obtaining multiaxial residual stress distributions from limited measurements

Materials Science and Engineering A303 (2001) 281–291

Obtaining multiaxial residual stress distributions from limitedmeasurements

D.J. Smith a,*, G.H. Farrahi b, W.X. Zhu c, C.A. McMahon a

a Department of Mechanical Engineering, Uni6ersity of Bristol, Bristol BS8 1TR, UKb School of Mechanical Engineering, Sharif Uni6ersity of Technology, Tehran, Iran

c BAe Systems, Bristol, UK

Received 7 February 2000; received in revised form 20 June 2000

Abstract

Knowledge of the complete multiaxial residual stress distribution in engineering components is essential for assessing theirintegrity. Often, however, only limited measurements are made. Here, an analysis is presented for determining the multiaxialdistribution from a limited set of measurements. These measurements are used with an assumed plastic strain distribution.Residual stress measurements were made on hot forged and shot blasted steel bars using X-ray and neutron diffraction techniques.The residual stresses were measured on the surface and at selected interior points of the specimens. The predicted multiaxialdistributions were compared with experimental measurements obtained using an incremental hole drilling technique developed forcomponents with curved surfaces. Agreement between measurements and predictions was found only if the elastic results fromhole-drilling were corrected for plastic strain. © 2001 Elsevier Science B.V. All rights reserved.

Keywords: Multiaxial residual stresses; X-Ray diffraction; Neutron diffraction; Incremental hole-drilling

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1. Introduction

Residual stresses can significantly affect the deforma-tion and failure of materials. All manufacturing pro-cesses introduce states of residual stress to differingdegrees. These stresses can have a beneficial or detri-mental effect depending on the direction of loading andconsequently knowledge of the state of residual stress isimportant.

For the manufacture of steel parts for the automotiveindustry, hot forging is a common manufacturing pro-cess particularly for the production of large quantitiesof engineering components. Internal and surface resid-ual stresses are developed in forged components [1,2] asa result of various aspects of the manufacturing pro-cess, including subsequent cooling, and heat treatment.Shot blasting is then used to clean the surface of theforged components. In general, forging and thermalloading introduce long-range residual stresses and sur-

face treatments such as shot blasting introduce short-range or near-surface residual stresses. Many of thecomponents are then used as safety critical parts sub-jected to fatigue loading. An accurate distribution ofthe residual stresses in the component is then requiredto investigate the interaction between residual stressesand fatigue cyclic loading.

Experiments have often been the best way to studyresidual stress in forged components. Bashun et al. [3]and Appleton et al. [4] studied residual stresses inforged components using X-ray diffraction method andboth studies indicate that the forging process introducesresidual stresses in components. In this paper, residualstresses measured in hot forged steel bars are presented.Three measurement methods were used: X-ray diffrac-tion, neutron diffraction and incremental centre holedrilling method.

X-ray diffraction is one of the most popular tech-niques for residual stress measurement, [5]. It is non-de-structive when it is used to evaluate only surfaceresidual stress. The technique becomes destructive whenlayer removal is used for determining in-depth residualstress distribution. Neutrons have greater penetrating

* Corresponding author. Tel.: +44-117-9288212; fax: +44-117-9294423.

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

0921-5093/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved.PII: S0921-5093(00)01837-2

D.J. Smith et al. / Materials Science and Engineering A303 (2001) 281–291282

power than X-rays. Therefore, measurements usingneutrons are not restricted to surface locations, [6].However, the disadvantages of such a technique are thehigh cost and low spatial resolution. Using experimen-tal measurement techniques such as neutron and X-raydiffraction, we can usually obtain limited informationabout residual stresses in a sample. Finally, the incre-mental centre hole drilling technique is employed toobtain the residual stress distributions for limiteddepths below the surface.

However, the complete multiaxial distribution ofresidual stresses in the specimen is usually required. Ananalysis is presented in this paper that permits thecomplete residual stress distribution to be determinedusing a limited set of measurements. A general solutionbased on unknown plastic strains is derived. The non-uniform plastic deformation field, which is generated inthe interior of the specimen by forging and thermalloading and at the surface by surface treatment, is thekey factor in determining the distribution of the resid-ual stresses. This plastic deformation field, related tothe strain history during the forging process and subse-quent heat and surface treatment is difficult, even im-possible, to completely determine. However,relationships between the residual stresses, elastic strainand plastic strains can be used to estimate residualstresses from limited measurements.

