A review of multiaxial fatigue failure criteria based on the critical plane approach Aleksander KAROLCZUK Ewald MACHA Opole University of Technology, POLAND.

A review of multiaxial fatigue failure criteria based on the critical plane approach

Aleksander KAROLCZUKEwald MACHA

Opole University of Technology, POLANDDepartment of Mechanics and Machine Design, e-mail: [email protected]

Colloque National MECAMAT- Aussois 2007 - 21-26 Janvier 2007Ecole de Mécanique des Matériaux

2

Plan of the presentation

• Introduction• Critical plane approach

- definition- assumptions- range of application- general expressions

• Multiaxial fatigue failure criteria based on the critical plane approach- stress based criteria- strain based criteria- energy based criteria

• Algorithm of the fatigue life calculation• Determination of the critical plane orientation

- damage accumulation method- variance method- weight function method

• Exemplary application of simple energy based criterion in fatigue life calculation

Part I

Part II

3

Introduction

In material science, fatigue is the progressive, localised, and permanent structural damage that occurs when a material is subjected to cyclic stresses that have maximum values less than (often much less than) the static yield strength of the material.

Issues:Many mechanical and structural components are subjectedto uniaxial or multiaxial fatigue loading that could lead to catastrophic failures (aircrafts, ships, trains).

Proper determination of fatigue life of components and structures is important issue at the designing and operating stages.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-400

-200

0

200

400

Time, s

xx

, MP

a

Yield strength

Fatigue failure

4

Real service loading often generates random and multiaxial stress/strain state, which complicates the analysis.

Introduction

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01-400

-200

0

200

400

Time,s

xx, M

Pa

Many researchers have attempted to reduce multiaxial stress/strain state to uniaxial one, which is used in fatigue life calculation.

Such uniaxial parameter is often called ‘equivalent’ and it means that the same fatigue life is obtained under uniaxial (‘equivalent’) and multiaxial stress/strain state.

The reduction is based on the multiaxial fatigue failure criterion. Numerous multiaxial fatigue failure criteria have been proposed in recent decades.

5

Critical plane approach

Among these criteria, one type called the critical plane approach can be distinguished.

Another possible criterion (...), was that in which the two components of stress acting across any plane, i.e. shear and direct stress, might be taken as each contributing a definite quota to ”disruption” combined by a simple arithmetical relation. (...). The planes on which such effect were maximum would not be the principal planes, (...)

G. Stanfield, 1935

G. Stanfield. Discussion on ”The strength of metals under combined alternating stresses”, by H. Gough and H. Pollard. Proc. Institution of Mechanical Engineers 131, (1935)

This approach dates back to 1935 when Stanfield suggested a new criterion but without calling it „critical plane criterion” and without making any further research on this subject.

6


This concept was not developed until the Fifties when Findley (1956, 1959), Stulen and Cummings (1954) introduced the phrase ”critical plane” and verified fatigue stress criteria based on the critical plane approach.

The critical plane approach assumes that the fatigue failure of the material is The critical plane approach assumes that the fatigue failure of the material is due to some stress or/and strain components acting on the critical plane. due to some stress or/and strain components acting on the critical plane.

In this approach, the aspect of microdamage or even short crack propagation are not considered.

It is based upon the experimental observation that in metallic materials fatigue cracks initiate and grow on certain planes.

7


The critical plane approach concerns the crack initiation process that is usually (in most papers) related to fatigue failure at high cycle fatigue regime. However, it was successfully used also at low cycle fatigue regime.

Transformation

Arbitrary plane orientation

jiijn nntt )()(

jiijns sntt )()(

8


Summary:

• critical plane criteria reduce the multiaxial state of stress/strain to the

equivalent-uniaxial state,• this single parameter (often called damage parameter) is used to

calculate fatigue life or damage degree on a plane using the standard S-

N curves (a - Nf , a - Nf , a - 2Nf , a - 2Nf),

• any phenomena regarding to the crack propagation are not considered,• fatigue life to crack initiation is therefore (usually) estimated,• orientation of the fatigue fracture plane (crack orientation) could be

determined in some cases .

9


Significance of the critical plane approach has increased during last years, because of its effectiveness and broad application range (proportional, non-proportional, cyclic and random loading). The phrase „critical plane” is included in around 350 articles in databases of Elsevier Journals and Springer Journals during the last 5 years.

1),,,),(( kTsntf iiij

1),,,),(( kTsntg iiij

1),,,),(),(( kTsntth iiijij

The problem is stated as follows: - construct fatigue failure criterion for multiaxial cyclic/random states of stress- determine the critical plane orientation

The general form of a critical plane-type failure criterion, that determines the fatigue life T after which crack initiation occurs, can be expressed as follows:

Stress based -

Strain based -

Energy based -

k – array of material coefficients

10

In general, the critical plane is the plane for which fatigue life will be determined.The problem is: which plane orientation is critical?


Dozens (or more) of multiaxial fatigue failure criteria based on the critical plane approach were proposed. 32 of them are described in paper:

KAROLCZUK A., MACHA E.: A review of critical plane orientations in multiaxial fatigue failure criteria of metallic materials, International Journal of Fracture, vol 134, 2005, pp. 267-

304

11

Multiaxial fatigue criteria based on the critical plane concept usually apply different loading parameters in the critical plane whose orientation is determined by

(a) only shear loading parameters (crack Mode II or III),

(b) only normal loading parameters (crack Mode I) or sometimes

(c) mixed loading parameters (mixed crack Mode).

