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Comput. Methods Appl. Mech. Engrg. 196 (2007) 35773584

On the estimation of residual stresses by the crack compliance method

Sebastian Nervi, Barna A. Szabo *

Center for Computational Mechanics, Washington University, St. Louis, MO 63130, USA

Received 1 November 2005; accepted 30 October 2006Available online 24 March 2007

Abstract

The crack compliance method is a destructive experimental method used for the estimation of residual stress profiles in thick metalplates. Simplifying assumptions, such as dimensional reduction, are generally used in applications of this method. The question of howthe simplifying assumptions affect the estimated residual stresses is addressed in this paper. 2007 Elsevier B.V. All rights reserved.

PACS: 81.40.Jj; 81.70.Bt

Keywords: Residual stress; Crack compliance; Destructive test; Generalized plane strain; Mathematical model; Inverse problem

1. Introduction

This paper is concerned with an investigation of anexperimental method, known as the crack compliancemethod, with respect to its application to the determinationof residual stresses in 7050-T7451 aluminum plates. Theseplates are widely used in the aerospace industry in the man-ufacture of various airframe components. Estimation ofresidual stresses is necessary for the prediction and man-agement of distortion of complex airframe components fol-lowing machining operations. These plates are hot rolled,quenched, stretched and over-aged. The stretching opera-tion imposes a strain of 1.53% in the rolling direction.This reduces the magnitude of residual stresses butincreases the complexity of their distribution.

The crack compliance method was initially developed byVaidyanathan and Finnie [1] in 1971, subsequently refinedby Cheng and Finnie [2]. For information on the crackcompliance method and other experimental techniques werefer to [38].

We denote the material points of an elastic body byX0 2 R3 and its boundary points by oX0. The domain X0

0045-7825/$ - see front matter 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.cma.2006.10.037

* Corresponding author. Tel.: +1 314 935 6352; fax: +1 314 935 4014.E-mail address: szabo@me.wustl.edu (B.A. Szabo).

is the reference configuration. The stress-free configurationof the body is not known. The unit normal to the boundaryis denoted by n0i . The residual stresses, denoted by r

0ij, must

satisfy the equations of equilibrium and the stress-freeboundary conditions. Assuming that the body is notconstrained,

r0ij;j 0 on X0; r0ijn0j 0 on oX0: 1

Since stresses cannot be observed directly, the magnitudeand distribution of residual stresses must be inferred fromtheir effects on the deformation of test articles. In destruc-tive testing methods test articles are systematically alteredby mechanical, chemical or electrical means i.e., X0 !X1 ! X2 . . . (where Xi1 Xi) and the resulting displace-ments and/or strains are measured in one or more points.Estimation of the residual stress distribution involves solv-ing an inverse problem. To this end certain assumptionshave to be made. These assumptions are essential in thesense that were they not justified, the residual stress statecould not be determined by destructive methods. Addi-tional assumptions are usually made for convenience andexpediency.

The essential assumptions are that (a) the material islinearly elastic and remains linearly elastic as the bodyis altered in the destructive testing process, hence the

mailto:szabo@me.wustl.edu

Fig. 1. Notation.

3578 S. Nervi, B.A. Szabo / Comput. Methods Appl. Mech. Engrg. 196 (2007) 35773584

destructive testing process does not introduce residualstresses, (b) the residual stresses are smooth functions ofthe spatial coordinates and (c) the grips used for constrain-ing the test article do not introduce stresses that wouldsignificantly affect measurement of strains caused be theresidual stresses. Assumption (a) implies that the principleof superposition is applicable throughout the destructiveprocess and, in any given configuration of the body, theresidual stress state depends on r0ij defined on X0 andthe current configuration Xjj 1; 2; . . . ; n but not onthe intervening configurations.

Commonly made non-essential simplifying assumptionsare that (a) the orientation of the principal stresses isknown a priori, (b) the distribution of the residual stresseswas uniform over that part of the body where the test cou-pon was obtained, (c) the material is isotropic and (d) inplanes of symmetry plane strain conditions exist.

