Elasto-Visco-plastic Analysis of Welding Residual Stress

Elasto-visco-plastic analysis of weldingresidual stress

X. L. Qin and P. Michaleris*

In the past three decades, extensive research has focussed on the application of numerical methods

for the computation of residual stress. Most commonly, the simulations involved performing weakly

coupled thermal mechanical finite element analyses in Lagrangian reference frames assuming rate

independent elasto-plastic material response. Nearly all approaches assumed rate independent

elasto-plastic material response, which is most appropriate at low to moderately elevated

temperatures. At, the high temperatures near the fusion zone, the material response becomes

rate dependent and an elasto-visco-plastic model would be more suitable. In 1989, Brown et al. (Int.

J. Plast., 1989, 5, 95130) proposed a rate dependent constitutive equation (commonly known as

Anands model) to describe the plastic evolution of metals at high temperatures. The objective of this

work is twofold: evaluate the suitability of Anands elasto-visco-plastic model in computing welding

residual stress and investigate the feasibility of an Eulerian implementation of Anands model in

modelling welding residual stress. Such an implementation has the potential to reduce

computational cost in modelling welding processes, since it is a steady state analysis as compared

to the common time incremental Lagrangian analyses. An Eulerian reference frame is also more

advantageous in modelling processes with large deformation such as friction stir welding, rolling and

extrusion since excessive mesh distortion and re-meshing are no issues as the case of Lagrangian

models (Int. J. Mater. Form., 2008, 1, 12871290).

Keywords: Elasto-plastic, Elasto-visco-plastic, Eulerian, Lagrangian, Mixed finite element analysis, Anand, Residual stress, Gas metal arc welding

IntroductionIn thermomechanical processes involving moving heatsources, such as welding, laser forming and cladding,large tensile stresses remain near the pass of the torch,balanced by lower compressive residual stresses else-where. Tensile residual stresses may reduce the perfor-mance or cause fatigue and brittle failure ofmanufactured structures. Compressive residual stressesmay cause excessive buckling distortion. Therefore, it isimportant to know the distribution of residual stress ofwelded structures in detail.In the past three decades, welding residual stress have

been extensively studied by both experimental andnumerical methods. Several experimental methods havebeen developed to measure residual stresses. Theyinclude non-destructive methods, such as X-ray diffrac-tion,1 neutron diffraction,2 and destructive methods, forexample slicing,3 boring and hole drilling.4 Numericalmethods for the estimation of residual stress haveinvolved performing weakly coupled thermal mechanicalfinite element methods.512 Nearly all researchers haveassumed rate independent elasto-plastic material

response, which is most appropriate at low to moder-ately elevated temperatures. At high temperatures, thematerial response becomes rate dependent and an elasto-visco-plastic model would be more suitable. Argyriset al.7 were among the first researchers to use a ratedependent plasticity model in residual stress computa-tion. Lindgren13 states that creep effects, which are timedependent, do not affect residual simulations becauseduring welding the material experiences creep tempera-tures for a very short time. Bergheau et al.14 demon-strate that consideration of rate dependent plasticitydoes not effect the computed residual stress, but it has asignificant effect on the computed distortion.

Recenlty significant effort has been devoted intomodelling friction stir welding (FSW). When aLagrangian reference frame, excessive mesh distortionis occurring, therefore, some investigators have consideronly the thermal loading and pressure loading from thetool.1517 To resolve mesh distortion adaptive re-meshing18 and explicit arbitrary Eulerian Lagrangian(ALE) methods have also been investigated.19 However,numerical errors have been reported due to re-meshing.20 When Eulerian models have been used,typically, visco-plastic response has been assumed whichresults into no residual stress.21 A two step visco-plasticfollowed by an elasto-plastic analysis along the stream-lines is proposed in Refs. 22 and 23.

Department of Mechanical and Nuclear Engineering, Pennsylvania StateUniversity, University Park, PA 16802, USA

*Corresponding author, email [email protected]

2009 Institute of Materials, Minerals and MiningPublished by Maney on behalf of the InstituteReceived 1 October 2008; accepted 19 May 2009DOI 10.1179/136217109X456988 Science and Technology of Welding and Joining 2009 VOL 14 NO 7 606