The experiments were carried out using hot forgedcylindrical steel bars, shown in Fig. 1. First, the basicprinciples of the three measurement techniques aredescribed briefly. Then the materials and the experi-mental results are presented. This is followed by ananalysis for determining the complete multiaxial resid-ual stress distribution from the limited measurements.Finally, the distributions are compared to the nearsurface residual stresses obtained from incremental cen-tre-hole measurements.

2. Residual stress measurement method

2.1. X-ray diffraction

The X-ray method measures strains in the surfacelayers of a material. The basic principle for obtainingthe strain is simple [7,8]: the interplanar spacings of aspecific form of planes are obtained from grains atdifferent orientations to the surface normal. This isachieved by tilting and rotating the specimen withrespect to the incident beam. These spacings are thenconverted into strains using:

ofc= (dfc−d0)/d0= −1/2(cotg u)D2u (1)

where d0 is the interplanar spacing for a stress freematerial, and dfc is the spacing in the direction fc,defined by the angles f and c. The spacing dfc in Eq.(1) is obtained from the angular value of the peakcorresponding to the diffraction planes using Bragg’sequation:

2d sin u=l (2)

where l is the wavelength of the radiation. with D2u=2u −2uo, where uo is the angular position of thediffraction for a stress-free material. The strain ofc inthe direction fc relates to the stress sf for the biaxialstate using:

ofc= [(1+n)/E ]sf sin2 c − (6/E)(s1+s2) (3)

or for a multiaxial state:

ofc=�1+n

E�

[(s11f −s33

f )sin2 c−s13f sin 2c+s33

f )

−�n

E�

(s11f +s22

f +s33f ) (4)

where E is Young’s modulus and n is Poisson’s ratio forthe material. The following expression gives the rela-tionship between D2u and sin2 c.

Fig. 1. Hot forged steel bar.

D.J. Smith et al. / Materials Science and Engineering A303 (2001) 281–291 283

D2u= − [(360/p)tan u ]{[(1+n)/E ]sf sin2 c} (5)

In applying the above relations to X-ray diffractionmeasurements, the following assumptions are necessary.1. The crystallites must have a linear elastic mechani-

cal behaviour;2. The polycrystalline material must be macroscopi-

cally isotropic, so the diffracting crystallites musthave a small enough size and no preferentialorientation;

3. The macroscopic state of stresses and strain must behomogenous in the X-ray irradiated volume.

By measuring the value of 2u at different incidentangles c for a fixed angle of f, if a linear relationshipis found for an isotropic material of D2u against sin2 c,then the slope gives the value of the stress sf from Eq.(3) and Eq. (5). It is not then necessary to use Eq. (4).

2.2. Neutron diffraction

Neutron diffraction is a non-destructive technique ofmeasuring residual stresses. In the interior of a speci-men there are six independent components of strain,which require at least six strain measurements in differ-ent directions to completely determine the strain state.Measurements of strains in three orthogonal orienta-tions are sufficient for determining three normal stressesin these three directions. Because of geometric symme-try of the forged samples (Fig. 1) it was assumed thatthe direction of the principal strains and stresses coin-cided with the axial, radial and hoop directions in thesample bars. Further details of the measurement tech-nique are given by Bourke et al. [9], and Smith et al.[10]. The elastic lattice strain o e, in the direction of thescattering vector Q, which bisects the angle between theincoming and detected beam, is given by

o e= −Du cot u (6)

where u is peak angle of the scattering and Du is a shiftof the peak angle corresponding to the lattice spacingchange. To obtain the absolute strain, either the latticespacing or the scattering peak angle of the unstrainedmaterial must be known first.

The stresses in the three directions, axial, radial andtangential, are given by

sr=E

(1+n)(1−2n)[(1−n)o r

e+n(oue +o z

e)]

su=E

(1+n)(1−2n)[(1−n)ou

e +n(o re+o z

e)]

sz=E

(1+n)(1−2n)[(1−n)o z

e+n(oue +o r

e)]

ÂÃÃÃÌÃÃÃÅ

(7)

where o re, ou

e and o ze represent radial, tangential and axial

strains measured by neutron diffraction and sr, su and

sz represent radial, tangential and axial residualstresses, respectively.