In this paper we examine the effects of simplifyingassumption (d) on the estimated residual stress state in con-nection with the crack compliance method applied to 7050-T7451 aluminum plates. In the crack compliance method asample is cut by electric discharge machining (EDM) andthe additional simplifying assumption is made that thewidth of the slot created by EDM is negligible.

The plan of the paper is as follows: The properties ofresidual stresses are described for a large plate and a sam-ple cut from the plate in Section 2. The crack compliancemethod, its interpretation by two- and three-dimensionalanalysis and the boundary layer effects are discussed in Sec-tion 3. Interpretation of the experimental data obtained byPrime and Hill [9] on the basis of two- and three-dimen-sional models is presented in Section 4 and conclusionsare presented in Section 5.

2. Residual stresses

When residual stresses r0ij are present in an elastic bodythen the stressstrain law is of the form

rij r0ij Cijklkl; 2

where Cijkl is the tensor of elastic constants and kl is thetensor of infinitesimal mechanical strain, correspondingto displacements with respect to the reference configurationX0. Implied is the assumption that the magnitude of thestress components is below the proportional limit:

f rij 6 r; 3

where f(rij) is a yield function and r is the proportionallimit. The residual stress rij0 can be simulated by ther-mal loading: The stressstrain law in the presence ofthermal loading is given by:

rij Cijklkl aklDT; 4

where akl is the tensor of the coefficients of thermal expan-sion and DT is the temperature change. It is seen thataklDTcan be defined to represent any r0ij. When the mate-rial properties are isotropic then;

rij kkkdij 2Gij 3k 2GadijDT; 5

where k and G are the Lame constants, defined by the mod-ulus of elasticity E and Poissons ratio m:

k : Em1 m1 2m ; G :E

21 m 6

In the following it will be assumed that the elastic proper-ties are isotropic but the coefficients of thermal expansionare orthotropic functions and the principal material axesare aligned with the coordinate directions. Therefore

rij kkkdij 2Gij kakkdij 2GaijDT; 7

where aij is a diagonal matrix of the coefficients of thermalexpansion which are functions of the spatial variablexk 2 X0.

2.1. Residual stress in a large plate

We will consider a large plate of constant thickness h.The coordinate axes x1, x2 lie in the mid-surface of theplate, aligned with the rolling and transverse directions.It is assumed that the rolling and transverse directionsare coincident with the principal residual stress directionsand the principal stresses are functions of x3 only. Theprincipal stresses (resp. strains) will be represented with asingle subscript: ri (resp. i), i 1; 2; 3. The notation isshown in Fig. 1.

Considering the equations of equilibrium, r3;3 0.Therefore r3 is a constant and since on the top and bottomsurfaces x3 h=2 the plate is stress free, it follows thatr3 0. Consequently Eq. (7) can be simplified to:

r1 E

1 m2 1 m2 EDT1 m2 a1 ma2; 8

r2 E

1 m2 m1 2 EDT1 m2 ma1 a2: 9

It is seen that the residual stresses in a large plate can berepresented as

rR1 EDT1 m2 a1 ma2;

rR2 EDT1 m2 ma1 a2: 10

We introduce the dimensionless variable g 2x3=h andwrite:

S. Nervi, B.A. Szabo / Comput. Methods Appl. Mech. Engrg. 196 (2007) 35773584 3579

a1DT XN1j1

A1jP j1g; a2DT XN2j1

A2jP j1g; 11

where P j1g j 1; 2; . . . are the Legendre polynomials.Because of the orthogonality of the Legendre polynomialswe have:Z h=2h=2

rRi x3dx3 Z h=2h=2

x3rRi x3dx3 0 for i 1; 2:

