I. Finnic, W.Cheng Department of Mechanical Engineering, … · 2014-05-17 · In this case, a strain gage is located on the back face of the strip. Typically the strain readings

Residual stress measurement by the introduction

of slots or cracks

I. Finnic, W.Cheng

Department of Mechanical Engineering, University of California,

Abstract

A relatively new approach for the experimental measurement of residual stressis outlined and extended. In essence, it makes use of strain measurement as aslot of progressively increasing depth is introduced into a body. For near-surface residual stress measurement the Nisitani "body force method" is usedto determine residual stresses from strain measurement In cases in whichthrough-the-thickness stress measurement is desired it is shown that manysolutions may be obtained from procedures based on linear elastic fracturemechanics. Although the method requires more extensive prior numericalcomputation than traditional methods, it involves a simple experimental pro-cedure and in many cases leads to much greater precision in prediction of resi-dual stresses than traditional methods.

1 Introduction

Residual stress is a topic of major importance in any discussion of the mechan-ical behavior of materials. In many important applications residual stresseshave led to premature failure. By contrast, compressive residual stresses areoften introduced deliberately to extend the life of parts. As a result there is anextensive literature on the measurement of residual stress. More recently, withenhanced computational ability, the numerical simulation of residual stress hasreceived increasing attention. However, experience teaches that any simula-tion requires experimental validation before being applied in practice.

In this paper we review and extend an approach to residual stress meas-urement which we have developed over the past decade and originally referredto as the "crack compliance method." In many, but certainly not all applica-tions this procedure is experimentally more convenient and more accurate forrapidly varying stresses than traditional methods such as "hole drilling" or

Transactions on Engineering Sciences vol 13, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533

38 Localized Damage

methods which require layer removal. As we will point out, the crack compli-ance method requires an extensive amount of numerical computation beforeexperimental measurements can be converted to residual stresses. This maywell be the reason why this approach, which appears obvious once explained,has only been implemented in the present era of generally available inexpen-sive high speed computation.

2 The crack compliance method

This approach to residual stress measurement had its roots in linear elasticfracture mechanics. In essence, instead of using known stresses and thegeometry of a cracked part to compute a stress intensity factor or displace-ments due to a crack, an inverse approach is taken. Residual stresses are deter-mined from measurements of strain, displacement, or, less conveniently, stressintensity factor as a cut of progressively increasing dimensions is introducedinto a part.

The first attempt to implement this approach appears to be due toVaidyanathan and Finnie [1] in 1971. They showed that measurements ofstress intensity factor as a function of crack length, using a photoelastic coat-ing, could be used to deduce the residual stresses due to a butt-weld betweentwo flat plates. However, the experimental technique required special equip-ment, was time consuming and was unsuited to general application. A moregenerally useful procedure, which was subsequently extended to a variety ofconfigurations was introduced by Cheng and Finnie [2] in 1985. This involvedmeasurements of strain as a function of crack depth to deduce the axial resi-dual stress distribution in a circumferentially welded cylinder. A similar pro-cedure was later proposed by Fett [3] in 1987. Other approaches whichinvolved introducing a cut to measure residual stresses were proposed byRitchie and Leggatt [4] in 1987 and Kang et al. [5] in 1989. However, thesetwo papers presented procedures which, in essence, followed a "layer removal"approach in estimating the stresses in layers as a slot was extended into thematerial. This procedure can be shown to lead to the accumulation of errorwhich is minimized in the approach proposed by the authors and Fett whichinvolves a "least squares fit" of all the data for different slot depths.

To explain the basis of our present approach we consider the plane bodyshown in Fig. 1 which contains normal residual stresses on the plane y' = 0. Ifonly near surface residual stresses are of interest, then the part may be treatedas a semi-infinite body and the dimensions x',y' and a' are normalized by thefinal depth of cut a^, i.e., x = x'/af, y = y'/a , a = a'/af. Alternatively, forthrough- the-thickness measurement in a beam-like member the thickness twould be used to normalize the dimensions a',x',y' and s'. In this case, a straingage is located on the back face of the strip. Typically the strain readings e(a,s' = 0) allow the stress <jy(x) to be obtained for values 0.025 < x < 0.975.For other shapes, such as cylinders, other strain gage locations may be moreconvenient. For near surface stress measurement, one or more strain gages are


Localized Damage 39

located close to the mouth of the cut as shown on the left side of the figure.There are important differences which must be considered in carrying outthrough-the-thickness or near-surf ace stress measurement but to outline themethod of analysis our discussion can be quite general.