There are also criteria based on few critical plane orientations and criteria based on critical plane orientations determined by a weighted averaging process of rotating principal stress axes.


12

Multiaxial fatigue criteria based on the critical plane approach can be divided according to the fatigue parameter into three groups:

- stress criteria, High Cycle Fatigue regime, HCF

- strain criteria, High and Low Cycle Fatigue regimes, HCF-LCF

- energy criteria, HCF and LCF


13


The present review of the multiaxial fatigue failure criteria based on the critical plane approach is focused on

(i) presentation of the large spectrum of the damage parameters resulting from the critical plane approach and

(ii) survey of the critical plane orientation used in the fatigue critical plane criteria

14

Stress criteria

15

Generalised hypothesis of material strength

Critical planes in criteria based on stresses

Adaptation of the static hypotheses of material strength to fatigue as the replacement of stress static values in these hypotheses by amplitudes or range of fatigue loading

The criteria most frequently verified with experimental data, were the criteria of:

• maximum normal stress,

• maximum shear stress,

• octahedral shear stress

(1)

(2)

(3)

16


Findley criterion (1956)

max,n

ans,

f

kf

1fN

2fN

21 ff NN

where: f and k are material coefficients, (for ductile materials k 0.2, 0.3)

(4)or

Form:

17



(5)

The critical plane is a plane with the maximum value of linear relationof shear and normal stress

The critical plane

50 60 70 80 90 100

Angle , deg

maximum value

critical plane

18



Findley did not defined a mathematical formula for coefficient f . Someresearchers (Park and Nelson, 2000; Backstrom and Marquis, 2001) assume that it can be determined from the shear-mode cracking

(6)

19


Findley criterion (1956) Experimental verification

(7)

This criterion was effective for proportional bending with torsion with non-zero mean stress value under the same ratio of normal to shear stress amplitudes for variable loading and static loading

20


McDiarmid criterion (1972)

Form:

(8)

where: af – shear fatigue strength for Case A or Case B of fatigue cracks,u – ultimate tensile strength

The criterion distinguished type A crack (along the surface) and type B crack (into the material)

From the criterion (8) damage parameter can be deduced (Park and Nelson, 2000) as follows

(9)

21

The critical plane


McDiarmid criterion (1972)

The critical plane is a plane with the maximum shear stress amplitude

(10)

Proposed criterion correlated experimental data for proportional and non-proportional bending with torsion loading with zero and non-zero mean value. Only for one case of loading a/a = 0.5 and phase shift /2 where all planes are planes with maximum shear stress range, the criterion was ineffective.

Experimental verification

22


Dietman et al. criterion (1974) Dietman et al. were among the first researchers who paid attention to the interaction between changes of principal stress directions and fatigue life. They proposed to modify the criterion of octahedral shear stress to take into account the changes of direction of principal stress axes.

This criterion assumes that material fatigue failure occurs when the shear stress amplitude, ns,a, in the critical octahedral plane reaches the criticalstress value, ns,a,c, characteristic for a given material

(11)

23

The critical plane is the octahedral plane at time t, for which octahedral shear stress oct,max achieves the maximum value.


Dietman et al. criterion (1974)

The critical plane

Unfortunately, this criterion was used only to determine the fatigue limit and the results were not compared to any standard fatigue characteristic.


24


Simbürger and Grubisic criterion (1976)

Simbürger and Grubisic proposed a criterion including mean stress value and rotation of principal stress directions. In plane stress state, all possible orientations of the considered plane can be described by the angle α. The proposed fatigue parameter S is formulated as follows

where:

Material coefficients a1 and a2 are functions of fatigue limits af and af , whereas a,c represents the critical stress amplitude for a given number of cycles to failure. Coefficients m allows to take into account the mean stress value eq,m.

(12)

25


Simbürger and Grubisic criterion (1976)

In principle, this criterion does not belong to the critical plane approach because the parameter Sn is independent of a specific plane orientation. However, Simbürger and Grubisic determined the position of fatigue fracture plane as a plane with maximum value of Sn parameter.

The critical plane


Simbürger and Grubisic did not define a fatigue characteristic (Nf −S) which should be used to calculate fatigue life.

26


Matake criterion (1977)

Form:

(13)

where: af – fatigue limit for fully torsion loadingk – material coefficient

The critical plane

(14)

The critical plane is one of two planes of maximum shear stress ns with a higher value of normal stress n.

27


Matake criterion (1977)

This criterion was created to analyze cyclic torsion, bending and proportional torsion with bending. The constant position of principal stresses direction were assumed.


28


Dang Van criterion (1982)

This criterion is based on the concept of micro-stresses in the critical volume of material.

(15)

where: is the microscopic shear stress in grain area, ,h is the microscopic hydrostatic stress,a1, a2 are constants determined from cyclic uniaxial fatigue tests.

29


Dang Van criterion (1982)

The critical plane


The critical plane is a plane with the maximum microscopic shear stress

Many researchers have simplified the Dang Van criterion by replacement of micro-stresses by macro-stresses.

30


Robert et al. criterion (1992)

Robert et al. proposed a criterion which takes into account the shear stress ns(t), the normal stress n(t) and the mean value of the normal stress n,m in

the critical plane

(16)

where a1(Nf ), a2(Nf ) are criterion parameter depending on uniaxial fatigue characteristic: fully reversed axial and torsion loading (R =−1), and tensile test (R=0). The fatigue criterion is defined by

(17)

where a3 is the third criterion parameter.