Typically, the residual stress computations are per-formed on Lagrangian reference frames, as illustrated inFig. 1a, where the reference frame is stationary andthe welding torch travels in time. With the exception ofthe work of Brown et al.,10 most investigors assume thatthe deformation has very small effect on the temperaturedistribution. Therefore a heat transfer analysis is per-formed first, followed by a mechanical analysis using thetemperature results as a thermal loading. In Lagrangianreference frames both thermal and mechanical analysesare performed in a time incremental scheme to model thetraveling welding torch. For large three-dimensionalmodels hundreds or thousand of time increments24 maybe required, resulting in a very high computational cost.In 1989, Stuart et al.25 proposed a rate dependent

constitutive equation to describe the plastic evolution ofmetals at high temperatures. The model is commonlyknown as Anands model and unlike the rate independentelasto-plastic model it does not exhibit a sharp transitionfrom elastic to plastic response. In Anands model, bothelastic and plastic deformation simultaneously evolve atall stress levels. At low stresses compared to an internalvariable analogous to the yield stress in elasto-plasticity,the deformation is mostly elastic. As the stress increasesthe rate of plastic deformation increases following a sinhcurve of the ratio of equivalent stress to internal variable.Weber and Anand also developed a radial return iterativemethod suitable for a time incremental Lagrangianimplementation of their model.26,27

Nguyen et al.28 proposed that several industrial metalforming processes, such as welding, rolling and extru-sion can be treated as a quasi-steady state process in anEulerian frame. As shown in Fig. 1b, an Eulerian frameis attached to the torch and the material is entering theanalysis domain through the inlet surface. Applicationof an Eulerian reference frame in modelling weldingresidual stress has the potential to reduce computationalcost in modelling arc welding processes, since it is asteady state analysis as compared to the common timeincremental Lagrangian analyses. An Eulerian referenceframe is also more advantageous in modelling materialmovement in friction stir welding processes, sinceexcessive mesh distortion and re-meshing are no issuesas the case of Lagrangian models.20

Shanghvi and Michaleris29 developed a displacemntbased Eulerian elasto-plastic model for thermo-mechanical processes. The model was used to compute

both distortion and residual stress in laser forming.However, convergence was not stable in modellingwelding. Yu and Thompson30 proposed an Eulerianmethod using the rate of equilibrium equation which cansolve pure elastic cases. However, the model does notconsider the plastic material flow during active plasti-city. Maniatty et al.31 developed an elasto-visco-plasticEulerian method using Anands model to solve largestrain rate problems, such as rolling and extrusion.However, this model does not handle low strain rateproblems such as arc welding, and can not computeresidual stress. Recently, Qin and Michaleris32 devel-oped a Galerkin Eulerian formulation with fourunknown fields (velocity, stress, deformation gradientand internal variable) to predict residual stress of elasto-visco-plastic materials using Anands model. The modelhas been demonstrated in material rolling applications.

The objective of this work is twofold: to evaluate thesuitability of Anands elasto-visco-plastic model incomputing welding residual stress and to investigatethe feasibility of an Eulerian implementation of Anandsmodel in modelling welding residual stress. A gas metalarc welding (GMAW) case with available experimentalresidual stress measurements is used as test case. Toaddress the first objective, a Lagrangian implementationof Anands elasto-visco-plastic model is performed andthe computed residual stress are compared againstresults from the conventional rate independent elasto-plastic model implemented in the same Lagrangianreference frame. To address the second objective, theEulerian implementation of Anands model as devel-oped in Ref. 32 is applied to the same GMAW test caseand computed residual stress are compared againstresults computed by the Lagrangian models of the firstobjective. Implementation of the Anands model inmodelling friction stir welding is the focuss of currentresearch. Preliminary results are available in Ref. 33.Detailed results will be available in future publications.

Lagrangian analysisIn this section, the governing equations for thermal andmechanical analyses using both elasto-plastic and elasto-visco-plastic constitutive models for a Lagrangianimplementation are briefly summarised. Detail presenta-tions are available in Refs. 26 and 34.

Thermal analysisThe thermal analysis in a Lagrangian frame isperformed in an incremental scheme. For a stationaryreference frame r fixed to the material at time t, thegoverning equation for transient heat transfer analysis is

rCpdT

dt(r,t)~{

Lqi(r,t)Lri

zQt(r,t) (1)

where r is the density of the flowing body, Cp is thespecific heat capacity, T is the temperature, Qt is theinternal heat generation rate and qi is the heat flux vectorwhich can be obtained by

qi~{kLTLxi

(2)

where k is the temperature dependent thermalconductivity.

1 a Lagrangian and b Eulerian reference frame

Qin and Michaleris Elasto-visco-plastic analysis of welding residual stress

Science and Technology of Welding and Joining 2009 VOL 14 NO 7 607

The initial temperature field is given by

T~T0 in the entire volume V (3)

where T0 is the prescribed initial temperature.The boundary conditions applied on the surface are

T(r,t)~_T(r,t) on the surface ATr (4)

qp(r,t)~_qp(r,t) on the surface Aqr (5)

where, T and qp represent the prescribed temperatureand temperature dependent surface flux respectively.The surface flux qp is evaluated by the projection of theheat flux qi, normal to the surface

qp(r,t)~qi(r,t):ni(r) on any surface A (6)

where n is the unit outward normal to the surface A.