2.3. Incremental hole drilling

The hole-drilling method is a semi-destructivemethod used to measure near-surface residual stressesin components. The method consists of drilling a 2-mmdiameter hole to a depth of less than one millimetre.The residual stresses are calculated from the measuredsurface strain change caused by relaxation of the resid-ual stresses. The conventional method of hole-drillingassumes that the stress field does not vary with depth[11,12]. However, non-uniform residual stresses can bedetermined by using an incremental hole-drillingmethod together with finite element (FE) analyses todetermine the incrementally relaxed strains for which anumber of methods are available, [13–16]. These recentdevelopments have been confined to flat specimensurfaces.

To measure the near-surface residual stresses in theround bars an analysis, similar to that used by Schajer[13], was developed for interpreting relaxed strain datafrom the incremental hole drilling method. For ndrilling steps, with h as the corresponding depth of thehole, the stress distribution is assumed to be

s (h)�hi−15h5hi=s i=

ÁÃÍÃÄ

sx, i

sy, i

txy, i

ÂÃÌÃÅ

i=1, ···, n andh0=0hn=H

", (8)

The stresses sx,i, sy,i, txy, i are the in-plane residualstresses.

An integral equation that relates the measured strainso at increment depth hj is given by

o (hj)= %j

i=1

A0 jis i i5 j5n, j=1, ···, n (9)

where

o (hj)=

ÁÃÍÃÄ

oa(hj)ob(hj)oc(hj)

ÂÃÌÃÅ

(10)

and

A0 ji=& hi

hi−1

A. (h,hj)dh. (11)

The strains oa, ob, ocare measured from the surfacestrain rosette at each increment depth. A0 ji and A. (h,hj)are 3×3 matrices. These matrices were determinedusing finite element analysis by Zhu and Smith [17] forincremental hole drilling on the curved surface of 8.2mm diameter forged steel bars. If A0 ji is known and o (hj)is measured, then the unknown residual stresses s i canbe determined by solving the linear algebra Eq. (9).


Fig. 2. Residual stress measurements in forged round bars.

3. Experimental details and results

3.1. Material and specimens

Hot forged round En15R steel bars of composition(wt.%) 0.4 C, 0.17 Si, 1.59 Mn, 0.07 Cr, 0.17 Ni, 0.02Mo, 0.004 S, 0.022 P remainder Fe were used in theexperiments. The forging process used in the produc-tion of these bars simulated the manufacture of wheelsuspension arms for automobile components. Theprocess consists of an initial induction heating of asteel billet to about 1200–1250°C then moulding in ahot forging press, followed by finish pressing. Afterclipping the excess material, the forged bars were al-lowed to cool in air to room temperature. Then thecomponents were heat treated (harden and tempered),followed by shot blasting to clean the surface of ox-ide scale. As will be seen later this process introducedcompressive residual stresses on the surface of thebars.

The diameter of the finished bars was 8.2 mm (Fig.1). Residual stress measurements were made using X-ray and neutron diffraction methods on a set of foursamples. The X-ray measurements were confined tothe surface of the specimens and interior measure-ments made using neutron diffraction. The incremen-tal hole drilling method was then applied on twofurther specimens. The cylindrical co-ordinate systemwas employed for the round bar specimens as shownin Fig. 2.

Subsequent analysis of the results used Young’smodulus E and Poisson’s ratio n of 208 GPa and 0.28respectively. The tensile yield stress and tensilestrength of the material are approximately 700 and740 MPa respectively.

3.2. X-ray diffraction

A Philips horizontal diffractometer with achromium X-ray source (lKa=0.22895 nm; 40 kV, 40mA) was employed to examine the 211 peak (2u=156°). A computer monitored all measurements andanalysed the experimental data.

Before carrying out the principal measurements, theaccuracy and reproducibility of our X-ray measure-ments was checked. The magnitude of residualstresses on a stress free sample at seven random posi-tions was (−30915) MPa. Repeat measurements atone location of a bar showed that the value of theaxial residual stress was (−300915) MPa.

Experimental measurements made on four samplesobtained hoop and axial strains. Residual stresseswere determined using Eq. (5). Fig. 3 shows theseresults in terms of the radius divided by the outerradius of the bar. The mean axial and hoop residualstresses were −287 and −242 MPa, respectively.

3.3. Neutron diffraction

Neutron diffraction measurements were carried outat the Risf neutron diffraction facility in Denmark.The ‘TAS8’ instrument in single detector mode, withan incoming neutron beam at a wavelength of 3 nmwas used for neutron diffraction measurements. Allmeasurements were obtained from the 110 reflectionwith the diffraction angle 2u close to 90°. For radialand tangential strains the sample volume was1×1×10 mm, and for axial strains the volume was 2×2×2 mm.

