load levels. Low cycle fatigue analysis is best treated by a strain based procedures which account for, rather than ignore, the effects of plasticity.

3.5 Bastenaire model

Since the work done by Wöhler, who first established a relationship between applied stress and number of cycles to failure, given the well known S-N curve, several models have been proposed to describe this curve.

Bastenaire is one of the most general formulations and was proposed in 1974, based on the analysis of thousands of tests made on steel specimens.

The relation between the life and the applied stress is given by the equation :

N = A/(S-E) * exp[-((S-E)/B)^C]

Where :

N is the number of cycles to failure

S is the applied stress

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E is the endurance limit of the material

A,B,C are material parameters

Statistical methods were proposed by Bastenaire to calculate the parameters of the model from raw test data and these methods have been implemented in ESOPE* software.

Bastenaire curves can also be modified to calculate lives at certainties of survival other than 50% based on the assumption that the stresses are normally distributed for a specified life. This is basically achieved by shifting the mean curve parallel to the stress axis based on the calculated standard deviation and the chosen probability:

Np% = A/(S±m*s-E) * exp[-((S±m*s-E)/B)^C]

The figure below gives an example of Bastenaire curve fitted to real data.

Figure 10 Bastenaire Curve fitted to real data

*ESOPE is a software developed by ARCELOR and distributed by nCode International.

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4.0 The Influence of Mean Stress.

As mentioned above, most basic fatigue data are collected in the laboratory by means of testing procedures which employ fully reversed loading, i.e. R = -1. However, most realistic service situations involve non zero mean stresses. It is, therefore, very important to know the influence that mean stress has on the fatigue process so that the fully reversed laboratory data can be usefully employed in the assessment of real situations.

Figure 11 High cycle fatigue data showing the influence of mean stress.

Fatigue data collected from a series of tests designed to investigate different combinations of stress amplitude and mean stress are characterised in Figure 11 above for a given number of cycles to failure. The diagram plots the mean stress, both tensile and compressive, along the x-axis and the alternating constant stress amplitude along the y-axis. This kind of representation was first proposed by Haigh and is, therefore, commonly referred to as the Haigh diagram.

The stress amplitude at zero mean stress, Sn, corresponds to the stress amplitude at N cycles to failure as measured by the fully reversed fatigue test. The failure data points tend to follow a curve which if extrapolated would pass through the ultimate tensile strength, Su, on the mean stress axis. Notice that the influence of mean stress is different for compressive and tensile mean stress values. Failure appears to be more sensitive to tensile mean stress, than compressive mean stress. When available, data of the type illustrated above are collated into what are commonly referred to as master diagrams for a particular material. Figure 12 illustrates the master diagram for SAE 4340 from the US Department of Defence MIL Handbook-5.

σ N

-σ N (Compressive mean) O (Tensile mean) σ Nσ m

N=constant for all points

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Figure 12 Master diagram for SAE 4340.

Since the tests required to generate a Haigh or master diagram are quite expensive, several empirical relationships which relate alternating stress amplitude to mean stress have been developed. These relationships characterise a material through its ultimate tensile strength, Su, and so are very convenient. For infinite life design strategies, the methods use various curves to connect the endurance limit, Se, on the alternating stress axis to either the yield stress, Sy, ultimate strength, Su, or true fracture stress, sf, on the mean stress axis. Of all the proposed relationships two have been most widely accepted, i.e. those of Goodman and Gerber.

Goodman :(Sa / Se) + (Sm / Su) = 1

Gerber :(Sa / Se) + (Sm / Su)2 = 1

Experience has shown that actual test data tend to fall between the Goodman and Gerber curves, (Goodman joining Se to Su by means of straight line and Gerber by means of a parabola). For most design situations where R < 1, i.e. small mean stress in relation to the alternating stress, there is little difference between the two relationships. However, when R approaches 1, i.e. nearly equal mean and alternating stresses, the two relationships show considerable differences. Unfortunately, little or no experimental data exist to support one approach over the other and so typically, the recommendation would be to select the approach which provides the most conservative lives in a given situation.

4.1 Example 2. Correcting for Mean Stress Effects.

A component is subjected to a maximum cyclic stress of 750 MPa and a minimum of 70 MPa. The steel from which it is manufactured has an ultimate tensile strength, Su, of 1050 MPa and a measured endurance limit, S6, of 400 MPa. The fully reversed stress at 1000 cycles is 750 MPa. Using both the Goodman and Gerber mean stress correction procedures, calculate the component life.

The first step is to calculate the stress amplitude, Sa and the mean stress, Sm

Sa = (Smax - Smin) / 2 = (750 - 70) / 2 = 340 MPa

Sm = (Smax + Smin) / 2 = (750 + 70) / 2 = 410 MPa

A Haigh diagram, for the Goodman correction procedure can now be constructed for constant lives of 106 and 103 cycles. This is done by connecting the endurance limit, S6, and the stress at 1000 cycles, S3, respectively on the alternating stress axis with the ultimate tensile strength, Su, on the

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AISI 4340Su= 158ksi, Sf=147ksiat 2000 cpm Un-notched notchedKf=3.3, p=0.010

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mean stress axis, see Figure 13. The stress conditions on the component calculated above, Sa = 340 and Sm = 410, can be plotted on the diagram and a line drawn from Su to the alternating stress axis. This line represents the constant life line for the component at all combinations of stress amplitude and mean stress. The line intersects the fully reversed axis at a stress Sn = 558 MPa.

Figure 13 The Haigh Diagram.

It should be noted that this stress can also be calculated directly from the Goodman equation:

(Sa / Sn) + (Sm / Su) = 1

(340 / Sn) + (410 / 1050) = 1

Sn = 557.8 MPa.

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It will be recalled that the S-N curve is given by

N = No (S/So)1/b

and so for the conditions defined at 103 and 106 cycles,

b = - (logS - logSo) / (logNo - logN)

b = - 1/3 log(S3 / S6)

and so b = - 1/3 log(750 / 400) = -0.091

and for Sn = 557.8, the life can be calculated from:

N = No (S/So)1/b

N = (558 / 400)-11 x 106

N = 26,000 cycles

The Gerber correction can be used in a similar way, i.e.

(Sa / Sn) + (Sm / Su)2 = 1

(340 / Sn) + (410 / 1050)2 = 1

Sn = 401.2 MPa.

and N = (401 / 400)-11 x 106

N = 973,000 cycles

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5.0 Factors Influencing Fatigue Life.

A standardised rotating bend test such as the R.R. Moore test is used to determine a base-line S-N relationship for a polished specimen of approximately, 6 mm diameter loaded under conditions of fully reversed bending. If the fatigue or endurance limit measured by these means be denoted by S'e then the actual limit for a real component, Se, must reflect all the modifications that come about in moving from a laboratory specimen to a component. For steels in particular, several empirical relationships have been developed which can account for the variation in Se as a result of the following:

- Component size- The type of loading- The effect of notches- The effect of surface finish- The effect of surface treatment.

The usual way to account for these effects is through the calculation and application of specific modifying factors so that

Se = S'e Cnotch Csize Cload Csur . . .

where reciprocal of the product, Cnotch Csize Cload Csur, is collectively known as the fatigue strength reduction factor Kf, i.e.,

Kf = 1 / (Cnotch Csize Cload Csur . . )

The approach tends to be conservative and the corrections are usually only applied at the endurance limit, the modifications required for the rest of the S-N curve are ill-defined. Typically the procedure is to pivot the S-N curve about the 1000 cycles point, see Figure 14.

Figure 14 Modification of the S-N curve.

It is very important to remember that all the modification factors are empirical, conservative and mostly only applicable to steels. They provide little or no fundamental insight into the fatigue process itself other than providing approximate trends. In particular they should not be used in areas outside

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their measured applicability.

5.1 The influence of component size.

Fatigue in metals results from the nucleation and subsequent growth of crack-like flaws under the influence of an alternating stress field. This view leads to the concept of failure commencing from the weakest link, the most favorably orientated metal crystal for example, and then growth through less favorably orientated grains until final failure. Intuitively, it would seem reasonable to suppose that the larger the volume of material subjected to the alternating stress, the higher the probability of finding the weakest link sooner. Actual test data do confirm the presence of a size effect particularly in the case of bending and torsion.

The stress gradient built up through the section, in bending and to a lesser extent in torsion, concentrates more than 95% of the maximum surface stress to a thin layer of surface material. In large sections, this stress gradient will be less steep than in smaller ones, and so the volume of material available which could contain a critical flaw will be greater leading to reduced fatigue strength. The effect is quite small for axial tension where the stress gradient is absent. The value for Csize can be estimated from one of the following,

if the diameter of the shaft is < 8 mm:

Csize = 1

if the diameter is between 8 mm and 250 mm:

Csize = 1.189 d-0.097

The effect of size is particularly important for the analysis of rotating shafts such as might be found in vehicle powertrains.

For situations where components do not have a round cross section, an equivalent diameter, deq, can be calculated for a rectangular section width, w and thickness, t, undergoing bending from:

deq2 = 0.65 w t

5.2 The influence of loading type.

Fatigue data measured according to one regime, axial tension for example, may be "corrected" to represent the data that would have been obtained had the test been carried out in some other loading methodology such as torsion or bending. Recall that the R.R Moore test calls for tests to be carried out under conditions of fully reversed bending.

The values of Cload to be used in conjunction with the endurance limit, Se, in moving from one loading condition to another are detailed below :

Measured Target CloadLoading Loading

Axial to Bending 1.25Axial to Torsion 0.725Bending to Torsion 0.58Bending to Axial 0.8Torsion to Axial 1.38Torsion to Bending 1.72

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Table 2: Modification factor at 106 cycles for various loadings.

In addition to influencing the endurance limit, loading conditions can also influence the Basquin slope, b. This effect is usually taken into account by modification of the stress at 103, S3, as well as Se. The following factors can be used to define C'load, the S3 modification factor,

Measured Target C'loadLoading Loading

Axial to Torsion 0.82Bending to Torsion 0.82Torsion to Axial 1.22Torsion to Bending 1.22

Table 3: Modification factor at 103 cycles for various loadings.

5.3 The influence of surface finish.

A very high proportion of all fatigue failures nucleate at the surface of components and so surface conditions become an extremely important factor influencing fatigue strength. The usual standard by which various surface conditions are judged is against the polished laboratory specimen. Normally, scratches, pits, machining marks influence fatigue strength by providing additional stress raisers which aid the process of crack nucleation. Broadly speaking, high strength steels are more adversely affected by a rough surface finish than softer steels, for this reason the surface correction factor, Csur, is strongly related to tensile strength. The surface finish correction factor is often presented on diagrams that categorise finish by means of qualitative terms such as polished, machined or forged, see Figure 15.

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Figure 15 Schematic surface finish correction factor for steel components (actual values are hard coded into nSoft).

Figure 16 Schematic diagram showing the effect of surface roughness on surface finish factor (actual values are hard coded into nSoft).

It is worth noting that some of the curves presented in Figure 15 include effects other than just surface

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finish. For example, the forged and hot rolled curves include the effect of decarburisation.

Other diagrams present the surface factor in a more quantitative way by using a quantitative measure of surface roughness such as RA, the root mean square, or AA the arithmetic average, see Figure 16. Values of surface roughness associated with each of the manufacturing processes are readily available in handbooks, as an example consider the following :

Type of finish Surface roughness(microns)

Lathe-formed 2.67Partly hand polished 0.15Hand Polished 0.13Ground 0.18Superfinished 0.18Ground and polished 0.05

Table 4: Values of surface roughness for various processes.

5.4 The qualitative influence of surface treatment.

As in the case of surface finish, surface treatment can have a profound influence on fatigue strength, particularly the endurance limit. Surface treatments can be divided broadly into mechanical, thermal and plating processes. The important point to note with all three is that the net effect of the treatment is to alter the state of residual stress at the free surface, in the first two processes by providing a compressive layer and in the case of plating by providing a tensile residual stress.

Figure 17 Residual stress in a beam.

Residual stresses arise when plastic deformation is not uniformly distributed throughout the entire cross-section of the component being deformed. Figure 17 represents a metal bar whose surface has been deformed in tension by bending so that part of it has undergone plastic deformation. When the external force is removed, the regions which have been deformed plastically prevent the adjacent elastic regions from complete elastic recovery to the unstrained condition. In this way the elastically

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deformed regions are left in residual tension, and the plastically deformed regions must be in a state of residual compression. For many purposes residual stress can be considered identical to the stresses produced by an external force. Thus the presence of a compressive residual stress at the surface of a component will have the effect of decreasing the likelihood of fatigue failure.

Figure 18 Superposition of applied and residual stresses.

Figure 18 illustrates the effect schematically. Figure 18(a) shows an elastic stress distribution in a beam with no residual stress. The typical residual stress distribution associated with shot peening is detailed in Figure 18(b). Note that the compressive stress at the surface must be compensated by an equivalent tensile stress over the interior of the cross section. In Figure 18(c) the distribution due the algebraic summation of the residual and applied stresses is shown. Observe that the maximum tensile stress at the surface has been reduced by the amount of the residual stress. Furthermore, note that the peak tensile stress has now been moved to the interior of the beam. The magnitude of this stress will depend on the gradient of the applied stress and the residual stress distribution. Also note that under these conditions, sub-surface crack initiation becomes a possibility.

5.4.1 Mechanical treatments.

The main commercial methods for introducing residual compressive stresses are cold rolling and shot peening. Although some alteration in the strength of the material occurs as a result of work hardening, the improvement in fatigue strength is due mainly to the compressive surface stress. Surface rolling is particularly suited to large parts and is frequently used in critical components such as crankshafts and the bearing surface of railway axles. Bolts with rolled threads typically possess twice the fatigue strength of conventionally machined threads.

Shot peening, which consists of firing fine steel or cast iron shot against the surface of a component, is particularly well suited to processing small mass produced parts.

It is important to remember that cold rolling and shot peening have their greatest effect at long lives. At short lives they have little or no effect.

As with other modifying factors, the effect of these mechanically induced compressive stresses can

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be accounted for by the use of correction factors which can be used to adjust the endurance limit Se. Typically the factor associated with peening is about 1.5 - 2.0

5.4.2 Plating.

Chrome and nickel plating of steel components can more than halve the endurance limit due to the creation of tensile residual stresses at the surface. These tensile stresses are a direct result of the plating process itself. As in the case of mechanically induced surface stresses, the effect of plating is most pronounced at the long life end of the spectrum and also with higher strength materials.

The deleterious effects of plating can be reduced by introducing a compressive residual stress prior to the plating process by either shot peening or nitriding. An alternative approach might be to anneal components after plating and thereby relieve the tensions.

5.4.3 Thermal Treatments.

Thermal treatments are processes which rely on the diffusion of either carbon, carburising, or nitrogen, nitriding, onto and into the surface of a steel component. Both species of atoms are interstitial, i.e. they occupy the spaces between adjacent iron atoms, and thereby both increase the strength of the steel and through volumetric changes, cause a compressive residual stress to be left on the surface. Carburising is commonly carried out by packing the steel components within boxes which contain carbonaceous solids, sealing to exclude the atmosphere and heating to about 900 degrees centigrade for a period of time which depends on the depth of the case required. Alternatively components may be heated in a furnace the presence of a hot carburising gas such as natural gas. This process has the advantage that it is quicker and more accurate. In addition, the carburising cycle may be followed up by a diffusion cycle, with no carburising agent present, which allows some of the carbon atoms to diffuse further into the component and so reduce gradients.

The nitriding process is very similar in nature to gas carburising except that, in this case, ammonia gas is used and the components are soaked at lower temperatures. Typically 48 hours at about 550 degrees centigrade will provide a nitrided case depth of about 0.5 mm. Nitriding is particularly suited to the treatment of finished notched components such as gears and slotted shafts, the effectiveness of the process is illustrated in Table 5:.

Endurance limit (MPa)

Geometry not nitrided nitrided

Un-notched 310 620Semi circular notch 175 600V notch 175 550

Table 5: The effect of nitriding on endurance limit.

5.5 The quantitative effect of surface treatments on the endurance limit (steels)

The effect of surface treatment depends on the surface finish. The increase in endurance limit with stress due to the surface treatment is given below.:

FINISH SHOT PEENED COLD ROLLED NITRIDED

Polished +15% +50% +100%

Ground +20% +0% +100%

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Table 6: Matrix of treatments and finishes on the endurance limit for steels

Whatever correction was made by the surface finish, then applying a surface treatment will have a subsequent effect taken from the table above. For example if machining reduces the endurance limit by 50% , and it is wished to recover the loss, then from the table it can be seen that cold rolling will increase the limit by +70%.

5.6 Example 3. To Shot Peen or Not ?

From a particular vehicle, as heat treated low alloy leaf springs, manufactured to a tensile strength of 1500 MPa, are failing in service, albeit after some considerable time, but nevertheless within the lifetime of the vehicle. Service measurement has shown that the loading is of the type from zero to a maximum, stress of about 770 MPa, i.e. R = 0. The question to be answered is, will shot peening the springs prior to installation make the problem go away ?

The springs have a section measuring 40 mm by 5 mm. From tables of surface roughness Csur has been determined to be about 0.75 for the as heat treated condition and about 0.58 for the shot peened condition. Measurements have shown that shot peening introduces a compressive residual stress of about 550 MPa.

From the ultimate strength, Su = 1500 MPa, S'e can be estimated to be 750 MPa.

For the as heat treated condition the modification factor required to correct for the section size of the springs can be estimated by first calculating the equivalent diameter, deq, associated with the rectangular cross section from :

deq2 = 0.65 w t = 0.65 x 40 x 5 = 130

deq = 11.4 mm

and since deq > 8 mm, Csize can now be computed from:

Csize = 1.189 x 11.4-0.097 = 0.94

Since, in operation, the leaf springs are loaded only in bending, CL is equal to 1.0.

We are now in a position to calculate the modified magnitude of endurance limit,Se.

Se = S'e x the product of all the modifying factors

Se = 750 x 0.94 x 1.0 x 0.75

Se = 529 MPa

For the loading condition defined by R = 0, the amplitude of the alternating stress, Sa, and the mean stress, Sm, must be equal. By setting Sa = Sm, the allowable stress level, S, for the as heat treated material can now be calculated from the Goodman equation.

(Sa / Se) + (Sm / Su) = 1

Machined +30% +70% +100%

Hot Rolled +40% +0% +100%

Cast +40% +0% +100%

Forged +100% +0% +100%

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(S / 529) + (S / 1500) = 1

S = 391 MPa

and the maximum allowable stress for infinite life would be

Smax = Sa + Sm = 2 S = 782 MPa

This figure is uncomfortably close to the operational maximum stress of 770 MPa and so clearly we do not have an infinite life design, we must expect to have some failures in service.

The procedure so far has been useful since it has helped us verify that the calculation procedure being used and provided some confidence in the methodology. We are now in a position the estimate the effect of shot peening.

In the shot peened condition, the modified magnitude of endurance limit,Se is given by :

Se = S'e x the product of all the modifying factors

Se = 750 x 0.94 x 1.0 x 0.58

Se = 409 MPa

As in the as heat treated case, for R = 0, the amplitude of the alternating stress, Sa, and the mean stress, Sm, are equal and the Goodman equation can be used to calculate the allowable stress. By subtracting the compressive residual stress from the mean stress the peening operation is taken into account,

(Sa / Se) + (Sm / Su) = 1

(S / 409) + (S - 550 / 1500) = 1

S = 439 MPa

and the maximum allowable stress for infinite life would be

Smax = Sa + Sm = 2 S = 878 MPa

The calculated maximum working stress for shot peened springs is significantly higher than both the equivalent as heat treated stress and the operational stress. From this we could justifiably conclude that shot peening to the extent which produces residual compression of 550 MPa would be sufficient to provide the leaf spring with an infinite service life.

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Section 3 - The Local Strain Approach

1.0 Introduction.

The nominal stress approach has been used extensively in the study of premature failures of components subjected to fluctuating loads. Traditionally, the magnitude of the observed cyclic stresses were observed to be less than the tensile elastic limit and the lives long, i.e. greater than about 105 cycles. This pattern of behaviour has classically been referred to as high-cycle fatigue.

As duty cycles have became more severe and components more complicated, another pattern of fatigue behaviour has emerged. In this regime, the cyclic loads are relatively large and have significant amounts of plastic deformation associated with them together with relatively short lives. This type of behaviour has been commonly referred to as low-cycle fatigue or more recently strain-controlled fatigue. The transition from low-cycle to high-cycle fatigue behaviour generally occurs in the range 104 to 105 cycles. A more precise definition of this transition will be developed later.

The analytical procedure evolved to deal with strain-controlled fatigue is called the strain-life, local stress-strain or critical location approach.

2.0 The Microscopic Aspects of Fatigue Failure.

It is almost universally agreed that fatigue failures start at the surface of a fatigue specimen or component. This is true whether the test is made in a rotating-beam machine where the maximum stress is always at the surface, or in a push-pull machine which gives a simple tensile-compressive stress cycle. Furthermore, fatigue failures start at small microscopic cracks and accordingly are very sensitive to even minute stress raisers. It is quite apparent from these considerations, that a fatigue specimen will give results which are representative of the metal tested only if its surface is free of defects. Tool or grinding marks left on the surface make the formation of fatigue crack easier and may result in low apparent values of fatigue limit or strength.

Early research into the mechanisms of fatigue clearly demonstrated that the failure process is linked to reversed plastic flow, i.e. the forward and backward motions of dislocations* along the slip planes of metallic crystals. Under cyclic loading, the direction of the strain is reversed over and over again, the slip lines that appear on the surface reflect this fact.

When the strain is monotonic, the slip steps that appear on a crystal surface have a relatively simple topology, see Figure 19(a). On the other hand, under cyclic loading the slip bands tend to group into packets or striations. The surface topology of these striations is more complex and is indicated schematically in Figure 19(b). Note that both ridges and crevices tend to be formed. There is good evidence that the crevices are closely associated with the initiation of cracks. Whether or not crevices are formed in a particular specimen or a specific grain of a particular specimen is largely a function of the crystal orientation.

* Note: Dislocations are imperfections in the crystal lattice of a metal and are responsible for nearly all aspects of plastic deformation.

If the shear direction in the striation is nearly normal to the surface, crevice formation will be favoured. However, if the slip direction is parallel to the surface crevices are not normally formed. It is possible, even in this case for damage to occur inside a striation since pores or holes can also open up inside the slip-band packet.

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A further consequence of these repeated dislocation movements is that small localised deformations called extrusions may occur in the slip bands. An extrusion is a small ribbon of material which is apparently extruded from the surface of a slip band. The inverse of extrusions, which are narrow crevices called intrusions have also been observed, Figure 19(b). Typically these surface disturbances are approximately 1 to 10 microns in height and constitute embryonic cracks.

Figure 19 Surface contours where slip bands intersect a surface.

Another way of describing the above observations is that, without some plastic deformation there can be no fatigue damage ! Given the this assertion , how could it be that the nominal stress approach, which specifically overlooks plasticity, could have been so successfully applied for so many years ?

The reason is that, in the high-cycle regime loading levels are low, and so stress and strain are almost linearly related. Under these circumstances, load and strain-controlled cycling are equivalent. Consequently, in the area of the endurance limit, load-controlled rotating beam fatigue data based on notionally elastic nominal stresses can be used to adequately describe fatigue behaviour. However, as the load levels become larger the method breaks down and a more generalised, strain-based approach, which accounts for plasticity, must be adopted. Indeed it should be clear by now that the nominal stress approach is in reality a special case of strain based analysis as the plastic strain tends to zero.

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3.0 Accounting for Plasticity.

3.1 The Strain-Life Methodology.

The strain life methodology is based on the observation that in many critical locations such as notches the material response to cyclic loading is strain rather than load controlled. This arises from the fact that whilst most components are designed to confine nominal loads to the elastic region, stress concentrations such as notches often cause plastic deformation to occur locally. The material surrounding the plastically deformed zone remains fully elastic and so the deformation at the notch root is considered to be strain-controlled.

The strain-life method assumes similitude between the material in a smooth specimen tested under strain-control and the material at the root of a notch, Figure 20. For a given loading sequence, the fatigue damage in the specimen and the notch root are considered to be similar and so their lives will also be similar.

Figure 20 Similitude between a laboratory specimen and the root of a notch.

The cyclic stress-strain response of the material at the critical location is determined by characterising the behaviour of smooth specimens subjected to similar loading, the local stress-strain history. The local stress-strain history must be determined, either by analytical or experimental means. Stress analysis procedures such as finite element modelling, or experimental strain measurements are usually required.

In performing smooth specimen tests which characterise fatigue performance, it must be recognised that fundamental material properties are being measured which are independent of component geometry. Phenomena such as cyclic hardening or softening, cycle dependent stress-relaxation, loading sequence effects are all taken into consideration.

Notch

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3.2 Monotonic Stress-Strain Behaviour.

The engineering tension test is widely used to provide basic information on the strength of materials and also as an acceptance test for the specification of alloys. In this test, a cylindrical specimen is subjected to a continually rising, monotonic, uniaxial load while simultaneously its elongation is measured.

The so-called engineering stress-strain curve can be constructed from the measured values of load and elongation, Figure 20.

The stress plotted on the engineering stress-strain curve is the average longitudinal stress in the test specimen and is obtained from:

S = P / Ao

where:

S is the nominal or engineering stressP is the applied loadAo is the original, unloaded, cross-sectional area of the specimen

Figure 21 The engineering stress-strain curve.

The engineering strain is the average linear strain obtained from:

e = Dl / lo = ( l - lo ) / lo

where:

e is the engineering strainlo is the unstrained specimen gauge lengthl is the strained gauge length

The shape and magnitude of the engineering stress-strain curve goes a long way towards characterising any particular material. The curve itself may be considered to be made up of three distinct regions. Uniform straining where stress and strain are linearly related, uniform straining where stress and strain are no longer linearly related, and a non uniformly straining region where

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plastic instability, necking, sets in.

Certain parameters may be measured directly from the engineering stress strain curve. For example, yield strength, tensile strength, percentage elongation, and reduction in area.

The yield stress is used to define the limit of elastic behaviour. In some steels it may be easily defined as a sharp transition between elastic and elastic-plastic behaviour. More commonly, it is defined, as in Figure 21 above, by a fixed offset i.e. the stress corresponding to 0.2% plastic strain for example:

S0.2 = P0.2 / Ao

The tensile strength, sometimes referred to as the ultimate tensile strength or UTS or Rm, corresponds to the stress resulting from the maximum load carried by the specimen. It is calculated by dividing the maximum load by the original cross-sectional area of the specimen:

Su = Pmax / Ao

The tensile strength defines the limit of uniform plastic deformation within the tensile test, it is not actually a fundamental material property, but rather it is a function of the test itself. However, it has been and remains the most widely quoted material property, indeed entire national standards are based around it.

Both percentage elongation and reduction in area are measure of ductility, that is the extent to which a metal will deform prior to fracture. Typically these properties are measured by putting a fractured tensile specimen back together and measuring its final length, lf, and final area, Af thence:

Percent elongation, EL% = [ (lf - lo) / lo ] x 100

Percent reduction in area, RA% = [ (Ao - Af) / Ao ] x 100

3.2.1 True Stress and Strain.

All the parameters derived above have been based on the original dimensions of the test specimen, and as such are normally referred to as engineering stress and strain parameters and denoted by S and e respectively.

A true stress is defined as the load divided by the instantaneous area, P / A, and since the specimen is stretching and its cross-sectional area decreasing, the true stress is always larger than the engineering stress.

Up to the onset of necking, a true strain is based on instantaneous gauge length, and is defined as the integral from the original length, lo to the instantaneous length, l, of 1 / l,otherwise:

e = ln (l / lo)

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Figure 22 Original and instantaneous specimen geometries.