31


Robert et al. criterion (1992)

The number of cycles to failure Nf is the solution of the Equation (17) and it is obtained from an iterative process.

The critical plane


The critical plane is the plane in which the equivalent stress eq(t) reaches the maximum value.

Advantages: (i) the criterion parameters a1, a2, a3 were identified with the use of three uniaxial S–N curves and (ii) the criterion could be applied for random loading.This criterion was successfully used under random, proportional loading.

If we assume that a1 =a2 =0.5k, then we will obtain the criterion similar to criteria proposed by Findley, McDiarmid and Matake.

32


Papadopoulos criterion (1993)

Form:

(18)

where: , are material parameters,Ta is a generalised shear stress amplitude

is hydrostatic stress.(19)

33



(21)

Generalised shear stress amplitude:

(20)

34



The critical plane

The critical plane is plane where generalized shear stress amplitudeTa achieves maximum value.

This criterion was analyze under cyclic multiaxial proportional and non-proportional loading.


35


Carpinteri and Spagnoli criterion (2001)

Form:

(22)

where: af – fatigue limit for fully torsion loading af – fatigue limit for fully reversed axial loading

The critical plane orientation is correlated with the averaged principal stress directions The averaged principal stress directions are computed using a weight function which depends on the maximum principal stress σ1(t) and two material parameters

The critical plane

36


Carpinteri and Spagnoli proposed to compute the critical plane orientation n with respect to the averaged maximum principal stress direction 1 in the plane of 1,3 by the following relationship

(23)

where the angle α is expressed in degrees.

According to Eq. (23), the angle α is equal to 0◦ for af/σaf = 1 (hard metals) and α = 45◦ for af /σaf = √3/3 (between hard and mild metals).

This criterion was analyze under cyclic proportional and non-proportional loading and under proportional random loading.


n

37


Summary:

Among stress criteria based on the critical plane approach we can distinguish criteria which assume that fatigue failure is due to:

(i) the linear combination of shear ns and normal n stresses acting on the

critical plane;

(ii) the linear combination of shear parameter (ns or Ta), acting on the critical

plane, with hydrostatic stress h;

(iii) the nonlinear combination of shear ns and normal n stresses acting on the

critical plane.

38

Critical plane criteria based on stresses

Conclusions

• The promising fatigue criteria seem to be the criteria which can be used under the most general loading, i.e. multiaxial random loading. Unfortunately, only a few stress criteria were experimentally verified (very little) under random loading.

• The fatigue failure criteria based on stresses are not able to take into account the effect of cyclic hardening or softening. If the fatigue tests are carried out under stress/force controlled system, the effect of cyclic hardening or softening is visible only in strain history, which is not taken into account in the fatigue failure criteria based on stresses.

39

Strain criteria

40

Critical planes in criteria based on strains


Adaptation of the static hypotheses of material strength to fatigue as the replacement of strain static values in these hypotheses by amplitudes or range of dynamic loading

The criteria most frequently verified with experimental data, were the criteria of:

• maximum normal strain,

• maximum shear strain,

• octahedral shear strain

(24)

(25)

(26)

41


Brown-Miller criterion (1973)

Form:

or (27)

(28)

where: S is a coefficient determined by experiment

Kandil-Brown-Miller modification (1982):

Wang-Brown modification (1993):

(29)

42


Brown-Miller criterion (1973)The difference between the two criteria above (Equations (28) and (29)) is based on different definitions of the normal strain range. The normal strain εn

* (called normal strain excursion by authors) in Equation (29) is calculated in the plane of maximum shear strain range ns.

(30)

(31)

Fatigue life is calculated based on the following expression

43

Brown-Miller criterion: Maximum shear strain planeWang-Brown criterion: The critical plane of maximum damage


Brown-Miller criterion (1973)

The critical plane

This criterion was analyze under torsion, tension-compression and their combination for proportional, non-proportional constant-amplitude and variable-amplitude loading.The calculated fatigue life obtained in the maximum shear strain plane and in the critical plane of maximum damage according to the Brown–Wang criterion were comparable.


44

Lohr and Ellison proposed a criterion to calculate fatigue life in low-cyclic fatigueregime. This criterion assumes that fatigue life and crack growth rate can be assessed by a linear combination of shear ns,a and normal strain n,a amplitudes in the critical plane


Lohr and Ellison criterion (1980)

(32)

where k is material coefficient (k = 0.2 for 1Cr–Mo–V steel).

45

The critical plane

Correlation of the experimental test results (fatigue life, Nf ) with thecalculated equivalent strain parameter eq,a based of the proposed criterion was satisfactory (under cyclic, proportional loading).



Lohr and Ellison criterion (1980)

The critical plane is the plane inclined by 45◦ to the free surface of material.

46


Socie-Fatemi et al. criterion (1985)

Socie et al. observing fatigue fractures came into conclusion similar to those byBrown and Miller, that is, the normal strain n in the plane of maximum shearstrain accelerates the fatigue damage process through crack opening.

Crack opening (by maximum normal stress) decreases the friction force between slip planes.

max,, nn ns

ns

crack

max,, nn

(33)

47


Socie-Fatemi et al. criterion (1987)

Form:

(34)

where: y - yield stressn is an experimental coefficient

- maximum normal stress (35)

(36)

48


Socie-Fatemi et al. criterion (1985, 1987)

The plane experiencing the maximum value of shear strain amplitude ns,a

The critical plane

This criterion was analyze under torsion, tension-compression and their combination for proportional, non-proportional constant-amplitude loading.