£(a',S)

rt

S

o (x) = I A,P(x)' '

s' e(a',s'=0)

/

Figure 1: A thin cut is made in a semi-infinite body or strip.The strain is measured as a function of the depth of cut. The

fictitious forces F will be discussed later in the paper.

3 Residual stress estimation

The unknown residual stress distribution which is to be determined may beexpressed in terms of the polynomial series shown in Fig. 1

O(x) =j=o

(1)

where Aj is the amplitude factor to be determined for the j^ order polynomial.The choice of Pj(x) is dictated primarily by a desire to reduce the influence oftruncational errors as the order of the polynomial series is increased. We havefound that Legendre polynomials are superior to a power series in this respect.


40 Localized Damage

They also have the advantage for through-the-thickness measurement in aplane body of satisfying the equilibrium conditions of zero net force andmoment for n > 2 while P<> which is a constant equal to unity and Pj whichvaries linearly from -1 at x = 0 to 1 at x = 1 should have coefficientsAQ = AI = 0. To obtain the coefficients Aj of eqn (1) it is necessary to calculatethe strain at the location of the strain gage due to a residual stress Pj(x) for mvalues of the dimensionless depth of cut a% where 1 < k < m. From the classicpaper of Bueckner [6] it is known that this calculation may be carried out byapplying the stress Pj(x) to the faces of the cut as shown in Fig. 1. The direc-tion of the loading shown corresponds to a tensile residual stress. We refer tothe strain produced by Pj(x) acting on the crack of length a% as the "compli-ance" Cj(a%) and return later to discuss the calculation of these quantities. Thestrain may now be expressed as

(2)j=o • •

In principle if the number of strain measurements m equals n+1, the unk-nown coefficients Aj may be obtained. In practice improved estimates areobtained if the number of cut depths m for which strain is measured is muchgreater than n+1.

The problem is now over-determined (m > n + 1) and the method of leastsquares is used to determine the Aj from experimental values of e(a%) and thecomputed compliances Cj(a%). This leads to

= 0 (3)o- k=l j=0

where i = 0,1, ...,n.

The subscript i is used to emphasize the fact that the partial derivativeinvolves only one term in the second summation of eqn (3).

After differentiation, eqn (3) becomes

= 0 (4)k=l k=l j=0

where i = 0,1,.. .,n.

These (n+1) equations may now be solved for the unknown Aj.

One can also write these equations in matrix form, which is convenientfor computation. Equation 2 may be rewritten as

[C]A = e (5)

where A is an (n+1) x 1 column vector of the coefficients Aj, [C] is the m x(n+1) compliance matrix C%j, and e is the m x 1 column vector of the measuredstrains e%. To find the A as shown by Mason [7] the equation is expressed as

[Cf[C]A = [Cfe (6)


Localized Damage 41

Solving for A yields:

A={[Cf[C]}-M[Cf e) (7)

Once the compliance matrix is obtained and strain measurements are available,manipulation can readily be carried out using software such as Mathcad orMathlab.

As shown by Gremaud et al. [8] a modification of the procedure based oncontinuous polynomials Pj(x) leads to improved predictions when stresses varyrapidly as in shot peening or may be discontinuous as in some cladding pro-cedures. In these cases we use piecewise functions which are a low order poly-nomial series to fit the data in regions. These regions may overlap if thematerial is continuous as in shot peening. Alternatively, non overlappingregions may be appropriate for a clad material in which there is no physicalreason for the stress to be continuous. The only difference in computationalprocedure using piecewise functions is that the strains due to the stresses in thefirst layer have to subtracted from measured strains before the stresses in thesecond layer can be computed and so on for successive layers.