On a reference sample of the same material thediffraction peak angle 2u for the 110 reflection wasapproximately 95.3°, which was taken as the reference


peak angle 2u0. Strains in three directions, axial, radialand tangential, were measured in the bars. Strain mea-surements were carried out at the centre and 2 mmeither side of the centre ensured that the sampling area(represented by a rhomboid of 2 mm equal sides for theaxial strains) was entirely within the material. These

locations were chosen to avoid possible near surfaceeffects using neutron diffraction. Residual stresses werecalculated from measured strains using Eq. (7). Fig. 3shows the experimental results at the two locations. Atthe centre of the bars, the mean hoop and radialresidual stresses were −25 and −75 MPa respectively.The mean axial residual stress was 119 MPa. At 2 mmeither side of the centre of the bars the mean hoop,radial and axial residual stresses were 25, 19 and 156MPa. Since measurements were taken 2 mm either sideof the centreline of each bar, eight results at r/R=0.5are shown in Fig. 3.

3.4. Incremental centre hole drilling

For the application of the incremental hole drillingmethod the TEA-06-062RK-120 type strain gaugerosette, manufactured by the Measurements GroupInc., was used. The strain gauge was carefully wrappedand bonded around the cylindrical part in the centre ofthe bar. The centre hole diameter was 1.926 mm. Thehole was drilled incrementally using a high-speed airturbine. Ten increments were made with the depth ofincrement being about 0.08 mm. The measured strainsare then converted to residual stresses by inverting Eq.(9). This procedure was carried out using twospecimens.

It was found that near surface results using thiselastic analysis were greater than those measured usingX-ray. Previous studies [18] on the mechanical be-haviour of the En15R steel during cyclic loadingdemonstrated that the steel exhibited a significantBauschinger effect. The surface stress state after shotblasting is close to equi-biaxial compression (see Fig. 3),and close to the compressive yield surface after priorplastic deformation. During incremental hole drilling itcan be shown that complete elastic recovery does notoccur, and further loading in compression can takeplace. In the other words, the increment of equivalentstress (seq) is greater than zero.

Many investigators have noticed the effects of plas-ticity and local yielding around drilled holes, [19,20]. Totake account of the incomplete recovery of elastic strainan approximate analysis was carried out [21] based onanalysis given by Gao [22]. This analysis generates acorrection curve where the initial elastic residual stressis corrected from knowledge of the measured strain.This analysis was applied to each step in the holedrilling. For the analysis, we assume that: (i) the resid-ual stress is uniform in the surface layer; (ii) the mate-rial is uniform and isotropic and stress–plastic straincurve satisfies s=A�n.

The hole drilling operation can be idealised as beingin two steps: first by drilling the hole but keeping theradial residual stress as a pressure on hole surface andsecond by removing the pressure from the hole surface.

Fig. 3. Combined X-ray and neutron measurements of residualstresses.


Fig. 4. Incremental centre hole drilling measurement of residualstresses.

stresses in the specimen is usually required. An analysisis presented that allows a complete residual stress distri-bution to be determined using a limited data set.

4.1. Go6erning equations

The residual stresses in the specimens are producedduring the various manufacturing processes, that createnon-uniform plastic deformation. The plastic deforma-tion is related to the strain history experienced duringmanufacture and it is difficult to determine its magni-tude and distribution. The deformation in the speci-mens will be partly elastic and partly plastic. It isgenerally assumed that total strain tensor oij in a com-ponent can be divided into elastic o ij

e and plastic o ijp

tensors, so that

oij=o ije +o ij

p. (12)

If all residual stresses and strains are independent of zand u, Fig. 2, the round bar can be treated as ageneralised plane strain problem and the equation ofequilibrium is [23]

(sr

(r+

sr−su

r=0. (13)

The boundary conditions include free traction on theouter surface

sr �r=R=0 (14)

where R is the outer radius of the bar. The sum of allforces in the axial direction must be zero&

A

szdA=0 (15)

The area A is pR2. Eq. (15) indicates that the summa-tion must be carried out over the entire cross-sectionalarea of the bar.

The equilibrium equations and the boundary condi-tions (Eqs. (13)–(15)), together with the limited set ofmeasurements and with an assumed residual stress dis-tribution, can provide a solution to the residual stressstate in the bar.