Mechanical analysisFinite element analysis of manufacturing processes inLagrangian reference frames has typically used eitherelasto-plastic34 or elasto-viscoplastic constitutive mod-els.26,27 Both sets of the formulations are presented inthis section for comparison purposes.For each time increment, the following equilibrium

equation is solved

LsijLxizbj~0 in Vr (7)

where s is the stress, and b is the body force. Theboundary conditions are

ui(r,t)~_ui(r,t) on surface A

ur (8)

sij(r,t):ni(r,t)~_tj(r,t) on surface A

tr (9)

where u(r, t) are the prescribed displacements on surface

Aur , t are the prescribed tractions on surface Atr, and n is

the unit outward normal to the surface Atr.

Elasto-plastic model

The stress strain relationship for the elasto-plastic model is

sij~Cijkl(Ekl{Epkl{E

tkl) (10)

where Cijkl is the fourth order elasticity tensor, Eij, Epij and

Etkl are the total, plastic and thermal strain res-

pectively. Using associative J2 plasticity,35 the yield

function fyeild and the evolution equation for equivalentplastic strain ~ep are

f yield~~s{sY(~ep,T) (11)

:~ep~(2=3)1=2

:Epij

(12)where ~s is the Mises stress. Active yielding occurs whenfyield50. The evolution equation for active yielding is

:f yield~0 (13)

Anands elasto-viscoplastic model

For Anands rate-dependent model, the stress evolutionequation is

s+ij~Cijkl(Dkl{Dpkl{D

tkl) (14)

where Dij, Dpij and D

tij are the rate of total, plastic and

thermal strain tensor respectively. The Jaumann deriva-

tive of Cauchy stress s+ij , rate of strain tensor Dij, and

rate of plastic strain tensor Dpij can be derived as

s+ij~:sij{WikskjzsikWkj (15)

Dij~1

2(LijzLji) (16)

Dpij~(3=2)

1=2:~epNij (17)

Nij is the direction of plastic flow tensor derived by

Nij~(3=2)1=2(sij=~s ) (18)

where sij is the deviatoric stress and ~s is the Missesstress.The velocity gradient tensor Lij, spin tensor Wij and

equivalent plastic strain rate _eep are

Lij~LviLxj

(19)

Wij~1

2(Lij{Lji) (20)

:~ep~f(~s,s) (21)

The evolution equation for the internal variable s inequation (21) can be expressed as

:s~g(~s,s) (22)

Functions f(~s,s) and g(~s,s) in equations (21) and (22) areconstitutive functions, which are defined in the sectionon Constitutive law and material properties.

Eulerian analysisThe Eulerian expressions of the quasi-steady statethermal and elasto-viscoplastic mechanical analysis aresummarised in this section. Detailed explanations ofthese equations are presented in Refs. 32 and 36.

Thermal analysisFor a quasi-steady state problem, the transient heattransfer analysis of equation (1) reduces to a boundaryvalue problem in an Eulerian frame3740

rCpviLTLxi~{

LqiLrizQt (23)

along with strongly enforced boundary condition

T~T0 on the inlet surface (24)

Mechanical analysisA coupled four-field (velocity v, stress s, deformationgradient F and internal variable s) Eulerian formula-tion32 using Anands model is used to transform theelasto-viscoplastic initial boundary value problem of thesection on Anands elasto-viscoplastic model to a staticboundary value problem. A Galerking (mixed) formula-tion is used rather than the streamline integrationmethod to compute the strain history dependent terms(see Ref. 41). The governing equations of this methodare:



(i) rate equilibrium equation

d

dt

LsijLxi

~

LPijLxi~0 (25)

(ii) strain integration equation

:Fij~vk

LFijLxk~LikFkj (26)

(iii) internal variable evolution equation

:s~vk

LsLxk~g(~s,s) (27)

(iv) stress strain relationship

:sij~vk

LsijLxk~Cijkl

:Eeklz

:CijklE

ekl%Cijkl

:Eekl (28)

In equation (28) above, the rate of elasticity tensor:Cijkl

is assumed to be negligible compared to the other terms.This simplification relaxes the need to compute theelastic strain Eekl which can be computed by using aGalerking method to compute the visco-plastic portionof the deformation gradient and then the elastic portionof the deformation gradient as shown in Ref. 33.However, as shown in the section on Evaluation ofmechanical analysis in Eulerian frame, the computedresidual stress with constant elastic properties is com-parable to that with temperature dependent properties.Therefore, the

:Cijkl term was omitted in equation (28)

to minimise any additional computational overhead.Finally, it is also noted that the omission of the

:Cijkl

term in equation (28) does not result in a pure visco-plastic model since the strain evolution is accounted forwith equation (26) and the plastic evolution withequation (27).