49

Among strain criteria based on the critical plane approach we can distinguish criteria which assume that fatigue failure depends on:

(i) the linear combination of shear ns and normal n strains acting on the critical plane;

(ii) the nonlinear combination of shear strain ns and different kind of tensile parameters acting on the critical plane.


Summary:

50

Critical plane criteria based on strains

Conclusions

In Table is clearly visible that the critical plane of maximum shear strain dominates. The reason of this is that the fatigue failure criteria based on strains are usually applied for non-brittle materials, where crack Mode II and III dominate.

51

Energy criteria

52


Critical planes in criteria based on energy

The first criteria basing on strain energy came from static hypotheses of material strength.

The most often applied criterion from this group is Huber-Mises-Hencky’s hypothesis.

In case of elastic strain state, Huber-Mises-Hencky’s hypotheses can be describe be means of shear stress or shear strain in octahedral plane.

53


Smith-Watson-Topper criterion (1970)

Form: (37)

Smith and co-authors proposed a simple form of a damage parameter, SWT,described as stress and strain product σmaxεa for fatigue life determination under uniaxial tension-compression.

54


Smith-Watson-Topper criterion (1970)

The critical plane

The plane of maximum normal strain range 1

The criterion is applied under multiaxial cyclic proportional and non-proportional loading for materials with crack Mode I.


Socie modification (1987)

(38)

55


Nitta-Ogatta-Kuwabara criterion (1988)

Form:

for crack Mode I

for crack Mode II

(39)

(40)

Under non-proportional loading, cracks with mixed Modes I and II were noticed. So, they proposed to calculate fatigue life Nou for non-proportional loading in the following form

(41)

56

The critical plane

- the plane of maximum range of normal strain energy density WI - the plane of maximum range of shear strain energy density WII

The criterion is applied under multiaxial cyclic proportional and non-proportional loading (under low cycle fatigue regime).



Nitta-Ogatta-Kuwabara criterion (1988)

where A1, A2 and β1, β1 are material constants.

57


Liu criterion,VSE (1993)

Form:

(42)

(43)

Proportional loading

Liu proposed an energy method to estimate the fatigue life, based on virtual strain energy (VSE). The parameters of virtual strain energy are associated with two different Modes of fatigue cracks.

58


Liu criterion,VSE (1993)Despite formal similarity of presented formulas for WII,A and WII,B,shear and normal stresses and strain ranges are calculated according to Mohr’s cycle in a different way: for Case A, from maximum and minimum normal stresses and strains σ1, σ3 and ε1, ε3; for Case B, from σ1, σ2 and ε1, ε2.

In uniaxial stress state, the parameter VSE can be written as follows:

The parameter VSE contains the elastic We and plastic strain Wp energy density. It can be described by the area of the rectangles defined by the hysteresis loop ranges.

σ

(44)

59



In uniaxial stress state, the energy density W is equivalent to the Smith–Watson–Topper parameter and, as a function of fatigue life Nf , it is written as follows:

(45)

where B1, B2, a1, a2 are materials constants.

60



IMode)(ˆ)(ˆˆ BWAWW III

IIMode)(ˆ)(ˆˆ BWAWW IIIIII

Non-proportional loading

(45)

(47)

Virtual pathsNon-proportional loading

61



Comments:

- The criterion proposed by Liu is limited only to some special kind of loading.- We cannot agree that all non-proportional paths of loading are represented by a rotated ellipse. - Liu did not define the fatigue parameter for random loading or even for triaxial state of stress.

62

The critical plane

- the plane of maximum range of normal strain energy density WI - the plane of maximum range of shear strain energy density WII



The criterion is applied under multiaxial cyclic proportional and non-proportional loading.


63


Glinka and et al. (1994, 1999)

Form:

(48)

Glinka et al. proposed an energy parameter, being a part of total strain energy density, expressed by stress and strain in the critical plane

In order to take into account the mean value, authors have modified the above parameter,

(49)

64



Pan and et. al. modification (1999)

(50)

In 1999 (Pan et al., 1999) noticed that the influence of strain energy in shear direction, (ns/2)(ns/2), on fatigue life is different than the influence of strain energy calculated in normal direction, (εn/2)(σn/2). For this reason, they proposed to modify Glinka criterion by applying two coefficients determined by experiments

where the coefficients are equal to

65



The critical plane

The plane of maximum range of shear strain ns

The criterion was applied under cyclic torsion, tension, bending and combined proportional bending and torsion loading with zero and non-zero mean stress. The fatigue parameters (48)– (50) were used to correlate experimental fatigue life Nf . However, any formula to determine relation between fatigue life Nf and the proposed strain energy parameters was not proposed.


66


Rolovic and Tipton (1999)

Rolovic and Tipton proposed a criterion for multiaxial cyclic proportional and non-proportional fatigue loading including the mean value of normal stress. In general, the criterion is written as follows

(51)

The specific form of Equation (51) was proposed as

(52)

67

The critical plane

The critical plane is determined a plane with the highest calculateddamage level.

Uniaxial and biaxial in phase and out-of-phase fatigue data from three materials were used to verify the proposed model. The proposed model can be used under multiaxial random loading. Unfortunately, this very interesting model was very little verified (only by authors). The problem is to determine the formula of functions: f1(σn,max) and f2(σn,max) for other materials.