An alternative approach is to include into the analy-sis additional information about the plastic strain dis-tribution. Plastic strain can be measured indirectlyusing X-ray and neutron diffraction methods. This iswill be discussed later. The analysis is extended toinclude compatibility conditions.

For axisymmetric conditions the compatibility equa-tion is:

rdou

dr=or−ou. (16)

The relationship between stresses and strains in cylin-drical co-ordinates is

In the first step, there is no strain change. The strainchange is only caused by removing the radial residualstress as a pressure from the hole surface. The analyti-cal solution for the strain release after a hole drillingwas obtained based on Gao’s [22] work. The solutionincludes a set of equations, which can be solved byusing iterative procedures.

The measured residual stresses modified for plasticityare shown in Fig. 4. Results from two specimens areshown.

4. Determination of residual stress distribution

Using experimental measurement techniques such asneutron diffraction and X-ray diffraction, without de-stroying the sample by layer removal, we can usuallyobtain limited information about residual stresses in asample. However, the complete distribution of residual


or=1E

[sr−n(su+sz)]+o rp

ou=1E

[su−n(sr+sz)]+oup

oz=1E

[sz−n(su+sr)]+o zp

and

o rp+ou

p+o zp=0

ÂÃÃÃÃÃÌÃÃÃÃÃÅ

(17)

By solving Eqs. (13)–(17), each residual stress compo-nent can be determined as a function of the plasticstrains, so that

sr= −E

2(1−n2)� & R

r

o rp−ou

p

r %dr %

−1−2n

R2

& R

0

r %(o rp+ou

p)dr %

+1−2n

r2

& r

0

r %(o rp+ou

p)dr %n

(18)

su= −E

2(1−n2)[2(1−n)ou

p−2no rp]

+E

2(1−n2)� & R

r

o rp−ou

p

r %dr %

−1−2n

R2

& R

0

r %(o rp+ou

p)dr %

+1−2n

r2

& r

0

r %(o rp+ou

p)dr %n

(19)

sz= −E

2(1−n2)[−2(1−n)ou

p−2o rp]

+E

2(1−n2)�

2n& R

r

o rp−ou

p

r %dr %

+2(2−n)

R2

& R

0

r %(o rp+ou

p) dr %n

(20)

where r % is an integral variable.

4.2. Plastic strain distribution

The analysis in the previous sections shows that onemethod for a complete solution for the residual stressdistribution requires information about the plasticstrain distribution. For a given residual stress distribu-tion, a unique plastic strain distribution cannot befound. For example, for zero residual stresses, thesolution of plastic strain distribution could be in theform,

o rp=a1, ou

p=a2, o zp= − (a1+a2) (21)

where a1, and a2 are arbitrary constants. Therefore, thesolution for plastic strain distribution is not unique.

From Eq. (17), it can be seen that the plastic strains areequal to the total strains when there are no residualstresses. Therefore, for the axisymmetric case the plasticstrains satisfy the compatibility Eq. (15), as do the totalstrains.

We can use this non-uniqueness to make a number ofassumptions regarding the nature of the plastic straindistribution. In the following, we consider an interpola-tion technique where limited information about theresidual stresses is known and is used with an assumedplastic strain distribution.

In the round bar specimens the true plastic strainsand their history are very difficult to obtain. For agiven residual stress state the corresponding plasticstrains are not unique. Therefore, we can choose anapproximate plastic strain field. It is very important tochoose a suitable plastic strain distribution that consid-ers both short-range and long-range residual stresses.We assume a third order polynomial equation for theradial distribution of the plastic strains o r

p and oup, given

by

oup=C1r̃

2+C2r̃3, o r

p=C3r̃2+C4r̃

3 (22)

where C1, C2, C3, and C4 are constants to be deter-mined and r̃ is the non-dimensional radius r/R. Theequations are chosen for two reasons: they representthe expected plastic strain distribution and when substi-tuted into Eqs. (18)–(20) they provide analytical solu-tions for the residual stress distribution.