4 Some practical considerations

Ideally, to minimize disturbance to the residual stress field and to facilitatecomputation of compliances the slit shown in Fig. 1 should be a flat crack.However, a crack is not easily controlled and a machining process has to beemployed. Our first experiments about a decade ago were made with thin mil-ling cutters. Later researchers [3,4,5] used saws. It was reported [4] that errorsof ±20 MN/irr could arise depending on the condition of the saw. As a resultof an extended visit to Japan by one of the authors in 1990 we became awareof the advantages of electric discharge wire machining (EDWM) as well asconventional electric discharge machining (EDM) for making thin slots. Wenow use these methods extensively for conducting materials. CommercialEDWM can be used with wires as small as 0.002 inch (50 \im) diameter. Wehave used 0.001 inch (25 |im) wire for some tests and cutting with 10 |im wirehas been reported by Kinoshita and Hayashi [9] although not in connectionwith residual stress measurement. In addition to providing thin slots EDWMgenerally is carried out with a copious flow of temperature controlled deion-ized water. This essentially eliminates any problems of temperature control forthe foil resistance strain gages we use for strain measurement. For cutting inremote locations, such as the valve seat inside a valve or on a curved surface,EDWM is not practical so conventional EDM with a thin electrode in kerosenecan be used. In this case the wear of the electrode has to be considered inestimating the depth of cut. One concern with EDWM or EDM is that residualstresses may be produced by the cutting process. Cheng et al. [10] havedeveloped a method by which the strains produced by cutting may be elim-inated from the measurements before computing residual stresses. Forthrough-the-thickness stress measurement or when the residual stress levels are


42 Localized Damage

high we have found the strain induced by cutting to be insignificant. Mostmachines provide settings for "fine cutting" which minimize the "recast" layerand residual stresses at the expense of reduced cutting speed which is not anissue in our experiments. Also the recent development of "anti-electrolysis"machines for EDWM is said to reduce the heat affected layer. In testing a car-burized steel with such a machine and a 0.002 inch (50 jLim) wire it was foundby Prime [11] that no correction had to be made for strains induced by cutting.

The displacements or strains induced by cutting could be measured by awide variety of techniques. We chose to use strain gages because of the gen-eral availability of gages and associated instrumentation.

5 Near surface stress measurement

Stresses near the surface are of interest since failures often initiate in thisregion. Strain measurements from a gage such as shown on the left side of Fig.1 cease to be useful when the depth of cut a' reaches the distance S from thecenterline of the strain gage to the edge of the cut. For this reason severalgages may be installed at different distances from the cut. As shown by Chengand Finnie [12] it is necessary to consider the finite width of cut to calculatethe compliances when the depth of cut is less than about ten times its width.After consideration of different computational techniques we decided to usethe body force method proposed in 1967 by Nisitani [14] which he used toobtain stress fields near notches [15]. This approach makes use of stress solu-tions for point forces acting in an unnotched semi-infinite body. The surfacetractions on the vertical faces of the slot shown in Fig. 2 are, as in Fig. 1, takento be the same as those in the unnotched body but with an opposite sign. Therest of the boundary is taken as traction free which is always satisfied by thepoint force solutions. It is therefore particularly convenient to apply Nisitani'smethod to a body with only residual stresses. The body force method is similarto the boundary integral method, but its formulation is simpler for prescribedloading on the slot faces. It is preferable also to the widely used finite elementmethod for the following reasons: (1) the body force method only modelsstresses on the boundary rather than throughout the volume which is a muchsimpler approach; (2) the boundary conditions on the free surface and atinfinity are satisfied exactly while these are only approximated in a finite ele-ment program; (3) the discretization of the loading conditions on the slot facesis easily modified in the body force method; and finally (4) it is simpler toapply the calculations to each depth of cut unless a finite element programwhich automatically generates a new mesh is used.


Localized Damage 43

Figure 2: Configuration of a rectangular slot loaded by normalstresses. The center of the strain gage is located a distance S

from the edge of the cut whose final depth is a^.

In essence, Nisitani's approach requires the introduction of distributedforces on the contour of the cut in the x and y directions so that the resultingstresses coincide with the desired surface tractions on the faces of the slot. Adetailed discussion of the computation of strain on the free surface as a resultof this procedure was given by Cheng and Finnic [12]. As an example, Fig. 3shows the ratio of the strains on the surface for a slot relative to a crack for uni-form surface loading. It should be noted that the distance S corresponds to thatfrom the edge of the slot to the center of the strain gage. At least for uniformloading of the slot surface, it is seen that the finite width may be neglectedwhen the slot depth is about ten times its depth. In practice an EDWM cut hasa semi-circular rather than a flat bottom. Cheng et al. [15] analyzed thisgeometry, again using Nisitani's method and have shown that taking a rec-tangular slot with a width which gives the same area as a slot of the same depthwith a semi-circular bottom leads to a very accurate estimate of compliances.