The rate equilibrium equation (25) is presented byThompson and Yu.30 sij in equation (25) is the Cauchystress tensor and Pij can be defined by

Pij:vkLsijLxk{

LviLxk

skjzLvkLxk

sij (29)

The tensor Fij in equation (26) is the deformationgradient tensor and the velocity gradient tensor Lij isdefined in equation (19). The internal variable evolution

equation (27) is proposed in Ref. 26. In equation (28),:Eekl is the rate of elastic strain tensor:

The corresponding boundary conditions are

vi~vpi on LBv (30)

ni:Pij~:tpj on LBP (31)

sij~spij on LBs (32)

Fij~Fpij on LBF (33)

s~sp on LBs (34)

where vpi , s

pij, F

pij , s^

p and:tpi are the velocity, stress,

deformation gradient, internal variable and rate oftraction specified on the boundary respectively; ni isthe unit outward normal vector on the boundary B. Formore detail of this method, please refer to Ref. 32.

Gas metal arc welding modelA single pass GMAW process, which has the samewelding conditions as the experimental weld in Ref. 42,is modeled under both Eulerian and Lagrangian framesto verify the usability and accuracy of the Euleriananalysis. Two 2869 in (711?26228?6 mm) HSLA-65steel plates with 0?25 in (6?35 mm) thickness are weldedtogether, as shown in Fig. 2. The moving velocity of thetorch is 2?54 mm s21 along the weld centreline and theheat input of the torch is 28?6 KJ in21. A two-dimensional plane strain model is used for bothLagrangian and Eulerian analyses. To eliminate theend effects, the Lagrangian model is longer (L51300 mm) than the actual plates. The Eulerian modelis even longer (L53000 mm) to allow for sufficientcooling. Figures 3 and 4 show the Lagrangian andEulerian model, respectively. The Lagrangian GMAWmodel is discretised into 1500 four node quadrilateralelements with biased smaller element size at the weldingregion and the same element size along the weldingdirection. The Eulerian GMAWmodel is discretised into1776 four node quadrilateral elements with refined mesharound the welding torch.

The heat flux distribution Qt of the electric arc ismodeled as a Gaussian distribution10

2 Geometry of measured plate3 Gas metal arc welding model (Lagrangian)



Qt~3gQwpR2

e{3xR 2z yR 2

(35)

where g is the torch efficiency, Qw is the input power, Ris the torch radius, and x and y are the distances of apoint from the centre of the torch. The radius R is10 mm and the torch efficiency g is taken as 0?8. A freeair convection of Fig. 5 is applied on the entire model dosimulate air cooling.

Four cases of GMAW are simulated in this section, aslisted in Table 1. Case 1 is the traditional Lagrangiananalysis using an elasto-plastic model, which has beenvery widely used and has been shown to effectivelypredict residual stress. In case 2, the Anands elasto-visco-plastic model is implemented in a Lagrangianframe and compared to the elasto-plastic model toevaluate the usability of the model at low strain rates(,1023 s21). Because the Eulerian analysis assumes thatthe elastic material properties do not change withtemperature, case 3 is a Lagrangian implementation ofAnands model with constant elastic properties toestimate the effect of elastic property temperaturedependence on the residual stress computation. Case 4is the Eulerian analysis of Anands elasto-visco-plasticmodel.

The same welding heat input model is used in allcases. The thermal analysis for cases 13 is the same andit is performed in a Lagragnian reference frame. Thethermal analysis of case 4 is performed in an Eulerianreference frame.

Constitutive law and material propertiesThe thermal properties for HSLA65 are assumed to bethe same as mild steel (A36) due to the low alloy contentof HSLA steels. They are illustrated in Fig. 5 and areobtained from Refs. 43 and 44. The elastic propertiesare also assumed to be the same as mild steel are areillustrated in Fig. 6.4548 In cases 3 and 4, the elasticproperties at room temperature (T520uC) are used.The yield strength for the rate independent elasto-

plastic analysis and hardening are illustrated in Fig. 7and are obtained from Ref. 44.

For the elasto-visco-plastic analysis, the functionf(~ss,s) in equation (21) is taken to be1

f~A sinh j~s

s

1m

(36)

Table 1 Analysis models for GMAW

Reference frame Constitutive law Elastic properties

Case 1 Lagrangian Elasto-plastic Temperature dependentCase 2 Lagrangian Anands model Temperature dependentCase 3 Lagrangian Anands model ConstantCase 4 Eulerian Anands model Constant

4 Gas metal arc welding model (Eulerian)

5 Conductivity k, specic heat Cp and air convection h

assumed for HSLA-65

6 Elastic modulus E, Youngs modulus n and thermal

expansion coefcient a assumed for HSLA-65

7 Yield strength of HSLA-65



where A~A0:expS{Qe= R 300z(T{300)=Ta f gT, Qeis the thermal activation energy, R is the ideal gasconstant, T is the absolute temperature and Ta is atemperature dependent parameter, which can beexpressed as