4.3. Interpolation method

After inserting Eq. (17) into Eqs. (14)–(16) andintegrating, the distributions of the residual stressesbecome a function of a number of constants, and canbe expressed as

sr=C1 f11(r̃)+C2 f12(r̃)+C3 f13(r̃)+C4 f14(r̃)su=C1 f21(r̃)+C2 f22(r̃)+C3 f23(r̃)+C4 f24(r̃)sz=C1 f31(r̃)+C2 f32(r̃)+C3 f33(r̃)+C4 f34(r̃)

ÂÃÌÃÅ

(23)

where fij(r̃) are given in Appendix A.At several points at r̃= r̃m, there are measured values

of the residual stresses sr, su and sz. These are denotedas skm where k=1, 2, 3 and represents r, u and z,respectively. Since there are three measured values ofresidual stress at three locations, the least squaresmethod is used to determine the constants Cj with j=1,2, 3 and 4. The total sum of squares of the errors e isexpressed as

e(C1, C2, C3, C4)= %3

k=1

%n

m=1

� %4

j=1

Cj fkj(r̃m)−skm

�2

(24)

where n is the number of measurement points. It can beshown that the minimum of the error is attained atunique values of C1, C2, C3 and C4, which are thesolution of the simultaneous equations,


(e(Cj)(Cj

=0, j=1,2,3,4 (25)

Eq. (25) is a set of simultaneous linear equations for C1,C2, C3 and C4, expressed as

%4

j=1

aijCj=bii, j=1, 2, 3, 4 (26)

where,

aij= %3

k=1

%n

m=1

fki(r̃m)fkj(r̃m)

bi= %3

k=1

%n

m=1

fki(r̃m)skm

ÂÃÌÃÅ

(27)

In the following, we use experimental results ob-tained from X-ray and neutron measurements to deter-mine the constants Cj. The average values of themeasured axial, tangential and radial residual stressesat r̃m=0, 0.5 and 1, are shown in Table 1. In the caseof the radial residual stress at r̃m=0 it was assumedthat this was equal to the measured hoop residualstress. This assumption is discussed later. These averagevalues correspond to values of skm (see Table 1) wherethe number of measurement points n is 3. Eq. (21)together with fij(r̃m) given in Appendix A was used todetermine the constants C1 to C4 where C1= −2.4×10−3, C2=2.8×10−3, C3=3.0×10−3, C4= −5.2×10−3. Using these constants therefore provides acomplete solution for the distribution of the residualstresses.

The distributions of the axial, tangential and radialresidual stresses, obtained from Eq. (23), are shown inFig. 5. Also shown are the results obtained from theincremental centre hole measurements. In general, thelargest residual stresses in the bulk of specimens were inthe axial direction. Finally, Fig. 6 shows the plasticstrain distributions for the hoop and radial directionsobtained from Eq. (22), and illustrates that the largestplastic strain occurs near to the surface of the bar.

Fig. 5. Interpolated residual stresses compared with centre holedrilling results.

Table 1Average of measured axial, hoop and radial residual stresses

Measured residual stress skm Normalised radial distance,r̃=r/R

0 0.5 1

Radial, s1m 019−25

−25Hoop, s2m 25

−242119Axial, s3m 156

−287

5. Discussion

The forging process would not be expected to giverise to axisymmetric deformation since the bars werepressed between two dies. We would expect also thatthis would lead to long-range residual stress


distributions in the centre of the gauge length of thebar that are not precisely axisymmetric. There wasevidence from neutron diffraction measurements for anon-axisymmetric field. For example, at the centre ofthe bar the measured mean hoop and radial residualstresses were −25 and −75 MPa. Our analysis fordetermining the multiaxial residual stress distributionassumes axisymmetry. Consequently, it has beenassumed that the radial residual stress is equal to themeasured hoop stress. A slightly different residualstress distribution at the centre of the bar wouldoccur if it was assumed that the hoop residual stresswas equal to the radial stress.

The shot blasting gives rise to short-range(near-surface) residual stresses that would be expectedto be axisymmetric. The plastic strain distributionsshown in Fig. 6 also illustrate that localised plasticstrain, arising from shot blasting, is confined near tothe surface of the bar. These distributions also revealthat shot blasting introduces localised radialcompressive plastic strains for distances up to 0.4 ofthe normalised radius.

It is observed in Fig. 5 that the residual stressmeasurements using the incremental centre hole aregenerally very similar to the distribution obtainedfrom the interpolation analysis. However, the resultsfrom the incremental hole drilling were also modifiedto take account of partial plastic deformation duringmeasurements. Without correcting for plasticity, themeasured residual stresses would have beensubstantially greater in compression than estimatedfrom the interpolation analysis. Both surfacemeasurements and the determination of the residualstress distribution indicate that this correction forplasticity to the results obtained from incrementalhole drilling is essential.