12

We have assumed that rR1 and rR2 are functions of x3

only. Consequently the equations of equilibrium are satis-fied. We have noted that the stress-free boundary condi-tions are satisfied on the top and bottom surfaces, i.e.,r3h=2 0. On the other boundary surface (side surface)SC C h=2; h=2 the tractions should be zero, that is:

T 1 rR1 n01 0; T 2 rR2 n

02 0 on SC;

where n01, n02 are the components of the unit normal to the

boundary curve C. This boundary condition cannot be sat-isfied exactly, given the assumption that the residual stres-ses are functions of x3 only. Nevertheless, this condition issatisfied in the sense of Eq. (12). In other words, theorthogonality of Legendre polynomials guarantees thatthe stresses on the boundary surface SC have zero resul-tants, that is, the membrane force and bending momentvanish at the boundaries. The shear force and twisting mo-ment are zero by virtue of the stated assumptions. BySaintVenants principle, their effects decay exponentiallywith distance from the boundary. Therefore the residualstresses satisfy the condition of Eq. (1) in the interior ofthe plate, on the top and bottom surfaces, and on SC inthe sense of Eq. (12).

The distribution of residual stresses at the boundaries of7050-T7451 aluminum plates depends on the quenchingprocess. In general it will not be the same as in a plateloaded by the temperature distribution given by Eq. (11).Nevertheless, the differences decay exponentially with dis-tance from the boundary. From the point of view of esti-mating the magnitude and distribution of residual stressesin an aluminum plate, it is assumed that the samples willbe obtained at distances greater than about 2h from theboundary SC.

2.2. Residual stresses in a sample

Samples typically have length dimensions similar to thethickness of the plate. As a sample is removed from the infi-nite plate, the existing stresses change so as to satisfy theequations of equilibrium and the traction-free boundaryconditions on the surface. In the following we assume thatthe process by which a sample is removed from a largeplate does not introduce additional residual stresses.

We denote the material points of the sample by Xs andits boundary by oXs. The residual stress distribution inthe sample, denoted by rij, satisfies the equilibrium equa-

tion rij;j 0 on Xs and the traction-free boundary condi-tion rijnj 0 on oXs. Therefore, by the principle ofvirtual work;Z

Xs

rijvi;j dV Z

oXs

rijnjvi dS 0; 13

where vi is an arbitrary virtual displacement function. Sincerij rji, Eq. (13) can be written asZ

Xs

rijvij dV 0 where

vij

1

2vi;j vj;i: 14

Eq. (14) must hold for all vi for which the integral expres-sion is finite-valued. We write rij as the sum of the residualstress in the plate rRij plus a correction r

Cij and writeZ

Xs

rRij rCij

vij dV 0; 15

where

rRij rR1 for i j 1;rR2 for i j 2;0 otherwise:

8>:

Letting rCij CijklCkl , where Cijkl is the material stiffness

tensor of isotropic elasticity and Ckl uk;l ul;k=2, andusing Eq. (10), this is equivalent to solving a thermal stressproblem on Xs. The resulting stress,

rij CijklCkl aklDT;satisfies the equations of equilibrium and the stress-freeboundary conditions.

This formulation for the computation of the displace-ment field ui and the corresponding correction to the stressfield rCij is equivalent to applying the tractions

T Ri rRij nj 16

to the surface of the sample oXs. To show this, we writeZXs

rCij vij dV

ZoXs

rRij njvi dS: 17

Since rRij satisfies the equations of equilibrium: rRij;j 0.

Applying the divergence theorem, we have:Z

oXs

rRij njvi dS Z

Xs

rRij vij dV : 18

Therefore Eq. (17) is equivalent to Eq. (15).Any further modification of the sample by cutting can

be treated analogously. We observe that, whereas the initialresidual stress field rRij does not have to satisfy the com-patibility conditions, the stress field rCij does.

The displacement ui and the strain ukl corresponding to

the correction are measurable in surface points and areindicative of the initial residual stress field rRij .

