experimental measurement and finite element simulation of the interaction between residual stresses...

International Journal of Fatigue 23 (2001) 293–302www.elsevier.com/locate/ijfatigue

Experimental measurement and finite element simulation of theinteraction between residual stresses and mechanical loading

D.J. Smitha,*, G.H. Farrahib, W.X. Zhu c, C.A. McMahona

a Department of Mechanical Engineering, University of Bristol, Bristol BS8 1TR, UKb School of Mechanical Engineering, Sharif University of Technology, Tehran, Iran

c Structure, Technology, BAe Systems, Bristol, UK

Received 7 April 2000; received in revised form 23 November 2000; accepted 27 November 2000

Abstract

Residual stresses, which can be produced during the manufacturing process, play an important role in an industrial environment.Residual stresses can and do change in service. In this paper, measurements of the statistical distribution of the initial residualstress in shot blast bars of En15R steel are presented. Also measured was the relaxation of the residual stresses after simple tensileand cyclic tension–compression loading. Results from an elastic–plastic finite element (FE) analysis of the interaction betweenresidual stresses and mechanical loading are given. Two material hardening models were used in an FE analyses: simple linearkinematic hardening and multilinear hardening.

It is shown that residual stress relaxation occurs when the applied strains are below the elastic limit. Furthermore, the resultsfrom the simulations were found to depend on the type of material model. Using the complex multilinear model led to greaterresidual stress relaxation compared to the simple linear model. Agreement between measurements and predictions was poor forcyclic loading, and good for simple tensile loading. 2001 Elsevier Science Ltd. All rights reserved.

Keywords:Cyclic loading; Residual stress relaxation; Finite element analysis; Hardening model; X-ray diffraction

1. Introduction

Residual stresses can be produced during many manu-facturing processes such as quenching, welding or dueto the different response of particles and matrix in amultiphase material. These stresses can have a beneficialor detrimental effect and play an important role in anindustrial environment. Crack initiation and propagationin static or fatigue loading, or in stress corrosion can beimpeded by compressive stresses normal to the crack andgreatly accelerated by tensile stresses.

Residual stresses are included in the advanced designand fatigue failure analysis of components in the aeros-pace, nuclear and automotive industries. Even in themicroelectronics industry residual stresses are taken intoaccount for dimensional stability of electronic packag-ing. Residual stresses can and do change in service.

* Corresponding author. Tel: +44-(0)1179-28-82-12; fax:+44(0)1179-29-44-23.

E-mail address:[email protected] (D.J. Smith).

0142-1123/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved.PII: S0142-1123 (00)00104-3

Under cycling loading, which generates cyclic plasticity,we would expect some relaxation of the residual stresses.After many cycles the residual stresses may no longerchange and become stabilised. Therefore, a knowledgeof the stabilised stress is important.

The behaviour of residual stresses during fatigue hasattracted the attention of many researchers. A number ofcombinations of load and material have been investi-gated and some relaxation models have been proposed.James [1] proposed a model for relaxation based on aneffective shear stress acting on primary slip planes ori-ented at an angle to the surface. Kodama [2] found theresidual stress relaxation of an annealed mild steel variedlinearly with the logarithm of the fatigue cycles whilethe relaxation rate was proportional to the stress ampli-tude. Iida and Takanashi [3] found that the relievedstress caused by repeated cyclic loading was larger thanthat caused by the reversed cyclic loading. Farrahi et al.[4] showed that fatigue stressing produces a decrease inresidual stresses and in micro-strain on shot peenedspring steel specimens. This reduction was a function ofthe applied stress and the depth of the plastically

294 D.J. Smith et al. / International Journal of Fatigue 23 (2001) 293–302

deformed layer. However, residual stress remained com-pressive even after final fracture.

In principle, residual stress relaxation occurs when thelinear superposition of applied stress and residual stressreaches the yield point of material. The onset of stressrelaxation is delayed by the presence of sufficient stableobstacles to dislocation movement [5]. Overall, underthe cyclic loading, the residual stress relaxation dependson the combined stress state of applied stress andresidual stress. James [6] proposed that the mechanismsof relaxation can be separated into three regimes: (i) thecombined stress above the macroscopic yield strength;(ii) the combined stress between the yield strength andendurance limit and (iii) the combined stress below ornear the endurance limit.