Errors in the calculated residual stresses using theincremental hole drilling method is also discussedelsewhere [17]. It is demonstrated that the error in theresidual stresses is accumulated for each increment inthe cutting depth. Therefore, more increments do notimply improved results, and measurements arerestricted to depths equal to about one half of thehole diameter.

A theoretical analysis based on an assumedthermo-mechanical history of the specimens togetherwith a least squares method, has provided a means ofdetermining the multiaxial residual stress distributionsfrom limited sets of measurements. The analysis isonly considered for the axisymmetric case. This is aspecial case since the compatibility condition for theaxisymmetric case leads to compatibility of the plasticstrains. This is not the case for all conditions sinceonly elastic strains are necessarily compatible.However, it may be possible to apply this method toother cases. The general plane strain condition oz isconstant and can be modified to oz=oz(r), which maybe pertinent to different manufacturing process, suchas the cold-drawing process. The assumptionregarding the plastic strain distribution is also flexiblesince for a given residual stress distribution it ispossible to have a number of plastic strain histories.Therefore, the functional forms of the radial andhoop plastic strains, Eq. (22), are somewhat arbitrary.Nevertheless, they are chosen because knowledge ofthe prior loading history (e.g. forging and surfaceshot blasting) provides an indication of the expectedplastic strain distribution.

Another feature is that the analysis ensures that theerrors are minimised. Clearly, if the residual stressesappear to be more localised, such that the short-rangeresidual stresses dominate, the functional form of theplastic strains is no longer arbitrary since the plasticstrain should be considered to be localised. Eq. (22)should therefore reflect the extent of the plastic straindistribution.

Although we have assumed the distributions of theplastic strains, the X-ray and neutron diffractionmethods can be used to measure indirectly the plasticstrain. Farrahi and Lebrun [24] show that a measureof the hardness and the extent of plastic strain can beinterpreted from the breadth at half-maximumintensity of the diffracted beam. Smith and Webster[25] also demonstrated that the change in peak widthfrom neutron diffraction measurements is related tochanges in plastic strain. However, it was found thatthe measurements were insensitive to the direction ofprior straining. Also the measurement method wasonly sensitive to relatively large changes inplastic strain. Fig. 6 shows that the maximumpredicted plastic strains were about 0.25%,whereas measurements by Smith and WebsterFig. 6. Predicted plastic strain distribution.


examined peak width changes for plastic strains greaterthan 1%.

6. Conclusions

A theoretical analysis has been developed to estimatethe multiaxial residual stress distributions in forged andshot blast bars, using limited neutron and X-ray exper-imental results.

X-ray surface measurements have shown that resid-ual stresses in hot forged and shot blast round barswere in biaxial compression.

The axial residual stresses measured using the neu-tron diffraction method in the interior of the bars werepredominantly tensile, although there was some scatterfrom bar to bar.

Measured residual stresses using an incremental holedrilling technique, to a depth of about 1 mm, anddeveloped for components with curved surfaces werefound to agree with the interpolated residual stresses.

Agreement with the predicted residual stress distribu-tion was only found after correcting the elastic resultsfrom incremental hole drilling for plastic strain.

The results and analysis provided in this paperdemonstrate that limited residual stress measurements,together with assumed plastic strain distributions,provide a method of determining complete multiaxialresidual stress distributions.

Acknowledgements

For this work we are grateful for the financial sup-port by the UK Government through a EPSRC GrantNo. GR/H 46 800 and with Rover Group plc and JohnStokes and Sons Ltd as collaborating Industrial Part-ners. The neutron diffraction work was carried at RISf

Denmark and supported by the CEC LIP. Programmeadvice in conducting the neutron diffraction measure-ments was provided by Torben Lorentzen at RISf.G.H. Farrahi who is on sabbatical leave would like tothank Sharif University of Technology for theirsupport.

Appendix A. Functional forms of fij(r̃) in InterpolationMethod

The plastic strains are assumed to be:

oup =C1r̃2+C2r̃3, o r

p=C3r̃2+C4r̃3, (A1)

where C1, C2, C3, and C4 are constants to be deter-mined. After inserting Eq. (A1) into Eqs. (14)–(16), theresidual stress distribution can be obtained as a func-tion of a number of constants,

sr=C1 f11(r̃)+C2 f12(r̃)+C3 f13(r̃)+C4 f14(r̃)su=C1 f21(r̃)+C2 f22(r̃)+C3 f23(r̃)+C4 f24(r̃)sz=C1 f31(r̃)+C2 f32(r̃)+C3 f33(r̃)+C4 f34(r̃)