When engineering stress and strain are corrected to their true values and the data re-plotted the nature of the stress strain diagram changes, Figure 23. True stress and strain may be related to engineering stress and strain, i.e.

True stress, up to necking

s = P / Ao (1 + e) = S (1 + e)

where:

S is the engineering stresse is the engineering strain

Figure 23 True stress true strain diagram.

Beyond necking true stress must be calculated from actual measurements of load and instantaneous

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True strain, up to necking:

e = ln (1 + e)

Beyond necking true strain can be calculated from:

e = 2 ln (Do / D)

where:

Do is the original diameterD is the instantaneous diameter

The fracture stress, corresponds to the highest stress carried by the specimen and is calculated by dividing the fracture load, Pf, by the final area, Af.

3.2.2 Stress-Strain Relationships.

Figure 24 below illustrates various regions of a true stress true strain curve. The curve has been generated by monotonically loading a specimen up to point B and then unloading down to point C and then re-applying the load up point to D.

The portion of the curve defined by O-A represents the elastic, reversible, behaviour of the material where stress and strain are linearly related.

Figure 24 Various regions of the true stress true strain curve.

The constant of proportionality is called the modulus of elasticity or sometimes the stiffness. In this region:

E = s / e

and any elastic strain, ee may be written as:

εe = σ / E

Stre

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The region B-D defines the elastic-plastic part of the true stress true strain curve, and for most metallic materials this region can be represented by a simple power law of the form:

σ = K (εp)n

where:

K the constant of proportionality is known as the strength coefficient.n is known as the work hardening coefficient.εp is the plastic strain.

The plastic strain, εp, can now be written as:

εp = ( σ / K )1/n

3.2.3 Reversing the Loading Direction.

If during the course of a monotonic tension test, point B in Figure 23 for example, the loading direction is reversed and the specimen unloaded, the stress-strain relationship will follow the line defined by B-C. The slope to the line will be the same as O-A that is to say equivalent to the modulus of elasticity, E. Indeed this is the most accurate way of measuring E during the course of a tension test.

The distance O-C represents a permanent extension of the specimen as a result of loading it up to point B. This extension, the plastic strain, is sometimes referred to as the permanent set.

The distance C-C' represents the recoverable extension of the specimen as a result of loading it up to point B, and is known as the elastic strain.

The total strain induced into the specimen by loading it up to point B, therefore, is the sum of the elastic and plastic components:

εt = εe + εp

and utilising the expressions derived above the total strain can be written as:

εt = σ / E + ( σ / K )1/n

which represents a good description of the true stress, true strain curve.

At fracture, point D in Figure 24, two further parameters can be defined, the true fracture strength and the true fracture ductility. True fracture strength, σf, is the true stress at final fracture and is defined as:

σf = Pf / Af

The true fracture ductility, εf, is the true strain at final fracture and is defined in terms of the initial and the area at fracture:

εf = ln(Ao /Af) = ln (1 / (1 - RA))

where the reduction in area RA is defined as:

RA = ( Ao - Af ) / Af

3.2.4 Reversing the Loading Direction, Again.

If from point C in Figure 24 above, the loading direction is again reversed, i.e. the specimen is loaded back towards point B and beyond, the true stress true strain curve will follow the trace C-B'-D. As in the case of unloading most of the trace will be a straight line with slope equal to the modulus of

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elasticity, E. The interesting thing to note, however, is that the new stress strain curve C-B'-D rejoins the original stress strain curve at B'. It is almost as though the material remembered the nature of the original curve and continued to deform along it. Conversely, once beyond B' the material has apparently totally forgotten the excursion B-C-B' and goes on to fracture at D as though it had never happened ! The excursion B-C-B' is, therefore, only an interruption in the primary loading process O-A-B-D, and in reality represents a fatigue cycle. These observations are central to the local strain life methodology and we shall return to them later.

3.2.5 The Bauschinger Effect.

As long ago as 1886, Bauschinger was carrying out and reporting experiments of the sort described above. In reality he went further in that he did not stop the unloading process at the zero stress level, point C in Figure 24, but rather continued on down into full compression, Figure 25 illustrates what he found.

In Figure 25(a) a specimen of material is loaded in tension to beyond the yield stress to a maximum stress of σmax, point A. At this point the loading direction is reversed, and the specimen is unloaded from point A through the zero stress level and on into compression to a stress equal to -σmax, point B, The interesting point to note is that the material appears to yield at a stress level before -σy is reached, Figure 25(b). This observation has nothing to do with the fact that the yield strength in compression might be different from that in tension. The whole experiment could be repeated, but this time loading in compression first by going to -σmax, yield would take place at -σy but would again occur before σy was reached when the loading direction was reversed. Unsurprisingly this behaviour is known as the Bauschinger effect and its importance cannot be overstated.

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Figure 25 Reversed loading into compression, the Bauschinger effect.

A this point a valid question to ask would be, if reversed yield does not take place at -σy, then at what point does it take place? Figure 25(b) provides the necessary clue, i.e. at a stress level equal to 2σy from σmax. It appears as though the stress strain curve defined by A-B is geometrically twice the initial true stress true strain curve O-A. This observation was first made by Massing in 1926 and is today known as Massing's hypothesis.

3.3 Cyclic Stress-Strain Behaviour.

If the loading process started in Figure 24 is continued, and at -σmax the loading direction is again reversed and the specimen is loaded back up to σmax then a complete loop will be defined, Figure 26. The stress strain loop illustrated in Figure 26 is called a hysteresis loop and defines, in stress strain space, a single fatigue cycle. Typically such loops are generated in the laboratory under strain control, that is by straining to specific strain limits. During the initial loading the stress-strain response is according to the curve O-A-B.

2σy

σmax

σmax

σy

σy

σmax

σy

Α.

Β.

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Figure 26 A complete stress-strain cycle, a hysteresis loop.

On unloading, yielding begins in compression at point C due to the Bauschinger effect. In reloading in tension the hysteresis loop develops. The dimensions of the loop are described by its total width, the total strain range ∆ε, and its height the total stress range ∆σ.

The total strain range is made up of the elastic and plastic components:

∆εt = ∆εe + ∆εp

which may be written as:

∆εt = ∆σ / E + ∆εp

3.3.1 Cyclic Loading Under Strain Control.

If a material is repeatedly cycled between fixed strain limits one of several things may happen depending on its nature and initial conditions of heat treatment.

∆σ

σ

Β.

∆ε2

∆ε2

Ο

2

C

B

A

∆σ 2

∆εe ∆εp

ε

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Figure 27 Cyclic hardening and softening under strain control.

The material could respond in any one of the following ways,

- it could cyclically harden

- it could cyclically soften

- it could remain stable

- it could soften or harden depending on strain range.

Figure 27 illustrates the effects of cyclic softening and hardening where the first two hysteresis loops of two different materials are plotted. In both cases, the cycling of the specimen is between fixed strain limits and the load is allowed to find its own level. It is clear that in the case of hardening, the maximum stress reached in each successive strain cycle increases with number of cycles and in the case of softening, the maximum stress decreases with number of imposed cycles. This process does not continue indefinitely. In both cases the stress will find a constant level and remain stable at that level until the first emergence of a fatigue crack.

The reason materials cyclically harden or soften is thought to be associated with dislocation substructures within the metal crystal lattice. Broadly speaking, soft materials such as aluminium, which contain low dislocation densities, tend to harden whereas hard materials such as steels tend to soften. As a rule of thumb, the ratio of the ultimate tensile strength, Su to the 0.2% proof strength, S0.2 can be used to gauge whether a particular material will cyclically harden or soften, if

Su / S0.2 > 1.4 then the material will harden

Su / S0.2 < 1.2 then the material will soften

For ratios between 1.2 and 1.4 the estimate is difficult and anything may happen, hardening, softening or both.

3.3.2 Determining the Cyclic Stress Strain Curve.

As mentioned above, under strain control and early in life, the stress-strain response of most materials changes significantly with applied cyclic strains. However, after a relatively small number

531

135

1 3 5

2 4

1 3 5

2 4

1 3 5

2 4 24

24

Control condition

Response variable

Response variable

Hysteresis loops

Hysteresis loops

Cyclic hardening

Cyclic softening

Stre

ss σ

Stre

ss σ

−

+

−

+

−+

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of cycles, typically no more than about 10% of total life, the hysteresis loops tend to stabilise so that the stress amplitude remains reasonably constant over the remaining portion of fatigue life. If the stress, strain co-ordinates of the tips from a number of stable hysteresis loops, of differing strain amplitudes, are plotted in stress-strain space, then the locus of these points defines the cyclic stress-strain curve., see Figure 28.

Figure 28 Definition of the stable cyclic stress strain curve.

Figure 29 Comparison between cyclic and monotonic stress-strain curves.

The cyclic stress-strain curve defines the relationship between stress and strain under cyclic loading conditions, just as the tensile true stress true strain curve defines the conditions under monotonic loading. The cyclic stress-strain curve can be compared directly with the monotonic stress-strain curve to quantitatively assess the cyclically induced changes in material behaviour.

Note that in Figure 29(a), when a material cyclically softens the cyclic yield strength is considerably lower than the monotonic value. Using monotonic properties in cyclic applications can result in the prediction of fully elastic strains when in reality considerable plastic strains are being generated.

140 MPa

0.01

Loops are approximateshape only

cyclic σ−ε curve

Strain

Stress

MonotonicCyclic Monotonic

Cyclic

Monotonic

Monotonic

Cyclic

Cyclic

Mixed behaviourCyclically stable

Cyclically softening Cyclically hardening

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It is also interesting to note that whilst stabilisation of hysteresis loops can consume 10 - 20% of total life in a constant amplitude test, a single large overload in service can produce an immediate change from the monotonic curve to the cyclic. Sobering to think that assembly, or using a machine or component just once, could cause an instant reduction of yield strength of up to 50%.

Figure 29(d) illustrates a common behaviour for low strength steels, whereby the material both hardens and softens depending on the strain amplitude. This phenomena is often known as mixed mode behaviour.

As with the monotonic stress strain curve, the cyclic stress-strain curve consists of both elastic and plastic strains :

εt = εe + εp

and can be fully characterised by an expression equivalent to that developed for the monotonic stress-strain curve. However, in this case, primes appended to the values of n and K to distinguish them from their monotonic counterparts:

εt = σ / E + ( σ / K' )1/n'

where:

K' is called the cyclic strength coefficient.

n' is called the cyclic strain hardening exponent.

Typically, K' takes values in the range 1000 to 3000 MPa and n' varies between about 0.1 to 0.2. In general, metals with a high monotonic strain hardening exponent, n, will harden whilst those with a low monotonic strain hardening exponent will cyclically soften.

Table 7: below compares the cyclic stress-strain behaviour of a number of well known alloys.

Material Heat 0.2% Yield n / n' Cyclic Treatment mono/cyclic mono/cyclic Behaviour

OFHC CopperAnnealed 20/138 0.40/0.15 HardensPartial anneal 255/200 0.13/0.16 StableCold worked 345/235 0.10/0.12 Softens

Aluminium alloys.2024 T4 305/450 0.20/0.11 Hardens7075 aluminiumT6 470/520 0.11/0.11 Hardens

Titanium alloy.Ti-8Al-1Mo-1V Annealed 1000/790 0.078/0.14 Mixed

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Continued...Material Heat 0.2% Yield n / n' Cyclic

Treatment mono/cyclic mono/cyclic BehaviourSteels.SAE 4340 Q+T 1206 UTS 1170/760 0.066/0.14 SoftensSAE 1045 Q+T 2050 UTS 1860/1720 0.071/0.14 Stable

Q+T 1725 UTS 1690/1275 0.047/0.12 SoftensQ+T 1550 UTS 1517/965 0.041/0.15 SoftensQ+T 1345 UTS 1275/758 0.044/0.17 Softens

SAE 4142 Q+T 1930 UTS 1690/1725 0.092/0.13 StableQ+T 1640 UTS 1725/1345 0.048/0.12 SoftensQ+T 1550 UTS 1585/1070 0.04/0.17 SoftensQ+T 1310 UTS 1380/825 0.051/0.18 Softens

Table 7: Cyclic stress strain behaviour of some common alloys.

3.3.3 Cyclic Yield and the Fatigue Limit.

The close association between plasticity and fatigue damage has already been mentioned. This point can be further illustrated through a consideration of the cyclic stress strain properties of some typical steels.

Monotonic properties.Material Yield Stress Ultimate Tensile Stress

MPa MPaLow strength 350 540Medium strength 650 790High strength 1090 1160

Cyclic properties.Material Strength coefficient Hardening exponent

K', MPa n'Low strength 1340 .226Medium strength 1062 .123High strength 1620 .112

Table 8: Monotonic and cyclic properties of some common alloys.

From the cyclic properties it is possible to use the equation for the cyclic stress-strain curve,

εp = ( σ / K' )1/n'

σ = K' εpn'

to calculate say an he 0.02% cyclic yield stress, i.e. the stress corresponding to 0.02% plastic strain. Note that this is one tenth the amount of plastic strain usually used to determine an offset yield stress. Broadly speaking, this stress represents a limit below which the level of plasticity is sufficiently low to ensure little or no fatigue damage, a fatigue limit. Values of the calculated cyclic yield together with the ratio of cyclic yield to ultimate tensile strength are tabulated below for the three steels.

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Material Cyclic Yield Cyclic Yield / UTSMPa

Steel:Low strength 195 0.36Medium strength 372 0.47High strength 624 0.53

Table 9: Cyclic yield stress and ratio to UTS

It is interesting to note that the ratios of cyclic yield to UTS are in the region of 0.5. This is consistent with the observation from the nominal stress approach that the endurance limit for steels is about one half the UTS and also supports the view that endurance limit and cyclic yield are closely related if not the same.

3.3.4 Hysteresis Loop Shape.

Massing's hypothesis states that the limb of a stress-strain hysteresis loop is geometrically similar to the cyclic stress strain curve but numerically twice it. Consequently, the equation for the limb can be derived directly from the equation of the cyclic stress strain curve.

Consider any point on the cyclic stress strain curve with co-ordinates (ε1,σ1), it follows that:

ε1 = σ1 / E + ( σ1 / K' )1/n'

From Massing's hypothesis, the same point can be located on the hysteresis loop curve and it will have co-ordinates (∆ε1,∆σ1) where:

∆ε1 = 2 ε1

∆σ1 = 2 σ1

Substituting into the equation for the cyclic stress strain curve:

∆ε1 / 2 = ∆σ1 / 2E + ( ∆σ1 / 2K' )1/n'

which in the general case reduces to:

∆ε = ∆σ / E + 2 ( ∆σ / 2K' )1/n'

The relationship between the cyclic stress strain curve and hysteresis loop shape is illustrated in more detail in the following example.

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4.0 Example 1. Calculating Cyclic Stress-Strain Response.

A test piece with the following cyclic properties,

cyclic strength coefficient, K' = 1200 MPacyclic strain hardening exponent, n' = 0.2modulus of elasticity, E = 210,000 MPa.

is to be cycled under strain control with a fully reversed strain range of 0.03. Calculate the expected stress-strain response.

In calculating the stress strain response it will be necessary to make two assumptions.

Firstly, it will be assumed that the material will follow the cyclic stress strain response rather than the monotonic one on initial loading from zero to the maximum strain amplitude of 0.015. Whilst there is a case for the use of the monotonic response for the first half cycle, in practice after repeated loading, the resulting hysteresis loop tip always migrates towards the cyclic stress strain curve.

Secondly, it will be assumed that the material will exhibit cyclically stable response from the initial loading. A more precise analysis would require accounting for the cyclic hardening or softening characteristics of the material. As in the case of the first assumption above, the cyclic modelling process is used to model the stable hysteresis behaviour since in most cases this occupies the majority of fatigue life.

Figure 30(a) illustrates a portion of the strain time sequence that is to be used to cycle the test specimen. The value of the stress corresponding to the first turning point, 1, can be calculated directly from the equation for the cyclic stress strain curve,

ε1 = σ1 / E + ( σ1 / K' )1/n'

0.015 = σ1 / 210,000 + ( σ1 / 1200 )1/0.2

Figure 30 (a) the driving strain history. (b) the stress-strain response.

The value of σ1 can be calculated from the above equation by using an iterative technique such as Newton Ralphson or interval halving. Such a procedure will provide a value of

σ1 = 500 MPa.

The value of the stress at the next turning point, 2 in Figure 30(a) can be calculated by considering

0.015

ε

-0.015

500

−500

0 ε

1 3

22

13

σ

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the total strain range, 0.03, together with the equation for twice the cyclic stress strain curve,

∆ε = ∆σ / E + 2(∆σ / 2 K')1/n'

0.03 = ∆σ / 210,000 + 2(∆σ / 2400)1/0.2

Solving this equation by iteration leads to a value of ∆σ = 1000 MPa. The co-ordinates of the second turning point in stress-strain space can now be calculated from:

ε2 = ε1 − ∆ε = ( 0.015 - 0.03 ) = -0.015

σ2 = σ1 − ∆σ = ( 500 - 1000 ) = -500 MPa

The co-ordinates of the third turning point can be calculated in the same way as for the second point, and for the fully reversed loading being considered here, they must be the same as for the first turning point. Figure 30 illustrates the stress-strain hysteresis loop corresponding to the induced cyclic loading.

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5.0 Strain-Life Characteristics.

5.1 The Strain-Life Curve.

A number of years ago, Basquin observed that stress-life data may be represented by a straight line relationship when plotted using log scales. The relationship could be expressed in terms of true stress as:

σa = σf' (2Nf)b

where:

σa is the true cyclic stress amplitudeσf' is the regression intercept called fatigue strength coefficient2Nf number of half cycles, reversals, to failureb is the regression slope called the fatigue strength exponent.

σf' and b are considered to be material properties with the fatigue strength coefficient being approximately equal to the monotonic fracture stress, σf, and b varies between -0.05 and -0.12.

The Basquin equation may be re-written in terms of elastic strain amplitude:

εa = σa / E = σf' (2Nf)b / E

where:

εe is the elastic strain amplitudeE is the modulus of elasticity.

In the 1950's Coffin and Manson independently proposed that the plastic strain component of a fatigue cycle may also be related to life by a simple power law:

εp = εf' (2Nf)c

where:

εp is the plastic strain amplitudeεf' is the regression intercept called fatigue ductility coefficient2Nf number of half cycles, reversals, to failurec is the regression slope called the fatigue ductility exponent.

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Figure 31 The total strain-life curve.

εf' and c are considered to be material properties with the fatigue ductility coefficient being approximately equal to the monotonic fracture strain, εf, and c varies between -0.5 and -0.8.

More recent work, notably by Morrow, has indicated that the total strain amplitude, that is the sum of the elastic and plastic components, may be better correlated to life. Figure 30 illustrates schematically the nature of the total strain-life curve.

Mathematically, this curve can be described by summing together the Basquin and the Coffin-Manson component curves:

εt = εe + εp

εt = σf' (2Nf)b / E + ef' (2Nf)c

Note that there is no fatigue limit defined by this equation and so it is usual to define a cut-off in terms of reversals which specifies an endurance limit. Typically, the cut-off is set to about 5 x 107 reversals.

5.2 Strain-Life vs. Stress-Life.

In the realm of high cycle fatigue, the equation for the S-N curve is usually written as:

Nf = No (S/So)1/b

which may be rearranged as

S = So / No (Nf)b

and at one cycle So is the intercept between the S-N line and the stress axis, approximately, the fracture stress, Sf and so

S = Sf (Nf)b

The Basquin equation may be rearranged so that:

σa = σf'2b (Nf)b

Stra

in A

mpl

itude

(log

Sca

le)

Reversals to Failure (log Scale)

b

1

c

1

Transition Life

plastic

elastic

σf’

εf’

total

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It is clear that these two equations are the same, particularly since magnitude of b is about -0.1 and 2b is close to unity. The thing to note from this comparison is that the classical S-N curve is actually a subset of the total strain life curve and also that the fundamental material S-N curve, as opposed to a component S-N curve, can be derived from the low cycle fatigue parameters, σf' and b. It is important to remember that whilst the elastic S-N curve is defined all the way up to, σf', it should only be used within the high cycle regime, i.e. above the transition life.

5.3 Transition Life.

In Figure 32, the point where the plastic and elastic life lines intersect is called the transition life. The transition life represents the point at which a stable hysteresis loop has equal elastic and plastic components. At lives less than the transition, plastic events dominate elastic ones and at lives longer than the transition elastic events dominate plastic ones. From this point of view, therefore, the transition life represents a very convenient and important way of delineating between the low and high cycle fatigue regimes.

This distinction is important because the solutions which may be proposed to a particular fatigue problem depend entirely on the dominant loading regime. Problems of high cycle fatigue are usually tackled through the selection of stronger, higher UTS, materials, or through the application of compressive surface stresses through shot peening or nitriding etc. These solutions would be totally ineffective for the treatment of a low cycle fatigue problem. Indeed, the selection of a material with a higher UTS, and presumably a lower ductility, could well make the situation worse !

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Figure 32 Idealised representation of several strain-life curves.

These observations are further emphasised in Figure 32 where three idealised materials are shown, one very high strength material (strong), one ductile material (ductile), and one whose properties lie between the two extremes (tough). The curve cross at about 2 x 103 reversals, 1000 cycles, at a strain amplitude of about 0.01. Thus, one would select the "strong" material for a design life requirement greater than 1000 cycles, pick the "ductile" material for design life requirements shorter than 1000 cycles, and "optimise" with the "tough" material for spectrum loading of a more complicated nature. It is interesting to note that all types of materials seem to have about the same fatigue resistance for a strain amplitude of about 0.01, corresponding to a failure life of about 1000 cycles.

The transition life can be derived directly from the equation of the strain-life curve. At the transition the elastic and plastic strain amplitudes are equal so that at Nf = Nt

εe = εp

σf' / E (2Nt)b = εf' (2Nt)c

and

2Nt = (εf' E / σf')1/(b-c)

2 x 103

Ductile

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6.0 Example 2. Transition Life and Shot Peening.

A component is manufactured from a steel with the following cyclic properties:

fatigue strength coefficient,σf' = 1100 MPafatigue strength exponent,b = -0.1 MPamodulus of elasticity,E = 210,000 MPafatigue ductility coefficient,εf' = 0.6fatigue ductility exponent,c = -0.5

If the component were to be cycled under a fully reversed constant strain amplitude of 0.004, would you expect shot peening to influence observed fatigue life ?

The first step is to calculate the transition life from:

2Nt = (εf' E / sf')1/(b-c)

2Nt = (0.6 x 210,000 / 1100)1/(-0.1+0.5)

2Nt = 114.5 2.5 = 140,425 reversals.

The next step is to calculate the total strain amplitude at the transition life. This can be done by calculating either the elastic or plastic strain components at the transition life:

εp = εf' (2Nt)c = 0.6 x 140425 -0.5

εp = 0.0016

and so εt is given by et = 2 x 0.0016 = 0.0032

The working strain amplitude of 0.004, is significantly larger than the strain amplitude at the transition and so the loading can be considered to fall within the low cycle regime. Under these circumstances shot peening should not be considered are providing any real benefit.

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7.0 Determination of Cyclic Fatigue Properties.

7.1 Laboratory Testing.

The cyclic material properties required to define the cyclic stress strain curve and the strain life curve are usually determined by carrying out tests, under strain control, on a series of smooth highly polished hour glass specimens. Typically, about 15 tests need to be performed at differing strain amplitudes. The properties can then be calculated by regression analysis on the following curves:

K' and n' : from a log stress vs log plastic strain regressionσf' and b : from a log elastic strain vs log 2Nf regressionεf' and c : from a log plastic strain vs log 2Nf regression

7.2 Estimation from Tensile Properties.

It is often difficult to gain access to measured cyclic properties. For this reason, a lot of effort has been put into finding ways of relating monotonic properties, of which there is an abundant supply, to cyclic properties. The approaches have all been empirical but have provided some approximations which are useful.

The first method of approximating the strain life relationship from monotonic properties was proposed by Manson and later modified by Muralidharan. The procedure is usually referred to as the method of universal slopes and can be applied to any metal.

Parameter Universal Slopes Modified Universal Slopes(Manson) (Muralidharan)

σf' 1.9 Rm 0.623 Rm0.823 E0.168

b -0.12 -0.09

εf' 0.76 εf0.6 0.0196 εf

0.155 (Rm / E)-0.53

c -0.6 -0.56

K' (σf' / εf')0.2 (σf' / εf')

0.2

n' 0.2 0.2

where :

Rm is the ultimate tensile strengthεf is the true fracture strain calculated from ln (1 / (1 - RA)) and

RA is the reduction in area.

Table 10: The method of universal slopes

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More recently, Baumel Jr. and Seeger have compiled an alternative approach based on the results of more than 1500 fatigue tests. Currently the approach is limited to plain carbon and low to medium alloy steels, aluminium and titanium alloys.

Parameter Uniform Material Law Uniform Material Lawplain and low alloy Aluminium and Titaniumsteels alloys

σf' 1.5 Rm 1.67 Rmb -0.087 -0.095εf' 0.59 α 0.35c -0.58 -0.69K' 1.65 Rm 1.61 Rmn' 0.15 0.11

Table 11: The uniform material law of Baumel and Seeger

The ductility factor α is calculated from:

α = 1.0 for values of Rm / E < 3 x 10-3

α = (1.375 - 125 Rm / E) for Rm / E > 3 x 10-3

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8.0 The Effect of Mean Stress.

As mentioned above, most basic fatigue data are collected in the laboratory by means of testing procedures which employ fully reversed loading, i.e. R = -1. However, most realistic service situations involve non zero mean stresses. It is, therefore, very important to know the influence that mean stress has on the fatigue process so that the fully reversed laboratory data can be usefully employed in the assessment of real situations.

Figure 33 Effect of mean stress on the strain-life curve.

Figure 33 illustrates schematically the effect mean stress has on the strain-life curve. Typically the effects are concentrated at the long life end of the diagram, with compressive means extending life and tensile means reducing it. At high strain amplitudes, mean stress relaxation occurs and any mean stress present tends towards zero. This effect is similar to the "washing out" of surface compressive stresses in the low cycle regime.

8.1 The Morrow Mean Stress Correction.

Morrow was the first to propose a modification to the base-line strain-life curve which can account for the effect of mean stress.

He suggested that mean stress could be taken into account by modifying the elastic part of the strain life curve by the mean stress, σo.

εe = (σf' - σo) / E (2Nf)b

the entire strain life curve becoming:

εa = (σf' - σo) / E (2Nf)b + ef' (2Nf)c

The Morrow equation is consistent with observation that mean stress effects are significant at low values of plastic strain and little effect at high plastic strains.

8.2 The Smith Watson Topper Mean Stress Correction.

Smith, Watson and Topper have proposed a slightly different approach to account for mean stress through a consideration of the maximum stress present in any given cycle. In this case the damage parameter is taken to be the product of the maximum stress, σmax, and the strain amplitude, εa of a cycle.

Compressive mean Stress

Fully reversed (zero mean stress)

Tensile mean stress

Log 2Nf

Log

∆ε /2

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For fully reversed loading the maximum stress is given by

smax = sf' (2Nf)b

and by multiplying the strain-life equation by this term

smax ea = sf'2 / E (2Nf)2b + sf'ef' (2Nf)b+c

The Smith Watson Topper equation predicts that no fatigue damage can accrue when the maximum stress becomes zero or negative, which is not strictly true.

8.3 Morrow vs Smith Watson Topper.

It is difficult to categorically select one procedure in preference to the other. However, for loading sequences which are predominantly tensile in nature the Smith Watson Topper approach is more conservative and is, therefore, recommended. In the case where the loading is predominantly compressive, particularly for wholly compressive cycles, the Morrow correction can be used to provide more realistic life estimates.