The relaxation of residual stress in a specimen is notalways in one regime. For example, in the first regime,the residual stress can be relaxed rapidly and the com-bined stress could be changed below the yield strengthin which the relaxation mechanism belongs to the secondregime. For cyclic softening or hardening materials, theyield strength changes with number of cycles and there-fore the relaxation could be between (i) and (ii).

It is not clear how the cyclic deformation and materialhardening characteristics can influence residual stressrelaxation due to mechanical loading. There have beenrelatively few studies investigating how cyclic defor-mation can be modelled and then used to predict relax-ation of residual stresses.

The main purpose of this paper is to determine themain features that influence the interaction of residualstress with mechanical loading. First, residual stresses,measured in forged and shot-blast steel bars, arepresented. Measurements were made using X-ray andcentre-hole drilling methods. The residual stresses weremeasured before and after tensile and cyclic loading.Then elastic–plastic finite element analyses were carriedout to predict the residual stress relaxation using differ-ent material models. A linear kinematic hardening modelwas used for tensile and cyclic loading cases. Animproved material model, using multilinear kinematichardening characteristics to simulate the cyclic stressstrain behaviour was also developed. This model waswritten in a user subroutine in the ABAQUS finiteelement code and used to simulate relaxation of residualstresses under cyclic mechanical loading. There is goodagreement between predictions and experiments forresidual stress relaxation following tensile loading. Forrelaxation from cyclic loading, surface residual stressmeasurements using the X-ray method revealed pooragreement with predictions. In contrast, centre-holemeasurements showed that complete relaxation hadoccurred.

2. Experiments

2.1. Material and specimens

Hot forged round En15R steel bars were used in theexperiments. The steel composition (wt%) was 0.4 C,0.17 Si, 1.59 Mn, 0.07 Cr, 0.17 Ni, 0.02 Mo, 0.004 S,0.022 P and remainder Fe. Young’s ModulusE and Pois-son’s ratio n of the material are 208 GPa and 0.28respectively. The dimensions of the steel bars are shownin Fig. 1. The forging process used in the production ofthese bars simulated the manufacture of wheel suspen-sion arms for automobile components. The process con-sists of an initial induction heating of a steel billet toabout 1200–1250°C then moulding in a hot forgingpress, followed by finish pressing. After clipping theexcess material, the forged bars were allowed to cool inair to room temperature. Then the components wereheat-treated (hardened and tempered), followed by shotblasting to clean the surface of oxide scale. Shot blastingalso introduces compressive residual stresses on the sur-face of the bars.

2.2. Tensile and cyclic mechanical loading

For the mechanical loading tests, strain gauge rosetteswere bonded on both sides of the manufactured bars.Two calibrated displacement transducers were used tomeasure displacements over the gauge length. The aver-age of the displacements on both sides was used as theaverage for control and output. To apply the loading,male screw threads were machined at both ends of thespecimens to enable them to be fitted into the testmachine. Tensile loading was applied to three speci-mens. Axial loading was first applied to each specimento produce a chosen plastic deformation. Then the speci-men was unloaded and removed to measure surfaceresidual stress using X-ray. Cyclic loadings were appliedto two specimens, and a representative stress–surfacestrain cyclic curve is shown in Fig. 2. First, a small strainrange was applied. After ten cycles, each test wasstopped and each specimen was taken for X-raymeasurement. Then the strain range was increased to agiven value and retained for ten cycles. The specimenwas removed and X-ray measurements were taken again.The same procedure was carried out for larger strain

Fig. 1. Hot forged En15R steel bar specimen.

295D.J. Smith et al. / International Journal of Fatigue 23 (2001) 293–302

Fig. 2. Cyclic stress–strain history for En15R steel at room tempera-ture.

ranges. Finally, these two specimens were centre holedrilled.

2.3. Residual stress measurement

Residual stress measurements were made using the X-ray diffraction method on the surface of many additionalsamples. This method is well known and is describedelsewhere [7]. The (211) peak (2q=156°) was examined.The axial residual stresses from many specimens wereused to obtain a sample to sample distribution. Residualstresses below the surface were also measured using theX-ray method. Chemical etching was used for materiallayer removal on a specimen. Corrections for residualstress relaxation caused by layer removal were alsointroduced. Measurements on this specimen were essen-tial for determination of initial distribution of residualstress which will be used in FE analysis.