44 Localized Damage

1.6

C/)(/)0)-t-»CO

EJ5

"E

C/)c

CO

-—CtfOC

1.4

1.2

1.0

T ^ I ^- |

S = 1.27 mm

S = 1.78mm

S = 2.29 mm

0 10

Depth to width ratio (a/2w)

Figure 3: The ratio of the strains due to a cut of aspect ratio(a/2w) to those for a line crack of the same depth when the

crack and cut faces are loaded by a uniform stress.

As a final point in connection with the Nisitani body force method, weshould mention that this is a central feature of the correction procedure wehave developed [10] for the effect of residual stresses introduced by EDM orEDWM.

Two examples which have been discussed in more detail [15] illustratethe results which may be obtained for near surface stresses. Figure 4 showsresidual stress measurements on a shot-peened specimen of Ti-6Al-4V. Theagreement with X-ray measurement is very satisfactory. Figure 5 showsresults for a carbon steel clad with Stellite 6 using laser melting of powder.X-ray measurements and an approximate numerical simulation are also shown.An interesting feature of this test is that metallography shows a layer of mar-tensite in the base material adjacent to the cladding. The volume expansionassociated with the transformation to martensite would be expected to producecompressive stresses. This is revealed by our experimental procedure and thenumerical simulations but was not detected by X-ray measurements.


Localized Damage 45

200

05CL

CDO03

0)

CO0)

CD-4—'(/)"cd

CDtr

-200 -

-400 -

-600 -

-800

-1000

Slit No. 1

Slit No. 2

X-ray results

200 400

Depth (jim)

600

Figure 4: Residual stress distributions measured by the presentmethod and those obtained by the X-ray method for a shot-peened part.The data points correspond to the depth at which strains are measured.

6 Through the thickness stress measurement

For measurements through the thickness of a part, the finite width of cut mayusually be neglected. Also the body force method is much less convenient forcomputation than for the case of a semi-infinite solid. Fortunately, we havebeen able to obtain the compliance for many two dimensional or axisymmetricconfigurations using solutions based on linear elastic fracture mechanics. Asummary of the solutions obtained is given in the following table.


46 Localized Damage

1000 -

030_

"cti13

CDoc

500

X-ray (a)A X-ray (b)

Compliance methodNumerical result

-500 -

200 400 600 800

Distance (pm)

1000

Figure 5: Comparison of the results obtained by the X-ray methodfrom two independent laboratories (data points), the present method(bold data line) and numerical simulation (dotted line). The datapoints in the curves correspond to the locations where strains

were recorded.


Localized Damage 47

Table 1 Configurations for which residual stress measurementshave been carried out. See Cheng and Finnic [15] for a list of

references.

Axisymmetric stresses

Thin-walled rings hoop stressesThick-walled rings hoop and radial stressesThin-walled cylinders axial stresses (non-uniform in the

axial direction)Thick-walled cylinders axial, hoop and radial stresses

(uniform in the axial direction)Solid cylinders axial, hoop and radial stresses

(uniform in the axial direction)

Stresses in plane stress/plane strain

Plates or beams with rectangularcross-section longitudinal stresses

Disks normal stresses on a diametricalplane

A fillet weld or a T-butt weld normal stresses on a plane at thetoe of the weld

The approach is based on Castigliano's theorem which may be used to obtainthe displacement due to loads applied to a cracked body. For two dimensionalparts of unit dimension perpendicular to the x-y plane subjected to mode Iloading, a general expression for the displacements v on the surface at a dis-tance s from the crack plane can be obtained by introducing a pair of virtualline forces F as shown in Fig. 1. This leads to

,v(a,s) = -- = - K,(a) - da (8)

b 0 dF

where E' = E for plane stress and E/(l-|i ) for the plane strain with E and |ibeing the elastic modulus and Poisson's ratio, respectively, U is half of thechange of strain energy due to the crack, Kj and K/ are the stress intensity fac-tors for an arbitrary stress on the crack faces and the virtual line force F respec-tively. The normal strain e(a,y) at a location y = s produced by introducing acrack of depth a is given by differentiating eqn (8) to obtain

, , 9v(a,s) a%U 1 , \ ^e(a,s) = ' = — - = — J Kj(a) — — - da (9)

os dFdS|p=o E 3F9s

Solutions are available for K/ and for Kj(a) for stresses corresponding to thecrack face loading Pj(x). With these available, compliance may be obtainedfrom eqn (9).