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9.0 Factors Influencing Fatigue Life.

A standardised, fully reversed, strain controlled, fatigue test is used to determine the base-line strain-life relationship for a polished specimen of approximately, 6 mm diameter. If the fatigue or endurance limit measured by these means be denoted by σ'e then the actual limit for a real component, σe, must reflect all the modifications that come about in moving from a laboratory specimen to a component. For steels in particular, several empirical relationships have been developed which can account for the variation in σe as a result of the following:

- Component size- The type of loading- The effect of surface finish- The effect of surface treatment.

The usual way to account for these effects is through the calculation and application of specific modifying factors so that

σe = σ'e Csize Cload Csur . . .

where reciprocal of the product, Csize Cload Csur, is collectively known as the fatigue strength reduction factor Kf, i.e.,

Kf = 1 / (Csize Cload Csur . . )

The approach tends to be conservative and the corrections are usually only applied at the endurance limit, the modifications required for the rest of the elastic portion of the strain-life curve are ill-defined. Typically the procedure is to pivot the elastic Basquin curve about the 1000 cycles point.

The magnitude of the modification factors required to account for the various effects can be taken to be the same as that required to modify the basic nominal stress life, S-N, curve. The manner in which the modifications are implemented is the same as that required for the nominal stress approach.

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Section 4 - Mult iaxial Considerations

1.0 Introduction.

Most engineers, if they consider the possibility of fatigue failure at all, adopt the simple S-N approach developed during the 19th century. Modern software design tools incorporate fatigue concepts such as Miner's rule (1920's), the strain life approach (1950's) and linear elastic fracture mechanics (1960's). These ideas are well tried and tested and their limitations well understood. Universities and research departments in some major industries, notably power generation and aerospace, are engaged in active research into other areas. Theories in these areas are much less well established and understood and predictions based on these ideas should be treated with some caution. Included in this category are elastic-plastic fracture mechanics, short crack approaches and multiaxial fatigue.

2.0 The Multiaxial Stress-Strain State.

Up until now, uniaxial cyclic loading has been implicit in the discussion of both high and low cycle fatigue. However, most real design situations, including rotating shafts, connecting links, automotive and aircraft components and many others involve a multiaxial state of cyclic stress. This often means that, at any point, the directions of the principal stresses can vary during the loading cycle and, therefore, as a function of time. Furthermore, the magnitudes of the principal stresses themselves may no longer be proportional to each other. Both these effects complicate the analysis required for the prediction of fatigue behaviour. To some extent the situation can be clarified through an understanding of the differences in the stress-strain states that prevail under simple uniaxial loading and non-proportional multiaxial loading.

Figure 34 The stress-strain state in uniaxial tension.

2.1 Separate Tensile or Torsional Loading.

A snapshot, in terms of Mohr's circle representation, of the stress-strain state prevailing under a static load in both tension or torsion loading is presented in Figures 30 and 31 . For uniaxial tension, σ = σ1, σ2 = σ3 = 0 and ε = ε1, ε2 = ε3 = 0 the Mohr's circles for stress and strain appear as in Figure 34.

σ3 σ1

σ

τ

ε3 εn ε1 ε

γ /2

(ε3 = −vε1)

γmax2

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Figure 35 The stress-strain state in torsion.

For pure shear as in the case of a torsion test, σ = σ1 = -σ3, σ2 = 0 and ε = ε1 = -ε3, ε2 = 0 the Mohr's circles for stress and strain appear as in Figure 34. During cyclic loading, the diameter of the Mohr's circle varies depending on the magnitude of the cyclic stress or strain. Figure 36 below illustrates this effect for bar subjected to a constant amplitude uniaxial strain cycle.

Figure 36 Mohr's circles for a uniaxial strain cycle.

Note that in this case the direction of the principal stresses, points x and y in Figure 36, do not stray away from the strain axis during the loading cycle. This means that, the normal strain, ε, and the shear strain, γ, remain linearly related to each other, that is to say the loading is proportional.

2.2 Combined Tensile and Torsional Loading.

It often happens in practice that separately induced load cases can result in non proportional loading environments. Figure 37 below can be used to illustrate the point by detailing the variation of axial

σ3 σ1

σ

τ

ε3 εn ε1 ε

γ /2

(ε3 = −vε1)

γmax2

Tmax

γ /2γ /2 γ /2

y x

CA B

Timey

xA

B

C

x y

e

x y

ε

ε

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and torsional strain with respect to time during the course of a complete fatigue cycle.

Figure 37 Non proportional loading sequences.

Note that the loading path is no longer a straight line between normal and shear strain but rather a rectangular function. Note also that the direction of the principal strain is no longer fixed i.e. the line x-y astounds an angle a with the strain axis.

2.3 Mechanisms of Fatigue Damage Accumulation.

Fatigue damage can be understood to be a process in which cracks initiate and then propagate until final failure. Crack initiation has been demonstrated to be the result of reversed plastic flow, i.e. the forward and backward motions of dislocations along the slip planes of metallic crystals. Slip occurs along planes of maximum shear, and will first appear in those grains which are most favourably oriented with respect to the maximum applied shear stress. In 1961, Forsyth, proposed a two stage model for early crack growth.

Stage I is a period of nucleation and crystallographically orientated growth following immediately after initiation and is confined to shear planes. In this phase, both the shear stresses and strains and the normal stresses and strains are the moduli which control the rate of crack extension.

Stage II growth is growth which occurs on planes which are orientated perpendicular to the maximum principal stress range. In this phase, the magnitude of the maximum principal stresses and strains dominate the crack growth process.

The proportion of life spent within each of these modes has been shown to depend on material type, ductile versus brittle, loading mode and amplitude. It is interesting to note that even in the case of

yx

γ /2

x yy

x

y

x

y

x

x

y

x

y

x

y

y xεε ε ε

εεε ε

D C B

A

HGF

E A B

C

D E F

G

H A

A

B C D

E

F G H

A

Point A Point DPoint CPoint B

Point E Point F Point G Point H

γ /2γ /2γ /2γ /2

γ /2γ /2γ /2

ε

γ√3

γ√3

αα

α α

SpecimenLoading HistoryLoading Path

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non proportional loading, Stage I fatigue cracks nucleate and grow on the plane of maximum shear strain range which is subjected to the largest amplitude of normal strain.

Since initiation life refers to the time taken to develop an engineering crack, this so-called initiation phase may include both stages of crack development. In practice, Stage I or Stage II may dominate lifetime. In uniaxial fatigue, this is not so much of a problem, as the controlling parameters in both cases are directly related to the uniaxial stress or strain. In multiaxial fatigue this is no longer the case. Furthermore, the Stage I to Stage II transition is sensitive to the material and the precise nature of the loading. It follows that it is very difficult to apply general rules, and a model that produces good predictions for a given material and loading may no longer hold true when applied to a different situation. To add further confusion, there may be interaction between damage on different planes; cracks growing on a particular plane may actually impede the progress of those growing on a different plane and lead to an increase in fatigue life.

Any theory of multiaxial fatigue must attempt, to a greater or lesser extent, to incorporate some of these physical observations if it is to have any chance of being successfully applied to real situations

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3.0 Equivalent Stress-Strain Approaches.

Traditionally, the approach to the design of components subjected to multiaxial loading is to make the following fundamental assertion:

Failure under a multiaxial loading is predicted to occur, according to the theory associated with a particular modulus, if and when the cyclically induced magnitude of that modulus, is sufficiently large that failure would occur in the uniaxial state for an identical magnitude of the same modulus.

The mechanical modulus referred to above, is a measurable quantity such as principal stress, principal shear stress or distortion energy. The range of the modulus may be defined by any two appropriate and independent values of it such as σmax, and σm or σmax and σmin.

This philosophy has lead to what is usually referred to as equivalent stress-strain approach, where an equivalent stress or strain modulus is calculated under multiaxial loading and then applied to uniaxial data.

A problem with all these methods is that they do not take into account the fact that fatigue is essentially a directional process, with damage, cracking, taking place on particular planes. In addition there are serious problems in applying any of these methods to situations of non-proportional loading, i.e. where the number of load inputs is greater than one, and these loads have a non-constant ratio or phase relationship.

This has lead to much greater research emphasis being placed on understanding the underlying mechanisms of fatigue damage accumulation under multiaxial loading. This has given rise to a somewhat different approach based on predicting the extent of damage in specific directions and planes within the component. This methodology is usually referred to as the critical plane approach.

3.1 Equivalent Stress-Strain Combination.

These approaches are based on extensions to static yield theories. They assume that lifetimes for fatigue under multiaxial loading can be predicted by substituting combined stress or strain parameters in the uniaxial stress-life or strain-life equations, i.e. calculating an equivalent uniaxial stress or strain for a given multiaxial loading. The main stress and strain parameters used are the maximum principal, the maximum shear, related to the Tresca criterion, and the von Mises or octahedral shear stress. The big advantage of this kind of approach is that it enables the large amount of uniaxial fatigue test data available to be applied to multiaxial situations.

The von Mises method has gained widest acceptance, but all of them have drawbacks. In the Tresca criterion, the median principal stress does not affect the equivalent stress or strain, and neither of the von Mises or Tresca criteria varies with the application of a hydrostatic stress, contrary to experimental evidence.

Before proceeding further with a more detailed account of the various equivalent stress-strain theories evolved to account for dynamic multiaxial stresses, it is worth examining the basis of the stress-strain combination procedures developed to account for static failure under multiaxial stresses.

3.1.1 Combined Stress Theories of Static Failure.

Predicting failure and establishing a geometry that will avoid failure is relatively straightforward if a component is subjected to a static uniaxial stress. It is only necessary to refer to a uniaxial stress-strain curve for the material of interest and ensure that the service loading is less than some critical value. For example, if yielding is the failure mode to be avoided, then failure would be predicted to occur when the maximum principal stress exceeds the yield stress.

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However, if the component under consideration is subjected to a multiaxial loading, failure prediction is much more complicated. It is no longer possible, for example, to predict when yielding will occur through a consideration of the maximum principal stress alone, because the other principal stresses will also have an influence.

A large number of multiaxial tests would be required in which all of the stress components would have to be varied over their entire range of values in all possible combinations in order to adequately describe the above situation. Such a testing programme would be prohibitively expensive and perhaps not even possible for certain stress states. When faced with such complex situations, engineers always attempt to develop a theory that relates behaviour in the complex situation to behaviour in a simple easily evaluated test through some characteristic modulus. Specifically, when it is required to predict the failure of a component subjected to multiaxial loading, it is usual to utilise a theory that relates failure in the multiaxial state to failure in a tension test through a well chosen modulus such as stress, strain or energy. To be useful such moduli must be calculable in the multiaxial state of stress and readily measurable in the uniaxial evaluation test. A combined stress failure theory, therefore, must contain three essential ingredients.

Firstly, it must provide a model that relates external loading to the stresses and strains at the critical point in the multiaxial state of stress.

Secondly, it must be based on physical properties of the material that are measurable.

Thirdly, it must relate the calculable mechanical modulus in the multiaxial stress state to a measurable criterion of failure based on the critical physical properties determined in a simple tension test.

A number of combined stress failure theories that contain these ingredients have been postulated. Numbered amongst these are the following; the maximum principal stress theory, the maximum principal strain theory, the maximum shear stress theory, the total strain energy theory, the distortion energy theory, Mohr's failure theory. Two of these theories, the distortion energy and the maximum shear stress, will be developed in more detail with failure being considered to be yielding.

3.1.2 Yielding Criteria for Ductile Metals.

As mentioned in the example above , yielding under uniaxial loading begins at the yield stress, σy. Under combined loading, it is expected that yielding will occur at some particular combination of principal stresses. At present there is no theoretical way of deducing the relationship between the stress components to correlate yielding for a multiaxial stress state with yielding in the uniaxial tension test. Therefore, yield criteria are empirical relationships. However, a yield criterion must be consistent with a number of experimental observations, the main one being the fact that pure hydrostatic pressure does not cause yielding in a continuous solid. This for example, rules out the use of the maximum principal theory since it would predict yielding failure when the magnitude of the maximum principal stress σ = σ1 = σ2 = σ3 becomes equal to the yield stress σy.

Since the hydrostatic stress component of a complex stress state does not influence the stress at which a material yields, the yield criterion must be based on the stress deviator. Moreover, for an isotropic material the yield criterion must be independent of the choice of axes, i.e. it must be an invariant function. These considerations lead to the conclusion that the yield criterion must be some function of the invariants of the stress deviator. At present there are two generally accepted criteria for predicting the onset of yielding in ductile metals.

3.1.2.1 Distortion-Energy, Von Mises.

In 1913 Von Mises proposed that yielding will occur when the second invariant of the stress deviator, J2 exceeds a critical value i.e.

J2 = k2

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where

J2 = 1/6 [ (σ1 − σ2)2 + (σ2 − σ3)2 + (σ3 − σ1)2 ]

The constant k can be evaluated and related to yielding in a tension test by realising that σ1 = σo , and σ2 = σ3 = 0 and by substituting this for J2,

k2 = σο2 / 3

and so the Von Mises' prediction of yield in terms of the principal stresses becomes,

σο = 0.7071 [ (σ1 − σ2)2 + (σ2 − σ3)2 + (σ3 − σ1)2 ]0.5

and yielding will occur when so exceeds the monotonic yield stress. In terms of the (x,y,z) component stresses it may be written as:

σo = 0.7071 [ (σx - σy)2 + (σy - σz)2 + (σz - σx)2 + 6 (τxy2 +τyz

2 +τxz 2) ]0.5

By considering the state of stress for pure shear as in a torsion test, it is possible to identify the constant k. In torsion,

σ1 = −σ3 = τ and σ2 = 0

at yield

σ12 + σ1

2 + 4σ12 = 6 k2

σ1 = k

This means that k represents the yield stress in pure shear, torsion. Therefore, the Von Mises' criterion predicts that the yield stress in torsion will be less than that for uniaxial tension according to:

k2 = σo2 / 3

k = 0.577 σo

In 1924 Henkey derived the Von Mises' yield criterion through a consideration of distortion energy and so it is sometimes also referred to as the distortion energy criterion.

3.1.2.2 Example 1 To Yield or not to Yield.

Stress analysis of a component, manufactured from a material with a yield stress of 500 MPa, indicates that at some point in its life it will be subjected to the following state of stress:

σx = 200 MPa, σy = 100 MPa, σz = -50 MPa, and τxy = 30 MPa. The other shear stresses are all zero.

The question is will it yield and if not what is the safety factor ?

Using the above component definition of the Von Mises' criterion,

σo = 0.7071 [ (200 - 100)2 + ((100 - (-50)))2 + (-50 -200)2 + 6(30)2 ]0.5

σo = 224 MPa

Since the value of σo is less than 500 MPa yielding will not occur and the safety factor is 500 / 224 = 2.2

3.1.2.3 Maximum Shear Stress, Tresca .

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The Tresca yield criterion suggests that yielding will occur when the maximum shear stress under multiaxial loading reaches the value of the shear stress under uniaxial tension test. Since according to convention σ1 is the algebraically greatest principal stress and σ3 the smallest, the maximum shear stress, τmax is given by:

τmax = ( σ1 − σ3 ) / 2

For uniaxial tension, σ1 = σo, and σ2 = σ3 = 0 and the shearing yield stress to is equal to σo / 2 and so

τmax = ( σ1 − σ3 ) / 2 = το = σo / 2

and

σ1 − σ3 = σο

For the state of pure shear, σ1 = -σ3 = k and σ2 = 0, the maximum shear stress criterion would predict yielding will occur when

σ1 − σ3 = 2k = σο

and so the maximum shear stress criterion predicts that yield in torsion will be less than in uniaxial tension according to:

k = σο / 2

It is worth noting that the Tresca criterion appears much simpler than that proposed by Von Mises and for this reason it is often used in engineering design. However, the maximum shear criterion does not take into account the intermediate principal stress. Furthermore it suffers from the major difficulty that it is necessary to know in advance which are the maximum and minimum principal stresses.

3.1.2.4 Example 2 To Yield or not to Yield.

Use the maximum shear stress criterion to establish whether yielding will occur in example 1 above.

τmax = ( σx - σz ) / 2 = σo

200 - (-50) = so

σo = 250 MPa

And so again, the Tresca criterion predicts that yielding will not take place.

3.2 Stress-Strain Theories of Multiaxial Fatigue.

In the descriptions that follow, each theory will be presented in terms of an equivalent stress or strain which may then be used to access uniaxial damage curve. As with the uniaxial approaches, stress based theories are usually confined to the high cycle fatigue regime and strain based ones to the low cycle end where plasticity becomes increasingly important.

3.2.1 Maximum Principal Stress and Strain.

The maximum principal stress or strain models are analogous to the use of applied amplitudes in the uniaxial case. The amplitudes of the maximum principals, on the maximum principal plane, are considered to be the appropriate moduli to describe fatigue damage.

The equivalent multiaxial stresses and strains are taken to be the maximum principal stress or strain, i.e.:

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σe = σ1

and

εe = ε1

The main benefit of this approach is its simplicity, however, its main difficulty is that no account is taken of the other two principals, σ2,σ3 and ε2, ε3. The direction of the maximum principal is also assumed to be fixed. In addition, the method is really limited to the prediction of Stage II type cracks which are by definition perpendicular to the principal directions. This method effectively describes the uniaxial stress state. It can be successfully applied in situations where the direction of the maximum principal stress has been demonstrated to be relatively fixed. The model is unsuited to non proportional loading conditions.

Damage accumulation can be summed directly from the appropriate uniaxial damage laws and so:

ε1 = σf' / E (2Nf)b + εf' (2Nf)c

The model could be applied equally well to a stress-based damage law.

3.2.2 Maximum Shear Stress and Strain, Tresca.

In this approach, equivalent stresses or strains are taken to be the maximum shear components respectively. This represents an extension to the Tresca yield theory and assumes that multiaxial shear stress or strain amplitude will correlate with the shear stress or strain amplitudes under uniaxial tension. The required moduli are given by the Tresca formulation:

τe = | ( σ1 - σ3 ) | / 2

and

γe / 2 = | ( ε1 - ε3 ) | / 2 = ( 1 + µ ) εe / 2

This model is a little more complicated to apply than the maximum principal stress approach. However, it does take into account the magnitude of the minimum principal stress or strains and also accumulates damage of the plane of maximum shear and so the method should be more applicable to the prediction of Stage I type cracks. As with the maximum principal approach, the maximum shear approach is limited to situations where the orientation of the plane of maximum shear is relatively fixed.

Since the equivalent moduli are shear stress or strain, a torsional stress or strain-life damage curve has to be used with this approach. This curve can be derived from the uniaxial strain life curve by recalling that:

γ = ε ( 1 + µ )

where:

µ is Poisson's ratio

and applying appropriate values of Poisson's ratio to the elastic and plastic parts of the uniaxial strain life curve:

γmax = 1.3 σf' / E (2Nf)b + 1.5 εf' (2Nf)c

3.2.3 Von Mises' Effective Stress and Strains.

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The Von Mises' effective stress or strain model is an extension of the Von Mises' yield criterion. In each case the Von Mises' effective strain (or stress) may be thought of as the root-mean square of the maximum principal shear strains (or stresses) normalised to uniaxial loading conditions. Recall from section 3.1.2.1 that the Von Mises' stress amplitude, also known as the octahedral shear stress, is given in terms of the principal stresses by:

σe = 0.7071 [ (σ1 − σ2)2 + (σ2 − σ3)2 + (σ3 − σ1)2 ]0.5

The equivalent strain amplitude can be written in terms of the principal strains as:

εe = A [ (ε1 − ε2)2 + (ε2 − ε3)2 + (ε3 − ε1)2 ]0.5

where A is defined by

0.7071 ( 1 / ( 1 + µ* ))

and m* is the elastic-plastic Poisson's ratio defined by

m* = ( me ee + mp ep ) / et

where

µe is the elastic Poisson's ratio, about 0.3µp is the plastic Poisson's ratio, about 0.5εt is the total strain amplitudeεe is the elastic strain amplitudeεp is the plastic strain amplitude

As opposed to the maximum principal and shear models, the Von Mises' model does take into account the influence of the median principal. The effective Von Mises' stresses and strains are scalar quantities and offer no information on the physical damage observed during the fatigue process. The model, therefore, does not account for non proportional loading.

An alternative interpretation based on octahedral shearing would suggest that damage may be observed on octahedral shear planes.

The calculated values of effective stress or strain can be used directly with the uniaxial damage laws, for example:

εe = σf' / E (2Nf)b + εf' (2Nf)c

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4.0 Critical Plane Approaches.

Fatigue cracks normally initiate on planes of maximum shear and grow initially in a shear-dominated mode, so-called mode II type growth, giving rise to Stage I cracks. Subsequently, growth changes to an opening mode, referred to as mode I, where growth is perpendicular to the direction of the maximum principal, Stage II cracks. In brittle materials or where initiation occurs at inclusions, growth may be in mode I from the outset.

4.1 The Miller Brown Parameter.

In addition to the importance of the plane of maximum shear, the stress and strain normal to this plane has also been recognised to strongly influence the development of fatigue cracks.

These ideas, whilst recognised by Mohr and others were first formulated by Miller and Brown who presented a two parameter formulation as:

γmax = f ( en )

where:

γmax is the maximum shear strainεn is the strain normal to the above shear.

They proposed that the maximum shear strain is the primary force in crack initiation and that the strain normal to the plane of maximum shear is a modifying factor. The above equation has been modified by Kandil and later by Fash to develop it into a damage law based on uniaxial cyclic properties:

γamax + S εa

n = α σf' / E (2Nf)b + β εf' (2Nf)c

where

γamaxis the maximum shear strain amplitude

εan is the strain normal to the plane of maximum shear.

The constants α and β can be calculated from:

α = ( 1 + µe) + S ( 1 - µe ) /2 = 1.65

β = ( 1 + µp) + S ( 1 - µp ) /2 = 1.75

where µe and µp are respectively, the elastic and plastic Poisson's ratios and the constant S is taken to be unity. The damage law now becomes:

( γamax + εa

n ) = 1.65 σf' / E (2Nf)b + 1.75 εf' (2Nf)c

Note that this is similar to the Tresca formulation but with the normal strain component added, and as it stands takes no account of the fact that the plane of maximum shear, like the principal stresses and strains could rotate.

4.2 Rotation of the Plane of Maximum Shear.

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In situations of non proportional loading, the direction of the principal stresses or strains can vary with time. The plane of maximum shear has a fixed relationship, to the plane on which the maximum principals are located and so the plane of maximum shear itself will rotate. Under these circumstances, even the Miller Brown damage parameter will fail to account for this effect. The net result of this is an averaging or smoothing out of the damage estimate and almost certainly an underestimate of the actual damage.

A strategy for accounting for these effects is to calculate fatigue damage across a range of orientations of shear plane in an effort to identify the most damaging. The effect of the applied loading on a particular plane is determined by a tensor rotation of the stresses and strains to that plane and estimating the damage that accumulates on it. This process is continued until a number of planes have been analysed, and the most damaging plane identified. Typically, under conditions of plane stress, possibly 18 calculations 10 degrees apart, would need to be made.

An alternative approach might be to carry out an analysis of the variation of the orientation of either the principals or the plane of maximum shear. For example, a time at level at a particular angle, analysis might be used to gauge just how long the plane was orientated in a particular direction. From such analyses more specific damage calculations could be carried out thus reducing the number of such calculations required and so greatly reducing the total computational time.

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Section 5 - The Stat ist ical Nature of Fatigue.

1.0 Background.

A considerable amount of time and effort have been expended in the statistical analysis of fatigue data and in reasons for the variability in fatigue-test results. Since fatigue life and fatigue limit are statistical quantities, as are most other measured parameters, it must be realised that considerable deviation from the average curve determined by only a few specimens is to be expected, Figure 38.

Figure 38 Plot of stress-cycle (S-N) data as it might be collected by laboratory fatigue testing of a new alloy

Figure 39 Distribution of fatigue specimens failed at a constant stress level as a function of logarithm of life.

It is necessary to think in terms of the probability of a specimen attaining a certain life at a given stress or strain level or the probability of failure at a given level near the fatigue limit. To do this requires the

100

80

60

40

20

0104 105 106 107 108

Stre

ss σ

, psi

x 1

0-3

Cycles to failure, N

xxx xx xxxxx x

xxx xx xxxxx xxx x x x

xxx xx xxxxx x

xxx xx xxx x xx

xxx xx xx xx xx x x x

x x

o→

o→o→o→o→o→o→

stress s = constant

Num

ber o

f spe

cim

ens

faile

d

Log N (cycles to failure)

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testing of considerably more specimens than might be expected so that the statistical parameters for estimating these probabilities can be determined.

2.0 Representation of Fatigue Data on a Statistical Basis.

The basic method for expressing fatigue data should, therefore, be a Wöhler line which has been expanded to form a surface representing the relationship between stress (or strain), number of cycles to failure and probability of failure. A distribution of fatigue life at constant stress is illustrated schematically in Figure 40 and based on this, curves of constant probability drawn. Thus at σ1, 1 percent of samples would be expected to fail after N1 cycles, 50 percent after N2 cycles, etc.

Figure 40 Representation of fatigue data on a probability basis

The figure also indicates a decreasing scatter in fatigue life with increase in stress, which is usually found to be the case.

3.0 The Statistical Distribution Function.

The statistical distribution function which describes the distribution of fatigue lives at a constant stress or strain level is not accurately known, for this would require the testing of over 1000 identical specimens under identical conditions at a constant stress. Workers have tested up to 200 steel specimens at a single stress and found that the frequency distribution of cycles to failure, Nf, followed the Gaussian, or normal, distribution if the fatigue life were expressed as log(Nf). For most engineering purposes it is sufficiently accurate to assume a logarithmic normal distribution of fatigue life at constant stress in the region of the probability of failure of P = 0.10 to P = 0.90. However, it is frequently important to be able to predict fatigue life corresponding to a probability of failure of 1 percent or less. At this extreme limit of the distribution the assumption of log-normal distribution of life is no longer justified, although it is frequently used. Alternative approaches have been the use of the extreme-value distribution or the well known Weibull distribution.

4.0 Probability of Failure at a Finite Life.

p=0.99

p=0.01p=0.

p=0.90p=0.99

p=0.10p=0.01

σ1

N1 N2 N0

Mean Fatigue Limit

Average curve P=0.50

Number of cycles (log scale)

Stre

ss

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For a given stress or strain level, fatigue lives from a series of tests may be plotted on probability paper in order to estimate probability of failure. As mentioned above it is usual to use either log-normal or Weibull probability paper. To plot sample data in a meaningful way the following procedure may be used :

- Order the array of data with the smallest value of life listed first.- Assign a rank, r, to each data value, i.e. 1 to the first, 2

to the second and n to the last, the longest life.- Determine the plotting position for each data point by

dividing its rank, r, by n+1, where n is the total numberof measured lives.

- Plot the magnitude of the life at its corresponding plottingposition on the chosen probability paper, either log-normalor Weibull.

Having plotted the data, the probability plot is examined for linearity. If the lives fall on a straight line, it may be concluded that the chosen distribution is appropriate.

As an example of the procedure, consider the following table of lives for constant amplitude fatigue tests at 345 MPa.

Rank Cycles to FailurePlotting Position

r x 10-5 100 r / (n+1)1 4.0 11.12 5.0 22.23 6.0 33.34 7.3 44.45 8.0 55.66 9.0 66.77 10.6 77.88 13.0 88.9

Table 12: Lives for fatigue constant amplitude tests at 345MPa

These data have been plotted in Figure 41, from which the mean, 50%, or characteristic , 63.2% lives can be read off directly. Extrapolation of the line to lower probabilities of failure are also possible.