Rolovic and Tipton (1999)

68


Chen et al. (1999)

Chen et al. proposed two criteria: the first one for materials characterised by Mode I crack, and the second one for materials characterised by Mode II crack.

For materials characterised by Mode I crack, the critical plane is theplane of maximum normal strain range εn

For materials with Mode II crack, the critical plane is the plane of maximum shear strain range γns

(53)

(54)

69

The proposed damage parameters correspond to the parameters proposed by Glinka and co-authors. In this criterion the different influence of the normal and the strain energy density on fatigue life is not included.


Chen et al. (1999)

The critical plane

The critical plane orientations depends on the observed dominating crack orientation.

The criterion was verified under cyclic, non-proportional loading.


70


Farvani – Farahani criterion (2000)

Varvani-Farahani proposed a fatigue parameter as the summation of the normalσnεn and shear ns,maxns,max/2 strain energy density ranges calculated in the critical plane of maximum shear strain.

(55)

The fatigue parameter (55) was used to correlate experimental fatigue life Nf (cyclic, non-proportional), but the function f(Nf ) was not formulated.

71

Critical plane criteria based on

strain energy densityConclusions

The fatigue failure criteria based on energy are able to take into account the effect of cyclic hardening or softening since they use both strain and stress histories.

72

Summary

Multiaxial fatigue criteria can be divided according to the critical plane orientation

Crack Mode I Crack Mode II Crack Mode III

73

Stress criteria

1. Maximum normal stress2. Macha (1979)3. Carpinteri-Spagnoli (2001)

Summary

Energy criteria

1. Smith - Watson – Topper (1970)2. Liu (1993)3. Lagoda-Macha (1998)4. Chen (1999)

Strain criteria

1. Maximum normal strain2. Macha (1988)

Crack Mode I

74

Summary

Stress criteria

1. Maximum shear stress2. McDiarmid (1972)3. Dietman-Issler (1974)4. Matake (1977)5. Macha (1979)6. Dang Van (1982)7. Papadopoulos (1993)8. Carpinteri - Spagnoli (2001)

Energy criteria

1. Liu (1993)2. Glinka - Shen – Plumtree (1994)3. Lagoda-Macha (1998)4. Chen (1999)5. Varvani-Farahani (2002)

Strain criteria

1. Maximum shear strain2. Brown-Miller (1973)3. Lohr-Ellison (1980)4. Fatemi-Socie (1987)5. Macha (1988)

Crack Mode II and III

75

Summary

Stress criteria

1. Octahedral shear stress2. Findley (1956)3. Simburger (1974)4. Macha (1979)5. Sempruch (1992)6. Vidal (1996)7. Zolochevski et al. (2000)8. Carpinteri – Spagnoli (2001)9. Kuppers-Sonsino (KoNoS) (2003)

Strain criteria

1. Octahedral shearstrain2. Macha (1988)

Energy criteria

1. Nitta - Ogata - Kuwabara (1988)2. Lagoda-Macha (1998)3. Rolovic-Tipton (1999)4. Hoffmeyera et al. (2001)5. Lee et al. (2003)

Mixed crack Mode

76

Generalized criteria

Form:

(56)

Main assumption:

• fatigue crack is controlled by normal strain n(t), stress n(t) or energy Wn(t) and shear strain ns(t), stress ns(t) or energy in Wns(t) the critical plane

Summary

Macha criterion (1988)

Macha criterion (1979)

Lagoda-Macha criterion (1998)

where: B, K, F,b, k, q,, , Q

are constants to select a particular form of criterion}

77

The criterion is a general form of many criteria and the particular form of the criterion depends on many conditions: material, temperature, loading etc.

For example: • for ductile materials the influence of shear strain, stress or energy is greater than the influence of normal strain, stress or energy, thus,

K,k, 0 (aluminum)• for brittle materials the influence of normal strain, stress or energy is greater than the influence of shear strain energy, thus,

B,b, 0 (cast iron)• for elastic-plastic materials,

B,b, 0 and K,k, 0 (alloys)


Summary

Generalized criteria

78

Summary

Through the literature review of the critical plane approach to multiaxial fatigue the following comments can be drawn:

1. The terminology applied in a case of static or proportional loading turned out to be false for non-proportional loading.

2. The rotation of principal stress and strain directions is not always taken into account, there are treated as scalar variables.3. In a few criteria of multiaxial fatigue failure it is assumed that the critical plane position is defined in relation to the averaged directions of principal stress and strain.4. There is not enough information concerning the influence of mean stress and strain on the critical plane and fracture plane orientation.5. Information about fatigue fracture plane positions under multiaxial random loading is very pure.6. One part of the multiaxial criteria applies a few critical plane orientations

for calculation of a single damage parameter.7. There are attempts to include different mechanism of fatigue process (Mode I, II, III, stage I, stage II etc.) to fatigue life calculation through:

79

Summary

• the application of several criteria based on different fracture mechanism and the fatigue life is established by criterion with highest

damage level, (Socie, Das),• summation of damage levels calculated according to several multiaxial criteria, (Socie, Nitta).