Several specimens were centre hole-drilled to deter-mine the residual stresses at the end of cyclic loading.This method, taking into account the influence of thecurved surface of the bar, is described elsewhere [8].The TEA-06-062RK-120 type strain gauge rosette,manufactured by the Measurements Group Inc., wasused. The strain gauge was carefully wrapped andbonded around the cylindrical part in the centre of thebar. The centre hole diameter was 1.926 mm. The depthof each drilling increment was about 0.08 mm. Inaddition, two specimens, which had not been subjectedto mechanical loading and chosen as references, werehole drilled.

2.4. Experimental results

Fig. 3 shows the initial distribution of residual stressmeasured using the X-ray method. Fig. 3(a) shows a fit-ted distribution to the sample-to-sample variation of thesurface axial residual stresses. The X-ray measurements,corrected for successive removal of surface layers of a

Fig. 3. Initial residual stress: (a) fitted distribution of the sample tosample variation of the axial surface residual stress; (b) X-ray measure-ment using successive layer removal and calculated initial distribution.

bar, are also shown in Fig. 3(b). Based on these experi-mental results, the initial distribution of the residualstresses, across the section of the bar, was calculated.This is explained in Section 3. The initial distribution isalso shown in Fig. 3(b).

The relaxed residual stresses, after tensile and cyclicmechanical loading, are shown in Fig. 4(a) and (b)respectively. In Fig. 4(a), the initial surface residualstresses are shown. This corresponds to when the appliedtensile strain is zero. With increasing tensile strain, theaxial surface residual stress reduced. For large tensilestrains, the residual stresses were relaxed completely.Similarly, for cyclic loading the axial surface residualstress also relaxed as shown in Fig. 4(b). Compared to


Fig. 4. Experimental results and FE simulations of residual stressrelaxation in the round bar specimen: (a) after tensile loading usingkinematic hardening model and also (b) after cyclic mechanical load-ing using multilinear kinematic loading.

results for tensile loading, the residual stresses, measuredusing the X-ray method, did not relax as much.

For large plastic cyclic strain ranges, the centre holemeasurements of residual stress showed that the speci-mens were essentially stress free. This is in contrast tothe X-ray measurements. The residual stresses were notrelaxed completely. The larger surface residual stressesmeasured using the X-ray diffraction method comparedto the results from the centre hole drilling method mayindicate that the residual stress relaxation on the speci-men surface was different from that under the specimensurface. This was found to occur for several specimens.We could not find any reasons for this discrepancyalthough a possible explanation is offered in the dis-cussion.

3. Finite element modelling

For the finite element modelling three aspects wereexamined prior to simulation of residual stress relax-ation; material models, initial residual stress-state, anddefinition of initial state for the material models. Thesethree topics are presented in this section, and in Section4 the results from the FE simulations are given.

3.1. Material models

Most structures under cyclic loading with plasticdeformation exhibit multiaxial stresses and strains whichare strain-history dependant. Many incremental plasticitymodels have been developed to describe the stress–strainbehaviour for general cases. Because of the complexityof the constitutive equations, few analytical solutionshave been found and only approximate results can beobtained by numerical techniques such as the finiteelement (FE) method. For the analysis, the ABAQUSfinite element code was used. To simulate relaxation ofresidual stresses when a component is subjected tomechanical loading, linear and multilinear kinematichardening material models were used.

3.1.1. Linear kinematic hardening modelFor many materials, compressive yielding is often

much lower after plastic deformation in tension. This iscalled the Bauschinger effect. Isotropic hardening cannotmodel the Bauschinger effect, and kinematic hardeningwas developed to deal with this kind of hardening behav-iour. The model for kinematic hardening assumes thatthe shape of loading surface is fixed while the centre ofthe loading surface moves around. This kinematic hard-ening model can be expressed as

f5fY(Sij2Xij )2sy50 (1)

whereSij is the deviator stress tensor andXij is a tensorthat indicates the present position of the loading surface.If linear hardening is assumed, the relationship betweenXij and the plastic strain tensorepij is

Xij52/3Hepij (2)

whereH is the linear hardening modulus measured in auniaxial tension test. The factor 2/3 occurs because theaxial deviator stress in a uniaxial tension test equals 2/3of the Cauchy stress.