Ta~25{

T{300

100

2" #1=2z1 T800 K

1 Tw800 K

8>>>: (37)

The values of material parameters A0, j and m are listedin Table 2.The internal variable evolution equation, defined by

the function g(~ss,s), is1

g(~s,s)~

h0 1{s

s

a sign 1{ s

s

:A sinh j~s

s

1=m(38)

s~~s:~e vp

A

n(39)

where h0, a, s and n are the material parameters listed inTable 2.The HSLA-65 steel with strain rate between 1023 and

1024 is considered in this work. The material propertieslisted in Table 2 are determined based on the experi-mental stress-strain plot of the HSLA-65 steel shown inRef. 49. Using the radial return algorithm presented inRef. 50, the computed stressstrain relationships of theHSLA-65 under different temperatures using Anandsmodel are shown in Fig. 8. It is noted that that theAnands model produces similar results as the rateindependent elasto-plastic model, with a smothe (nokink) transition from elastic to actively plastic response.

Recrystalisation in Anands model can be simulatedby resetting the internal variable s to its initial value s0 atthe recrystalisation temperature. However, this feature isnot implemented in this study to maintain consistencywith the rate independent elasto-plastic analysis.No element rebirth techniques have been implemented

in any of the analyses. Physically, this conditionrepresents autogenous welding and it is a simplification.However, the objective was to main consistency andfacilitate comparison of the various models.

Evaluaton of Anands modelTo compute the residual stress in a Lagrangian frame,the plate should be cooled down to room temperature,therefore a sufficiently long analysis time (t51200 s) isperformed. Figures 9 and 10 show the contour plotsof the longitudinal component of the residual stress

Table 2 Material parameters for HSLA-65 steel

Material parameter Value Material parameter Value

A0 6?3461011 s21 Qe 352?35 KJ mol

21

j 3?25 R 8?314m 0?1956 a 1?5n 0?06869 ~ss 125?1 MPas0 80 MPa h0 3093?1 MPa

8 Computed stressstrain curves of HSLA-65 steel under

different temperature

9 Lagrangian longitudinal residual stress for elasto-

plastic model (case 1)

10 Lagrangian longitudinal residual stress for Anands

model (case 2)



computed by the elasto-plastic model (case 1) andAnands model (case 2) respectively. The results are verysimilar. To compare the results more quantitatively,lines 1 and 2 of Fig. 11 plot the Longitudinal componentresidual stress of cases 1 and 2 along line P in Fig. 3. Thedifference between the two lines is very small. Theexperimental results, which are listed in Ref. 42, are alsoplotted in Fig. 11. Line 5 shows the top side stress, line 6plots the bottom side stress, and line 7 plots the averagethrough thickness stress of the plate. The tensile stress ofboth computational and experimental results decreasesharply from the weld centreline toy62 mm away fromthe centreline and then become compressive. At the freeedge of the plate the computed compressive stress for allcases are close to the measured stress. As seen in Fig. 11,although the computational results are in good agree-ment, they have some discrepancy compared to theexperimental measurements. The difference is attributedto the limitation that an in-plane two-dimensional modelis used in the computations. It is noted however, that themain objective of this work is to compare elasto-visco-plastic models (cases 24) to the conventional rateindependent elasto-plastic models (case 1).

The longitudinal component of the residual stress ofAnands model with constant elastic properties (case 3) isalso plotted in Fig. 11 (line 3). The residual stresscomputed using constant elastic properties is very closeto that computed using temperature dependent elasticproperties (line 2). Therefore, the residual stress computedby Anands model is not sensitive to the temperaturevariation of the elastic material parameters (a, m and E).

Evaluation of mechanical analysis in EulerianframeSimilarly to the Lagrangian analysis, an Eulerianthermal analysis is performed first, and the temperature

result is used by the mechanical analysis. To comparethe Eulerian analysis with the Lagrangian analysis, theheat torch is located at 2540 mm from the outlet surface,Thus the distance between the torch and the outletsurface is the same as the distance between the torch andthe plot line P in Lagrangian analysis at t51200 s. Asshown in the contour plot of temperature result ofEulerian analysis in Fig. 12, the temperature at theoutlet surface has dropped to room temperature, thusthe computed stress at the outlet surface denotes theresidual stress of the welded plate. The temperaturedistribution resulting from the Lagrangian analysis isploted in Fig. 13, where it can be seen that a particleexperience similar heating and cooling cycles down toy350uC).The following boundary conditions are applied to the

inlet surface of the model

11 Comparison of longitudinal residual stress

12 Temperature distribution of GMAW in Eulerian frame



vx~vin~2:54 mm s{1 (40)

vy~0 (41)

sij~0 (42)

Fij~dij (43)

The material properties used in the Eulerian analysis are:elastic modulus E50?2066106 MPa, Poissons ratiom50?3 and thermal expansion a50?11761024. Thestressstrain response still varies with temperature, asshown in Fig. 8. The computed internal variabledistribution from the Eulerian analysis is shown inFig. 14. For comparison purposes, the internal variabledistribution from the Lagrangian analysis (case 3) aftercooling is shown in Fig. 15, along with a line plot ofcomparison in Fig. 16. The close agreement of theresults demonstrates the effectiveness of the Galerkingmethod used in this work to compute the materialevolution in an Eulerian reference frame.