80

Algorithm of the fatigue life calculation

81


Algorithms of fatigue life calculation based on the critical plane approach have in general the same structure. A generalised scheme of this algorithm applicable under multiaxial random loading is as follows:

1) Recording, generation, computingij(t), ij(t), (i, j = x, y, z)

2) Determination of the critical plane orientation: n, s

3) Computation of the equivalent damage parameter course

4) Cycle counting

5) Damage degree computationS(T0)

6) Fatigue life calculationTcal

Based on the fatigue criterion0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

-500

0

500

Time,s

Eq

uiv

ale

nt s

tre

ss c

ou

rse

, MP

a

Random loading

0 50 100 150 200 250 300 350 400 450 5000

100

200

300

Amplitudes, MPa

Nu

mb

er

of c

ycle

s

Histogram of amplitudes after rainflow cyclic extracting

Fatigue damage accumulation hypothesis

Tcal = T0/S(T0)

82


2) Determination of the critical plane orientation: n, s

The critical plane

The position of the critical plane is determined by normal vector n and shear components are determined by shear vector s on that plane. The direction n and s may be computed by three methods:

• damage accumulation method,

• variance method,

• weight function method.

83

Damage accumulation method

The damage accumulation method is the most often used for determination of the critical plane orientation. This comes from the fact that this method refers directly to the fatigue life.

The critical plane orientation is this method is the plane experiencing the highest damage degree, this means the plane of the theoretical fracture and the lowest fatigue life.

Unfortunately, the damage accumulation method is the most time consuming. In the damage accumulation method the algorithm of fatigue life calculation is repeated once per each considered plane orientation.


84

Damage accumulation method - simulations


The aim of the simulation is the determination of the critical plane orientation for different plane stress states.

Normal and shear stresses in an arbitrary plane orientation for plane stress state can be presented as a function of the angle α

85



Cyclic loading

1) a = 200 MPa; a = 200 MPa = a / a = 1 = 0

86



-100-50

050

100

0

0.01

0.02

0.03

0.04

0.05-400

-200

0

200

400

, o

time, s

ns(

,t),

MP

a

-100 -80 -60 -40 -20 0 20 40 60 80 1000

50

100

150

200

250

, o

ns,

a, M

Pa

1) a = 200 MPa; a = 200 MPa; = 0

-100-50

050

100

0

0.02

0.04

0.06-400

-200

0

200

400

, otime, s

n(

,t),

MP

a

-100 -80 -60 -40 -20 0 20 40 60 80 1000

100

200

300

400

, o

n

,a,

MP

a

87



eq(t) = n(t) + k ns(t)

k=0.2

-100

-50

0

50

100

0

0.01

0.02

0.03

0.04

0.05-400

-200

0

200

400

, otime, s

e

q(

,t),

MP

a

-100 -80 -60 -40 -20 0 20 40 60 80 10050

100

150

200

250

, o

e

q,a

, M

Pa

88



2) a = 200 MPa; a = 200 MPa; = /2

-100-50

050

100

0

0.01

0.02

0.03

0.04

0.05-400

-200

0

200

400

, o

time, s

ns(

,t),

MP

a

-100 -80 -60 -40 -20 0 20 40 60 80 100100

120

140

160

180

200

, o

ns,

a, M

Pa

-100

-50

0

50

100

0

0.01

0.02

0.03

0.04

0.05-400

-200

0

200

400

, otime, s

n(

,t),

MP

a

-100 -80 -60 -40 -20 0 20 40 60 80 1000

50

100

150

200

250

, o

n

,a,

MP

a

89



eq(t) = n(t) + k ns(t)

k=0.2

-100

-50

0

50

100

0

0.01

0.02

0.03

0.04

0.05-400

-200

0

200

400

, otime, s

e

q(,t

), M

Pa

-100 -80 -60 -40 -20 0 20 40 60 80 100100

150

200

250

, o

e

q,a

, M

Pa

90

Variance method

The variances of stress, strain or strain energy density are important parameters in the fatigue process.

The variance method assumes that the plane experiencing the maximum variance of equivalent damage parameter (e.g. stress, strain or energy) is critical.


In the case of linear fatigue failure criteria the equivalent e.g. stress may be expressed as a function of: tensor components (in general as Yij(t))), the critical plane orientation and material constants, k)

91

Variance method


Variance of the equivalent course is nonlinear function of the direction cosines of n, s, but it is linear function of the covariance matrix components of Yij(t) tensor. Thus, the variance of equivalent course may be expressed by the sum of the products of the covariance matrix components μykl and nonlinear function of material constants and direction cosines of n, s vectors: ak, al

92

Variance method


93

Variance method - simulations


For a given plane stress state

with zero mean value of stresses the covariance matrix is as follows

where E[x] – mean the expected value of x

94



where

95



where

96

-100 -80 -60 -40 -20 0 20 40 60 80 1000

1

2

3

4

5

6x 10

4

, o

n, n

s, M

Pa2

n

ns



1) a = 200 MPa; a = 200 MPa; = 0

-100 -80 -60 -40 -20 0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3x 10

4

, o

n, n

s, M

Pa2

n

ns2) a = 200 MPa;

a = 200 MPa; = /2

97

Maximum principal stress direction 1 Averaged direction of maximum principal stress 1

Averaged directions of principal stresses 1, 2, 3

Are they related to fatigue fracture plane position?

Weight function method

The weight function method consist in weighted through a suitable function average process of instantaneous principal axis directions. The critical plane orientation is determined in relation to the averaged principal axes directions.


98



Principal axis directions 1(t), 2(t), 3(t) could be described by Euler angles.