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Figure 41 Weibull plot for fatigue data from Table 9.10. (copyright ASTM; adapted with permission)

2x105 4 6 106 2 4 6 107

9590

8070605040

30

20

10

5

63.2

N.Cycles

Prob

abili

ty o

f fai

lure

(%)

NaNo

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5.0 Probability of Failure for Infinite Life.

Design for infinite life requires operating stresses which are all smaller than the fatigue limit. As with the main part of the Wöhler line, the fatigue limit itself can be treated as a statistical quantity. For a statistical interpretation of the fatigue limit we are concerned with the distribution of stress at a constant fatigue life. The fatigue limit of steel was once considered to be a sharp threshold value, below which infinite life would prevail. However, it is now recognised that the fatigue limit is really a statistical quantity which requires special techniques for an accurate determination. For example, in a heat treated alloy steel the stress range which would include the fatigue limits of 95% off all specimens tested could easily range from 280 - 360 MPa.

The statistical problem of accurately measuring fatigue limit is complicated by the fact that we cannot measure the individual value of fatigue limit for any given specimen. We can only test a specimen at a particular stress, and if it fails, then the stress was somewhere above the fatigue limit of the specimen. Since the specimen cannot be retested, even if it did not fail at the test stress, we have to estimate the statistics of the fatigue limit by testing groups of specimens at several stresses to see how many fail at each stress.

Several testing procedures have been evolved for making statistical estimates. Two of the most commonly used procedures are known as Probit analysis and the staircase method. It is beyond the scope of this text to detail these procedures, however, if the reader is interested, a good account may be found in "Guide for Fatigue Testing and the Statistical Analysis of Fatigue Data" ASTM STP 91-A, 2 edition, 1963.

6.0 Handling Statistics Under Random Loading Conditions.

The statistical concepts outlined above have been and remain mainly applicable to fluctuating stresses which have a constant amplitude, rather like the rotation of Wohler's railway axles. However, the nature of the fatigue problems being addressed today requires the treatment and handling of random loading situations. Clearly, the log-normal or Weibull probability approaches involving probability paper would be totally inappropriate for use in fatigue life estimation of real field data consisting of several thousand, if not millions, of data values. Several factors have come together to meet this challenge of providing statistical information during the process of fatigue life estimation.

In the late sixties, the switch away from the so-called nominal stress, S-N, based fatigue approach to the local stress-strain approach led workers to design fatigue testing procedures which are inherently much more precise. It became necessary to control and measure local stresses and strains during the fatigue test and so the "bubble of uncertainty" associated with these parameters in a nominal S-N test was dramatically reduced. Great improvements in laboratory testing procedures, compare Figure 42 and Figure 43, were required to achieve the necessary results, and this has gone a long way towards reducing the scatter in measured data and so mitigated the extent of the effect. That is not to say of course that it does not remain a problem which needs to be addressed, it's just

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less visible.

Figure 42 Schematic diagram of a computer-controlled closed-loop fatigue testing machine

Load

Strain

Readout instrumentation

Digitalprocessorinterface

Digitalcomputer

Analogcontroller

Mode selector

Displacement

Loading Frame

Hydraulic actuator

Servovalve Displacement

Strain

LoadSpecimen

Displacement

strain

load

Transducersignalconditioners

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Figure 43 Rotating-bending fatigue testing machine of the cantilever bending type

In addition to this, the need to process large amounts of data, together with the advent of low cost computing has meant that procedures have been developed for on-line statistical analyses, which whilst being firmly based on the principles outlined above, differ in their execution. Figure 44 illustrates the difference in approach.

Figure 44

Instead of considering the life at a particular stress or strain level to be distributed according to some

ω

ωt

Load bearingsSupport bearings

FA

Specimen

101

100

10-1

10-2

101 102 103 104 105 106 107 108 Ne

N

εa [%]Ck 45 σ1’ = 1326 N/mm2 b= -0.1387

εf’ = 0.4020 c= -0.5283ε2 = 0.1285% 25 exp. results

εa p, ε′f

·2N( )c 10q Sp⋅••=

εa e,σaE------

σ′fE----- 2N( )b 10q Se⋅••= =

εa = εa,e + εa,p

Sp=0.0667

Se=0.0228 N NE≤

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chosen distribution such as log-normal or Weibull, it is convenient to think of the stress or strain being distributed, at any life, according to the same distributions. In Figure 44 for example, the strain life curve is made up of two components; the plastic strain εp, and the elastic strain, εe.

The equation of the strain life curve may then be written as :

ε = εe + εp

otherwise

ε = σf' / E (2Nf)b + εf' (2Nf)c

Where 2Nf represents the number of reversals, half cycles to failure, and the other parameters are material constants. This equation can now be expressed statistically in terms of probability of survival :

ε = σf' / E (2Nf)b.10q.e + εf' (2Nf)c.10q.p

Where :

q - is the number of standard errors from the mean lifee - is the standard error of 2Nf for the elastic life linep - is the standard error of 2Nf for the plastic life line

Typically, the elastic and plastic component standard errors can be calculated directly from the results of the log elastic and log plastic vs log reversals regression analyses, generally :

Std. Error = SDe ( 1 - r2)0.5

Where :

SDe - is the standard deviation of log(e) either elastic or plasticr - is the regression correlation coefficient.

For a normal distribution, the following values of 'q' can be assumed :

No. of Standard Errors from mean, q Probability of Survival %-3 99.87-2.33 992 97.7-1.28 90-1 840 501 161.28 102 2.32.33 13 0.13

Table 13: Values of q and corresponding probability of survival

For example, if the following material parameters are assumed :

εf' - 0.4σf' - 1326c - -0.528

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b - -0.139E - 205000SDe - 0.023SDp - 0.067

Then the following values of reversals to failure, 2Nf, can be calculated for a 10% and 90% probability of survival.

Probability of survival %50 10 90

Plastic εf' 0.4000.456 0.350Elastic σf' 1326 1417 1240Reversals2Nf 1769 23071355

Which means that there is a 90% chance that the specimen will survive 1355 reversals and only a 10% chance it will survive 2307 reversals.

In the above examples, we have assumed values of q, the number of standard deviations from the mean, for a log-normal distribution. It would be equally valid to use values which represent the Weibull distribution. Indeed, for confidence levels greater than 90% it is recommended to use Weibull.

This approach is by no means confined to a treatment of strain life curves, S-N curves can also be considered by these means see for example Figure 42.

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7.0 The Absolute Accuracy of Fatigue Life Estimation.

From the above discussion, it has been indicated that fatigue life is a rather statistical quantity to which it is very difficult, if not impossible to ascribe a unique value. It has been shown that the material law used to describe the fatigue response of a material, the Wöhler line (stress or strain), is itself not unique and must be treated as a statistical quantity. However, Wöhler lines are amenable to statistical treatment and so a measure of the variability resulting from the determination of material properties can be defined. In addition to this variability, other sources of error can and most often do arise.

Errors in the measured loading environment can also lead errors in calculated life. As a rule of thumb, a 10% error in strain or stress measurement can either half or double life, all else being equal. Similarly, errors in the description of component geometry can lead to significant variations. For this reason, and for complicated geometries in particular, finite element analysis is being used to accurately describe the state of stress at any point within a structure.

Taking all these factors into account, today the state of the predictive art with respect to fatigue life is to within between a factor of two of the actual, which itself you should now realise is not a unique number. Expressing this mathematically we can say, for any given calculated life, Lc, the largest scatter band we should expect is given by the expression :

( 2Lc - Lc / 2 )

If our calculations are significantly more widely scattered than this, then a major error in one or more of the inputs to the analysis has been made and must be sought out if the predictive process is to continue.

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Section 6- Fracture Mechanics Based Fatigue Crack Growth Analysis

1.0 Introduction

In general engineering, the possibility of crack like defects being present at the start of service life has to be recognised. This is particularly true in the sense of "what if we can't be sure its not cracked"? i.e. the safest assumption is that the biggest crack that non-destructive testing cannot reliably find is present. Even if the component is not cracked to start with, the remnant life after initiation may be significant or possibly dominant with respect to total life. Furthermore, for inspection related activities, the remnant life of a component discovered to be cracked during inspection needs to be determined so that decisions regarding serviceability can be supported by reliable predictions.

In low stress (linear elastic) situations, the growth of fatigue cracks governs useful life leading to the "defect tolerance approach" to design which necessarily also implies regular inspection and assessment. In service, fatigue loading is nearly always random and the operating environment is nearly always corrosive or aggressive in nature.

Diagnosis or life prediction can only be achieved in one of two ways:- laboratory testing under accurately simulated service conditions of geometry, loading and especially environment; or prediction from quantitative models of crack growth processes. Laboratory testing is very expensive and, because of the time dependent corrosion influence which prohibits test acceleration, it is also very time consuming. For example, an off-shore platform designed to withstand 108 cycles in a 30 year life would require a 30 year test at 0.2Hz to validate the design. Computer based prediction methodologies can perform the same validation with equal accuracy in a very small fraction of the time (e.g. less than two hours) at a very small fraction of the cost.

Such prediction software requires the crack growth damage to be accumulated on a cycle by cycle basis and must also incorporate the complexing effects of stress history, crack closure, static fracture modes and notches and be able to predict the environmental influence on fatigue crack growth rates (corrosion fatigue). Based on the fracture mechanics stress intensity parameter, K, the similitude concept allows material response measured in standard laboratory tests to be used with geometric K solutions for the structures under consideration to predict crack growth rates and propagation life.

The stress history input needs to be obtained from service recordings though in many engineering sectors such as off-shore, bridges, aerospace, power generation, ground vehicles etc., there are standard histories or duty cycles.

The crack growth software developed by nCode offers the engineer the facilities to predict defect tolerant fatigue lives based on all these aspects. These diagnostic techniques enable iteration and interpolation with different material, geometry and history inputs as well as probably the most important variable, the initial crack size i.e. the biggest crack that NDT cannot find.

1.1 Fracture Mechanics

The branch of engineering science on linear elastic fracture mechanics (LEFM) is by no means a new one. The earliest work in the UK dates back to Inglis (1913) but the major developments took place following the research by Griffith at RAE in 1920 and Irwin in the USA in 1956 and flourished from then onwards.

The driving force for a crack to extend is not the strain or stress but the stress intensity factor, universally known as K. This is not to be confused with the stress concentration factor, Kt or with the strain hardening exponents k or k'. The stress intensity factor embodies both the stress and the crack size and uniquely describes the crack tip stress field independent of global geometry. The

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relationship may be viewed as a triangle:-

Figure 45 The Fracture Mechanics Triangle

The triangle may be solved by knowing two of the three corners to derive the third. For example, the material toughness, KIC, and the NDT crack size limit may be known and hence the working stress may be derived to provide safety against fracture.

The relationship between K, stress and crack size is:

K = Y . stress ( π .crack size)1/2

The compliance function, Y, describes the geometry of the structure, component or specimen in which the crack exists. It takes the value 1 for an internal defect in an infinite body and 1.12 for a surface crack in a semi-infinite body. For finite sized bodies, i.e. real structures, Y may be seen as a geometric correction which takes a general form:

Y = C1 + C2 (a/T) + C3 (a/T)2 + C4 (a/T)3 + C5 (a/T)4

though it may take a much more complex mathematical description in some cases.

In fatigue, again the stress intensity factor provides the driving force for crack growth in the sense that the rate of crack growth, da/dN, is governed by the cyclic range of K, ∆K. The two are related by a crack growth "law", the best known being the Paris Law derived by Paul C. Paris at Lehigh University in 1960:

da/dN = C ( ∆ K)m

In terms of fatigue crack growth and overall life, a rectangular rather than a triangular representation is used:

FractureMechanics

Stress intensity

StressCrack size

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Figure 46 The Fatigue Crack Propagation Rectangle

Here we have a relationship between fatigue stress range and fatigue life, in the same way as in an S-N curve approach, but we now also have the initial and final crack sizes and indeed all the crack sizes in between these limits throughout the fatigue crack propagation period. In a similar way to solving the triangle, the fatigue crack propagation rectangle can be solved by declaring the three known corners to derive the unknown fourth.

Fatigue CrackPropagation Life

Final crack size Cycles to failure

Stress rangeInitial crack size

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2.0 Complexing Effects

In service, components and structures experience fatigue crack growth under complex situations and without proper modelling of these complexing effects, accurate prediction using fracture mechanics methods would be impossible. The effects that have to be taken into account include:

crack closureif the crack is closed for part of the fatigue cycle then the effective ∆K and hence growth rate are less.

historythe crack extension in a cycle actually depends on the previous historical sequence of cycles i.e. linear damage accumulation (Miner's Rule) is not appropriate and history effects such as retardation have to be modelled.

notchesfatigue cracks growing at a notch behave differently until they are large enough to escape from the influence of the notch.

static fracturegrowing fatigue cracks eventually reach a failure condition at which point the crack growth rate is essentially infinite and as this point is approached, the contribution from static fracture modes increases.

environmentmaterials data may be obtained in the relevant environment but each change of cyclic frequency, waveform or temperature constitutes a new dataset so that modelling the environment without relevant data becomes a necessity.

Until the recently, fatigue crack growth prediction was based on approximate methods even though it was known that all the effects described above made the approximating assumptions invalid. However, the advent of sufficiently powerful computers now makes the desired cycle by cycle approach possible and the future will be bring yet more powerful computers and speedier predictions of fatigue crack growth.

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3.0 Background to Fatigue Crack Growth Analysis

The basic premise of the fracture mechanics approach to fatigue relies on the "similitude concept" which adopts the simple belief that if the K is the same, the response is the same. In fatigue terms, the similitude applies to the cyclic range of K, ∆K, and the "response" is the rate of fatigue crack growth. It should be clearly understood that even though the growth of macroscopic fatigue cracks is governed by linear elastic fracture mechanics, the microscopic crack extension process of fatigue still necessitates local plasticity. At the tip of a fatigue crack, there are two plastic zones, one for monotonic shear the other for reversed shear, whose sizes depend on the ratio of (K/ σy )2. Hence, these crack tip plastic zones are negligibly small in relation to crack size, especially for high strength materials, but essential to the fatigue crack growth process.

To undertake a fatigue crack growth life prediction, it is necessary to adopt a cycle by cycle approach and to have a good knowledge of the service loading environment, the appropriate properties to describe materials response, a geometric description and the means of combining this information into a coherent model of the fatigue crack growth process. This is viewed as a set of interconnecting boxes as shown in Figure 47.

Figure 47 Schematic Illustration of Remnant Life Prediction Case

3.1 The service environment

The cycle by cycle fatigue crack growth modelling requires a sequence of cycles to be extracted from a representative stress-time history. Representative here means not only the sequence of stresses but also the durations, cyclic frequencies and waveforms. These help to define the environmental effects in corrosion fatigue and the corrosive medium itself should also be characterised in terms of

Stress-time

Geometry

Material

Cycle by cycle fatiguePost processing

Data

and interpolation crack growth

history

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composition, temperature etc.

Figure 48 Display of stress histories with different frequencies and waveforms.

For example, slower frequencies produce faster crack growth rates, on a da/dN basis, and sine waveforms give faster rates then square waveforms. A multi-file display of stress histories with different frequencies and waveforms is shown in Figure 47. Stress-time histories and cycle counting, including sequence re-ordering must be dealt with.

3.2 Geometric description

The geometric description of a component or engineering structure is accomplished by means of the appropriate stress intensity solution. This is a mathematical description of the relationship between K, stress (range) and crack size for the geometry in question and many solutions have been derived and documented.

3.3 Materials response

The material response is modelled by measuring fatigue crack growth rates as a function of ∆K in constant load amplitude laboratory tests using small scale specimens. These can be carried out under the correct environmental conditions and then the similitude concept is invoked to apply the data to large scale components and structures. Also the fatigue crack growth threshold characteristics and the material fracture toughness are measured using the same type of specimens and all these tests are standardised e.g. in BSI and ASTM Standards.

The parameters which characterise the materials response in terms of fatigue crack growth have been measured for many materials, in standard tests, reported in the literature and now built into computer databases.

3.4 Cycle by Cycle approach

The cycle by cycle approach has been adopted as the only way in which fatigue crack growth can be

SINESTRESSMPA

TRIANGULARSTRESSMPA

SQUARESTRESSMPA

’SAWTOOTH’STRESSMPA

220

220

220

220

220

TRI-2STRESSMPA

0

Time(s)

10

54

54

54

54

54

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accurately modelled. In each fatigue cycle in turn, (the correct order must be preserved), the apparent ∆K is calculated from:

∆Kapp = Y. ∆σ (πa)1/2

where ∆σ is the stress range of the cycle, a is the current crack size and Y is the compliance function obtained from the relevant look up table generated for the geometry in question. This value of ∆Kapp has to be modified, if necessary, to account for the effects of potentially complicating features until the effective ∆K, the driving force that the crack tip actually experiences, is derived. Only then can the crack extension, the amount of crack growth or damage, be calculated simply from the Paris Law as:

da = C ( ∆Keff)m

This is added to the current crack size and the process repeated until the crack growth life is exhausted.

Note that this approach disregards all other crack growth laws developed since the original Paris Law, to account for complexing effects such as Forman's Law for static fracture modes and the Klesnil and Lukas correction for threshold. The Paris Law holds if, and only if, the effective ∆K is used and all possible complicating features are modelled through effective ∆K.

For each of the features of mathematical models, a description is given in the following sections:

3.5 Initial crack size

The initial crack size, to which individual cyclic crack extensions are cumulatively added, may be preset but often is unknown. A minimum initial crack size is modelled to ensure that:-

a) the crack size is not too small to be handled by the concepts of fracture mechanics i.e. it does not violate the assumption of isotropic continuum by being smaller than a metallurgical feature such as grain size.

b) the crack size at the start of the propagation phase is equal to the crack size at the end of the initiation phase i.e. critical location fatigue analysis can be combined with fracture mechanics crack growth analysis with ensured continuity to give total fatigue life.

The minimum crack size is derived from the so called "short crack parameter" which is calculated from the "Kitagawa Diagram". An example of a Kitagawa Diagram, which relates threshold stress

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and threshold ∆K to fatigue crack size is given in Figure 49.

Figure 49 An example of a Kitagawa Diagram for Fatigue

The short crack parameter, lo, is given by:

lo = (∆Kth / ∆σo)2 / π

and is taken to be the minimum initial crack size. It is in fact calculated in each cycle using the value of ∆Kth for the stress ratio, R, of that cycle for other purposes, but for the minimum initial crack size it is calculated for R=0.

3.6 Crack closure and thresholds

Crack closure can be caused by a number of mechanisms including plasticity, oxide, roughness, corrosion product, metallurgical transformation and incompressible viscous fluids. Starting from the viewpoint that a crack cannot grow when it is closed, then if crack closure occurs at a level, Kcl, that is above Kmin in the cycle, the effective range of stress intensity becomes:

∆Keff = Kmax - Kcl

The crack growth rate, da/dN, will be reduced as the driving force that the crack tip experiences is reduced from ∆Kapp to ∆Keff. In the limit, as ∆Kapp is reduced, Kmax reaches Kcl and ∆Keff becomes zero; crack growth ceases but there remains a positive applied ∆ K, ∆Kapp, identified with the threshold for fatigue crack growth, ∆Kth.

Log

(str

ess r

ange

)

Log (crack size)

Fracture mechanicsis valid

Fracturemechanicsnot valid

lo

stress range α threshold x (1/crack size)0.5

stress range = fatigue limit

No fatigue FatigueKey

}

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Experimental measurements of crack closure (or opening) level are often made by use of crack mouth displacement gauges or back face strain gauges. In threshold testing, the measured closure level is subtracted from the apparent ∆Kth to still leave a positive result, sometimes termed the 'intrinsic' threshold. In fact, this is erroneous as only the mechanical closure level is remotely measured (plasticity induced closure) but not the crack tip oxide induced closure. At high stress ratio, R (=Kmin/Kmax), the mechanical closure mechanisms produce Kcl below Kmin but thresholds induced by oxide lead to an approximately constant value of 2MPam1/2 for steels. Applying the logic that at threshold (zero growth rate), the effective ∆K is zero then the overall closure level can be obtained directly from:

Kcl = ∆KRth / (1 - R)

It must be appreciated that measured ∆Kth values are linked to a cut off growth rate (10-11 m/cycle) not zero, though the error is small due to the steepness of the growth rate curve in the rear threshold region. If the closure level, Kcl, is a constant for a particular stress ratio, then in ∆K increasing tests (and service situations) the closure effect peters out to nothing as Kmin rises Kcl above so that ∆Kapp and ∆Keff are equal.

The closure level, which defines the threshold ∆K, manifests itself as a stress ratio effect on ∆Kth: higher R (higher mean stress) gives lower ∆Kth values. It has been found from a wide ranging survey of measured threshold values that the relationship can be described by the equation:

∆Kth = D0 - (D0 -D1).R/Rcrit

where

∆Kth is the threshold ∆K at a particular R value

D0 is the threshold ∆K at R=0

D1 is the threshold ∆K as R->1

R is the stress ratio for the ∆Kth in question

Rcrit is the value of R above which ∆Kth is constant and equal to D1

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The appearance of the dependence of ∆Kth on R is as shown in Figure 50.

Figure 50 The dependence of ∆Kth on stress ratio

3.7 Short cracks

A logical progression from the previous section on crack closure is to consider the apparently anomalous behaviour of short cracks. It is well established that short cracks grow at faster rates and at lower ∆K values than would be projected by long crack data and thresholds. Broadly speaking, this is because short cracks have less closure (especially plasticity induced closure) associated with them and indeed may have no threshold at all.

A definition of what is 'short' and what is 'long' arises from the Kitagawa Diagram, (see previous section on initial crack size). Clearly, if the cyclic stress range is above the fatigue limit, ∆σo, then fatigue failure will eventually occur implying zero threshold for growth whatever the defect size. For crack sizes smaller than lo, conventional fracture mechanics is unsafe in its prediction of no growth and the use of K is dubious as the crack size may well be less than the micro structural features such as grain size, pearlite colony spacing etc. The parameter lo thus defines the boundary between short and long cracks and takes values typically in the range 0.01 to 1mm for steels where high strength steels take the smallest values.

Obviously, because short cracks grow faster, they very quickly become long cracks and as they do so, the plastic deformation wake builds up the crack closure level to that pertaining to long cracks. Mechanistically, it could be suggested that short cracks are not anomalous; long cracks are, because of the closure phenomena associated with them.

Short crack growth is not normally modelled in current software tools. However, the crack length dependence of the crack closure level can be taken into account. The derivation of the formula is described elsewhere, but it describes how the closure level, Kcl, depends on Kmax (because of the plasticity effect) and on crack size in relation to lo.

Kcl = Kmax - Kmax[(a+ lo )/a]1/2 + ∆Kth/(1-R)

Delta K Threshold-Ratio Plot

7

6

5

4

3

2

1

00 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

Stress Ratio

delta

K T

hres

hold

MPa

m1/

2

EN30B Environment : Air D0:4.7 D1:2.9 Knee Ratio:0.57PD6493 lb : D0 :5.35 D1 : 2 Knee Ratio : 0.5

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Note that it reduces to the long crack value of Kcl when a is very much greater than lo. An example of the build up of crack closure level with increasing crack size is shown in Figure 51.

Figure 51 The build up of crack closure level with increasing crack size

3.8 Notches

To account for cases where fatigue cracks grow from notches, it is necessary to modify the fracture mechanics description of the crack tip stress field. The presence of the notch overrides the crack tip K solution when the crack is close to the notch but, as the crack grows by fatigue, it escapes from the notch influence. Therefore it is inappropriate to simply multiply the stress range by the stress concentration factor and equally inappropriate to ignore the notch presence in the K solution. A mathematical model is required in which the notch correction factor varies with crack size and disappears as the crack escapes from the notch influence. Cracks near notches behave like short cracks and a model developed to meet these requirements incorporates the short crack parameter,

Cross Plot of Data : CA016

40 1 2 3 4 5 6 7 8 9 10

Crack size (mm)

K a

t cra

ck c

losu

re (M

Pa m

1/2

)

5.4

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lo (see previous sections).

Figure 52 Schematic Illustration of a Crack at a Notch

Referring to Figure 52. for definition of terms, the procedure is to calculate ∆Kapp assuming the notch is part of the crack, based on a notional crack size equal to a+D. Then this value is reduced by a correction factor to account for the fact that the notch is not a sharp crack but has a significant root radius, r. The full derivation is given elsewhere and the formula to account for notch effects is:-

correction factor = ∆Keff / ∆Kapp ={[a+(aD/ lo ).( ρ / ρc)0.5] / [a+D]}0.5

where

∆Keff is the effective ∆K at the crack tip

∆Kapp is the remotely applied or apparent ∆ K

a is the crack size

∆ is the notch depth

ρ is the notch root radius

ρc is the root radius of a sharp fatigue crack

lo is the short crack parameter

The correction factor is less than unity and the formula correctly reduces to unity for a sharp notch (ρ = ρc). The factor reaches unity when the crack escapes from the notch; the crack size at this point is given by:-

a = lo ( ρ/ρc)0.5

It should also be noted that a cannot be less than lo so that the minimum correction factor is:

minimum correction factor= { [ lo +D ( ρ/ρc)0.5]/[a+D] }0.5

rD

a

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A plot of the notch correction factor against crack size is shown in Figure 53.

Figure 53 Plot of notch correction factor against crack size

3.9 Static fracture

Fatigue crack growth rate increases with increasing crack size because it is driven by ∆K and ∆K depends on (a)0.5. Nevertheless, as the critical fracture condition is approached, growth rate increases even more rapidly due to the contribution of static fracture modes to the cracking process. Eventually, the crack growth rate becomes infinite (or at least as fast as the speed of sound) when Kmax in the cycle reaches the critical K usually identified with the material fracture toughness.

To model this increase in growth rate from static fracture modes, an additional K, Kfs, is added to Kmax so that ∆Keff is increased and growth rate is increased according the Paris Law. The formulae are:-

∆Keff = Kmaxeff - Kmineff

Kmaxeff = Kmax + Kfs

Kfs = Kmax / ( KIC-Kmax )

where

∆Keff is the effective ∆K

maxeff is the effective maximum K in the cycle

Kmineff is the effective minimum K in the cycle

Kfs is the additional K from static fracture modes

Kmax is the apparent maximum K in the cycle

KIC is the material fracture toughness


0 1 2 3 4 5 6 7 8 9 10Crack size (mm)

Not

ch C

orre

ctio

n Fa

ctor

0

1.1

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Note that Kfs is small at small values of Kmax and tends to infinity as Kmax approaches KIC.

A plot of Kfs with increasing ∆Kapp is shown as an illustrative example in Figure 54.

Figure 54 Contribution from static fracture modes, Kfs, increasing with ∆Kapp.

3.10 Cycles sequence

Under constant amplitude loading, the fatigue crack growth rate response has been modelled using ∆Keff so that crack closure and static fracture are modelled by:-

∆Keff = Kmaxeff - Kmineff

Kmaxeff = Kmax + Kfs

Kmineff = Kmin or Kcl

To account for the effect of cycles sequence, the so called "history effect", an additional term is incorporated into these equations as follows:-

Kmaxeff = Kmax + Kfs - KR

Kmineff = ( Kmin - KR ) or Kcl

where KR is termed the "residual K". This residual K arises whenever a previous cycle has produced a crack tip plastic zone that extends beyond the plastic zone of the current cycle. Prior deformation therefore causes the cyclic deformation to deviate from constant amplitude behaviour and a history effect results. History effects include retardation (possibly to crack arrest), acceleration, delayed retardation and what is known as "overshoot" or "lost retardation". These invalidate linear damage accumulation concepts (Miner's Rule) and necessitate cycle by cycle modelling.