The linear kinematic hardening model was applied tothe stabilised uniaxial cyclic stress–strain curve shownin Fig. 2. The constants for the model are 653 MPa and24.1 GPa for the radius of yield surfacesy and the hard-ening modulusH, respectively. Fig. 5(a) shows theexperimental cyclic stress-strain curve for the stabilisedstate and the kinematic hardening simulation.


Fig. 5. Simulation of uniaxial stress strain curve: (a) linear kinematichardening; (b) multilinear kinematic hardening.

3.1.2. Multilinear kinematic hardening modelThe material cyclic stress–strain curve is shown in

Fig. 2. The experimental result illustrates that there wasmovement of the yield surfaces with changes in numberof cycles. To take account of this behaviour we use thetheoretical multilinear kinematic model proposed byMroz [9]. This model consists of a number of surfacesin stress space, inserted one into another with linearkinematic hardening. The key point is that there is kine-matic movement of the surfaces. We assume that thereare n surfaces expressed by each function

fL5fY(Sij2Xij,L)2kL50 L51,2,%,n (3)

wherefY is a function described by the Von Mises yieldcriterion, Xij,L is the current centre of the surfaceL andkL is the radius of the surfaceL.

The translation of theLth surface is

dXij,L5dm(Sij,L+12Sij ) (4)

where Sij,L +1 is the stress point on the (L+1)th surface

and dm is a constant increment related to the stressincrement.

The plastic strain increments are given by the Prandtl–Reuss equation

depij5dl∂fL∂Sij

(5)

The projection of stress incrementdS on the normalto the surfacefL, is dF, and this contributes to the plasticdeformationdp. The relationship betweendF anddp is

dF5!23HLdp (6)

where the hardening modulusHL may depend on stressS. However, it is preferable to use a constant value thatmay be determined from a uniaxial test. Under uniaxialstress Eq. (6) becomes

ds5HLdep (7)

Four hardening surfaces were chosen, with four hard-ening moduliH1 to H4, taken as functions of maximumequivalent plastic strain history. The radiikL of surfaces1, 2 and 3 were kept constant, whilst the radius of thefourth surface,k4, equalled the maximum equivalentstress in the history.

All the material constants were determined from uni-axial cyclic stress–strain curves. For the initial yielding,the stress-strain curve is expressed as

s55sy1+h1ep ep#400me

sy2+h2ep 400,ep#8000me

sy3+h3ep ep.8000me

(8)

whereh1, h2 andh3 are hardening moduli for the expan-sion of the fourth yield surface. Therefore,H4 is equalto h1 or h2 or h3. The diameters of each surface weremeasured from the points of tensile maximum stress,shown in Fig. 5(b). The hardening moduliHL were nolonger constants and were plastic strain and historydependent. The material constantshi, syi andki are givenin Table 1.

3.2. Initial residual stress distribution

In ABAQUS, residual stresses were input as an initialcondition. The initial residual stress distribution, in theround bar, was obtained from the X-ray measurementscorrected for successive surface layer removal. The dis-tribution was obtained using a method where conditionsof equilibrium in the round bar are used. The equilibriumconditions are


Table 1Parameters of multilinear kinematic model applied to En15R steel atroom temperature

Young modulus,E 208 GPaPoisson ratio,n 0.28Initial yield stresssy1 in tension sy1=800 MPaSecond yield stresssy2 in tension and sy2=835 Mpa;corresponding tensile plastic strainep1, and ep1=400 µε;hardening modulush1 betweensy1 andsy2 h1=87.5 GPaThird yield stresssy3 in tension and sy3=870 Mpa;corresponding tensile plastic strainep2, and ep2=8000µε;hardening modulush2 betweensy2 andsy3 h2=4.6 GPaHardening modulus in tensionh3, whens.sy3 h3=0Maximum equivalent plastic strain in the straineeq,m

p

historyRadius of 1st surface,k1 470 MPaRadius of 2nd surface,k2 630 MPaRadius of 3rd surface,k3 750 MPaRadius of 4th surface,k4 seq

max corresponding to theeeq,m

p

Hardening modulus between surface 1 andsurface 2

(k2−k1)10%eeq,m

p


(k3−k2)20%eeq,m

p


(k4−k3)70%eeq,m

p

ER

0

sRqdr=0

ER

0

rsRzdr=0

6 (9)

together with the boundary condition

sr|r=R50 (10)

where r is the nondimensional radius of the round barr/R. Based on the measurements using the X-ray methodtogether with layer removal using chemical etching, thedistributions of residual stresses in the round bars takethe form

sRr =Ha

a+bx2

x#0

x.0

sRq=

ddr

(rsr)

sRz=Hc

c+dx

x#0

x.0

(11)

wherex=r2q anda, b, c andd are constants which weredetermined by using equilibrium and boundary con-ditions. The surface axial and tangential residual stressesare defined assz1 andsq1 and the constantsa, b, c andd are