Principal axes 123 described by Euler angles ϕ, θ, ψ

99



Weight functions, examples:

According to such a weight function, each position of the principal axes influences the mean position of the principal axes in the same degree, irrespective of the stress values. Application of this weight function leads to the arithmetic mean of Euler angles

This weight function includes only those positions of the principal stress axes for which the maximum principal stress σ1(t) is greater than the product of c and the fatigue limit σaf . Averaging process exponentially depends on the parameter mσ of the Wöhler curve

100



Algorithm of fatigue life calculation with the critical plane orientation determined by the weight function method

101


The critical plane orientations αcal according to the criterion of maximum normal stress

The critical plane orientation according to weight function W2

The critical plane orientation αcal according to the maximum variance of normal stress σn(t)

Comparison of methods

102

Exemplary application of simple energy based criteria in fatigue life calculation

103

Exemplary application of simple energy based criterion in fatigue life calculation

Lagoda - Macha criterion (1998)

Lagoda and Macha formulated a generalised criterion of normal Wn(t) and shear strain energy density Wns(t) in the critical plane.

Main assumption:

•Fatigue crack is formed by the part of strain energy density which corresponds to work of normal stress σn(t) on normal strain n(t) - Wn(t) and work of shear stress ns(t) on shear strain occurring in s direction in the critical plane with normal n - Wns(t)

QtWtW nns )()(max (1)

104

Exemplary application of simple energy based criterion in fatigue life calculation

The main aims of the example are:1. Verification of the energy criteria of multiaxial fatigue proposed by Lagoda

and Macha for low-cycle non-proportional loading 2. The analyze of the history of energy parameters in the critical plane

2)](sgn[)](sgn[

21 )()()( tttttW

where the strain energy density parameter is

(2)

105

The energy parameters Four particular versions of the generalized criterion of normal and shear strain energy density

are verified:

QtWtW nnst

)()(max (3)

)()()( tWtWtW nns (4)or

THE CRITICAL PLANE

Maximum normal strain energy density

Maximum shear strain energy density

The criterion of normal strainenergy density (CC11)

The criterion of shear strainenergy density (C2C2)

The criterion of normal and shear strain energy density (C3C3)

The criterion of normal and shearstrain energy density (C4C4)

106

The energy parameters

THE CRITERION OF MAXIMUM NORMAL STRAIN ENERGY DENSITY IN THE CRITICAL PLANE (C1)

)()()(2

1)( tWtttW eqxxxx

in uniaxial tension-compression tests right side of equation (5) is

(6)

Fatigue effort under multiaxial state Eq. (5) must be equivalent to fatigue effort under uniaxial state Eq. (6), thus

)()( tWtW neq (7)

cbfff

bf

ffnfaeq NN

ENWNW )2(

2

1)2(

2)2()2( ''2

2'

,

Fatigue life is computed from energy characteristic obtained from uniaxial tests Eq. (6)

(8)

)()()( tWtWtW nns generally:

for =0, = 1 (5))()( tWtWn )(max,

tWntn

in the plane of

107


THE CRITERION OF MAXIMUM SHEAR STRAIN ENERGY DENSITY IN THE CRITICAL PLANE (C2)

)()(2

1)( tttW nsns


(10)


for =1, = 0 (9))()( tWtWns )(max,,

tWnstns

in the plane of

108

Mohr’s circles for stress and strain state

)(4

1)()(

2

1

4

1

)(2

1)(

2

1

2

1)()(

2

1)(

tWtt

tttttW

eqxxxx

xxxxnsns

(11)

Tension-compression fatigue tests

109


Fatigue effort under multiaxial state Eq. (9)

must be equivalent to fatigue effort under uniaxial state Eq. (11),

thus,

)(1

4)( tWtW nseq (12)

cbfff

bf

ffaeq NN

ENW )2(

2

1)2(

2)2( ''2

2'

,

Fatigue life is computed from energy characteristic obtained from uniaxial tests Eq. (8)

(13)

)()( tWtWns

)(4

1)( tWtW eq

110


THE CRITERION OF MAXIMUM SHEAR AND NORMAL STRAIN ENERGY DENSITY IN THE CRITICAL PLANE (C3)

)()()(2

1)( tWtttW eqxxxx


(15)


for 1, = 1 (14))()()( tWtWtW nns )(max,

tWntn

in the plane of

Fatigue effort under multiaxial state Eq. (14) must be equivalent to fatigue effort under uniaxial state Eq. (15), thus

)()()( tWtWtW nnseq (16)

111


cbfff

bf

ffnfaeq NN

ENWNW )2(

2

1)2(

2)2()2( ''2

2'

,

Fatigue life is computed from energy characteristic obtained from uniaxial tests

(19)

The coefficient should be chosen to obtain the best correlation between uniaxial and multiaxial fatigue tests. For simplicity (criterion C2) the coefficient is chosen as:

1

4(17)

Thus, the equivalent parameter is:

)()(1

4)( tWtWtW nnseq

(18)

112


THE CRITERION OF MAXIMUM SHEAR AND NORMAL STRAIN ENERGY DENSITY IN THE CRITICAL PLANE (C4)

)()(2

1)()(

2

1)( tttttW nsnsnnnn


(21)


for 1, = 1 (20))()()( tWtWtW nns )(max,,

tWnstsn

in the plane of

113

)()(2

1)()(

2

1)( tttttW nsnsnnnn

(22)