ÂÃÌÃÅ

(A2)

f11(r)=E

2(1−62)3−2n

4(1− r̃2) (A3)

f12(r)=E

2(1−62)8−6n

15(1− r̃3) (A4)

f13(r)= −E

2(1−62)1+2n

4(1− r̃2) (A5)

f14(r)= −E

2(1−62)2+6n

15(1− r̃3) (A6)

f21(r)=E

2(1−62)3−2n

4(1−3r̃2) (A7)

f22(r)=E

2(1−62)8−6n

15(1−4r̃3) (A8)

f23(r)= −E

2(1−62)1+2n

4(1−3r̃2) (A9)

f24(r)= −E

2(1−62)2+6n

15(1−4r̃3)f14(r)

= −E

2(1−62)2+6n

15(1− r̃3) (A10)

f31(r)= −E

2(1−62)2−3n

2(1−2r̃2) (A11)

f32(r)= −E

2(1−62)6−86

15(2−5r̃3) (A12)

f33(r)= −E

2(1−62)2+6

2(1−2r̃2) (A13)

f34(r)= −E

2(1−62)6+26

15(2−5r̃3) (A14)

References

[1] M.T. Watkins, Metal Forming, Oxford University Press, 1975.[2] R. Myllymaki, Residual Stress in Design, American Society for

Metals, Metals Park, OH, 1987, pp. 137–141.[3] T.V. Bashun, L.A. Vasileva, T.P. Urban, A.M. Shterenzon,

Metal. Sci. Heat Treatment 30 (1986) 117–119.[4] E. Appleton, C.M. Day, I.M. Notter, J.R. Moon, J. Mech.

Working Tech. 15 (1987) 375–384.[5] J. Lu, D. Retraint, J. Strain. Anal. 33 (1998) 127–136.[6] G.A. Webster, A.N. Ezeilo, Physica B. 234–236 (1997) 949–955.[7] G. Maeder, Chem. Scripta 26A (1986) 235–247.[8] I.C. Noyan, J.B. Cohen, Residual Stress Measurement by Dif-

fraction and Interpretation, Springer, Berlin, 1987.[9] M.A.M. Bourke, D.J. Smith, G.A. Webster, P.J. Webster, Pro-

ceedings of the 9th International Conference on ExperimentalMechanics, 1990, pp. 1196–1206.

[10] D.J. Smith, R.H. Leggatt, G.A. Webster, H.J. Macgillivray, P.J.Webster, G. Mills, J. Strain. Anal. 23 (1988) 201–211.

[11] J. Mathar, Trans. ASME 56 (1934) 249–254.


[12] N.J. Rendler, I. Vigness, Exp. Mech. 6 (1966) 577–586.[13] G.S. Schajer, J. Eng. Mater. Technol. Trans. ASME 110 (1988)

338–343.[14] W.E. Nickola, Proceedings of 1986 SEM Spring Conference on

Experimental Mechanics, Brookfield Centre, Society for Experi-mental Mechanics, Connecticut, 1986.

[15] G.S. Schajer, J. Eng. Mater. Technol. Trans. ASME 103 (1981)157–163.

[16] M. Bijak-Zochowski, VDI-Berichte 313 (1978) 469–476.[17] W. Zhu, D.J. Smith, in: S. Gomes, et al. (Eds.), Recent Advances

in Experimental Mechanics, Balkema, Rotterdam, 1994, pp.777–782.

[18] G. Shatil, Ph.D. Thesis, Department of Mechical Engineering,

University of Bristol, 1990.[19] W.E. Nickola, Proceedings of 1984 SEM Spring Conference on

Experimental Mechanics, Society for Experimental Mechanics,Brookfield Centre, Connecticut, 1984, pp. 126–136.

[20] M. Beghini, L. Bertini, P. Raffaelli, J. Strain. Anal. 30 (1995)227–233.

[21] W. Zhu, Ph. D Thesis, University of Bristol, 1996.[22] X.L. Gao, Int. J. Press. Ves. Piping. 52 (1992) 129–144.[23] L.D. Landau, E.M. Lifschitz, Theory of Elasticity, Pergamon,

Oxford, 1986.[24] G.H. Farrahi, J.L. Lebrun, Int. J. Eng. Islamic Rep. Iran. 8

(1995) 159–167.[25] D.J. Smith, G.A. Webster, J. Strain Anal. 32 (1997) 37–46.

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obtaining multiaxial residual stress distributions from limited measurements

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