The contour plot of the longitudinal residual stress ofthe Eulerian analysis is shown in Fig. 17. The long-itudinal residual stress along the outlet surface is alsoplotted in line 4 of Fig. 11. The elasto-vsicoplasticEulerian results (case 4) show excellent agreement withthe results from Lagrangian analysis with both elasto-plastic model (case 1) and Anands elasto-vsicoplasticmodel (case 2). Only in the small region near the weldcentreline (|Y|,18 mm), the residual stress results have

noticeable difference between the Lagrangian analysis(case 3) and the Eulerian analysis (case 4). Themaximum difference is y50 MPa, which means therelative difference in this region is y10%. In the otherregions (18(|Y|(228?6 mm), the relative differencebetween Lagrangian analysis (case 3) and the Euleriananalysis (case 4) is ,5%.

Computational costBoth Lagrangian and Eulerian analyses are performedby an in-house finite element code which is capable to

13 Temperature distribution of GMAW in Lagrangian

frame

14 Eulerian internal variable for Anands model (case 4)

15 Lagrangian internal variable for Anands model (case

3) after cooling

16 Comparison of Lagrangian (line P in Fig. 15 ) and

Eulerian (outlet) internal variable results

17 Eulerian longitudinal residual stress for Anands

model (case 4)



analyse coupled thermo-mechanical problems. Themodels are tested on an IBM RS/6000 system with16 Gbs shared memory. The computational costs ofcases 1, 2 and 4 are measured according to the CPUrunning time and listed in Table 3.

The Lagrangian implementation of the elasto-plasticand Anands model require the same the computationaltime for both thermal and mechanical analysis (y300 s).The mechanical analysis in the Eulerian frame does notshow a computational advantage. This is because theEulerian mechanical analysis uses a time consumingprogressively stiffening method, which is described inRef. 32, to improve the convergence. Furthermore, thesize of the element tangent matrix (44644) for anEulerian analysis is larger than the size (868) for aLagrangian analysis. The computational cost of theEulerian thermal analysis is only 0?5% of the Lagrangianthermal analysis. Therefore, the total computationalcost for a combined thermomechanical analysis in anEulerain analysis is about a half of a Lagrangiananalysis. It is noted that most of the CPU savings areobtained in the Eulerian thermal analysis, while themechanical thermal analysis requires equivalent CPUwith the Lagrangian. However, an Eulerian mechanicalanalysis is also needed to take advantage of thecomputational savings of the Eulerian thermal analysis.Furthermore, in a three-dimensional implementation,the Eulerian mechanical analysis is also expected to befaster than the Lagrangian, due the fact that meshcoarsening in both longitudinal and transverse direc-tions are possible in the Eulerian analysis, while onlytransverse coarsening is permitted in the Lagrangiananalysis.

ConclusionsAn elasto-visco-plastic formulation based on Anandsconstitutive model has been developed for weldingresidual stress modelling. The formulation has beenimplemented in both Lagrangian and Eulerian referenceframes. Gas metal arc welding has been used as a testcase to verify the formulation by comparing computedresidual stress against measurements and simulationsusing the conventional Lagrangian elasto-plastic for-mulations. The investigation leads to the followingconclusions.

1. The Anands elasto-visco-plastic model is capablecapturing the rate independent material response ofmetals at low temperatures. In addition, the same modelis also cable to simulate the rate dependent materialresponse at high temperatures. Applications couldinclude modelling heat treating, friction stir weldingand coupled weld pool residual stress modelling.

2. An Eulerian implementation of Anands constitu-tive model is possible. The formulation can account forlarge material deformations without the need for adap-tive meshing as would be the case in a Lagrangianimplementation. To our knowledge, this paper is the first

successful implementation of an Eulerian approach inresidual stress modelling.

Acknowledgement

The authors would like to acknowledge the fundingfrom Office of Naval Research, awardno. N000140410175 and program managers JulieChristodoulou and Johnnie DeLoach.

References1. A. N. Fitch, C. R. A. Catlow and A. Atkinson: Measurement of

stress in nickel oxide layers by diffraction of synchrotron

radiation, J. Mater. Sci., 1991, 26, 23002304.

2. P. J. Webster: Strain scanning using X-rays and neutrons, Mater.

Sci. Forum, 1996, 228231, 191200.