As an example, consider the case of a single overload in a constant amplitude sequence. The crack


0 10 20 30 40 50 600

10K

from

sta

tic fr

actu

re (M

Pa m

1/2

)

Apparent delta K (MPa m1/2 )

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growing under CA has two plastic zones associated with its tip: a maximum zone based on Kmax and a reversed zone based on ∆K. When the overload occurs, it then has four plastic zones at the crack tip and, because of the confusion and complexity, transient response results. The measured crack size - cycles and growth rate - ∆K information appears as in Figure 55 and 55 with retardations occurring at each overload.

Figure 55 Crack Length - Cycles Response for Single Overloads

0 1E4 2E4 3E4 4E4 5E4 6E4

0.8

0.6

0.4

0.2

0

Cross Plot of Data : SAEBRAKT

Life (Cycles)

Cra

ck s

ize

(mm

)

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Figure 56 Growth Rate - K Response for Single Overloads

In a random sequence of cycles, the largest current overload governs the response of the crack to each cycle in turn.

The original formulation for KR was due to Willenborg and was subsequently modified by Johnson at NASA at the same time as the closure concepts of Elber were also being introduced to explain and model history effects. The original Willenborg expression was:-

KR = KOL(1 - a/ZOL)0.5 - Kmax

whereKOL is the K due to the overloada is the crack extension since the overloadZOL is the plastic zone size due to the overload and given by:-

ZOL = (KOL/ σy)2 / 3π

This has been substantially expanded by Austen to account for delayed retardation and overshoot as:-

KR = KOL(1- ∆a / { ZOL-Z } )0.5 - ( KOL-Kmax ).( 1- ∆a / Zrev )0.5 - Kmax( 1- ∆a / ZOL )0.5

whereZ is the maximum plastic zone size for the current cycleZrev is the reversed plastic zone size for the overload

These are given by:-

Z = (Kmax/ σy)2 / 3π

and

Zrev = ( { KOL - Kmin } / σy)2 / 12π

15 16 17 18

2E-7

1.5E-7

1E-7

5E-8

Apparent delta K (MPa m1/2)

Cross plot of data : CA02

Gro

wth

rate

(m/c

ycle

s)

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The first term describes the decay in KR which becomes zero as the boundary of the current maximum plastic zone crosses that due to the overload. The second term describes the delayed retardation such that KR is in fact zero at the point of overload and realises its maximum effect as the moving crack tip crosses the reversed plastic zone due to the overload. The third term controls the overshoot phenomenon where, during the crack extension phase, KR is negative and increases both Kmaxeff and Kmineff. An overshoot will occur if Kmineff is thus raised above the current closure level, Kcl. During this phase the crack behaves almost as a closure free short crack and it is only when it is fully clear of the overload zone that "normality" returns.

This model not only incorporates delayed retardation, overshoot and closure but also the significant crack extension caused by an overload are modelled via:-

∆a = C [ KOL + ( KOL/ { KOL-KIC } ) - Kmin ]m

It is notable that in single overload block loading, the growth rate is significantly retarded (which should increase life), but the overall life may in fact be reduced because of premature failure at an overload.

Clearly, history effects are complex but they are physically observed and it is only through cycle by cycle modelling of the crack tip response that the actual behaviour is reproduced in prediction methods and software.

3.11 Environment

Fatigue crack growth rates, and hence lifetimes, are strongly influenced by the environment such that corrosion fatigue is a synergistic phenomenon in which corrosion and fatigue are mutually enhanced. Cracks generally adopt that process which gives the fastest rate with the easiest or lowest energy path and this was termed "process competition" in corrosion fatigue modelling. To predict lifetimes accurately, environmental influences have to be accounted for. If materials data is available for the precise environment concerned, then they can be used for accurate predictions. However, each change of cyclic frequency, waveform, temperature, electrochemical potential or chemical composition of the environment produces a requirement for a new dataset and clearly this is impossible to accommodate with a database even if the test data were available.

Materials data for air can be combined with models for the environmental effects on fatigue crack growth rate for the most common environmental mechanism, that of hydrogen embrittlement. In this way, corrosion fatigue can be modelled in the absence of appropriate corrosion fatigue crack growth data for the material/environment system.

In aqueous environments, two electrochemical processes occur: the anodic reaction dissolves metal which makes the crack less sharp, the cathodic process generates hydrogen which, in the presence of cyclic deformation, concentrates in the crack tip plastic zone. Crack tip blunting reduces Keff and slows down the growth rate; localised hydrogen embrittlement introduces an alternative, faster growth rate.

The equation with governs the growth rate for local hydrogen embrittlement is:-

da/dN = greater of [ ( L - s ).( δ - s ) / ( Z - s ) and s]

whereL is the hydrogen penetration distance = 4(DH.t)0.5

s is the striation width = C ( ∆Keff)mδ is the maximum crack tip opening

= 0.12( DK / {1-R} )2 / sy.EZ is the plastic zone size given by Z = (Kmax/ σy)2 / 3πt is the cycle rise timeDH is the hydrogen diffusion coefficient

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The most important parameter is the hydrogen diffusion coefficient and published values very widely between 1e-7 and 1e-12 m2/sec. The effect of the hydrogen influence on crack growth rates and life is compared to behaviour in air in Figure 57. In non aqueous hydrogen bearing environments or if cathodic protection suppresses anodic dissolution in aqueous ones, then only the hydrogen effect needs to be modelled leading to the often observed plateau effect in growth rates (Figure 58).

Figure 57 Growth Rate Behaviour for Local Hydrogen Embrittlement

Cross Plot of Data : LHEAIR

___ Hydrogen Embrittlement___ Air

Apparent delta K (MPa m1/2)

Gro

wth

rate

(m/c

ycle

s)

1E-4

1E-5

1E-6

1E-7

1E-8

1E-920 40 60 80

AIR

LHE

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Figure 58 Crack Size - Life Behaviour for Local Hydrogen Embrittlement

The blunting effect is accounted for by reduction in ∆Keff according to:-

∆Keff = ∆K ( rsharp / rblunt)0.5

where

ρsharpis the sharp or air crack tip radius and

ρbluntis the blunt or corroded crack tip radius

A full derivation of the derivation of ρblunt from Faraday's Laws of electrochemistry may be found elsewhere. The most important parameter here is the anodic current density and again, published values vary widely from le-5 to 1 amp/sq cm. The effect of blunting alone, which may be experienced in square wave loading for example, is shown in comparison to air behaviour in Figure 59 and Figure 60. The overall combined effect of crack tip blunting and local hydrogen embrittlement is shown in Figure 61 and .

LHE Air

70

60

50

40

30

20

100 0 5E4 1E5 1.5E5 2E5

Cra

ck s

ize

(mm

)------ Air ----- Hydrogen Embrittlement

Cross plot of data : LHEAIRDK

Life (cycles)

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Figure 59 Growth Rate Behaviour for Crack Tip Blunting

Figure 60 Crack Size - Life Behaviour for Crack Tip Blunting

Figure 61 Growth Rate Behaviour for Combined Corrosion Fatigue Processes

AIR

CTB

------ Air ----- Crack BluntingCross plot of data :AIRCTB

20 40 60 80 100Apparent delta K (MPa m1/2)

E-4

E-5

E-6

E-7

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Cycles101 102 103 104 105

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Crack Size - Life Behaviour for Combined Corrosion Fatigue Processes

AIR

CTB

------ Air ----- Combined CTB/LHECross plot of data :AIRCOMBI

20 40 60 80 100Apparent delta K (MPa m1/2)

1E-4

1E-5

1E-6

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70

60

50

40

30

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100 0 5E4 1E5 1.5E5 2E5

Cra

ck s

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(mm

)

------ Air ----- Combined CTB/LHE

Cross plot of data : AIRCOMBIAN

Life (cycles)

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Section 7 - Fatigue Modeling And Analysis Within The FATIMAS System

1.0 Introduction

The purpose of this section is to provide a broad overview of the software analysis system and to discuss, in particular and in detail, the functions and capabilities of the fatigue analysis suites.

First, a short description of the fatigue technologies which underlie the FATIMAS analysis systems is given. This is followed by a detailed discussion on the materials properties required for use in fatigue analyses and how these are determined by means of standardised laboratory tests. The modules which comprise the fatigue analysis systems are then described individually. Finally, three "case studies" or "worked examples" are presented which illustrate the use of each of the fatigue analysers: Stress-Life (S-N), Local Strain and Fracture Mechanics.

2.0 An Overview of Fatigue Analysis Methodologies

2.1 Introduction to the fatigue process

In general terms, the fatigue process can be viewed in terms of the following expression:

Nf = Ni + Np

where: Nf =Total Fatigue Life

Ni=Life to Crack Initiation

Np=Fatigue Crack Propagation Life

In a sense, the three main methods for fatigue durability analysis also follow this expression. The traditional Stress-life (S-N) method models total fatigue life (Nf) and, thus, encompasses both the crack initiation and crack propagation phases simultaneously but models neither of these accurately. The Local-Strain method considers fatigue crack initiation only (Ni) and Fracture Mechanics is used to model fatigue crack propagation (Np). Both of these latter two methods, because they each consider the driving force for fatigue using proven materials models, are highly accurate and reliable given high quality inputs to the analysis.

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Regardless of the particular analysis technique applied, similar types of inputs are required; these can be described by means of the "5-Box Trick" shown in Figure 62. In each case, the loading environment of the structure under consideration must be defined: the Loading History box. Also, some form of geometry factor or description must be made: the Geometry box; this may take the form of a component S-N curve, a fatigue strength reduction factor (Kf) or a compliance (Y) function, depending upon the type of analysis used. Finally, the response of the material to cyclic loading must be defined: the Material Data box; an S-N curve, Strain-life and cyclic stress-strain curve or Paris relationship as appropriate. These three inputs are then combined in a cycle-by-cycle fatigue analysis and an initial result presented. This result has to be considered as preliminary as each of the inputs (load history, geometry and materials data) are subject to statistical variation and manipulation. It is essential, therefore, that the initial fatigue analysis result be post-processed to determine the sensitivity (scatter) of the result due to both minor and major changes to the inputs. This helps engineers to gain a "feel" for how the fatigue performance of the structure under consideration can be modified, by design changes, say, or what the range of expected fatigue lives may be for a population of structures. These "sensitivity" factors are considered in more detail in the "case studies" section below.

Figure 62 The fatigue analysis "5-Box Trick"

2.2 Nominal Stress (S-N) Fatigue Life Analysis

Within the fatigue analysis software system, Nominal Stress, or S-N, fatigue life estimations are carried out with the program module called SLF which forms part of the FATIMAS program suite.

Loading

Geometry

Material

Cycle by cycle fatiguePost processing

Data

and interpolation analysis

history

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2.2.1 Basic approach

SLF makes fatigue life estimations using the Nominal Stress or S-N method and models total fatigue life; i.e. no differentiation is made between the initiation and propagation phases of fatigue crack "life". This approach takes no account of localised cyclic plasticity, the real cause of fatigue damage, such as might be found at a geometric stress raiser. Account for such effects must, therefore, be made through the S-N fatigue damage curve; i.e. use a component-based S-N curve for the "material properties".

2.2.2 Fatigue damage (S-N) curves

SLF uses as "material properties" Stress-Life (S-N) fatigue damage curves which have been produced from laboratory tests conducted under elastic loading. This loading may be "simple" (constant amplitude sine waves, say) or "random". The important factor is that both the S-N curves and the analysis methods are based around considerations of nominal elastic stresses. By convention, S-N curves are defined as Log Stress Range vs Life in Cycles. A typical S-N curve is shown in Figure 63.

Figure 63 Typical S-N curve showing 3SD scatter lines

2.2.3 Statistical factors and scatter

It is possible, if known, to include within the S-N damage curve parameters a statistical "Standard Error" factor. This factor is used to define the scatter in the S-N data and, hence, the confidence limits which may be placed on the fatigue analysis results, generally 3 standard deviations from the mean life line. In general, the S-N method will give significantly less accurate fatigue life estimates than either the Local Strain (CLF) or Fracture Mechanics (LEFM) techniques also available within the

SRI1 : 1.201E4 b1 : -0.3333 b2 : -0.2 E : 2.07E5 UTS : 500

_____ class F

Stre

ss R

ange

(MPa

)

1E0 1E1 1E2 1E3 1E4 1E5 1E6 1E7 1E8 1E9 1E10

1E5

1E4

1E3

1E2

1E1

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software system. This is because the S-N method does not model cyclic plasticity and only considers net section elastic loads.

2.2.4 The significance of turning points

The Nominal Stress (S-N) method only needs information about the amplitudes of the test signal. Sequence and frequency effects have no real significance when calculating fatigue durability using this technique. By extracting just the Peak-Valley or Turning Points from the signal file, the input data is reduced (by about a factor of ten) and the analysis runs much faster.

2.2.5 S-N method applications

SLF can be used when considering the fatigue performance of:

Welds (including spot welds) and other structures which may contain cracks or crack-like defects.

Machined components

Forgings

Castings

Pressings

Non-metals; e.g. plastics, composites etc.

2.2.6 A note of caution

SLF should not be applied when considering the fatigue performance of safety-critical components where fatigue crack initiation (CLF) or fatigue crack propagation (CRG) considerations could be used as alternative and more rigorous modelling techniques.

2.3 Local Strain (e-N) Fatigue Life Analysis

Within the fatigue analysis software system, Local Strain, or e-N, fatigue life estimations are carried out with the program module called CLF which forms part of the FATIMAS program suite.


The Local Strain method is based around empirical 'rules' first defined in a unified form by Manson & Coffin in the mid-1950s.

The method models the response of a material to cyclic plastic strains at a fatigue critical location; i.e. the deformation of a small volume of material which is subject to cyclic plasticity and which is completely surrounded (constrained) by material which is deforming elastically, as illustrated in Figure 64.

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Figure 64 Constrained plastic deformation under cyclic loading due to stress concentration feature

The Local Strain technique requires both a strain-life fatigue damage curve (Figure 65) and a cyclic stress-strain curve (Figure 66) to be defined for each material. Both of these sets of material properties are determined from strain-controlled fatigue tests conducted using a number, usually 8-15, of carefully prepared smooth specimens.

Figure 65 A typical Strain-Life fatigue damage curve showing 'elastic' and 'plastic' life lines

STRESS CONCENTRATION INDUCES ZONE OF CONSTRAINED

PLASTIC DEFORMATION

APPLIED CYCLIC LOADING

BULK OF STRUCTURE DEFORMS ELASTICALLY

Strain Life PlotSAE 45_390_QTsr’ : 1408 b: -0.07 Ef’ : 1.51 c: -0.85

Life (Reversals)

Stra

in A

mpl

itude

(m/m

)

1E0

1E-1

1E-2

1E-31E0 1E1 1E2 1E3 1E4 1E5 1E6 1E7 1E8 1E9

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Figure 66 A typical cyclic Stress-Strain curve

NOTE: 'Failure' in Local Strain fatigue terms is not fracture but, rather, the generation of a small, growing fatigue crack which would almost certainly, under continued loading, propagate to a size at which fracture would occur.

The Local Strain method is used to model the life to crack initiation; i.e. the appearance of a growing fatigue crack perhaps no more than 1mm long. The method is widely used in, particularly, the ground vehicle industries to model the fatigue performance of 'safety critical' systems such as suspension, steering & chassis components - those items where the likelihood of a fatigue crack being produced is considered proper grounds for saying the component has 'failed' or that such an event must be avoided at (almost) all cost.

CLF uses the Manson & Coffin 'rules' which define fundamental material properties, plus the application of other techniques which include:

• Neuber's Rule to translate nominal, elastic stresses & strains to their local elastic-plastic equivalents.

• Massing's Hypothesis to model the cyclic stress-strain hysteresis response of the chosen material to the applied loading.

• Mean stress correction based on procedures defined by Smith, Topper & Watson (STW) and Morrow - this allows the modelling of components subjected to random loading sequences.

• Miner's Linear Damage Summation to determine the 'failure' life.• 'Rainflow' cycle counting to define the distribution of loading cycles and fatigue damage.

Cyclic Stress-Strain Plot_____ 150M19

n’ : 0.151 K’ : 1065 E: 2.07E5

Strain (M/M)0 0.005 0.01

900

Stre

ss (M

Pa)

0

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The combination of these, and other, techniques in CLF allow the modelling of material response to cyclic loading in a highly precise manner - identical to that actually experienced by the material itself in service or laboratory testing.

2.3.2 Strain-Life method applications

CLF should be applied (to metallic structures) when considering the fatigue performance of:

Machined componentsForgingsCastingsPressings

CLF should not be used when considering the fatigue performance of:

• Welds (including spot welds) and other structures which may contain cracks or crack-like defects.

• Non-metals; e.g. plastics, composites etc.

2.4 Linear Elastic Fracture Mechanics (LEFM) and Fatigue Crack Propagation Life Analysis

The crack growth software suite allows the rapid prediction of fatigue crack growth and propagation life by using the principles of LEFM to model crack extension, under the action of either simple or random loading sequences, on a cycle-by-cycle basis. The fatigue analysis procedures used enable the quantitative modelling of effects including: crack closure, notches, loading sequence effects, environmental influences (e.g. corrosion fatigue) and static fracture modes; all of which occur in real structures.


For many types of engineering structure, it has to be recognised that there is the possibility of crack-like defects being present at the start of service life. In such cases, it is reasonable and "safe" to assume that the largest defect which non-destructive testing can fail to find is present and that this is located in the most "critical" location. Additionally, many components develop (initiate) fatigue cracks during service and, in these cases, crack growth prediction software is very useful in being able to define the remnant life of the component; i.e. how long is the likely period between crack initiation and final fracture? The results from such analyses can also be used to define inspection periods and decisions regarding serviceability can be supported by reliable predictions.

The driving force for crack growth is not stress or strain but the stress intensity factor, K. This should not be confused with the Stress Concentration Factor Kt or with the Strain Hardening Coefficients K and K'. The Stress Intensity Factor (K) represents both the applied stress and the crack size and describes uniquely the stress field at the crack tip. The relationship between K, applied stress and crack size is:

K = Y . σ .( π a)1/2

The Compliance Function, Y, describes the geometry of the structure in which the crack exists and is related to crack size and section thickness by a power law.

The Stress Intensity (K) is the driving force for crack growth in the sense that the rate of crack growth, da/dN, is governed by the cyclic range of Stress Intensity, ∆K. These parameters are related by a crack growth "law" of which the best known is that defined by Paul Paris. The "Paris Law" can be stated as:

da/dN = C ( ∆K)m

It should be noted that the LEFM parameters C and m are defined from "materials" tests using

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standardised, small-scale specimens and testing procedures. A typical "Paris" curve is shown in Figure 67.

Figure 67 A typical 'Paris' curve showing the relationship between crack growth rate and applied stress intensity range

In service, structures experience fatigue crack growth under complex loading and environmental conditions. Without the correct modelling of these complex effects, accurate predictions using LEFM methods would be, at best, overly simplistic or, at worst, impossible. The complex effects which are taken into account within the crack growth analysis software include:

CRACK CLOSURE: if the crack is closed for part of the fatigue cycle, then the effective ∆K and, hence, the growth rate are reduced.

HISTORY: the crack extension in any one loading cycle actually depends upon the previous sequence of cycles; i.e. linear damage accumulation (Miner's Rule) is not appropriate and effects such as crack retardation must be modelled accurately.

NOTCHES: cracks growing "close" to notches behave differently until they are large enough to "escape" from the influence of the notch.

STATIC FRACTURE: growing fatigue cracks eventually reach a failure condition at which point the crack

Delta K Apparent Plot----- 2.25Cr1Mo. Ratio 0.5 Environment. Hydrogen gas

C: 8.767E-13 m: 4.171 Kc: 72.5 DO: 4.69 D1: 2 Rc:1

Delta K Apparent (MPa m1/2)

da/d

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ega/

cycl

es) 1E-6

1E-7

1E-8

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1E-111E0 1E1 1E2

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growth rate is, effectively, infinite. As this point is approached, the contribution to crack growth from static fracture modes increases.

ENVIRONMENT: materials data may be obtained by testing under the relevant environmental conditions (e.g. active corrosion) but each change in cyclic frequency, waveform or temperature constitutes a new data set. To re-test each material under every possible set of environmental and loading conditions is a huge and (prohibitively) expensive task and this is why the crack growth software has in-built features which allow the modelling of the environment without the absolute need for the relevant data set.

2.4.2 Applications

There are many potential applications for crack growth simulation by software, including:

PRE-PROTOTYPE DESIGN ANALYSIS: an assessment can be made of the "crack tolerance" of a design in terms of its safety and integrity throughout its design life. The use of crack growth software predictions may reduce the number of prototypes tested, thus saving considerable time and expense.

PROTOTYPE TESTING: by using crack growth software to "pre-predict" prototype test results, testing programmes can be optimised in terms of both time and cost. As an example, one crack growth software user applies this approach to ALL planned testing programmes and, as a result, now only conducts those tests which are deemed "necessary"; effectively using the tests to "validate" the crack growth software predictions.

INSPECTION STRATEGY: where cracks are discovered during service and decisions are required quickly concerning actions to be taken. For example:

Should the component be removed from service ?Can it be repaired and returned to service ?Is the structure "safe" until the next planned maintenance date ?If it is "safe" now, when should it be inspected again ?

FAILURE ANALYSIS: crack growth software can be used to re-create the service failure as a computer model to work backwards to the start point of the "known" life and reach some conclusions concerning the material, loading, environment and original design & manufacture. This process can assist in determining the probable cause of failure and to answer questions about the possible likelihood of failures in other, similar components.

DECISION SUPPORT: crack growth software (indeed, the fatigue analysis software system as a whole) provides a set of tools which can be used to derive information to support engineering decisions. Such decisions need not now rely entirely on experience and good sense but can be substantiated by data from "state-of-the-art" prediction methods.

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3.0 Materials Data for Fatigue Modelling

3.1 Introduction to Materials Data

One of the main inputs to any fatigue life estimation is a description of the material from which the component or structure has been manufactured. The fatigue properties of the material under consideration may be determined from Local Strain-, Nominal Stress- or Linear Elastic Fracture Mechanics-based cyclic testing procedures. The precise type of cyclic materials properties used will depend upon the type of fatigue modelling technique to be applied.

It is worth noting that the specimens used in defining the cyclic properties of materials are manufactured to high precision. In order to obtain the best quality materials data, it is essential that test specimens are machined accurately and are treated prior to testing to remove any residual surface stresses. In the case of strain-controlled tests, which are especially severe, it is vital that no out-of-plane bending occurs; this invalidates the test and can, in some circumstances cause damage to expensive test fixtures or the operator!

3.2 Strain Based Materials Data Parameters

To define the basic cyclic material properties, strain-controlled constant-amplitude fatigue tests are conducted, at a number of different strain levels, in a closed-loop servo-hydraulic testing machine. Typically, ten to twelve specimens are tested to fatigue lives up to 106 or 107 cycles. Testing under strain control, where the specimens are fully constrained, results in a very low degree of scatter in the test data points. It is not usual, therefore, to process these data by statistical means other than regression analysis. If data points are found to fall away from 'good' test points, then this is generally due to either a fault in the specimens, mis-alignment in the gripping fixtures or an incorrect setting on the test machine.

At the end of each test, the number of strain reversals (half cycles), 2Nf, is recorded. If the applied strain amplitude of each test is plotted against it's corresponding fatigue life, then a damage curve called the Strain-Life Curve will result, as shown in Figure 68.

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Figure 68 The Strain-Life Curve

The equation for the Strain-Life curve is:

εa = εf' (2Nf)c + [σf'(2Nf)b]/E

Where:

εa - Strain Amplitude

εf' - Fatigue Ductility Coefficient

c - Fatigue Ductility Exponent

σf' - Fatigue Strength Coefficient

b - Fatigue Strength Exponent

E - Modulus of Elasticity

The relationship of the various parameters above to the Strain-Life curve are shown in Figure 69, where:

b - Slope of the Elastic Strain-Life line

σf' - Intercept of the Elastic Strain-Life line

c - Slope of the Plastic Strain-Life line

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εf' - Intercept of the Plastic Strain-Life line

Figure 69 Annotated Strain-Life Fatigue Damage curve showing the definition of parameters

During the course of a strain-controlled fatigue test, an extensometer is used to control the applied strain amplitude and the corresponding stress response of the specimen is measured by a load cell. It is usual to record the stresses resulting from the applied constant-amplitude strain after the specimen has 'stabilised'; i.e. after the stress response has settled to a constant level. The initial variation in stress response at the commencement of a strain-controlled test results from the material either cyclically 'hardening' or 'softening'. Most plain-carbon and low-alloy steels cyclically 'soften' compared to their monotonic (tensile) properties, but austenitic-stainless steels exhibit cyclic 'hardening'.

By cross-plotting the stabilised stress and strain amplitudes from a set of companion fatigue tests, i.e. specimens tested at different strain levels, the shape of the Cyclic Stress-Strain Curve may be defined. The Cyclic Stress-Strain curve, therefore, describes the relationship between stress and strain under cyclic loading conditions, as shown in Figure 70.

The equation for the stable Cyclic Stress-Strain curve is:

εa = σa/E + (σa/K')1/n'

Where:

εa - Strain Amplitude

σa - Stress Amplitude

n' - Cyclic Strain Hardening Exponent

K' - Cyclic Strain Hardening Coefficient

E - Modulus of Elasticity

C

b

Log Life (reversals)

Log

Stra

in A

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ε’f

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100

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The relationship of the various parameters above to the Cyclic Stress-Strain curve are shown in Figure 70, where:

n' - Slope of the Stress Amplitude-Plastic Strain line

K' - Intercept of the ea-sp line at sp = 1

Figure 70 Annotated Stress Amplitude-Plastic Strain curve showing the definition of n' and K'

For strain-based data, therefore, six parameters are required to define the Strain-Life fatigue damage curve and a further two to complete the definition of the Cyclic Stress-Strain curve. The individual parameters may be further defined as follows:

Fatigue Strength Coefficient (σf')

The stress intercept of the elastic strain-life (Basquin) line. The Fatigue Strength Coefficient typically has values in the range: 500 to 2000 MPa.

Fatigue Strength Exponent (b)

The slope of the elastic strain-life (Basquin) line. The Fatigue Strength Exponent typically has values in the range: -0.05 to -0.15.

Fatigue Ductility Coefficient (εf')

The strain intercept of the plastic strain-life (Coffin) line. The Fatigue Ductility Coefficient typically has values in the range: 0.1 to 2.

Fatigue Ductility Exponent (c)

The slope of the plastic strain-life (Coffin) line. The Fatigue Ductility Exponent typically has values in the range -0.2 to -1.

Modulus of Elasticity

The Elastic Modulus of the material which, typically, has values in the range: 10000 to 250000 MPa.

Cyclic Strain Hardening Coefficient (K')

C

Log Life (reversals)

Log

Stre

ssK’

0.001 1

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If the elastic-plastic portion of the Cyclic Stress-Strain curve is plotted on a log-log basis, then the intercept of the resulting line is defined as the Cyclic Strain Hardening Coefficient which, typically, has values in the range: 500 to 2500 MPa.

Cyclic Strain Hardening Exponent (n')

If the elastic-plastic portion of the Cyclic Stress-Strain curve is plotted on a log-log basis, then the slope of the resulting line is defined as the Cyclic Strain Hardening Exponent which, typically, has values in the range: 0.1 to 0.2.

Material Endurance Limit or Cut-Off (Rc)

The Material Endurance Limit is the number of strain reversals beyond which no further fatigue damage is said to accumulate, or beyond which there is no data ! Normally, the value of this parameter would be set to 2x107 reversals (107 cycles), or greater.