)(2

1)(

2

1

2

1)(

2

1)(

2

1

2

1)( tttttW xxxxxxxx

)(4

)1(1)()(

2

1

4

)1(1)( tWtttW eqxxxx

Tension-compression fatigue tests M

ohr’s

circ

les

for

stre

ss a

nd s

trai

n st

ate

114


Fatigue effort under multiaxial state Eq. (20)

must be equivalent to fatigue effort under uniaxial state Eq. (22),

thus,

)()()( tWtWtW nns

)(4

)1(1)( tWtW eq

)()()1(1

4)( ,, tWtWtW ansaneq

(23)

115


cbfff

bf

ffnfaeq NN

ENWNW )2(

2

1)2(

2)2()2( ''2

2'

,

Fatigue life is computed from energy characteristic obtained from uniaxial tests

(26)

The coefficient should be chosen to obtain the best correlation between uniaxial and multiaxial fatigue tests. For simplicity of Eq. 29 the coefficient is chosen as:

1

3(24)

Thus, the equivalent parameter is:

)()(1

3)( tWtWtW nnseq

(25)

116

The energy parameters, summary

The equivalent parameters are:

)()(1

3)( tWtWtW nnseq

)()( tWtW neq C1:C1:

)(1

4)( tWtW nseq C2:C2:

C4:C4:

)()(1

4)( tWtWtW nnseq

C3:C3:

)(max,

tWntn

in the plane of

)(max,,

tWnstsn

)(max,,

tWnstsn

in the plane of

in the plane of )(max,

tWntn

in the plane of

cbfff

bf

faafaeq NN

ENW )2(

2

1)2(

22

1)2( ''2

2'

,

Fatigue lifeFatigue life::

117

Fatigue tests (M. Ohnami and N. Hamada)

Materials: SUS 304 steel

Al 6061 aluminum alloy

Specimens: cylindrical thin-walledLoading: combined tension-compression and torsion under controlled strainTests were performed under 14 different strain paths

Strain paths

%8.0%5.03/ or

%8.03/

118

Fatigue tests

Case 13

Strain path

Histories of energy parameters

Wn

s, M

J/m

3

Not taken into account

)()( tWtW neq

Wn,

MJ/

m3


)(1

4)( tWtW nseq )()(

1

4)( tWtWtW nnseq

SUS 304

%8.03/

)()(1

3)( tWtWtW nnseq

119

Fatigue tests

Case 12

Strain path

Al 6061

%8.03/


Wn

s, M

J/m

3

Histories of energy parameters

120

Calculated and experimental results

Amplitudes of strain energy density Weq,q according to criterion C1 against the energy fatigue characteristic for SUS 304

3(2Nf)

(2Nf)/3

C1:C1:

121



C2:C2:

122



C3:C3:

123



C4:C4:

124


Amplitudes of strain energy density Weq,q according to criteria C1-C4 against the energy fatigue characteristic for SUS 304

C1:C1: C2:C2:

C3:C3: C4:C4:

125


Amplitudes of strain energy density Weq,q according to criterion C1 against the energy fatigue characteristic for Al 6061

C1:C1:

126



C2:C2:

127



C3:C3:

128



C4:C4:

129


Amplitudes of strain energy density Weq,q according to criteria C1-C4 against the energy fatigue characteristic for Al 6061

C4:C4:C3:C3:

C1:C1: C2:C2:

130


Comparison of the calculated Ncal

and experimental lives Nexp

)()( tWtW neq C1:C1:

)(max,

tWntn

in the plane of

131




)(1

4)( tWtW nseq

C2:C2:

)(max,,

tWnstsn

in the plane of

132




)()(1

4)( tWtWtW nnseq

C3:C3:

in the plane of

)(max,

tWntn

133




)()(1

3)( tWtWtW nnseq

C4:C4:

)(max,,

tWnstsn

in the plane of

134


Histograms of scatter coefficients

C2:C2:

C3:C3:

C1:C1:

C4:C4:

135

1.1. For 6061 Al aluminum alloy, the best relation between the For 6061 Al aluminum alloy, the best relation between the experimental life and the energy parameter was obtained experimental life and the energy parameter was obtained according to the criterion C2 of shear strain energy density in the according to the criterion C2 of shear strain energy density in the critical plane.critical plane.

2.2. For SUS 304 steel, the best relation between the experimental life For SUS 304 steel, the best relation between the experimental life and the energy parameter was obtained according to the criterion and the energy parameter was obtained according to the criterion C3 of normal and shear strain energy density in the critical planeC3 of normal and shear strain energy density in the critical plane

3.3. In general, the criterion C2 of shear strain energy density in the In general, the criterion C2 of shear strain energy density in the critical plane can be applied for both materialscritical plane can be applied for both materials

4.4. For unstable materials and regimes the uniaxial fatigue tests For unstable materials and regimes the uniaxial fatigue tests should be carried out under energy control system to obtained theshould be carried out under energy control system to obtained the fatigue fatigue energy characteristic energy characteristic (W(Waa-2N-2Nff) ) used in criteria based on used in criteria based on energy parametersenergy parameters

Conclusions

A review of multiaxial fatigue failure criteria based on the critical plane approach Aleksander KAROLCZUK Ewald MACHA Opole University of Technology, POLAND.

Documents

critical plane criterion

fatigue failure slide

phrase critical plane

approach dates

multiaxial stressstrain

components of stress

matriaux slide

new criterion