It follows that, subject to the stated assumptions, theresidual stress in a part cut from a plate depends only onthe residual stress in the plate, the location of the cut,and the configuration of the part.

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3. The crack compliance method

In the crack compliance method a sample is removedfrom a large plate, usually by saw cut. The location ofthe sample must be sufficiently far from the boundariesof the plate to justify the assumption that the residual stres-ses in the large plate can be represented by Eq. (11). Typi-cal length dimensions of the sample L1, L2, indicated inFig. 2, are approximately 2h.

Strain gauges are attached at a small number of pointson the surface of the sample. The gauges are usuallyaligned with the expected principal stress directions.

A slot of small width is progressively cut by EDM.Because of its small width, the slot is usually idealized asa crack. For each slot length ak the strain readings arerecorded. We denote the strain reading in gauge locationm corresponding to ak by r

mk m 1; 2; . . . ;M ; k

1; 2; . . . ;K.The distribution of residual stress in the large plate can

be inferred from the strain readings. Two methods of inter-pretation are discussed in the following.

Remark 3.1. In the crack compliance method some yield-ing may occur at the crack tip. The assumption thatsuperposition can be applied does not hold in the smallregion affected by plastic deformation. There are othermethods for inducing deformation in the sample that donot cause yielding. For example the wide slot methoddiscussed in [10]. For a discussion of measurementtechniques we refer to [5].

3.1. Interpretation based on two-dimensional analysis

In most interpretations of the measured data in thecrack compliance method it is assumed that plane strainconditions exist on the cross-section x shown in Fig. 2,(see, for example, [9,6,7,11]), and a1 a2 a. In this caseEq. (10) is simplified to:

rR1 rR2

EaDT1 m : 19

We let

aDT XNj1

AjP j1g; 20

Fig. 2. Sample geometry. Notation.

where, in view of Eq. (19), a single index is used for label-ling the coefficients Aj.

A thermoelastic plane strain problem is solved corre-sponding to each slot length ak k 1; 2; . . . ;K and eachterm in the polynomial expansion in Eq. (20), using

aDTj P j1g; j 1; 2; . . . ;N

for the thermal load. We denote the corresponding solutionby ukjx1; x3 and the normal strain, computed fromukjx1; x3 at strain gauge location m, by mkj . Using theprinciple of superposition,

rmk XNj1

Ajmkj : 21

From these equations the coefficients Aj can be estimatedby least squares fitting. Therefore the estimated residualstresses in the plate are:

rR1 g rR2 g

E1 m

XNj1

AjP j1g: 22

It is necessary to have a substantially larger number ofobservations than N, that is, N K M . It is not possibleto measure very small strains accurately because the signalto noise ratio is small, so all observations are not equallyreliable. Therefore the strain measurements should beweighted by the magnitude of strains, or small strain mea-surements ignored.

As N increases, the strains mkN decrease. It is not usefulto have N larger than the value at which the size of mkN issimilar to the size of errors in strain measurement.

Remark 3.2. The assumption that the material is isotropicimplies that the coefficient of thermal expansion is the samein each direction, therefore rR1 r

R2 , see Eq. (19), which

is contradicted however by the results of experimentspresented in Section 4. This contributes to the errors ofinterpretation based on two-dimensional plane strainmodels.

3.2. Interpretation based on three-dimensional analysis

The assumption that generalized plane strain conditionsexist on the cross-section x, indicated in Fig. 2, would bejustified only if the dimension L2 would be much largerthan L1 and h. Given that the dimensions of the sampleare of similar magnitude, the stress distribution on X isinfluenced by the boundary layer effects, hence the actualstress distribution is three-dimensional.

Interpretation of the experimental information bymeans of three-dimensional analysis permits one to makeindependent assumptions about the distribution of residualstresses in the rolling and transverse directions. The proce-dure is described in the following.