5a=−sRq1(1−q)

2

b=sRq1

2(1−q)

d=sR

z1

1−q−t

c=−dt

(12)

where

t5(1−q)2/2+q(1−q)3/3

0.5+q−q2 (13)

The nondimensional distanceq was 0.84. Based on thestatistical distribution of the initial surface X-ray residualstresses, shown in Fig. 3(a), the initial surface axialresidual stresses,sR

z1, for the FE simulation were chosento be between2250 MPa and2450 MPa. The initialtangential residual stresses,sR

q1, were 50 MPa lowerthansR

z1.

3.3. Initial state for multilinear kinematic model

For the multilinear kinematic hardening model, theinitial plastic strains and their related historic variableswere introduced through a user subroutine in ABAQUS.It was assumed that the maximum equivalent plasticstrain eeq,m

p from the previous strain history, due to shotblasting, was known. The corresponding maximum equi-valent stresssm

eq was obtained from the uniaxial tensilecurve. This is also the radius of the 4th yield surface.The initial state was related to the initial residualstresses. The residual stress distribution given by Eqs.(11)–(13) were used as the initial stress-state.

3.3.1. Determination of initial plastic strainsInitial plastic strains were determined for the round

bar assuming a generalised plane strain condition, whereaxial strains,ez(r) are constants. It was also assumed thatthe tangential strains,eq(r) were constant. Whenr#q,there was no plastic deformation and axial and tangentialstrains were found using residual stresses atr=0,together with Hooke’s law

5eez(0)=

1E

(c−2va)

eeq(0)=1E

[a−v(a+c)]

(14)


where the constantsa and c are given in Eq. (19).The plastic strains are

5epz=eez(0)−

1E

[sRz−v(sR

q+sRr )]

epq=eeq(0)−1E

[sRq−v(sR

z+sRr )]

epr =−(epz+epq)

(15)

3.3.2. Determination of initial position of yieldsurfaces

The positions of first, second and third yield surfaceswere determined from the initial residual stress-state,where

s15sRr , s25sR

q, s35sRz (16)

It has been shown thats2 ands3 are in compressionand s1 is a small value. Considering that the type ofsurface loading from shot blasting was along the radialdirection, the maximum deviatoric stresses,Sm, corre-sponding tosm

eq can be expressed approximately by

Smr =−bSR

r

Smq=−bSR

q

Smz =−bSR

z6 (17)

whereSRr , SR

q andSRz are deviatoric residual stresses and

b is a constant. The relationship between the maximumequivalent stress and the deviatoric components is

!32(Sm

r Smr +Sm

qSmq+Sm

z Smz )5sm

e (18)

Similarly for the residual stress components

!32(SR

r SRr +SR

qSRq+SR

zSRz)5sR

eq (19)

The constantb is found by inserting Eq. (17) into Eqs.(18) and (19), and is given by

b5sm

eq

sReq

(20)

The locations,Xr,L, Xq,L, andXz,L of the centres for allsurfaces for the maximum loading history are given by

5Xr,L

Xq,L

Xz,L65sm

eq−kL

smeq 5

Smr

Smq

Smz6 (L51,2,3) (21)

4. Finite element analysis and results

4.1. Residual stress relaxation under tensile loading

To simulate the behaviour of the round bar specimens,a single row of axisymmetric elements was used anduniform displacement along the axial direction wasapplied. Because there was no reverse yielding duringtensile loading and unloading, the initial state of plasticstrain had little influence on the results from the simula-tions. Therefore, the multilinear hardening was notemployed for this loading case.