48 Localized Damage

As an illustration of the power of this method of measurement we meas-ured the residual stresses in a beam which was first fully stress relieved andthen had stresses introduced by bending. In this case the residual stress distri-bution may be predicted very accurately since the stress-strain curves in ten-sion and compression may be deduced from the bending test which inducesresidual stresses. Figure 6 shows the predicted residual stress distribution andthat predicted by LEFM computed compliances from a single strain gage onthe back face of the beam, as shown in Fig. 1. Very small strains are measuredfor normalized depths of cut less than about 0.025 while for large depths of cuta > 0.975 the small remaining ligament introduces problems. Nevertheless inthe central 95% of the beam the residual stresses, which are quite small, arepredicted with remarkable precision. In passing we note that confirmation ofthe predicted stresses by the X-ray approach was not possible because of thecoarse grain size of the stainless steel.

An interesting and useful feature of the approach to strain predictionbased on LEFM is that the stress intensity factor due to a residual stress fieldmay be obtained without determining the residual stresses. By taking a deriva-tive of eqn (9) with respect to the crack size we obtain

Thus, the stress intensity factor due to residual stresses can be estimateddirectly from the change of strain measured when a thin cut is introduced. Formore complex geometries such as a welded connection we have measured resi-dual stresses, e.g., at the toe of a fillet weld, using the crack compliancemethod. Since analytical solutions using LEFM are no longer feasible, we usefinite element computation of compliances. For finite dimensions, thisapproach now becomes more tractable than the body force method. To illus-trate that finite element computation can lead to precise prediction, we havealso used this approach for a bent beam as shown in Fig. 6.

7 Conclusions

The approach we have discussed is capable of measuring stresses on a plane inproblems in which the stresses vary rapidly in both the direction of cutting anddistance from the plane. That is, in the x' and y' directions of Fig. 1. Such asituation arises at the toe of a fillet weld and other welded junctions. Tradi-tional measurement techniques are not well suited to such a configuration. Asyet, we are not able to handle problems in which stresses vary rapidly in adirection perpendicular to the plane of Fig. 1. However, thickness variations inthis direction can be handled. An example in which this arises is the valvebody shown in Fig. 7. A strain gage is located in the circumferential directionon the surface of the hardfacing and a cut made as shown with EDM. In thisapplication other techniques would be impossible to implement without cuttingthe valve apart.


Localized Damage 49

100

05Q_

COCOQ)to

co0)DC

I ' I

Compliance functions

FEM computations

-100

Normalized distance

Figure 6: Residual stresses produced by bending a stress free beamof type 304 stainless steel. Circles show values deduced from knownapplied moment and the stress-strain curves in tension and compres-

sion. Solid and dashed lines show values measured experimentally usinga Legendre polynomial series with n = 7. Compliances were calculated

using linear elastic fracture mechanics solutions and finite elementcomputation.

In situations in which strain gage rosettes may be mounted on a surface,this measurement technique has advantages and disadvantages relative to theslitting technique. The rosette method may be applied in the field and providesthe biaxial stress field in the near surface region. The slitting technique pro-vides the normal stresses on a plane. Its sensitivity is considerably greater thanthat for hole drilling with rosettes. Also, its ability to resolve rapidly varyingstress gradients is greatly superior to hole drilling techniques. In hole drillingthe hole has to be located precisely at the center of the strain gage rosette. Inthe slitting technique the distance of the gage from the edge of the slot and thewidth of the slot may be measured precisely after cutting. We conclude thatthe approach we have presented is a valuable addition to traditional experimen-tal methods. The procedure appears obvious once presented but has only beendeveloped in the past decade. As a result it is not included in the new "Hand-book on Techniques of Measurement of Residual Stresses," which has beenmany years in preparation and is due to be published in 1996 by the Society forExperimental Mechanics in the USA.