We solve N 1 N 2 thermoelastic problems in threedimensions for each slot length ak. Specifically, we leta1DTj P j1g, j 1; 2; . . . ;N 1 and compute the strain

Fig. 3. Boundary layer effects: (a) notation and (b) solution domain andfinite element mesh.

S. Nervi, B.A. Szabo / Comput. Methods Appl. Mech. Engrg. 196 (2007) 35773584 3581

component corresponding to the strain measured in gaugelocation m for each slot length ak. We denote the computedstrain values by m1kj . Similarly, we let a2DTj P j1g,j 1; 2; . . . ;N 2 and compute the strain in gauge locationm for each slot length ak. We denote the computed strainvalues by

m2kj .

Using the principle of superposition,

rmk XN1j1

A1jm1kj

XN2j1

A2jm2kj : 23

The coefficients A1j, A2j are determined by least squaresfitting.

3.3. Boundary layer effects

In order to illustrate the differences between two- andthree-dimensional models, we examine a single term P2(g)in the polynomial approximation of the residual stress.The two-dimensional (plane strain) model is equivalent toa three-dimensional model where the normal displacementson the boundary surfaces x2 L=2 (shown in Fig. 3) areset to zero. Given that the residual stresses satisfy Eq. (12)in the large plate, the plane strain solution is a generalizedplane strain solution. Therefore the solution of the three-

0 2 4 6 8 10-0.325

-0.32

-0.315

-0.31

-0.305

-0.3

-0.295

-0.29

-0.285

-0.28

Cut 2

Cut 1

0 2 4 6 8 10

0

0.02

0.04

0.06

0.08

0.1

0.12

Cut 2

Cut 1

Plane strain solution

0 2 4 6 8 10-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 2 4 6 8 10-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05Cut 2

Cut 1

Cut 2

Cut 1

Fig. 4. Computed normalized strains for a 20 mm. (a) Computed strains 1 and 2 for rR1 CP 2g, rR2 0 and (b) computed strains 1 and 2 for

rR1 0, rR2 CP 2g.

Fig. 5. Schematic view of experiments performed by Prime and Hill [9].The x1 (resp. x2) coordinate axis corresponds to the transverse (resp.rolling) direction.

3582 S. Nervi, B.A. Szabo / Comput. Methods Appl. Mech. Engrg. 196 (2007) 35773584

dimensional problem with zero tractions on the boundarysurfaces x2 L=2 converge to the generalized plane strainsolution when the ratio L/h goes to infinity. It is knownthat the boundary layer effects decay very rapidly. For adiscussion on generalized plane strain problems we referto [12].

Given the constraints imposed by the plane strainassumption, the computed strains depend only on the stres-ses acting along the direction normal to the cutting plane.In three-dimensional analysis the dependence of the com-puted strains on both rR1 and r

R2 can be considered by

allowing different series expansions for a1DT and a2DT.To illustrate the boundary layer effect we consider two

cases. In the first case rR1 CP 2g and rR2 0, in the

second case rR1 0 and rR2 CP 2g where C is an arbi-

trary constant having the dimension of stress. These condi-tions were induced through appropriate selection of a1DTand a2DT. For the purposes of this analysis the size ofthe slot a is kept constant (a 20 mm) while the dimensionL of the sample is progressively increased.

The results are shown in Fig. 4. As expected, the contri-bution of the in-plane stress component vanishes as L/hincreases, and the three-dimensional solution convergesto the plane strain solution. In the figure the points labelledCut 1 and Cut 2 correspond to the first and second cuts in[9]. It is seen that the boundary layer effects are substantialfor the first term in the series given by Eq. (11).

4. Interpretation of the experimental data

Prime and Hill [9] investigated residual stress distribu-tions in 7050-T74 and 7050-T7451 aluminum plates1 bymeans of the crack compliance method. They made theirexperimental data available to the writers. The experimentswere conducted as follows:

A specimen, measuring 150 150 75:8 mm3, wasremoved from the central region of a 760 mm long,760 mm wide, 75.8 mm thick plate by saw cut.