3.3 Nominal (Elastic) Stress Based Material Data Parameters

In order to define a Nominal Stress-Life fatigue damage (S-N) curve, load-controlled constant-amplitude fatigue tests, at a number of stress (or load) levels, are carried out in a fatigue testing machine. The test pieces may be simple cylindrical specimens tested in a rotating-bending (Wöhler) machine, or be in the form of actual components or pseudo-components which are designed to simulate real structures. The tests themselves may even be considered as simulations in which external effects, such as corrosion, can be taken into account.

Typically, ten to twenty (more or less, depending on cost) specimens are tested to lives out to 5x107 cycles or, exceptionally, 108 cycles. At the end of each test, the number of cycles to failure, Nf, is recorded. If the stress range of each test is plotted against it's corresponding life, on a log-log basis, then a Nominal Stress fatigue damage (S-N) curve will result, as shown in Figure 71.

The equation for the S-N curve is:

S = SRI(Nf)b

or, alternatively, as:

log(S) = log(SRI) + b1 log(Nf)

Where:

S - Nominal Stress Range

SRI - Stress Range Intercept at 100 cycles

b1 - Slope of the Stress-Life line

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Note that in Figure 71, the slope of the Life Line changes, with the second slope, b2, being shallower than the first. A 'knee', or fatigue transition point, punctuates the change in slope and is typically defined to occur at around 107 cycles (BS5400:Part 10). The second slope can have any value between that of the first and zero. The transition point is sometimes used to represent an Endurance Limit below which no further fatigue damage will accumulate.

The equation of the second stress-life line is:

S = SRI'(Nf)b2

Where:

SRI' - Stress Range 'Intercept' at the Transition Point

b2 - Slope of the Stress-Life line

If b2 is non-zero, a stress-based endurance, or Fatigue Limit, may be defined to cut off any further fatigue damage accumulation.

The relationship of the various parameters above to the Stress-Life curve are shown in Figure 71.

Figure 71 Annotated Stress-Life curve showing the definition of parameters

The individual parameters, which apply to metals and alloys but not necessarily to plastics and composites, may be further defined as follows:

Stress Range Intercept (SRI)

On an S-N curve, the Stress Range Intercept is defined as the stress range at ONE cycle for the upper (main) Stress-Life line and typically has values in the range: 1000 to 25000 MPa.

b1

b2

Log Life (cycles)

Log

Stre

ss R

ange

SRI1

100

FL

Nt

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First Fatigue Strength Exponent (b1)

The slope of the upper (main) Stress-Life line, which typically has values in the range: -0.1 to -0.5

Fatigue Transition Point (Nc)

The Fatigue Transition Point is defined as the point on the Stress-Life line at which the slope changes from b1 to b2. The Fatigue Transition Point lies typically between 105 and 107 cycles.

Second Fatigue Strength Exponent (b2)

The slope of the lower (minor) Stress-Life line, which typically has values in the range -0.6 to zero.

Endurance or Fatigue Limit (FL)

The value of stress range below which no fatigue damage is to be counted. Typically, the Fatigue Limit has values between 0 and 100 MPa.

Standard Error of Log(Nf) (SE)

In fatigue tests conducted under Nominal (Elastic) Stress control, the test specimen is completely unconstrained. This can, and frequently does, result in a high degree of scatter in the test data points. It is usual, therefore, to apply statistical methods to account for this variability; generally through a consideration of the Standard Error of Log(Nf).

The Standard Error of Log(Nf) is calculated from:

SE = SDx (1-r2)0.5

Where:

SE - Standard Error of Log(Nf)

SDx - Standard Deviation of Log(Nf) from regression analysis

r - Correlation Coefficient from regression analysis

The application of the Standard Error statistic gives a family of Stress-Life fatigue curves, centered around the 'Mean' Stress-Life line. It is usual to consider a scatter of 3 standard deviations around the 'mean' life line, as shown in Figure 61. The lower limit (-3SD) represents 99.9% confidence that, for a given applied nominal (elastic) stress range, failure will not occur after fewer cycles than those which correspond to that stress range. The 'mean' life line represents 50% confidence and the upper line only 0.15% confidence that the sample will survive.

Typically, the Standard Error has values in the range: 0 to 10.

3.4 Fracture Mechanics Based Materials Data Parameters

Linear Elastic Fracture Mechanics (LEFM) is used to describe the resistance of a material to Fatigue Crack Propagation. Several LEFM-based parameters are required for a complete description of the crack propagation behaviour of a material.

Firstly, the Paris Law describes a material's fatigue crack growth response to applied loads in the form of a Stress Intensity Range, ∆K. The Paris Law is expressed as:

da/dN = C( ∆K)m

Where:

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da/dN- Rate of Crack Growth per cycle

C - Paris Coefficient

m - Paris Exponent

∆K - Applied Stress Intensity Range

Secondly, the Threshold Stress Intensity Range, ∆Kth, is defined as the stress intensity range below which the crack will not propagate. Finally, the fracture condition, which provides the end point to fatigue crack growth is defined by the Plane Strain Fracture Toughness, KIc, of the material.

Note that the LEFM fatigue crack growth parameters are specific to a particular test environment and that a change to the test frequency, waveform, stress ratio (R-ratio), temperature or test medium (air, sea-water etc.) can have a significant effect on the LEFM properties obtained.

LEFM-based fatigue properties are defined using standardised test techniques and specimens. Testing is conducted using servo-hydraulic testing machines. A typical test specimen is shown in Figure 72.

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Figure 72 A typical LEFM test specimen

Three types of test are used to define the LEFM properties of a material:

The Fracture Toughness Test: in which the specimen is fatigue pre-cracked and tested under a monotonic rising load to define the Plane Strain Fracture Toughness, KIc. It should be noted that there are many validation criteria which must be satisfied before KIc can be taken as a material constant. In particular, the specimen thickness must be > 2.5(KIc/sy)

2.

The Fatigue Crack Growth Test: in which a fatigue pre-cracked specimen is tested under conditions of constant cyclic load range and mean load and the crack length is recorded continuously to give a Crack Size-Cycles, a-N, dataset. From this, values of Crack Growth Rate, da/dN, are calculated at equally spaced crack sizes together with the corresponding value of Stress Intensity Range, ∆K, which are plotted in a log-log format. In this test, the applied load range remains constant but ∆K increases as the crack size increases.

The Fatigue Threshold Test: is similar to the Fatigue Crack Growth Test, except that the applied load range is reduced progressively until the rate of crack growth falls until no growth is detected for 50000 cycles. The Stress Intensity Range measured at this point is referred to as the Fatigue Crack Growth Threshold, ∆Kth.

The Paris Curves for one material tested at two different stress ratios (Kmin/Kmax) are shown in Figure 73.

Fracture Mechanics K Solution Library

Specimen type : Round compact tension specimen (RCTS)

Enter any changes (none)

B

W

Dimensions

Thickness, B (mm) : 25

Width. W (mm) : 50

a

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Figure 73 Comparison of 'Paris' curves showing the effect of stress ratio, R

The individual parameters may be further defined as follows:

Paris Law Coefficient (C)

The Paris Law Coefficient is defined as the crack growth rate intercept, on a Log(da/dN) vs Log( ∆K) graph, at a ∆K equal to unity, which typically have values in the range: 10-12 to 10-9 m/cycle.

Paris Law Exponent (m)

The Paris Law Exponent is defined as the slope of the Log(da/dN) vs Log(∆K) graph and typically has values in the range: 2 to 4 for samples tested in air, though the value may be greater in other environments.

Threshold ∆K at R=0 (D0)

For zero minimum stress cycling (R=0), the measured Threshold, ∆Kth, is typically around 8 MPa m1/2 for steels and 4 MPa m1/2 for aluminium alloys; both tested in air.

Threshold ∆K at R>0 (D1)

For higher and higher mean stresses, the stress ratio, R, tends to unity and the measured Threshold, ∆Kth, tends to an asymptotic value which is typically around 2 MPa m1/2 for steels and 1 MPa m1/2

for aluminium alloys; both tested in air.

Delta K Apparent Plot

10-11

10-9

10-8

100 101 102

[Delta K Apparent MPa m1/2]

[da/

dN m

/cyc

le]

2.25Cr1Mo: Ratio 0.3 Environment: Air

10-7

10-6

10-5

10-10

2.25Cr1Mo: Ratio 0.5 Environment: AirC: 8.02E-12 m: 3 Kc: 72.5 DO: 7.31 D1: 2 Rc: 0.97

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3.5 Concluding Remarks

From the above, it is clear that fatigue analysis has, in the past, been somewhat arbitrarily divided into three separate methodologies, namely: Nominal Stress-Life, Local Strain-Life and Fracture Mechanics. In fact, these techniques are all really describing the same phenomenon - cyclic plasticity leading to fatigue damage. The three approaches have developed independently, through traditional groupings of fatigue research workers, over many years: the Local Strain and LEFM approaches are both over 30 years old and the S-N approach a good deal older!

Nevertheless, an integrated approach to fatigue analysis is emerging, championed mainly by expert Durability and CAE Modelling companies. Integrated, intelligent fatigue modelling systems combined with sophisticated materials data management systems, now provide the full spectrum of fatigue modelling techniques to the Designer and Development Engineer alike.

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4.0 Overview of the Fatigue Analysis Software System

FATIMAS is a total system for fatigue analysis of components and structures by fast, flexible and accurate computer simulation. It takes in measured operating data by integration with data acquisition systems and has its own in built databases, expert systems and libraries for materials response parameters and geometric detail definitions. The package is based on over 10 years software engineering and over 150 years of fatigue expertise coming from nCode International of Sheffield UK, the "leaders in fatigue". FATIMAS can be used for:

• Durability assessments in the design/test loop to avoid costly and time consuming prototype repeat testing

• Durability Optimisation by "what if games" to achieve better fatigue solutions (loading, material, surface condition, local geometry)

• Failure analysis by mimicking the failure and providing substantiated remedial solutions

• Assessment of defects (cracks) in components and structures prior to and during operation• Optimisation of inspection and maintenance scheduling by fatigue assessments and

updates

• Computer simulation prior to structural testing to optimise test programmes from a fatigue content perspective and to optimise time, budget and testing resources

• Optimisation of materials selection and manufacturing process routes (casting, forging, machining, shot peening) from a fatigue durability viewpoint.

4.1 General Features

4.1.1 FATIMAS - Fast, flexible, accurate

FATIMAS is a product that has evolved from the leaders in fatigue, nCode. It provides seamless integration within the environment through the use of standard user interfaces such as Motif and nCode's own Mask interface.

4.1.2 A standard User Interface

The use of standard graphical user interfaces on PCs, VAXs, and UNIX workstations provides standardization across many applications and packages today. The interface can be operated by mouse driven menus or by quick commands, or a mixture of both to provide fast, flexible configuration of text inputs and graphics screen layouts. On-line help is provided at all input options and every window.

All the software modules operate in "batch mode" for unattended, end to end analysis procedures involving a chain of modules, functions or operations and multiple analyses using macro batch files or templates.

The FATIMAS environment allows the user to customise operational defaults and behaviour and also colour schemes and provides open access to integrate to external data and software. Direct access through shared memory mechanisms are provided for use in a server/client mode of operation.

4.2 System Functional Description

FATIMAS has six functional groups of modules that can carry out a sequence of steps (some optional) in a total fatigue analysis:

Acquired time series data interface

Data display

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Data manipulation

Data pre-processing

Fatigue Analysis

Materials response databaseLocal geometry definitionFatigue modelling

Results post-processing

4.2.1 Data Display and Hardcopy

FATIMAS gives the user graphical displays at every opportunity which are completely user controlled through a full complement of display options affecting the attributes of the displayed data allowing customizing and annotation of the view. The commands and menus are consistently standardised and when any display is satisfactory to the user, a hardcopy plot file can be initiated which can be subsequently plotted to an output device. The generation, maintenance, and output of all plot files is handled by a user friendly plot manager module which can:

update the plot index

preview all or selected plots

on-line select/deselect plots for output

select a hardcopy output device

select the orientation and number per of plots per page

convert plot files to standard file formats (HPGL, Postscript etc.) forinclusion in external DTP software

4.2.2.1 Quick Look Display (QLD)

QLD gives a display of a time series data file (or any single parameter file) which is completely user controlled through a full complement of display options affecting the attributes of the displayed data allowing customizing and annotation of the view. A hardcopy plot file can be initiated at any point. The options in QLD include:

Main

Statistics on/offTowers/join dataNew file inputData listing screen/file

Axes

Log/linear axesGrid/dashed grid/ticksZero lines on/offAxes transpose

Move

Full x/full y/full bothPage left/page rightZoom

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X window or Y window limits

Annotate

Set/delete titleAdd/move/delete text/delete all textTop/side labelling

Setup

Set coloursGridAxesAnnotationTextDataBackground

Grid styleSave/restore pensSave/restore all

Figure 74 The FATIMAS Quick Look Display of Time Series Data

File Display View Axes Annotate Preferences Full Plot Help

QLD

1500

-15000 0.5734

Stra

in (u

E)

Time (seconds)

Display of Signal ; SPIKES.DAC

8191 points741 pts/secondDisplayed:499 pointsfrom pt 1Full file dataMax = 1499at 7.105 secondsMin = -1445at 9.672 secondsMean = 40.82S.D. = 445.5RMS = 447.3

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4.2.1.2 Multifile Display (MFD)

MFD gives a multiple display of up to 32 time series data files (or any single parameter files) which is completely user controlled through a full complement of display options affecting the attributes of the displayed data allowing customizing and annotation of the view. The data files are individually displayed and can be customised individually or in tagged groups. In addition, MFD can display overlays of datafiles or cross plot datafiles. A hardcopy plot file can be initiated at any point. The options in MFD include:

Main

Tag/tag all/untag allFull plot/replotNew files inputData listing screen/file

Axes

Log/linear axesGrid/dashed grid/ticksBox on/offZero lines on/offAxes transposeSet axes limits

Move

Full x/full y/full bothPage left/page rightZoom in/out

Annotate

Add/move/delete text/delete all text

Setup

Point skipOverlay/quick overlayCross plot

Set coloursGridAxesAnnotationTextDataBackgroundSave/reset pens

Pages

Go to pageEdit pages

Round

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Round x/y/bothFull round x/y/both

Figure 75 The FATIMAS Multifile Display of Time Series Data

4.2.1.3 Two Parameter Display (TPD)

Gives a display of a two parameter (x and y) data file which is completely user controlled through a full complement of display options affecting the attributes of the displayed data allowing customizing and annotation of the view. A hardcopy plot file can be initiated at any point. The options in TPD include:

Main

Join/scatter/join with pointsNew file input

Axes

Log/linear axesGrid/dashed grid/ticksBox on/offZero lines on/offAxes transpose

Move

Full x/full y/full bothPage left/page rightZoom in/outX window or Y window limits

Annotate

Set/delete title

Editing test : DATA

Secs.

Secs.

Secs.

Secs.

0.4594

-0.3965

0.4594

-0.3965

1499

-1445

Test E-Data, Chan 1

Test G-Data, Chan 1

Test TES1T, Chan 1

Test TEST2, Chan 4

0 2 4 6 8 10 12

0 10 20 30 40 50 60

0 2 4 6 8 10 12

0 2 4 6 8 10 12

2.252E4

2.345E4

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Add/move/delete text/delete all textTop/side labelling

Setup

Plot skip

Set coloursGridAxesAnnotationTextDataBackground


Figure 76 The FATIMAS Display of Two Parameter Data

4.2.1.4 Histogram Display (P3D)

P3D gives a display of a three parameter (histogram of cycles or damage) data file which is completely user controlled through a full complement of display options affecting the attributes of the displayed data allowing customizing and annotation of the view. A hardcopy plot file can be initiated at any point. The options in P3D include:

Main

Full plot/replotNew file input

Axes

Log/linear z axis

R07R04R01R09

Cross Plot of Data : KRAKMAT

Delta K (MPa m1/2)

da/d

N (m

/c)

6E-7

5E-7

4E-7

3E-7

2E-7

1E-7

10 20 30 40

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x/y/z window

Plot Type

HistogramTowers/surfaceView left/right

View

Quadrant/rotate/tiltBack plane on/offHide lines on/off

Annotate

Add/move/delete text/delete all text

Setup

Set coloursAxesAnnotationTextDataBackground

Save/restore pens

Figure 77 The FATIMAS Display of Histogram Data

4.2.2 Data Manipulation

4.2.2.1 Graphical Editing (GED)

The GED program is a multi-channel interactive graphical editor for time series (single parameter file) data. It uses a customizable window for the file being operated upon plus a reference file window for comparison. Facilities are available for correcting, overwriting, appending, extracting, inserting or creating data with safe quit/exit protocol and restore options. A hardcopy plot file can be initiated at any point. The options in GED include:

Cycle Histogram Distribution for : AUTO.CYO

Maximum height : 16 Z Units :

1078.9

Mean uE

1629.7 -534.61

Range uE

00

Cycles

16

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Main

New file/new reference file input Restore

Statistics displayHardcopyExit/quit

Edit/Multi-edit

File insert/extractTruncateRescaleDrift correctionInsert/deleteSpike removal

Append/Overwrite

RampHalf sineStep up & along/ along & upExponential decay to x/ decay to y

Move

Page left/page rightcentreFull x/full y/locally fullGo to start/go to endX window/Y window Zoom in/out

Setup

PensGridAxesAnnotationTextDataBackgroundSurround Prompt/error message Save/restore pens

Grid style

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Figure 78 The FATIMAS Graphical Editor of Time Series Data

4.2.2.2 Arithmetic Manipulation (ART)

The ART program takes an input file (time series, x-y or histogram type file) and processes each data point on a particular axis through a user selected arithmetic manipulation. ART is very useful for general data manipulation and is optimised for speed of processing. Axes labels and units settings are available and operations can be performed on segments of data leaving the remainder unaffected. Arithmetic manipulation functions available are:

Multiplication/division by a constant

Addition/subtraction of a constant

Normalisation to a new mean value

Raise to a power

Sine/cosine/tangent

Absolute values

Y=mx + C

Log/antilog base 10/base e

4.2.2.3 Multifile Manipulation (MFM)

The MFM program allows the user to specify 2 or more input files which can be combined to produce

0 0.1 0.2 0.3 0.4

0.1 0.2 0.3 0.4

time (secs)

time (secs)

front

g2

(g)

front

g1

(g)

0.4

0.2

0

-0.2

0.4

0.2

0

-0.2

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a single output time series type file in a number of ways. Files with different sample rates, lengths and start values are handled automatically. The file combination options available are:

Add file1+file2+file3+...

Subtract file1-[file2+file3+...]

Multiply file1*file2*file3*...

Divide file1/[file1*file2*file3*...]

Vector add [file12+file22]0.5

Vector add [file12+file22+ file32]0.5

4.2.2.4 Extraction and Concatenation (LEN)

The LEN program allows users to carry out extraction, deletion operations across up to 32 files and concatenation of any number of files. It is very useful for extracting damaging sections (and re-joining them) from measured data or for deleting unwanted or suspect data. Data can be selected for operation on a data point or time value basis for accurate manipulation. The options available are:

Extract single/multiple filesDelete single/multiple filesConcatenate files

4.2.2.5 Sample Rate Adjustment (SRA)

The SRA program allows the user to adjust the sample rate of time series type data in a file, either upwards or downwards. Down sampling removes data points while up sampling interpolates additional points. Down sampling is particularly useful for reducing over-sampled measured data to save disk storage space and increase processing times for large data files.

4.2.2.6 File Header and Extra Details Manipulation (FILMNP)

The FILMNP program is a file "doctor" that carries out file header and file extra details area manipulations as well as header and data validation. The header is a fixed format 512 byte area containing such information as start value, sampling rate, max and min values and their location, mean and rms values, axes labels and units. The header of a file can be viewed, verified, updated or modified using FILMNP.

The extra details area of a file is of unlimited size containing relevant information in the form of keyword/value pairs. Extra details can be read from or written to the extra details area by many other programs in FATIMAS including the fatigue analysers and the report quality post-processor, RQP. The extra details of a file can be viewed, modified, added, deleted, copied from other data files, copied from text files or passed to and from the local environment file using FILMNP.

All header and extra details keywords can be used in arithmetic manipulations, for example in RQP.

4.2.3 Data Pre-Processing

4.2.4.1 Peak Valley Extraction (PVX)

The PVX program extracts turning points (maxima and minima or "peaks" and "valleys") from a time series. The output file from PVX has the same format and properties, with the extra details carried forward, and can be used as direct input to the fatigue analysers (see page 155) SLF, CLF and

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NIAGRA (option) or for further pre-processing via the cycle counting module CYC (see below) prior to fatigue crack growth analysis (option).

Acquired time series data for fatigue analysis purposes should be sampled at at least 10 times band width to ensure that the peaks and valleys are accurately defined. Consequently, the pre-processing function of extracting peaks and valleys and ignoring the intermediate data points, reduces the storage requirements of the fatigue analysis input file by typically a factor of ten.

Peaks and valleys can be extracted from a segment or the whole file and a "gate" can be applied to eliminate noise or non-damaging events; that is excursions from turning point to turning point smaller than the gate are excluded. The total number of peaks and valleys extracted is reported on screen and to the Notebook , if set.

4.2.3.2 Rainflow Cycle Counting (CYC)

The CYC program extracts cycles from a time series file (including PVX files - see above) according to a standard Rainflow Counting algorithm. The output files from this processing can of be several types and in various combinations:

1. a cycles histogram classified into ranges and means

2. a file of individual cycles in the extraction sequence

3. a file of individual cycles in the extraction sequence with associated time period information

4. a file of individual cycles with associated time period information re-ordered into the sequence of peak occurrence

5. 1 & 2

6. 1 & 3

7. 1 & 4

Cycles smaller than a user defined "gate" are excluded.

The histogram file (which is much smaller than the time series file) can be used as input to the fatigue analysers with only a slight loss of accuracy in life prediction. The time based cycles file is a required input to fatigue crack growth analysis using the CRG program (option).

Both types of cycles file can be listed using the program CYL (see page 147). Options for the cycle classification histogram are:

Range bin size 3-64 elements

Mean bin size 3-64 elements

Max and min range independent of file limits

Max and min mean independent of file limits

Setting max and min ranges and means inside the limits of the time series will not classify all cycles. An on screen report (and Notebook entry) summarises the histogram formation and reports the number of cycles classified and out of range. The default limits will always classify all cycles. Optionally, the histogram can be displayed, for viewing and hardcopy, by automatically launching the

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P3D program.

Figure 79 The FATIMAS Cycle Counting Options Menu

4.2.4 Fatigue Analysis

4.2.4.1 Materials Data Manager (MDM)

MDM is the materials data manager program that is linked to a supplied database containing the properties of over 120 engineering materials. The information stored for each material includes names and alternative names (standards from different countries), type category and parameters representing material or component stress-life response, cyclic stress/strain curve and strain life curve and fatigue crack growth behaviour. The database serves all the fatigue analysers in FATIMAS in which the user is asked to name a material so that the fatigue response properties can be used in the fatigue prediction analysis.

The material data manager also includes a material selection optimisation expert system.

Input Filename

X Cancel ? Help

CYC - Input Parameters

OK

X

Histogram

Window Type

Histogram File

TEST101.DAC

Cycles File Both

Time Points

Output Type

Gate

STARTStart Time

Store Cycle Time No Yes

Cycles Filename

No Yes SlowSort Cycles

ENDEnd Time

ENDWSR Exponent

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Figure 80 The FATIMAS Material Data Manager Main Menu Options

The main menu options are:

List the full database, or selectively on a combination of properties and attributes

View or edit data entries

Create or delete data entries

Graphical displays of property data

Materials optimisation expert system

User Preferences

The graphical display sub menu options are:

Strain life

Morrow life

Smith-Topper-Watson life

Cyclic/monotonic stress/strain curve(s)

Effective delta K/ growth rate

apparent delta K/ growth rate

Threshold delta K/ stress ratio

X Cancel ? Help

MDM

OK

X

Search and listTabulate ->

Full list

Unload ->Load ->

Edit ->Create ->Delete ->Weld Classifier ->Graphical display Preferences ->eXit

Set 1: MANTEN

Set 2: URANUS

Database in central directory

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to allow viewing or comparison of properties using standard display customisation and hardcopy facilities.

The user preferences sub menu options are:

Stress units system (MPa/ksi/psi)Strain units system (S or uS)Material parameter input range checking on/off

A user's local copy of the database can easily be made from the supplied, secure database for customising using create/delete/edit functions.

4.2.4.2 Local Geometry Definition

4.2.4.2.1 Stress Concentration Factor Library (KTAN)

The KTAN program is a database library of geometries from which the user can select the appropriate geometric detail and, by inputting the dimensions, obtain a calculated value for the stress concentration factor, Kt, for use as input to the fatigue analysers SLF, CLF, NIAGRA (option) and TCD (future option).

The value of Kt or Kf (fatigue strength reduction factor) has always been a hard question to provide an input for and users would formerly have to look up factors in text books from graphs or tables or equations. Now KTAN provides accurate calculations and passes the value via the environment to the fatigue analysers. It is easy to use in that the user only has to pick a general geometry type, then a particular detail (from a list or from a pictorial selection pick) and a graphic shows the selected detail for dimensional input, editing or calculation.

The supplied database library has 26 cases (which will be enhanced in future upgrades) in the following geometry type categories:

NotchesHolesFilletsMisalignmentKeywaysGearsSpringsCrankshaftsKf from KtOther

As well as calculation functions, KTAN allows the user to perform database functions on a users copy of the library database, that is edit, delete entries or add new entries. Each entry is composed of a set of general details, a plot file of the graphical representation (made for example using RQP) and the Kt data as a two parameter file, an ASCII file or as a polynomial. Multiple families of data are possible.

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KTAN has therefore two main menu functions: database operations and Kt calculation; the sub menu options are as follows:

Database Functions

Add entryDelete entryEdit entryShow entries, all or by geometry typeShow pictures in a geometry typeShow details of an entry

Calculate

Secure/users database selectionSelect geometry/new geometry typeSelect geometry/new geometry by description or by pictureEnter dimensionsCalculateEdit dimensions/re-calculateHardcopy

Figure 81 A sample from the FATIMAS Kt Calculation Library, in this case a crankshaft

r

d

M

t

M

S

X = t/d

Family = s/dt=50

d=100

s=10

Kt=3.88

Nominal stress = M / (3.142 d3 /32)In this case r/d = 0.1

Description : Crankshaft in bending r/d=0.1Comments : as a function oft/dLookup type : MDF file

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4.2.4.2.2 Stress Intensity Factor Solution Library (KSN)

The KSN program is a database library of LEFM stress intensity factor solutions (K solutions) from which the user can select the appropriate geometric detail for use in solving fracture/fatigue defect assessment calculations or for manufacturing a specific compliance function which is required for fatigue crack growth calculation using the CRG program (see page 150). Without such software, users would have to derive K solutions from the literature as graphs, tables or complex equations. Here, the user only has to identify the general geometry type and the specific solution to be presented with a graphical description which shows, unambiguously, the input dimensions required.

It is possible to generate any number of specific K solutions from the 36 cases supplied in the library and also it is possible to input experimental or numerical (FE analysis) results via a parametric definition option.