To simulate residual stress relaxation three initialvalues of axial residual stress at the surface were chosen,2200 MPa,2300 MPa and2400 MPa. The predictedresidual stress relaxation, from the FE simulations, as afunction of total strain, is shown in Fig. 4(a). It can beseen that relaxation as the tensile strain increases for dif-ferent initial residual stresses has the same trend as theexperimental results. There is an elastic range of appliedstrain where there is no relaxation because the sum ofresidual stress and applied stress is less than yield stress.There also appears to be good agreement with experi-mental results.

4.2. Residual stress relaxation under cyclic loading

In the analysis the two material models, describedearlier were used: linear and multilinear kinematic hard-ening.

4.2.1. Linear kinematic hardeningFor linear kinematic hardening, two initial residual

stress-states at the surface were considered. Case 1assumed that the initial axial residual stress was2300MPa, while for case 2 it was assumed it to be2400MPa. The results obtained from case 2 are shown in Fig.6, with the residual stress relaxation as a function ofnumber of cycles for different half cyclic strain rangesshown. In both cases, the simulations revealed that byusing the linear kinematic model, the relaxation of theresidual stresses occurred in the first cycle. Similarresults were obtained for an initial axial residual stressat the surface of2300 MPa.


Fig. 6. FE prediction of axial residual stress relaxation with differentcyclic strain range using linear and multilinear kinematic hardeningfor an initial surface axial residual stress2400 MPa.

4.2.2. Multilinear kinematic hardeningDifferent initial surface axial residual stresses were

chosen for the simulations, starting from2250 MPadecreasing to2450 MPa. For cyclic loading conditions,half total strain ranges from 1000µε to 3200µε wereapplied. For example, at an applied half total strain rangeof De/2=1000µε, first the axial load was applied to themesh to give a tensile straine=1000 µe, then unloadedand the mesh compressed to give a compressive straine=21000 µε. Finally the mesh was unloaded to zeroaxial load. For more cycles, the loading conditions wererepeated as for the first cycle. This was repeated for upto 7 cycles. Fig. 6 shows the axial residual stress relax-ation for an initial surface axial residual stress of2400MPa for different half-cyclic strain ranges. Fig. 7 showsthe redistribution of the residual stress across the sectionof the bar after the first cycle for several half-cyclicstrain ranges.

The results shown in Fig. 6 indicate that the surfaceaxial residual stress relaxation eventually becomesstabilised. It is assumed that the relaxation of the surfaceresidual stresses is given by

sR,N5sR91B exp(2CN) (22)

whereN is the number of cycles andsR,N, B andC arefitted constants andsR9 is the stabilised residual stressfor N=`. Consequently, the stabilised residual stressesfor each cyclic loading range was found without carryingout more calculations for many more cycles. The stabil-ised residual stresses, using Eq. (22) for different initialsurface residual stresses and cyclic strain ranges, areshown in Fig. 8.

In addition to the results shown in Fig. 8, there are

Fig. 7. Redistribution of axial residual stress in one specimen afterthe first fatigue cycle.

Fig. 8. FE prediction of stabilised surface axial residual stresses fordifferent initial residual stresses using the multilinear kinematic hard-ening model.

two boundary conditions for the relationship between thethree variablessR9, sR and De/2 wheresR is the initialresidual stress. WhenDe/2=0 there is no relaxation andsR9=sR. If sR9=0, thensR9 should be zero for anyDe.


5. Discussion

It is obvious that residual stress can be released bymechanical loading. But how does relaxation take place?For the round bar steel, the yield stress from tensile testresults was about 800 MPa, and the corresponding limitof elastic strain (the strain for onset of yielding) wasabout 3800µε. On the surface of the shot blasted bars,the average axial residual stress was about2380 MPa,and the corresponding residual elastic strain was about21800µε. If we consider that the initial yield stress onthe surface was about 800 MPa, then the surface elasticstrain was about 5200µε. For the tensile tests with sur-face strains from mechanical loading less than 5600µε,we would expect there to be zero or only limited residualstress relaxation. However, the experiments and the FEsimulations reveal, in Fig. 4(a), that the relaxation ofresidual stresses was a non-linear function of the surfacestrain. Furthermore, residual stress relaxation is pre-dicted to occur at applied strains less than the yieldstrain. To explain this, we should consider the Bausch-inger effect for the material. The effect is shown in Fig.2. The surface material had been subjected to compli-cated strain histories during the manufacturing process.The surface stress-state was in biaxial compression,which could be at the yield surface. When tensile loadingwas applied, the yield stress would be much less than800 MPa and the limit of elastic strain could be lessthan 3800µε. These values depend on the degree of theBauschinger effect for the material. Fig. 4(a) reveals thatthere was nonlinear relaxation of the residual stressesunder the tensile loading. There is a good comparisonbetween experimental results and results from FE analy-sis for tensile loading.