Cuts were made using EDM with a 0.3 mm diameterbrass wire. The first cut was in the plane of symmetryperpendicular to the transverse direction, the secondcut was in the plane of symmetry perpendicular to therolling direction. Note that, for the first cut, the widthto thickness ratio is given by L2=h 1:979 and for thesecond cut is given by L1=h 0:989. The machine wasset to skim cut in order to minimize the stress inducedby the cutting process. The slot was cut in 0.5 mm incre-ments to a depth of 12 mm and then in 1 mm incrementsfor the remainder of the test.

The location of the strain gauges is indicated in Fig. 5.One gauge was placed very close to each cut on the sur-face where the cut begins (top strain gauges, 1 and 5),

1 The nomenclature indicates that the plates were quenched and, in thecase of 7050-T7451, stretched through the imposition of 1.53.0% strain inthe rolling direction in order to reduce the magnitude of residual stresses.

and another was placed on the opposite surface centeredon the cut plane (bottom strain gauges 3 and 7). Gaugesparallel to the cut (transverse strain gauges, 2, 6, 4, and8) were also placed at each location. Micromeasure-ments CEA-13-125UT-350 constantan gauges with anactive gauge length of 3.18 mm were used. Only thestrains measured from gauges 3 and 7 (respectivelytransverse and rolling directions) were used in the com-putation of the stresses.

The differences between interpretations of the experi-mental data based on two- and three-dimensional analysesare examined as follows.

As in [9], we modeled the slot as an ideal crack ratherthan representing the actual geometry of the cut. Computa-tional experiments have indicated that there were no signif-icant differences between the computed strain values in thegauge locations when the 0.3 mm wide slot was idealized asa crack. In the finite element analysis a geometricallygraded mesh was used in order to control the errors ofapproximation, following the recommendations for opti-mal mesh layout in the neighborhood of singular points[13].

The computations were performed with StressCheck2.The errors of approximation were controlled by means ofp-extension. For all solutions the estimated relative errorin energy norm was less than 2%. StressCheck provides aframework for parametric studies in which the depth ofthe slot can be incremented automatically. For details werefer to [14].

2 StressCheck is a trademark of Engineering Software Research andDevelopment, Inc.

0 10 20 30 40 50 60 70 80 -25

-20

-15

-10

-5

0

5

10

15

20

25

Depth (mm)

Plane Strain3D

Res

idua

l Str

ess

(MP

a)

0 10 20 30 40 50 60 70 80 -20

-15

-10

-5

0

5

10

15

Res

idua

l Str

ess

(MP

a)

Depth (mm)

Plane Strain3D

Fig. 6. Comparison between interpretations based on plane strain and 3D analyses. (a) Rolling direction and (b) transverse direction.

0 10 20 30 40 50 60 70 80-150

-100

-50

0

50

100

150

200

Rel

ease

d S

trai

n (

)

Depth (mm)

ExperimentSimulation

Fig. 7. Released strain for the second cut (rolling direction).

S. Nervi, B.A. Szabo / Comput. Methods Appl. Mech. Engrg. 196 (2007) 35773584 3583

The results reported in [9] on the basis of a plane strainmodel were recomputed and verified. In the verification pro-cess only strain gauges 3 and 7 were used. The residual stressdistributions obtained by means of the plane strain modeland the three-dimensional model are shown in Fig. 6. Thedifference is approximately 10% in the rolling direction,and 20% in the transverse direction in maximum norm.

The significance of these differences must be evaluated inrelation to variations of residual stresses within a plate,variation of stresses between plates from different produc-tion runs, and variations that depend on the manufacturer.Using procedures where both the numerical errors andexperimental errors were carefully controlled, but only avery small number of experiments performed [10], wefound that the variation in maximum stress in the rolling(resp. transverse) direction was approximately 18% (resp.5.5) percent as compared with the results reported by Primeand Hill [9].