3. D. Rosenthal and J. T. Norton: A method for measuring triaxial

residual stresses in plates, Weld. J., 1945, 24, 295307.

4. Determining residual stresses by the hole drilling strain gage

method, Standard E837, ASTM, West Conshohocken, PA, USA.

5. H. Hibbitt and P. V. Marcal andA Numerical: Thermo-mechanical

model for the welding and subsequent loading of a fabricated

structure, Comput. Struct., 1973, 3, (11451174), 11451174.

6. E. F. Rybicki, D. W. Schmueser, R. B. Stonesifer, J. J. Groom and

H. W. Mishler: A finite-element model for residual stresses and

deflections in girth-butt welded pipes, J. Press. Vess. Technol.y,

1978, 100, 256262.

7. J. H. Argyris, J. Szimmat and K. J. Willam: Computational

aspects of welding stress analysis, Comput. Methods Appl. Mech.

Eng., 1982, 33, 635666.

8. J. Goldak, A. Chakravarti and M. Bibby: A new finite element

model for welding heat sources, Metall. Trans. B, 1984, 15B, 299

305.

9. P. Tekriwal and J. Mazumder: Transient and residual thermal

strain-stress analysis of GMAW, J. Eng. Mater. Technol., 1991,

113, 336343.

10. S. B. Brown and H. Song: Implications of three-dimensional

numerical simulations of welding of large structures, Weld. J.,

1992, 71, (2), 55s62s.

11. J. Wang and H. Murakawa: A 3-D fem analysis of buckling

distortion during welding in thin plate, Proc. 5th Int. Conf. on

Trends in welding research, Pine Mountain, GA, USA, June 1998,

ASM.

12. C. L. Tsai, S. C. Park and W. T. Cheng: Welding distortion of a

thin-plate panel structure, AWS Weld. J. Res. Suppl., 1999, 78,

156s165s.

13. L.-E. Lindgren: Computational weld mechanics: thermomechani-

cal and microstructural simulations; 2007, Cambridge, Woodhead.

14. J.-M. Bergheau, Y. Vincent, J.-B. Leblond and J.-F. Jullien:

Viscoplastic behaviour of steels during welding, Sci. Technol.

Weld. Join., 2004, 9, (4), 323330.

15. P. Dong, F. Lu, J. K. Hong and Z. Cao: Coupled thermo-

mechanical analysis of friction stir welding process using simplified

models, Sci. Technol. Weld. Join., 2001, 6, 281287.

16. Z. Feng, X.-L. Wang, S. A. David and P. S. Sklad: Modelling of

residual stresses and property distributions in friction stir welds of

aluminium alloy 6061-T6, Sci. Technol. Weld. Join., 2007, 12, 348

356.

17. C. Chen and R. Kovacevic: Thermomechanical modelling and

force analysis of friction stir welding by the finite element method,

J. Mech. Eng. Sci., 2004, 218, 509519.

18. G. Buffa, J. Hua, R. Shivpuri and L. Fratini: Design of the friction

stir welding tool using the continuum based fem model, Mater.

Sci. Eng. A, 2006, A419, 381388.

19. H. Schmidt and J. Hattel: A local model for the thermomechanical

conditions in friction stir welding, Model. Simul. Mater. Sci. Eng.,

2005, 13, 7793.

20. L. Fourment and S. Guerdoux: 3D numerical simulation of the

three stages of friction stir welding based on friction parameters

calibration, Int. J. Mater. Form., 2008, 1, 12871290.

21. P. Ulysse: Three-dimensional modeling of the friction stir-welding

process, Int. J. Mach. Tools Manuf., 2002, 42, 15491557.

22. A. Bastier, M. H. Maitournam1, K. Dang Van and F. Roger:

Steady state thermomechanical modelling of friction stir welding,

Sci. Technol. Weld. Join., 2006, 11, (3), 278288.

Table 3 Mechanical analysis CPU times

Case 1 Case 2 Case 4

CPU time for thermal analysis, s 300?3 300?3 1?5CPU time for mechanical analysis, s 236?5 280?2 333?9Total CPU time, s 536?8 580?5 335?4



23. A. Bastier, M. H. Maitournam, F. Roger and K. Dang Van:

Modelling of the residual state of friction stir welded plates,

J. Mater. Process. Technol., 2008, 200, 2537.

24. P. Michaleris, L. Zhang and P. Marugabandhu: Evaluation of

applied plastic strain methods for welding distortion prediction,

J. Manuf. Sci. Eng., 2007, 129, 10001010.

25. S. B. Brown, Kwon H. Kim and L. Anand: An internal variable

constitutive model for hot working of metals, Int. J. Plast., 1989, 5,

95130.