The main menu options are:

Units system selection

Solve for stress/K/crack size

Generate a compliance function

The general geometry types are:

Standard specimens

Cracks at holes

Elliptical, semi-elliptical cracks in plates

Cracks at corners

Cracks in solid cylinders

Cracks in hollow cylinders

Cracks in welded plate joints

Cracks in welded tubular joints

Cracks at spot welds

User parametric definition

In graphics display, the options are:

Define/change dimensional parameters

Calculate

Hardcopy

Setup PensSave/restore pensPrompt/error message coloursDataTextBackgroundSurround

Post processing options are:

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a. single/multiple calculations

Plot stress intensity against crack size

Plot compliance against crack size

Plot stress against crack size

Plot stress intensity against stress

Re-run

b. Compliance function generation

Output compliance as text to screen or file

Output compliance function to graphical display (launches QLD)

Re-run

Figure 82 A sample from the FATIMAS Stress Intensity Factor Library

4.2.4.3 Stress Life Fatigue Analyser (SLF)

The SLF program models fatigue life in software to predict durability based on constant amplitude test results for specimens or components; "S-N" curves. In this way, failure can be predicted in a total life sense for any component subject to any variable amplitude or service measured history using the Miner Rule of linear damage calculation. In SLF, a single fatigue life answer is not the end point, only the beginning of potentially very many "what if games" and sensitivity studies using multiple input parameters and back calculations with varying scale factor, mean stress offset, % certainty of survival, stress concentration factor, size effect, surface finish and surface treatment.


Specimen type : Crack at Spot Weld in tension

Enter Dimensions

DimensionsP

P

W

T

D

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Operating from a job file created in a previous analysis, the user can simply recalculate after quickly modifying one or more controlling parameters. Materials S-N parameters are stored in the database and can be picked from a list; Kt values can be determined within KTAN (see page 138) and passed into SLF as default selections. A full range of post processing and results display options are available in SLF.

Features of SLF are:

Loading

Time series/peak valley series/cycles histogram/constant amplitude and mean input

Calibration file/scale factor*/offset*/hysteresis gate*

Setting of equivalent units and units system

Modelling

% certainty of survival*

Material S-N/component S-N/BS5400 weld class

Mean stress correction Gerber/Goodman/none/all

Material Selection

Load/enter/generate/edit

Surface finish choice of 10 or all

Surface treatment nitrided/cold rolled/shot peened/none/all

Geometry

Stress concentration factor/additional factor e.g. size effect

Analysis

single shot/multiple analysis/back life calculation

Results

Cycles/damage matrices

Two parameter display of multiple analysis

Post-processing

Edit loading/S-N definition/geometry definition/model parameters/output definition

Cycles matrix/damage matrix display/damage analysis/matrix listing/results listing

Job file /new/save as/list

Preferences for material checking on/off, back life sensitivity, back life setting, units system

Recalculate

* target life back calculation is possible on any of these parameters

4.2.4.4 Critical Location Fatigue Analyser (CLF)

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The CLF program models fatigue life to crack initiation at a critical location in software to predict durability based on standard strain controlled test results for specimens; "strain-life and cyclic stress-strain" curves. In this way, failure can be predicted in an initiation life sense for any component subject to any variable amplitude or service measured history using the Neuber Rule, Massing's Hypothesis and the Miner Rule of linear damage calculation. In CLF, a single fatigue life answer is not the end point, only the beginning of potentially very many "what if games" and sensitivity studies using multiple input parameters and back calculations with varying scale factor, mean stress offset, % certainty of survival, Kf factor, size effect, surface finish and surface treatment.

Operating from a job file created in a previous analysis, the user can simply recalculate after quickly modifying one or more controlling parameters. Materials parameters are stored in the database and can be picked from a list; Kt and Kf values can be determined within KTAN (see page 140) and passed into CLF as default selections. A full range of post processing and results display options are available in CLF.

Features of CLF are:

• Loading

• Time series/peak valley series/cycles histogram/constant amplitude and mean input

• Calibration file/scale factor*/offset*/hysteresis gate*

• Setting of equivalent units and units system• Modelling

• % certainty of survival*

• Material strain-life• Mean stress correction Morrow/Smith-Topper-Watson/Haibach/none/all

• Material Selection

• Load/enter/generate/edit• Surface finish choice of 10 or all

• Surface treatment nitrided/cold rolled/shot peened/none/all

• Geometry• Stress concentration factor/additional factor e.g. size effect

• Analysis

• single shot/multiple analysis/back life calculation• Results

• Cycles/damage matrices

• Two parameter display of multiple analysis• Hysteresis loop display

• Post-processing

• Edit loading/Material definition/geometry definition/model parameters/output definition• Cycles matrix (nominal or local)/damage matrix display/Loop display/damage analysis/

matrix listing/results listing

• Job file /new/save as/list• Preferences for material checking on/off, back life sensitivity, back life setting, units system

• Recalculate

4.2.4.5 Crack Growth Fatigue Analyser (CRG)

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The crack growth analyser, CRG, models in software the extension of a crack by fatigue on a cycle by cycle basis. It contains sophisticated, accurate and realistic modifications to account for cycle sequence (retardations etc.), crack closure, notch effects, static fracture modes and even corrosion fatigue.

Materials response data are stored in the database as crack growth rate, threshold and fracture toughness parameters. Time series are pre-processed via CYC to re-order cycles in the correct sequence for history effect modelling. Modelling is based on the fracture mechanics similitude concept where, if the stress intensity on a crack in a specimen is the same as that on a crack in a structure, then the response will be the same. Stress intensity solutions for the structure being software modelled must be pre-prepared using KSN.

During the cycle by cycle crack growth analysis, status updates can be displayed graphically and results saved at preset intervals for subsequent post-processing using the CRL module (see page 151).

• Features of CRG are:• Loading

• Cycles filename listing

• Scale factor/offset• Plane stress/plane strain conditions

• History effect modelling on/off

• Results setup• Output filename/none

• File output interval

• Alpha/graphics display• Screen update interval

• Target design life

• Local geometry• K solution name

• Local notch details

• Maximum crack size• Initial crack size (can be zero)

• Material selection

• Material name list and details display• Environment name list

• Edit facilities for material parameters

• Environmental parameter setting• Online

• Alpha/graphics switching

• Pause/quit/restart• Post-processing

• Final situation details

• Final graphics and hardcopy• Fatigue life interpolation

• New job

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• Edit input parameters

• Material checking on/off• Recalculate

Figure 83 The FATIMAS Crack Growth Analyser On-line Display

4.2.4.6 Initiation and Growth Rate Analysis (NIAGRA)

The NIAGRA program may be seen as both a special fatigue analyser in its own right and as a fatigue analysis system manager. The special analysis it can perform is an integrated, seamless crack initiation plus crack growth total life prediction. Many instances of durability assessment in the engineering world consist of significant portions of initiation and growth; sometimes 50/50, sometimes 10/90 sometimes 90/10% and both parts have to be modelled.

As a fatigue system manager, NIAGRA can, on user choice, select to analyse for crack initiation life (by launching CLF), crack propagation life (CRG) or total endurance (SLF). During the integrated analysis, opportunities arise for the user to temporarily leave NIAGRA to operate within the materials data manager (MDM), the Kt library (KTAN) and the stress intensity library (KSN).

The options in NIAGRA include:

• Analysis

• Local strain /propagation/endurance

File Preferences Help

CRG

Modifying

Closure

Effects

History

Notch

Environ

Stat Frac

Repeats66

Size (mm)0.42682

DLKAPP6.0574

DLKEFF2.5597

dA/dN

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Cra

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Notch

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• Loading

• Time series filename (pick/list)• strain calibration by scale & offset or look up table

• stress calibration by scale & offset or look up table

• Plane stress/plane strain conditions• Material

• Material name list and details display

• Environment name list and pick• launch facility to MDM

• Geometry

• Kf value with launch facility to KTAN

• K solution filename with launch facility to KSN• Local notch details

• Final crack length

• Initiation setup• Expected total life in associated equivalent units

• Results output filenames' stem

• Propagation setup (after initiation calculation and report of life so far with crack size at initiation/propagation interface)

• Final situation report

• Post-processing• Edit loading/materials selection/geometry/Initiation/propagation input options

• Display cycle/damage histograms/crack growth results (using P3D/CRL)

• Recalculate/exit

4.2.5. Results Post-Processing

4.2.5.1 Damage Histogram Display (SLF,CLF,P3D)

The fatigue analysers CLF, SLF and NIAGRA have the capability to determine the fatigue damage for each class of stress or strain cycle and these damage results can be displayed as a range-mean-damage matrix using P3D for comparison with the corresponding cycles matrix. In this way, the most damaging cycles, and by contrast the non-damaging cycles, can be identified.

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Figure 84 The FATIMAS 3D Damage Matrix Display

4.2.5.2 Cycles and Damage Analysis (CDA)

The CDA program allows the user to compare the cycles and damage results as 2D overlaid histograms summarised either to the range or mean axis. It also allows the cycles content or the damage content of two cases to be compared.

The CDA options are:

Main menu

Plot 1st cycles/2nd cycles/both cycles filesPlot 1st damage/2nd damage/both damage filesPlot cycles and damage case1/case2

Graphical display options

Range/mean axisLog/linear cycle axisLog/linear damage axisSave as time series file formatX window limitsGrid/dashed grid/ticksZero lines on/off

Full plot/joined plot/tower plot/exceedance plotSet/delete titleAdd/move/delete text/delete all textTop/side labelling

Setup

Set coloursGridAxesAnnotationText

Mean uE

107

-534.611629.7

Range uE

00

Damage

3.9971E-5

Damage Histogram Distribution for : AUTO.DHH

Maximum height : 3.9971E-5 Z Units :

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DataBackground


Figure 85 The FATIMAS Cycles and Damage Display

4.2.5.3 Cycles Listing (CYL)

The CYL program allows the user to list the individual cycles in a rainflow count produced by CYC (see page 135) or the cycles and damage results from the fatigue analysers, CLF and SLF. The program automatically determines which types of information are available. In the case of the local strain analysis results from CLF, cycles can be listed either as local or nominal cycles.

Listing can be made as range/mean or as min/max, to the screen window or to a text file (for inclusion in a report). A cycles gate can be applied to exclude cycles smaller than the gate.

4.2.5.4 Hysteresis Loop Display (CLF,TPD)

The critical location fatigue analyser, CLF, tracks stress and strain hysteresis loops during the Neuber translation from nominal to local (using Kf) location and uses Massing's Hypothesis to define the loop shapes. This information is stored during analysis and one of the post-processing features of CLF is the ability to generate and display up to 100 of the biggest hysteresis loops.

The display summarises the main features of an analysis on one display including the material, Kf, scale factor and predicted fatigue life. The display shows the width of the loops, that is the plastic strain at the critical location that accumulates fatigue damage, and the user can select which loops of those generated to be displayed.

The display can be customised using all the functionality of the TPD program (see page 129) and

Dam

age

4.037E-5

1630

0

Range

DamageCycle

0

0

Cyc

les

145.4

Total plot of file AUTO

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can, of course, be hard copied for report purposes.

Figure 86 The FATIMAS Hysteresis Loop Display

-5000 0 5000Strain uE

Stress MPa

Material EN24VFactor = 7 Kf = 1Life = 3 repeats

-6000 -4000 -2000 0 2000 4000 6000

600

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4.2.5.5 Sensitivity Analysis (SLF,CLF,TPD)

A major feature of fatigue analysis by software modelling is the ability to play "what if games" and perform sensitivity studies of the predicted life as a function of input parameters, quickly and easily. This is particularly true in CLF and SLF where two parameter files are automatically created and displayed using the functionality of the two parameter display program, TPD (see page 129).

Possible parameters potentially taking multiple values for plotting against predicted fatigue life include:

Stress/strain amplitude/mean

scale factor/offset/hysteresis gate

% certainty of survival

Surface finish/surface treatment

Stress concentration factor/additional factor

Figure 87 he FATIMAS Sensitivity Analysis Results

Cross Plot of Data : AUTO

Life (laps)

Cer

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ty o

f sur

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l (%

)

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40

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1E2 1E3 1E4

10

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4.2.5.6 Crack Growth Results Listing (CRL,TPD) - Optional

The CRL program is dedicated to the post-processing of results files produced by the crack growth analysis program, CRG (see page 143). These results files contain information stored as 13 useful parameters saved at intervals during the analysis. CRL allows listing of any 6 from 13 parameters, plotting of pre-set two parameter displays or a user definable pairing, compression of a results file or fatigue life interpolation based on varying the starting crack size or the final crack size or both. Either the initial or final crack size can be a multiple input to obtain a plot of crack size against life for use in inspection and maintenance optimisation for example.

Figure 88 The FATIMAS Crack Growth Interpolation Display

4.2.5.7 Report Quality Post-Processing (RQP)

RQP is a complete report quality post-processing tool that the user can employ to design generic layouts for use and re-use with files, text, lines, boxes, file header and extra details values, user defined or query information, system date and time stamping facilities.

The layout, or template, can be generated graphically using mouse driven menus and then refined if necessary by text editing to add advanced features. All types of data files, including multiple files or even plot files, can be incorporated into the layout and customised as to axes definition, labels, ticks, limits etc..

The template can then be run with specified inputs to create a finished display that itself can be hard copied. The generation of report quality plots can be automated as the ultimate part of a procedure and run unattended in the powerful batch mode of operation of the software.

The functionality of the graphical generation facility of RQP is as follows:

Main menu

CROSS PLOT OF DATA - GRGTEST

Cycles

Cra

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(mm

)

151040

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25

30

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HardcopyPreviewExit/quitBoxesAdd/edit/delete/copyMove/adjust/resize /colour/outlineLinesAdd/edit/deleteMove/colourLink to box/groupTextAdd/edit/delete/copy/moveRotation/colour/spacing /aspect/size/justificationContentsDefine/edit/delete/copyQueryAdd/edit/deleteUser DefineAdd/edit/deleteSetupRedraw on/offSnap on/offGrid on/offSet gridSet precisionHelpConceptCreating plotsBoxes/text/linesDefining queriesPlotting datafilesSetupMetavariables

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Figure 89 The FATIMAS Report Quality Post-processor

0 1 2 3 4 5 6 7 8 9

1

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-1442.3682 Accel. G 1496.079

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0 Frequency Hz 16

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5. Fatigue Analysis Case Studies

5.1 Introduction

In order to illustrate more clearly the potential uses of the FATIMAS and KRAKEN software fatigue analysis modules, three "case studies" or "worked examples" are presented, one for each analysis type: Nominal Stress, Local Strain and Crack Growth.

In each case, all of the major inputs to the analyses and the corresponding results are given; mostly in the form of edited information stored in the "NOTEBOOK" files which were created automatically during the operation of the software. Additional notes and comments are also presented to guide the reader through the thought processes used during each analysis session and to demonstrate the analysis and post-processing potential of the FATIMAS and KRAKEN software systems.

None of these case studies is exhaustive in its use of the software system, or even the fatigue analysers themselves: multi-axial fatigue considerations have not been included, for example. However, all of the most significant functions of the FATIMAS and crack growth software systems are illustrated and examples of their use given.

5.2 Definition of the S-N Case Study problem - Welded Steel Bracket

The problem centers around the fatigue durability of welded steel brackets on production stamping presses, illustrated schematically in Figure 90. The presses are used to manufacture small automotive parts, at a rate of 4 to 5 pieces per minute each, and are operated on a continuous 3-shift system 24 hours per day.

Figure 90 Schematic of the welded steel bracket

Each welded bracket has a design life of 100,000 HOURS.

Strain measurements have been made on a prototype bracket during three (3) hours of simulated "normal" production.

These data have been measured at a nominal elastic location on the test component, about 100mm away from the weld, and stored in the file SLFDEMO.DAC.

The material selected for the manufacture of these components is a plain carbon steel and laboratory

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tests have been used to define component S-N curves for this bracket using two (2) different qualities of welding procedure. These S-N curves are called CLASSW and CLASSB and represent the limits of "worst" and "best" weld quality respectively.

First, we need to know if the bracket will survive for the required 100000 hour usage when manufactured using the "worst" (lowest quality) welding practice.

The analysis uses the 'five box trick' (Figure 62) to which there are three main inputs:

1. Strain-Time history ('Signal' SLFDEMO.PVX)

2. Material Data (CLASSW.SNC)

3. 'Local Geometry' which is accounted for in the S-N data sets which have been defined from tests on "real" welded components.

Figure 91 shows the S-N data plot for CLASSW including the scatter of 3 standard deviations (SDs) about the mean life in the data set.

Figure 91 S-N curve for BS5400 CLASSW weld type showing 3SD scatter lines

Because 'real' service/field data is generally broad-band random in nature, it is important that the fatigue analysis can handle variable mean stress effects correctly. As this particular case concerns a welded detail, we shall use the fatigue analysis procedure defined in BS5400:Part10 which has been designed specifically for use with welded structures. However, the SLF analyser will also allow other Mean Stress Correction procedures to be used in the analysis: particularly those defined by Goodman and Gerber.

5.2.1SLF ANALYSIS - RUN 1 : 'Basic' Inputs

A run of SLF with the relevant data (below) produced the following results:

Service Environment

Name of input strain history : SLFDEMO.PVX Type of data : Time Series

S-N Data Plot

104

103

102

101

100101 102 103 104 105 106 107 108 109 1010

[Life Cycles]

Stre

ss R

ange

MPs

classWSRI1: 7179 b1: -0.3333 b2: -0.2 E: 2.07E5 UTS : 500

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Equivalent units for this history : 3 Hours Number of data values in time series:1668 Maximum strain value in input data (uE):750 Minimum strain value in input data (uE):-723 Output cycles file name : SLF1.SLF

Damage Law Details

S-N data set name : CLASSW Stress RANGE InterceptSRI1 : 7179 Slope of first line b1 : -0.3333 Transition life NC : 1E7 Slope of second lineb2 : -0.2 Fatigue Limit FL : 10 Elastic modulus (MPa),E : 2.07E5 Standard error SE : 0.1844

Analysis Parameters

Analysis Type :BS5400 Welded Scaling factor : 1 % Certainty of Survival : 50 Cycles Hysteresis Gate (uE) : 0 Miners Constant : 1

Results of Analysis

Turning point history :SLFDEMO.PVX S-N Data Set :CLASSW Scaling factor : 1 % Certainty of Survival : 50 Cycles Hysteresis Gate (uE) : 0 Analysis Method :BS5400 Welded Output file name :SLF1.SLF Miners Constant : 1 Number of cycles used : 834 Fatigue life :5768 Hours

The calculated life for our welded bracket is 5768 Hours. This is well below our required Design Life of 100000 Hours. Obviously, the current design or material selection (Weld Classification) is inadequate to meet the stated performance requirement. Also, note that this first result is based on a mean life (50% survival) criterion and we do not yet know the "scatter" of lives, on a 3 Standard Deviations basis.

5.2.2 SLF ANALYSIS - RUN 2 : Effect of "material" data scatter

As the fatigue durability calculated in 5.2.1 above is less than that required, we must now re-run the analysis with different input parameters in order to assess how the fatigue durability of our welded

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bracket can be improved and the Design Life achieved. Please note at this point that we are attempting to achieve a practical and cost effective solution to the problem.

As a next step, we shall look at the effect of statistical variations in the "materials" data (S-N curve) to define the upper and lower bounds on life using our initial weld classification (CLASSW).

We could, at this point, run SLF repeatedly using different % certainty of survival factors for the analysis. A feature of SLF, however, is that it will accept multiple factors and calculate a fatigue life for each.

SLF can calculate multiple %certainties quite easily. The results of a multiple calculations are given below:

Service Environment

Name of input strain history : SLFDEMO.PVX Type of data : Time Series Equivalent units for this history : 3 Hours Number of data values in time series : 1668 Maximum strain value in input data (uE):750 Minimum strain value in input data (uE):-723 Output cycles file name : None

Damage Law Details

S-N data set name : CLASSW

Analysis Parameters

Analysis Type :BS5400 Welded Scaling factor : 1 % Certainty of Survival : ALL Cycles Hysteresis Gate (uE) : 0 Miners Constant : 1

Design criterion vs. Life analysis

Number of SD's% CertaintyDesign lifefrom mean of survival (Hours)

-3 99.9 1613 -2.5 99.4 1994 -2 97.7 2466 -1.5 93 3049 -1 84 3771 -0.5 69 4664 0 50 5768 0.5 31 7133

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1 16 8823 1.5 6.7 10913 2 2.3 13498 2.5 0.6 16698 3 0.15 20663

These results are shown graphically in Figure 95

Figure 92 'Design Criterion-Life' plot for the S-N analysis described in 5.2.2 above

Note that the results of this analysis follow a sinus curve and that the "scatter" in the calculated fatigue lives is somewhat greater than one order of magnitude.

Unfortunately, none of the lives calculated above, even the Lower Bound life, is even close to our required Design Life of 100000 Hours. Obviously, therefore, the lowest quality welding procedure, represented by the CLASSW S-N curve, is insufficient to produce this welded bracket successfully.

5.2.3 SLF ANALYSIS - RUN 3 : Effect of mean stress correction method

At this point, questions may be raised about the effect of different mean stress correction procedures on the values calculated for fatigue life. Before we go on with finding a solution to the case study example, therefore, it is worth making a short digression to show how fatigue life varies with Mean Stress Correction procedure.

Another run of SLF with a % certainty of survival set to 50, and a variety of analysis methods, produced the results below:

Service Environment

100

80

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Cross plot of Data TEST!

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1E4 2E4 3E4 4E4 5E4Life (hours)

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Name of input strain history : SLFDEMO.PVX Type of data : Time Series Equivalent units for this history : 3 Hours Number of data values in time series : 1668 Maximum strain value in input data (uE):750 Minimum strain value in input data (uE):-723 Output cycles file name : None

Damage Law Details

S-N data set name : CLASSW

Analysis Parameters

Analysis Type : ALL Scaling factor : 1 % Certainty of Survival : 50 Cycles Hysteresis Gate (uE) : 0 Miners Constant : 1

Analysis method vs. Life analysis

Analysis Method Fatigue life (Hours)

S-N 5768 BS5400 Welded 5768 BS5400 Non-Welded 11073 Goodman 5755 Gerber 5766

In this particular case, there is little real difference in the calculated fatigue life for most of the analysis method options available. Only the BS5400:Part10 method for use with Non-Welded features (bolts, rivets, flame-cut holes etc.) gives a significantly longer life estimate than the others. This situation, however, is not usually the case and often the Goodman or Gerber methods will give the most conservative life estimate. In general, it is recommended to base all post-processing calculations on the method which gives the most conservative result for safety reasons. However, we are considering here a welded structure and, in such cases, the BS5400:Part 10 Welded analysis method is generally recommended.

5.2.4 SLF ANALYSIS - RUN 4 : Effect of Changing Material S-N Properties

The analysis will now change direction again and assume that the 'best' welding practice will be adopted. Using the CLASSB S-N data set, shown in Figure 93, and BS5400 as the analysis method, produced the following results.

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Figure 93 S-N curve for BS5400 CLASSB weld type showing 3 SD scatter lines

Service Environment

Name of input strain history : SLFDEMO.PVX Type of data : Time Series Equivalent units for this history : 3 Hours Number of data values in time series : 1668 Maximum strain value in input data (uE): 750 Minimum strain value in input data (uE) : -723 Output cycles file name : SLF2.SLF

Damage Law Details

S-N data set name : CLASSB Stress RANGE InterceptSRI1 : 6955 Slope of first line b1 : -0.25 Transition life NC : 1E7 Slope of second lineb2 : -0.1667 Fatigue Limit FL : 10 Elastic modulus (MPa),E : 2.07E5 Standard error SE : 0.1824

Analysis Parameters

Analysis Type : BS5400 Welded Scaling factor : 1 % Certainty of Survival : 50 Cycles Hysteresis Gate (uE) : 0

S-N Data Plot

104

103

102

101

100101 102 103 104 105 106 107 108 109 1010

[Life Cycles]

Stre

ss R

ange

MPs

classWSRI1: 7179 b1: -0.3333 b2: -0.2 E: 2.07E5 UTS : 500

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Miners Constant : 1

Results of Analysis

Turning point history : SLFDEMO.PVX S-N Data Set : CLASSB Scaling factor : 1 % Certainty of Survival : 50 Cycles Hysteresis Gate (uE) : 0 Analysis Method : BS5400 Welded Output file name : SLF2.SLF Miners Constant : 1 Number of cycles used : 834 Fatigue life : 1.84E5 Hours

A summary plot of this result showing the input load history, Rainflow cycles and fatigue damage distributions is shown in Figure 94.

Figure 94 Summary of an analysis, showing input load history and 'Rainflow' cycles & fatigue damage distributions

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Mean uE

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-7231488

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Damage Histogram Distribution

Cycle Histogram Distribution


750

Mean uE

1488 -723

Range uE

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Cycles

70

Service Environment

Input strain : SLFBEND.DACType of data : Time seriesEquivalent units: 3 hours

Analysis Parameters

Analysis type : BS5400 WeldedScaling factor : 1% Certainty of survival 50Hysteresis gate : 0Miners constant : 1

Time Seconds

600

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This result shows that the Design Life of our welded bracket can be achieved by changing the welding practice from "CLASSW" to "CLASSB". However, it must be noted that this result only shows that fifty percent (50%) of the brackets will survive for 184000 Hours or, alternatively, there is a 50% chance that a single bracket will fail in this period of time.

5.2.5 SLF ANALYSIS - RUN 5 : Effect of "Material" data scatter (CLASSB)

Once more, therefore, we must post-process the results from 5.2.4 and now take into account the "SCATTER" inherent in the S-N Data Set (CLASSB, this time).

A fifth run of SLF with a % certainty of survival set to ALL produced the following results:

Service Environment

Name of input strain history : SLFDEMO.PVX Type of data : Time Series Equivalent units for this history : 3 Hours Number of data values in time series : 1668 Maximum strain value in input data (uE): 750 Minimum strain value in input data (uE) : -723 Output cycles file name : None

Damage Law Details

S-N data set name :CLASSB

Analysis Parameters

Analysis Type : BS5400 Welded Scaling factor : 1 % Certainty of Survival : ALL Cycles Hysteresis Gate (uE) : 0 Miners Constant : 1

Design criterion vs. Life analysis

Number of SD's% CertaintyDesign lifefrom mean of survival (Hours)

-3 99.9 5.14E4 -2.5 99.4 6.36E4 -2 97.7 7.86E4 -1.5 93 9.71E4 -1 84 1.2E5 -0.5 69 1.49E5 0 50 1.84E5 0.5 31 2.28E5 1 16 2.84E5

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1.5 6.7 3.53E5 2 2.3 4.4E5

2.5 0.6 5.5E5 3 0.15 6.88E5

These results are shown graphically in Figure 95.

Figure 95 'Design Criterion-Life' plot for the S-N analysis described in 5.2.5 above

The results from this analysis run show that the lower bound Life - 99.9% certainty of survival - is only 51400 Hours; a little more than half our Design Life requirement of 100000 Hours. Clearly, most of the welded brackets (over 90%) will survive to the required endurance and this may be an acceptable position. However, if complete security is required, i.e. 99.9% certainty of survival, then even using the "best" quality welding practice will not quite suffice.

100

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2E5 4E5 6E5 8E5 1E6Life (hours)

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5.2.6 SLF ANALYSIS - RUN 6 : BACK calculation of Scale Factor

One way to enhance the performance of the welded bracket would be to reduce the loading on the welded area. Let us try to quantify this reduction in the applied loading: essentially, increase the load-bearing section by some amount.

This next SLF run will define the reduction in loading and will be based on the lower bound (99.9% survival) criterion. We shall use the "BACK" life (iterative) calculation option to define the factor by which the loads must be reduced to achieve the Design Life of 100000 Hours at the lower bound (99.9% survival) level.

In other words, SLF can be used to determine a value for the scale factor that must be used to achieve a % certainty of survival of 99.9% for 100000 hours. The method is BS5400, the input data is CLASSB.