We note that neither estimates are reliable in the vicinityof the top and bottom surfaces (x3 h=2). This is due tothe fact that when the slot is small then the noise-to-signalratio in the strain gauges is large. When the slot is largethen the strain gauges nearest to the slot, where thenoise-to-signal ratio would be smallest, are compromisedby the effects of the cutting process.

In order to verify the estimates, a sequence of forwardproblems was solved in which the estimated residual stressdistribution was used and the cutting process was simu-lated for the second cut (perpendicular to the transversedirection). The computed and the measured strains areplotted in Fig. 7. It can be seen that the computed strainscompare well with the experimental measurements.

Remark 4.1. Standard material properties for 7050-T7451aluminum were used in the computations. Uncertainties inthe material data undoubtedly influence the results. For adiscussion on the variability of elastic modulus in 6061-T6drawn aluminum tube we refer to [15]. Assuming thatthe variation of the elastic modulus is not substantiallygreater in 7050-T7451 aluminum plates, we believe thatthe effect of uncertainties in material properties is not

greater than the effect of uncertainties in the measurementof strains.

Remark 4.2. The experimental data presented in Fig. 6 con-tradicts the assumption that rR1 r

R2 , see Remark 3.2.

One can view the two-dimensional data presented inFig. 6 as the first step in an iterative process in which rR1(resp. rR2 ) are successively re-interpreted as a new distribu-

tion rR2 (resp. rR1 ) is computed until convergence occurs.

5. Conclusions

Estimation of residual stresses in metals by destructivemethods, such as the crack compliance method, involvesthe solution of an inverse problem. The essential assump-tions incorporated in the mathematical model are that (a)the material remains linearly elastic in the entire process,hence the principle of superposition is applicable and (b)the residual stresses are smooth functions of the spatialvariables.

Even when these assumptions are fully justified, ourability to determine residual stresses with high accuracy islimited by errors in the strain measurements, the mathe-matical model used for the interpretation of experimental

3584 S. Nervi, B.A. Szabo / Comput. Methods Appl. Mech. Engrg. 196 (2007) 35773584

data and the errors in the numerical solution of the math-ematical model. The present investigation was concernedwith estimation of errors caused by the choice of mathe-matical model.

It is commonly assumed that generalized plane strainconditions exist in the plane of symmetry. This would betrue if the dimension of the sample normal to the planeof symmetry would be much larger than the other dimen-sions. Typical samples do not satisfy this condition how-ever. Therefore the experimental data are influenced bythree-dimensional boundary layer effects.

When a plane strain model is used for the interpretationof experimental data then only measurements in the planeof symmetry can be considered. Hence the orientation ofthe principal stresses must be assumed a priori and theeffects of residual stresses acting in the transverse directionon residual stresses acting in the rolling direction (and viceversa) cannot be considered. Such limitations are not pres-ent when a fully three dimensional model is used.

In the example considered the error in stresses, attribut-able to the choice of the mathematical model, was approx-imately 10% in the rolling direction, 20% in the transversedirection in maximum norm.

Acknowledgements

The authors are indebted to Dr. Michael B. Prime ofthe Los Alamos National Laboratory who provided de-tailed experimental data on crack compliance measure-ments and responded to queries concerning hisexperiments. This work was supported by the Air ForceOffice of Scientific Research through Grant No. F49620-01-1-0074 and Grant No. FA9550-05-1-0105. The viewsand conclusions contained herein are those of the authorsand should not be interpreted as necessarily representingthe official policies or endorsements, either expressed orimplied, of the Air Force Office of Scientific Research orthe US Government.

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On the estimation of residual stresses by the crack compliance methodIntroductionResidual stressesResidual stress in a large plateResidual stresses in a sampleThe crack compliance methodInterpretation based on two-dimensional analysisInterpretation based on three-dimensional analysisBoundary layer effectsInterpretation of the experimental dataConclusionsAcknowledgementsReferences