26. G. G. Weber, A. M. Lush, A. Zavaliangos and L. Anand: An

objective time-integration procedure for isotropic rate-independent

and rate-dependent elastic-plastic constitutive equations, Int. J.

Plast., 1990, 6, 701744.

27. G. Weber and L. Anand: Finite deformation constitutive

equations and a time integration procedure for isotropic hyper-

elastic-viscoplastic solids, Comput. Methods Appl. Mech. Eng.,

1990, 79, 173202.

28. Q. S. Nguyen and M. Rahimian: Mouvement permanent dune

fissure en milieu elastoplastique, J. Mec. Appl., 1981, 5, 95120.

29. J. Y. Shanghvi and P. Michaleris: Thermo-elasto-plastic finite

element analysis of quasi-state processes in Eulerian reference

frames, Int. J. Numer. Methods Eng., 2002, 53, 15331556.

30. E. G. Thomson and S.-W. Yu: A flow formulation for rate

equilibrium equations, Int. J. Numer. Methods Eng., 1990, 30,

16191632.

31. A. M. Maniatty, P. R. Dawson and G. G. Weber: An Eulerian

elasto-viscoplastic formulation for steadystate forming processes,

Int J. Mech. Sci., 1991, 33, 361377.

32. X. Qin and P. Michaleris: Eulerian elasto-visco-plastic formula-

tions for residual stress prediction, Int. J. Numer. Methods Eng.,

2009, 77, 634663.

33. X. Qin and P. Michaleris: Coupled thermo-elasto-viscoplastic

analysis of friction stir welding, Proc. 8th Int. Conf. on Trends in

welding research, Pine Mountain, GA, USA, June 2008, ASM.

34. J. C. Simo and R. L. Taylor: Consistent tangent operators for rate-

independent elasto-plasticity, Comput. Methods Appl. Mech. Eng.,

1985, 48, 101118.

35. J. Lubliner: Plasticity theory, 1st edn; 1990, New York,

Macmillan Publishing Company.

36. K.-J. Bathe: Finite element procedures; 1996, Upper Saddle River,

NJ, Prentice Hall.

37. D. Balagangadhar, G. A. Dorai and D. A. Tortorelli: A

displacement-based Eulerian steady-state formulation suitable for

thermo-elasto-plastic material models, Int. J. Solids Struct., 1999,

36, (16), 23972416.

38. J. Goldak M. Gu and E. Hughes: Steady state thermal analysis of

welds with filler metal addition, Can. Metall. Q., 1993, 32, (1), 49

55.

39. E. Ohmura, Y. Takamachi and K. Inoue: Theoretical analysis of

laser transformation hardening process of hypoeutectoid steel

based on kinetics, Trans. Jpn Soc. Mech. Eng. A, 1990, 56A, 1496

1503.

40. S. Rajadhyaksha and P. Michaleris: Optimization of thermal

processes using an Eulerian formulation and application in

laser hardening, Int. J. Numer. Methods Eng., 2000, 47, 1807

1823.

41. A. Agrawal and P. R. Dawson: A comparison of galerkin and

streamline techniques for integration strains from an Eulerian flow

field, Int. J. Numer. Methods Eng., 1985, 21, 853881.

42. S. R. Bhide, P. Michaleris, M. Posada and J. Deloach:

Comparison of buckling distortion propensity for SAW,

GMAW, and FSW, Weld. J., 2006, 85, 189195.

43. The British Iron and Steel Research Association (ed.): Physical

constants of some commercial steels at elevated temperatures;

1953, London, Butterworths Scientific Publications.

44. O. A. Vanli and P. Michaleris: Distortion analysis of welded

stiffeners, J. Ship Product., 2001, 17, (4), 226240.

45. W. H. Hill, K. D. Shimmin and B. A. Wilcox: Elevated-

temperature dynamic moduli of metallic materials, Proc. ASTM,

1961, 61, 890906.

46. R. Michel: Elastic consants and coefficients of thermal expansion

of piping materials proposed for 1954 code for pressure piping,

Trans. ASME, 1955, 77, 151159.

47. C. J. Smithells (ed.): Metals reference book, 5th edn, London,

Butterworths.

48. J. Wray: Mechanical, physical and thermal data for modeling the

solidificaton processing of steels, in Modeling of casting and

welding processes, (ed. H. D. Brody and D. Apelian), 245257;

1980, Rindge, NH, The Metallurgical Society of AIME.

49. S. Nemat-Nasser and W.-G. Guo: Thermomechanical response of

hsla-65 steel plates: experiments and modeling, Mech. Mater.,

2003, 37, 379405.

50. A. M. Lush, G. Weber and L. Anand: An implicit time-integration

procedure for a set of internal variable constitutive equations for

isotropic elasto-viscoplasticity, Int. J. Plast., 1989, 5, 521549.



Documents

Elasto-Visco-plastic Analysis of Welding Residual Stress