The results are as follows:

Service Environment

Name of input strain history : SLFDEMO.PVX Type of data : Time Series Equivalent units for this history : 3 Hours Number of data values in time series : 1668 Maximum strain value in input data (uE): 750 Minimum strain value in input data (uE) : -723 Output cycles file name : None

Damage Law Details

S-N data set name : CLASSB

Analysis Parameters

Analysis Type : BS5400 Welded Scaling factor : BACK % Certainty of Survival : 99.9 Cycles Hysteresis Gate (uE) : 0 Miners Constant : 1

Back Factor Results

Turning point history : SLFDEMO.PVX S-N Data Set : CLASSB % Certainty of Survival : 99.9 Cycles Hysteresis Gate (uE) : 0 Design life : 1E5 Hours Calculated life : 9.72E4 Hours

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Calculated back factor : 0.854

This final step in the case study now shows effectively that we can achieve the required Design Life by a combination of changing the welding procedure used to produce the bracket from "worst" to "best" and by increasing the section size such that the applied loading will be reduced by around 15%.

This solution almost certainly requires only minor changes to the geometry of the bracket (increased section size) but does require, perhaps, a more significant effort in terms of manufacturing quality control to ensure that the "best" welding practice is used in the production of these brackets.

By implementing these changes we have achieved what is probably the most cost-effective and simplest solution. Further development and testing work can now proceed with confidence.

Using the CLASSB S-N data set, there are fewer damaging cycles - this welding procedure is "better" than CLASSW at 'resisting' the measured loads therefore each cycle causes less fatigue damage.

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5.3 Definition of the Local Strain (CLF) Case Study problem - Vehicle Suspension

The problem centers around the fatigue durability of a vehicle suspension component which is produced as a steel forging. This component is illustrated schematically in Figure 96.

Figure 96 Schematic of the vehicle suspension component

This component has a design life of 500 000 km and the appearance of fatigue cracks during this period cannot be allowed as this is a 'Safety-Critical' component.

Strain measurements have been made on a prototype suspension over a distance of 25 km of carefully selected roads which are believed to be representative of 'normal' customer usage - whatever that is!

These data have been measured at a nominal elastic location on the test component and stored in the file DEMO.DAC.

The material selected for the manufacture of these components is a plain carbon steel which we shall call MAT1. (Actually, this is MANTEN steel).

First, we need to know if the component will survive for the required 500 000 km usage when manufactured from MAT1.

To do this we will use the 'five box trick' (Figure 62) to which the three main inputs are:

1. Strain-Time history ('Signal' DEMO.PVX)

2. Material Data (MAT1.MAT)

3. 'Local Geometry' factor (Kf)

The 'Local Geometry' factor (Kf) is similar in some ways to the Elastic Stress Concentration factor (Kt) and is used to 'convert' nominal elastic strains to local elastic-plastic strains using Neuber's Rule, Masing's Hypothesis and the material's Cyclic Stress-Strain properties. Kf is also used to take account of 'Surface Condition' and its value varies with surface finish: polished, machined, as-cast, corroded etc..

Because 'real' service/field data is generally broad-band random in nature, it is important that the

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fatigue analysis can handle variable mean stress effect correctly. The Smith-Topper-Watson and Morrow procedures in FATIMAS handle the effects of variable mean stress and, when combined with the best available materials modelling techniques (Neuber, Massing etc.), model the fatigue response of a material to a strain-time signal in precisely the same way that the 'real' material would respond in a physical test.

To make a fatigue life prediction, CLF was run with the following inputs:

Signal : DEMO.PVX(CLF will alert the user that the data is equivalent to 25 km)

Kf : 2.5 (Accounts for local geometry [Kt] and anAs-Forged surface condition)

Mean Stress Correction : Smith-Topper-Watson

Material : MAT1Results Filename:EX1

Figure 97 and Figure 98 respectively show the strain-life fatigue damage curve and the cyclic stress-strain curve for material MAT1.

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Figure 97 Strain-Life fatigue damage curve for material MAT1

Figure 98 Cyclic Stress-Strain curve for material MAT1

5.3.1 CLF ANALYSIS - RUN 1 : 'Basic' inputs

Strain Life PlotMAT1 sr’ : 917 b: -0.095 Ef’ : 0.261 c: -0.47

Life (Reversals)

Stra

in A

mpl

itude

(m/m

) 1E-1

1E-2

1E-3

1E-41E0 1E1 1E2 1E3 1E4 1E5 1E6 1E7 1E8 1E9

100

Cyclic Stress-Strain Plot_____ MAT1

n’ : 0.193 K’ : 1103 E: 203000

Strain (M/M)0 0.05 0.1

900

Stre

ss (M

Pa)

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The results obtained from this first CLF analysis run are summarised below:

Service Environment

Name of input strain history : DEMO.PVX Type of data : Time Series Equivalent units for this history : 25 km Number of data values in time series: 1708 Maximum strain value in input data (uE): 999 Minimum strain value in input data (uE): -495

A plane strain correction will not be applied. Output cycles file name : EX1.CLF

Materials Environment

Materials data set name : MAT1 Fatigue strength coefficient (MPa),Sf' : 917 Fatigue strength exponent,b : -0.095 Fatigue ductility exponent,c : -0.47 Fatigue ductility coefficient,Ef' : 0.26 Elastic modulus (MPa),E : 2.03E5 Cyclic strain-hardening exponent,n' : 0.193 Cyclic strength coefficient (MPa),K' : 1103 Cut-off (reversals), Rc : 1E9

Analysis Parameters

Mean Stress Correction Method :Smith-Topper-Watson Scaling factor : 1 Stress Concentration factor, Kf : 2.5 Cycles Hysteresis Gate (uE) : 0 Miners Constant : 1

Results of Analysis

Turning point history : DEMO.PVX Material : MAT1 Scaling factor : 1 Stress Conc. factor, Kf : 2.5 Cycles Hysteresis Gate (uE) : 0 Damage Parameter : Smith-Topper-Watson Output file name : EX1.CLF Plane Strain correction : N Miners Constant : 1 Number of cycles used : 854

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Fatigue life : 1.89E5 km

The calculated life for our suspension component is 189000 km. This is well below our required Design Life of 500000 km. Obviously, the current design or material selection is inadequate to meet the stated performance requirement.

Figure 99 shows a summary of the input strain history and the Rainflow cycles and fatigue damage distributions for the above analysis. In general, 95% of the damage is caused by the largest 5% of the cycles in the strain history.

Figure 99 Summary of the analysis above showing input load history and 'Rainflow' cycles & fatigue damage distributions

5.3.2 CLF ANALYSIS - RUN 2 : Effect of varying strain magnitude

As the calculated fatigue durability in 5.3.1 is less than that required, we must now re-run the analysis with different input parameters in order to assess how the fatigue durability of our suspension component can be improved and the Design Life achieved. Please note at this point that we are attempting to achieve a practical and cost effective solution to the problem.

As a first attempt, we shall look at the effect of reducing the strain levels in the component. This is equivalent to increasing the section size of the component.

We could, at this point, run CLF repeatedly using different scaling factors for the input strain history

nCode n

Mean uE

750

-7231488

Range uE

00

Damage

3.694E-5

Damage Histogram Distribution

Cycle Histogram Distribution


750

Mean uE

1488 -723

Range uE

00

Cycles

70

Service Environment

Input strain : SLFBEND.DACType of data : Time seriesEquivalent units: 3 hours

Analysis Parameters

Analysis type : BS5400 WeldedScaling factor : 1% Certainty of survival 50Hysteresis gate : 0Miners constant : 1

Time Seconds

600

400

200

0

-200

-400

-600

0 50 100 150

Stra

in u

E

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(DEMO.PVX). A feature of CLF, however, is that it will accept multiple scaling factors and calculate a fatigue life for each.

Multiple scaling factors can be entered into CLF like this:

Scaling Factor :0.8,0.9,1,1.1,1.2

The results of entering the above scaling factors are summarised below:

Service Environment

Name of input strain history : DEMO.PVX Type of data : Time Series Equivalent units for this history : 25 km Number of data values in time series : 1708 Maximum strain value in input data (uE):999 Minimum strain value in input data (uE):-495

A plane strain correction will not be applied. Output cycles file name : None


Materials data set name : MAT1

Analysis Parameters

Mean Stress Correction Method :Smith-Topper-Watson Scaling factor :0.8,0.9,1,1.1,1.2 Stress Concentration factor, Kf : 2.5 Cycles Hysteresis Gate (uE) : 0 Miners Constant : 1

Factors vs. Life analysis

Factor Fatigue life (km)

0.8 8.07E5 <-- Beyond Design Life 0.9 3.68E5 1 1.89E5 1.1 1.07E5 1.2 6.53E4

If it were possible to reduce the magnitude of the measured strains by around 15-20%, the Design Life of 500000 km could be achieved successfully.

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It should be noted that the relationship between Scaling Factor (strain magnitude) and fatigue life is not linear as is shown graphically for the above CLF analysis results in Figure 100.

Figure 100 The effect of varying the magnitude of the applied load history on calculated fatigue life

5.3.3 CLF ANALYSIS - RUN 3 : BACK Kf calculation & the influence of geometry on fatigue life

Unfortunately, it is not always possible to increase the section size of a component due to other constraints: for example, connection to other components then becomes difficult, the component becomes too heavy, manufacturing costs are increased to unacceptable levels etc..

In this case, our suspension component is failing due to the initiation of a fatigue crack at a stress raiser. It may be possible to reduce the effect of this stress raiser by increasing it's radius or improving the local surface finish by machining the as-forged surface.

Running CLF can tell the user what value of local stress concentration will achieve the desired design life of 500000 km.

The results obtained from this third CLF analysis run are summarised below:

Service Environment

Name of input strain history : DEMO.PVX Type of data : Time Series Equivalent units for this history : 25 km Number of data values in time series : 1708 Maximum strain value in input data (uE): 999 Minimum strain value in input data (uE) : -495



1E50.8

1.2

Life (Km)

Scale factor vs. fatigue life for data EX1

Scal

e fa

ctor

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Analysis Parameters

Mean Stress Correction Method: Smith-Topper-Watson Scaling factor : 1 Stress Concentration factor, Kf : BACK Cycles Hysteresis Gate (uE) : 0 Miners Constant : 1

Back Kf Results

Turning point history : DEMO.PVX Material : MAT1 Scaling factor : 1 Cycles Hysteresis Gate (uE) : 0 Damage Parameter : Smith-Topper-Watson Plane Strain correction : N Required life : 5E5 km <-- Design Life Calculated back Kf : 2.16 Actual life : 5.02E5 km

We can now see that if it were possible to reduce the effect of the local Stress Concentration (Kf) from its original value of 2.5 to a new value of 2.16, then the Design Life would be achieved.

This reduction in Kf could be achieved by either improving the surface finish in the critical area or by modifying the local geometry by, say, making the local radius larger.

5.3.4 CLF ANALYSIS - RUN 4 : Effect of Mean Stress Correction Method

Many different methods for handling the effects of variable mean stress in local-strain fatigue life calculations have been proposed over the past 30-35 years. nCode has investigated many of these and found that those proposed by Smith, Topper & Watson (STW) and Morrow give the most consistent and reliable results.

The RANKING of the calculated fatigue lives when using these different Mean Stress Correction methods varies and is related to the form of the input strain history. In general, the STW parameter will tend to give the most conservative (shortest) lives for signals which have a positive or tensile mean and the Morrow method will tend to give the most conservative lives for signals with negative or compressive means. Note that no correction (i.e. simply using the Strain-Life fatigue damage curve in the calculations) will always give the same life value regardless of the mean level of the input sinewave.

Running CLF with the original values (Scaling Factor = 1, Kf = 2.5) and Mean Stress Correction Method set to ALL illustrates the different lives that CLF will calculate using these various mean stress correction methods.

The results obtained from this fourth CLF analysis run are summarised below:

Service Environment

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Name of input strain history : DEMO.PVX Type of data : Time Series Equivalent units for this history : 25 km Number of data values in time series : 1708 Maximum strain value in input data (uE): 999 Minimum strain value in input data (uE) : -495




Analysis Parameters

Mean Stress Correction Method : ALL Scaling factor : 1 Stress Concentration factor, Kf : 2.5 Cycles Hysteresis Gate (uE) : 0 Miners Constant : 1

Damage Parameter vs. Life analysis

Damage ParameterFatigue life(km)

Smith-Topper-Watson 1.89E5<-- Most Conservative Morrow 2.63E5 Strain Life 3.09E5

These results show that the Smith-Topper-Watson (STW) method gives the most conservative life estimate (the signal has a high positive mean) and is, thus the 'safest' parameter to use in this particular case.

5.3.5 CLF ANALYSIS - RUN 5 : Effect of Changing Material

We have examined and quantified the effects of changing those input parameters which can be said to relate to the 'geometry' and 'manufacturing process' for the suspension component. It is also possible to assess the fatigue durability of our test structure should we make it using a different material.

Our original material choice was a Plain Carbon steel and this was shown to be inadequate when analysed using measured loads. We can use the same strain history and geometry (original settings for Scaling Factor [1] and Kf [2.5]) as in our first analysis but see now the effect on life of using an alternative material - in this case an alloyed steel, MAT2. The strain-life fatigue damage curve and the cyclic stress-strain curve for MAT2 are shown in Figure 101 and Figure 102 respectively.

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Figure 101 Strain-Life fatigue damage curve for material MAT2

Strain Life PlotMAT2 sr’ : 1227 b: -0.095 Ef’ : 1 c: -0.66

Life (Reversals)

Stra

in A

mpl

itude

(m/m

)

1E-1

1E-2

1E-3

1E-41E0 1E1 1E2 1E3 1E4 1E5 1E6 1E7 1E8 1E9

100

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Figure 102 Cyclic Stress-Strain curve for material MAT2

NOTE: The use of alternative material data sets in a CLF analysis should be performed with care. It is acceptable to look at, say, alternative steels if the original measurements were made on a steel component. Due to differences in Elastic Modulus (E), however, it is not advised that, for example in this case, an aluminium alloy be used in place of steel. Aluminium has an Elastic Modulus about a factor of 3 less than the value for steel and, hence, the component would need to be 3 times 'thicker' to give strain measurements equivalent to those made form a steel component. By very careful manipulation of the input strain history it may be possible to overcome this problem but we would always recommend that FATIMAS users discuss such cases with nCode first.

To illustrate the effect of a material change, CLF was run with the new material data set (MAT2). The scaling factor was set to 1, Kf to 2.5, and the correction method was STW.

The results obtained from this fifth CLF analysis run are summarised below:

Service Environment

Name of input strain history : DEMO.PVX Type of data : Time Series Equivalent units for this history : 25 km Number of data values in time series : 1708 Maximum strain value in input data (uE):999

Cyclic Stress-Strain Plot_____ MAT2

n’ : 0.18 K’ : 1344 E: 200000

Strain (M/M)0 0.05 0.1

1000

Stre

ss (M

Pa)

0

nnCode n




Minimum strain value in input data (uE):-495

A plane strain correction will not be applied. Output cycles file name : EX2.CLF


Materials data set name : MAT2 Fatigue strength coefficient (MPa)Sf' : 1227 Fatigue strength exponent,b : -0.095 Fatigue ductility exponent,c : -0.66 Fatigue ductility coefficient,Ef' : 1 Elastic modulus (MPa),E : 2E5 Cyclic strain-hardening exponent,n' : 0.18 Cyclic strength coefficient (MPa),K' : 1344 Cut-off (reversals), Rc : 2E8

Analysis Parameters

Mean Stress Correction Method : Smith-Topper-Watson Scaling factor : 1 Stress Concentration factor, Kf : 2.5 Cycles Hysteresis Gate (uE) : 0 Miners Constant : 1

Results of Analysis

Turning point history : DEMO.PVX Material : MAT2 Scaling factor : 1 Stress Conc. factor, Kf : 2.5 Cycles Hysteresis Gate (uE) : 0 Damage Parameter : Smith-Topper-Watson Output file name : EX2.CLF Plane Strain correction : N Miners Constant : 1 Number of cycles used : 854 Fatigue life : 6.7E5 km <-- Beyond Design Life

This result shows effectively that we can achieve the required Design Life (with a degree of safety) by simply changing the material from which the suspension component is made from a Plain Carbon Steel to an Alloy Steel.

This solution almost certainly requires no changes to the production tooling (forging dies) or processing. We can use the component geometry defined by the designer and, although the material costs will be a little higher, we have achieved what is probably the most cost-effective and simplest solution. Further development and testing work can now proceed with confidence.

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Figure 103 is a summary plot which shows the input time history and the Rainflow cycles and fatigue damage distributions obtained from this final analysis step. Note that the cycles & damage distributions are different to those obtained from the first CLF analysis in 5.3.1 above (Figure 99).

Figure 103 Summary of the analysis in 5.3.5 above showing input load history and 'Rainflow' cycles & fatigue damage distributions

The ten largest stress-strain hysteresis loops obtained from the analyses using MAT1 (5.3.1 above) and MAT2 are given in Figure 104 and Figure 105 respectively. Using MAT2, there are FEWER damaging cycles because this material is better than MAT1 at 'resisting' the measured loads; therefore each cycle causes less fatigue damage and the appearance of the hysteresis loops is more 'elastic' in nature.

999

-4950 1898Time (secs)

Stra

in u

E

Mean uERange uE

00

Damage

Damage Histogram Distribution Cycles Histogram Distribution

Mean uERange uE

Cycles

Material : MANTENScale factor : 1Life : 1.890E5

Stre

ss (M

Pa)

-247.3-1.237E-3 3.496E-3Strain (m/m)

328.4

Hysteresis loops for data EX1

Kf : 2.5

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Figure 104 The ten largest stress-strain hysteresis loops from analysis using material MAT1 (5.3.1 above)

Figure 105 The ten largest stress-strain hysteresis loops from analysis using material MAT2 (5.3.5 above)

5.4 Definition of the LEFM (crack growth software) Case Study - Highway Bridge

The problem centers around a highway bridge built across a river estuary. The supporting structure, some below, some above the water line and some in the tidal zone, is of a welded steel (50D) tubular construction as shown schematically in Figure 106. (Perhaps the designer did a technology transfer from wooden tree trunks tied together!). A crack tolerant design has been based on the UK Department of Energy and the Scandinavian DNV Rules for weld class F including a penalty factor of two on life for the sea-water environment; the sea provides the corrosion and the cyclic loading by waves passing through the structure every 5 seconds. The bridge is designed to last at least 50 years.

Material : 1044QTScale factor : 1Life : 6.702E5

Kf : 2.5

395.1St

ress

(MPa

)

-266.7-1.052E-3 3.087E-3Strain (m/m)

Hysteresis loops for data EX2

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Figure 106 Schematic of a welded tubular joint in the bridge structure

During the first four months after construction, a substantial growth of "marine fouling" (seaweed and mussels) has grown on the structure at and just below the water line. It also smells badly of "rotten eggs", i.e. hydrogen sulphide, and it is known that sulphate reducing bacteria in the fouling layer can produce H2S at the steel surface. The structure, rather foolishly, was neither painted nor cathodically protected.

We can use crack growth software to assess the effect of this nasty environment on the growth rate of any weld cracks and hence on remaining fatigue life. Using the 5 box trick approach (Figure 60), we need to know the local weld toe geometry, the material crack growth rate response in the environment, and some idea of the cyclic stresses from the wave loading: wind, tidal and resonance loadings are ignored; but should they be? All we have is a short section of a standard off-shore loading history (MARINE.DAC) which contains 983 cycles but is deemed to be equivalent to 6 days of typical operation in this location.

5.4.1 CRG ANALYSIS - RUN 1: Basic inputs

The first run of CRG (the KRAKEN software fatigue crack propagation module) is to assess the life of the most critical (highly stressed) joint in the structure in the sea-water/H2S environment. Note that we do not know the initial crack size at the weld toe so we can enter zero and let CRG work out a minimum size. Also, we know that all joints had their weld toes treated by grinding leaving a notch assumed to be 3mm deep, 3mm radius. This treatment is done during fabrication so that welds pass the "dime test" (an American durability assessment procedure involving placing a coin upright at the weld toe and checking that a 1mm wire, a paper clip, will not pass under the coin!).

All the input parameters and results for this first CRG run are given below. The Paris curve for 50D steel tested in a sea-water/H2S environment is shown in Figure 107. The K solution inputs for the welded tubular joint geometry are illustrated in Figure 108 and the variation in the resulting compliance (Y) function with crack ratio in Figure 109.

WELD WATER LINE

miaooww.o o

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Figure 107 The 'Paris' curve for steel 50D tested in a sea-water/H2S environment

Figure 108 The Stress Intensity (K) solution for the welded tubular joint geometry used in the crack growth software analyses


10-11

10-9

10-8

100 101 102


[da/

dN m

/cyc

le]

50D: Ratio 0.1 Environment: Seawater. H2S

10-7

10-6

10-5

10-10

C: 1.849E-10 m: 3.534 Kc: 60 DO: 5.5 D1: 3.5 Rc: 0.75


Specimen type : Crack in Welded Tubular Joints

Dimensions

T

D

d

t Diam ratio BETA : 0.5Thickness T : 32SCF : 8AVESCF : 2

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Figure 109 The relationship between Compliance (Y) function crack ratio (a/T) for the welded tubular joint geometry used in the crack growth software analyses

General Information

Input file name : MARINE.TCY Output file name : MARINE.CRG solution file name : MARINE.KSN

Plane stress correction not included.

Scale factor for cycles : 1 Mean offset for cycles : 0 Number of cycles between output : 500 Maximum number of cycles :100000000

Material Information

Material name : 50D Yield or proof strength (MPa) : 355 UTS (MPa) : 480 Elastic modulus (MPa) : 1.914E5 Fracture toughness (MPam1/2) : 73.8 Environment name : SEA WATER/H2S Unnotched fatigue limit (MPa) : 50 Paris law coefficient (m/cycle) : 1.849E-10 Paris law exponent : 3.534

0.1 0.2 0.4 0.6 0.8 1.0

Com

plia

nce

(Y)

Display of EXAMPLE.KSN

1.9

1.1

Crack ratio (a/T)

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Delta K threshold at R=0 (MPam1/2) : 5.5 Delta K threshold as R-1 (MPam1/2) : 3.5 Stress ratio at threshold knee : 0.8

Local Geometry Information

Notch depth (mm) : 3 Notch root radius (mm) : 3 Sharp crack root radius (mm) : 1E-2 Initial defect size (mm) : 3.852 Maximum defect size (mm) : 32

History Information

History effect included.

Result

Failure method: Crack size reached failure size. Cycles to failure: 26142

The remnant life of 26142 cycles is equivalent to about 5 months, i.e. only one month left.

5.4.2 CRG ANALYSIS - RUN 2: Crack growth after cleaning

The options now are either to close the bridge and inspect and replace all damaged joints or to clean off the marine fouling. The latter option now allows us to check in the software model what the remnant life would be if the H2S is absent. This was carried out in the second CRG analysis where an initial crack size of 16mm, 50% crack penetration, was used. It is assumed that such a crack could not possibly be missed on a non-destructive inspection: or could it, under water??

The K solution used in this second CRG analysis is the same as before , but different material

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properties were used: 50D steel tested under free corrosion in sea-water, as shown in Figure 110.

Figure 110 The 'Paris' curve for steel 50D tested in a sea-water + free corrosion environment

General Information

Input file name : MARINE.TCY Output file name : MARINE2.CRG K solution file name : MARINE.KSN

Plane stress correction not included.

Scale factor for cycles : 1 Mean offset for cycles : 0 Number of cycles between output:500 Maximum number of cycles : 100000000

Material Information

Material name : 50D Yield or proof strength (MPa) : 355 UTS (MPa) : 480 Elastic modulus (MPa) : 1.914E5 Fracture toughness (MPam1/2) : 73.8 Environment name : SEA/FC Unnotched fatigue limit (MPa) : 152 Paris law coefficient (m/cycle) : 2.22E-11 Paris law exponent : 3.28 Delta K threshold at R=0 (MPam1/2):6.17 Delta K threshold as R-1 (MPam1/2):3


10-11

10-9

10-8

100 101 102


[da/

dN m

/cyc

le]

50D: Ratio 0.1 Environment: Seawater. free corr.

10-7

10-6

10-5

10-10

C: 2.2E-11 m: 3.28 Kc: 72.7 DO: 6.17 D1: 3 Rc: 0.594

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Stress ratio at threshold knee : 0.8

Environment Information

Anodic current density in A/cm2 : 0 Hydrogen diffusion coefficient in m2/s : 0 Constant for metal dissolution calculation:0

Local Geometry Information

Notch depth (mm) : 3 Notch root radius (mm) : 3 Sharp crack root radius (mm) : 1E-2 Initial defect size (mm) : 16 Maximum defect size (mm) : 32

History Information

History effect included.

Results

Failure method : Crack size reached failure size. Cycles to failure : 144102

This life of 144102 cycles is equivalent to 28 months; i.e. 24 months from now. This gives time after the cleaning operation to consider the matter further. This result, however, critically depends on the assumption of 16mm as a starting crack size; so what is the actual crack size now after 4 months in the H2S environment? We can find this out by post processing the first results file, MARINE.CRG, using CRL.

5.4.3 POST PROCESSING WITH CRL - RUN 1

The cycle by cycle crack growth listing module, CRL, is used to post- process the results files created by the CRG module. CRL has three main functions:

1. To list the results to the screen, or to an ASCII file, for subsequent report generation.

2. To cross plot graphically, on screen or to plot files, any two parameters of the 13 saved in the CRG file.

3. To interpolate the crack size/cycles data for sensitivity studies, inspection strategy, or defect assessment purposes.

Running CRL and selecting the fatigue life interpolation option shows that the final crack size corresponding to 4 months operation is equivalent to 19933 cycles and gives a crack length of 16.4mm.

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5.4.4 POST PROCESSING WITH CRL - RUN 2

We can now run CRL again on the results from the second CRG run, (MARINE2.CRG), and find by interpolation, the remnant life from an initial defect size of 16.4mm. This remnant life is 132814 cycles, or about 27 months. If the structure is cleaned immediately, then the expected total life is around 31 months, of which 4 have already passed.

Further analysis shows that if the joint had not been allowed to become fouled, the life would be about 22 years. The bridge supports would still not meet their design life requirement of 50 year design life, but joints above the water line would last very much longer.

These crack growth software analyses have now directed the engineers' decision to an immediate expenditure on cleaning which gives them a further 27 months breathing space to investigate and effect more considered remedial action. This will of course mean major road-works and traffic holdups but at least the bridge has been rescued from potentially a complete and catastrophic loss.

As the critical joint was in sea-water, it would have been better to make it as a casting which could be assumed to be crack free during manufacture. So, an initiation life could be added to the propagation life in the style of Nf = Ni + Np, where Ni, the life to fatigue crack initiation, could be estimated using CLF or the total life could be estimated directly using the NIAGRA module.

Other sensitivity studies are possible in KRAKEN which can assist in solving this and other fatigue cracking problems. These include: changing the overall joint geometry (K solution) and local notch geometry, changing material as well as environment, excluding the environment by painting or assessing the effect of cathodic protection, assessing the effect of stress cycle errors using the scale factor, assessing the effect of an imposed mean stress induced by welding stress relief or "fit up" stresses and assessing the assumptions about initial and final defect sizes.

Crack growth software is most effective when run automatically in batch command mode for longer runs and a complete batch process or batch template can be easily developed.

KRAKEN, of course, applicable to the fatigue durability assessment of anything that it is cracked or, indeed, anything that cannot be proven to be crack free. These methods have also been used on plastics and crack growth software is used by durability managers in automotive, aerospace, safety, construction and power generation industries, and in educational institutes.

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Fatigue Theory

Documents

fatigue life

history of fatigue

computerised fatigue

cyclic stressstrain

multiaxial stressstrain

stress cycles

strainlife characteristics

influence of mean stress