50 Localized Damage

Hardsurfacing

(M<M

30

1.3 mm

electrode

63 mm -

E_ECDCD

Figure 7: The use of EDM for measurement of the residual hoop stressin the hardsurfaced seat of a valve body.

Acknowledgement

Most of our research on residual stress measurement has been supported by theElectric Power Research Institute of Palo Alto, California. We would also liketo express appreciation to the Japan Society for the Promotion of Science forsupporting an extended visit to Japan by I. Finnic in 1990. His host scientist,Professor Masaru Sakata, then with the Tokyo Institute of Technology and nowat Takushoku University, knowing his research interests introduced him to Pro-fessor Nisitani and to the EDWM process. We are grateful for his thoughtfulhelp which was of great value in advancing our research.

References

1. Vaidyanathan, S. & Finnic, I. Determination of Residual Stresses fromStress Intensity Factor Measurement, ASME, J. Basic Eng., 1971, 93,242-246.

2. Cheng, W. & Finnic, I. A Method for Measurement of Residual Stress inCircumferentially Welded Thin-Walled Cylinders, ASME, J. Eng. Mat. &


Localized Damage 51

Tech., 1985,106, 181-185.

3. Fett, T. Bestimmung von Eigenspannungen mittels bruchmechanischerBeziehungen, Materialpriifung, 1987, 29, 82-94.

4. Ritchie, R. & Leggatt, R. H. The Measurement of the Distribution ofResidual Stresses Through the Thickness of a Welded Joint, Strain, 1987,61-70.

5. Kang, K. J., Song, J. H. & Earmme, Y. Y. A Method for the Measure-ment of Residual Stresses Using a Fracture Mechanics Approach, /. ofStrain Analysis, 1989, 24, 23-30.

6. Bueckner, H. F. The Propagation of Cracks and the Energy of ElasticDeformation, Trans. ASME, 1958, 80, 1225-1230.

7. Mason, J. C. Basic Matrix Methods, Butterworths & Co. (PublishersLtd.), 1984.

8. Gremaud, M, Cheng, W., Finnic, I. & Prime, M. B. The complianceMethod for Measurement of Near Surface Residual Stresses — AnalyticalBackground, ASME J. Eng. Mat. & Tech., 1994,116, 550-555.

9. Kinoshita, H. & Hayashi, Y. Study in Micro Wire EDM, EDM Technol-ogy, 1994, 2, 24-29.

10. Cheng, W., Gremaud, M., Prime, M. B. & Finnie, I. Measurement ofNear Surface Residual Stresses Using Electric Discharge Wire Machin-ing, ASMEJ. Eng. Mat. & Tech., 1994,116, 1-7.

11. Prime, M. B. Private communication.

12. Cheng, W. & Finnie, I. A Comparison of the Strains due to Edge Cracksand Cuts of Finite Width with Applications to Residual Stress Measure-ment, ASMEJ. Eng. Mat. & Tech., 1993,115, 220-226.

13. Nisitani, H. Two Dimensional Problem Solved Using a Digital Com-puter,, /. Japan Soc. ofMech. Engrs., 1967, 70, 627-635.

14. Nisitani, H. Solution of Notch Problems by Body Force Method, StressAnalysis of Notch Problems, Mechanics of Fracture, ed. G. C. Sih, pp. 1-68, Noordhoff Int. Publishing, 1978.

15. Cheng, W., Finnie, I., Gremaud, M., Rosselet, A. & Streit, R. D. TheCompliance Method for Measurement of Near Surface Residual Stresses— Application and Validation for Surface Treatment by Laser and ShotPeening, ASME, J. Eng. Mat. & Tech., 1994,116, 556-560.

16. Cheng. W. and Finnie, I. An Overview of the Crack Compliance Methodfor Residual Stress Measurement, Proc. 4th Int. Conf. on Residual Stress,pp. 449-458, pub. by Soc. for Exp. Mechanics, Bethel, CT, 1994.


I. Finnic, W.Cheng Department of Mechanical Engineering, … · 2014-05-17 · In this case, a strain gage is located on the back face of the strip. Typically the strain readings

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