For cyclic mechanical loading, reverse yielding wasthe most likely because of the presence of the compress-ive biaxial residual stresses. The onset of yielding islower for compressive loading than under tensile loadingbecause of the Bauschinger effect. From the results ofthe FE analysis, shown in Fig. 4(b), it can be seen thatthere was more residual stresses relaxation under cyclictension–compression loading than under tensile loading.Therefore under cyclic loading there was more plasticdeformation and this resulted in greater residual stressrelaxation than for tensile loading alone. The FE resultsshow that residual stresses completely relax when thehalf-cyclic strain range is larger than the elastic limit.Furthermore, even small cyclic strain ranges lead tosome degree of residual stress relaxation as shown inFig. 8.

The multilinear kinematic hardening model is animproved, but more complex, model for the cyclic stressstrain behaviour compared to linear kinematic hardening,especially for small plastic strains. Fig. 6 shows resultsfrom the FE analysis for the same conditions and initialresidual stress distribution using the two different hard-

ening models. The results for the linear kinematic hard-ening model show that, compared to those from themultilinear kinematic hardening model, the linear modelalways leads to residual stress relaxation only in the firstcycle. In contrast, the multilinear model demonstratescontinuous relaxation of the residual stresses withincreasing number of cycles. The extent of relaxation isalso greater using the multilinear model compared to thelinear model.

From the results of the FE analyses for cyclic loading,shown in Fig. 6, it can be seen that, as the cyclic strainrange increased, there was more relaxation of theresidual stresses. When the strain range reaches the elas-tic strain range based on the initial tensile yield stress,residual stresses will be eliminated. Even for some strainranges that are regarded as within the elastic range acertain degree of relaxation will occur because the sur-face had already undergone plastic deformation due toshot blasting. This relaxation only takes place during thefirst cycle.

From the centre hole drilling results shown in Fig. 4,it can be seen that the residual stresses in specimens thathad undergone large strain cyclic loading had totallyrelaxed. This is in general agreement with the FE analy-sis. However, the X-ray measurements only indicatedthat about 40–60% relaxation of residual stresses undercyclic loading, shown in Fig. 4, had occurred. The reasonfor this requires further investigation. One explanationcould be associated with surface roughness. Forexample, the residual stresses at the peaks of the profilesmay not relax in the same way as those in the bulk of thematerial. The bulk material experienced complete stressrelaxation as shown in the results from the centre-holemeasurements. In contrast, there is good agreementbetween X-ray measurement of residual stress relaxationfor tensile and finite element predictions.

6. Conclusions

1. Residual stresses, in forged and shot-blasted steelbars, were measured using X-ray and centre-holedrilling methods. These measurements, together witha statistical distribution of surface residual stresses,were used to provide an initial condition for finiteelement simulations.

2. Residual stress relaxation following simple tensileloading and tension–compression cyclic loading wasmeasured. However, the different measurementmethods produced different results. For large plasticstrains, centre-hole measurements revealed completerelaxation. X-ray measurements indicated that surfaceresidual stresses were retained.

3. The FE analysis, together with a linear kinematic


hardening model, provided predictions that were ingood agreement with the experimental results for ten-sile loading.

4. Under tension–compression cyclic loading, the FEanalysis predicted, using a linear kinematic model ora multilinear kinematic material model, more rapidrelaxation of the residual stresses than measured usingthe X-ray method.

5. Even for low cyclic strain ranges, regarded as in theelastic range, residual stress relaxation took place.This is because the surface had undergone prior plas-tic deformation arising from shot-blasting.

Acknowledgements

For this work, we are grateful for the financial supportby the UK Government through an EPSRC Grant No.GR/H 46800 and with Rover Group plc and John Stokesand Sons Ltd as collaborating Industrial Partners. G.H.Farrahi who is on sabbatical leave would like to thankSharif University of Technology for their support.

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experimental measurement and finite element simulation of the interaction between residual stresses...

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