A New Thermo-elasto-plasticity Constitutive Theory for Polycrystalline Metals

Acta Mech Sin (2015) 31(3):338348DOI 10.1007/s10409-015-0462-1

RESEARCH PAPER

A new thermo-elasto-plasticity constitutive theoryfor polycrystalline metals

Cen Chen1 Qiheng Tang1 Tzuchiang Wang1

Received: 29 December 2014 / Revised: 17 April 2015 / Accepted: 23 April 2015 / Published online: 26 May 2015 The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag BerlinHeidelberg 2015

Abstract In this study, the behavior of polycrystallinemetals at different temperatures is investigated by a newthermo-elasto-plasticity constitutive theory. Based on solidmechanical and interatomic potential, the constitutive equa-tion is established using a new decomposition of the defor-mation gradient. For polycrystalline copper and magnesium,the stressstrain curves from 77 to 764 K (copper), and 77 to870 K (magnesium) under quasi-static uniaxial loading arecalculated, and then the calculated results are compared withthe experiment results. Also, it is determined that the presentmodel has the capacity to describe the decrease of the elasticmodulus and yield stress with the increasing temperature, aswell as the change of hardening behaviors of the polycrys-talline metals. The calculation process is simple and explicit,which makes it easy to implement into the applications.

Keywords Thermo-elasto-plasticity constitutive theory Yield stress Hardening behaviors Finite temperature

1 Introduction

Polycrystalline metals have been investigated widely formany years, as they are importantmaterials for the aerospace,energy, and chemical processing industries.Also, theirmater-ial response at different temperatures has drawn the extensiveattention of researchers. The results of experimental investi-gations [18] have determined that the yield stress and hard-

B Tzuchiang [email protected]

1 State Key Laboratory of Nonlinear Mechanics,Institute of Mechanics, Chinese Academy of Sciences,100190 Beijing, China

ening of polycrystalline metals decreases with the increasein temperature at the same strain rate.

In previous years, theoretical models for polycrystallineplasticity have been developed. For example, the classicTaylor [9], self-consistent [10], and crystal plasticity finite-element models [11] have made significant progress fromboth macro and micro angles. First of all, the classic Tay-lor model [9] assumes that all grains must accommodate thesame plastic strain, which is equal to the macroscopic strainand neglects the interaction between crystals. Therefore, it ismore applicable for the face centered cubic (FCC) and bodycentered cubic (BCC) metals due to their crystallographicsymmetry [1214]. The next models are those based on theself-consistent approach. These models have been applied inthe hexagonal close packed (HCP) [10,15] and other poly-crystalline materials [16] for many decades. These modelshave the ability to describe the stress and strain variationsfrom one grain to another and the interaction among thegrains for the low crystallographic symmetry in polycrystals[1719]. Moreover, new research has revealed that the self-consistent approach could potentially be implemented intothe complicated loading condition deformation process [20],thereby describing the visco-plastic deformation using thedislocation-density constitutive law [21]. Lastly, the crystalplasticity finite-elementmodels have been used to investigatethe effects of the dislocation creep [22,23], hardening behav-ior [24], and crystal orientation [25] on the plastic behaviorof metals at various temperatures. This model offers vari-ous constitutive formulations at the elementary shear systemlevel, and can be applied easily into complicated boundaryconditions [11].

In addition to the above mentioned theories, some newmodels have been proposed in recent years [2632]. Forexample, the JohnsonCook model [27], ZerilliArmstrong

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A new thermo-elasto-plasticity constitutive theory for polycrystalline metals 339

model [28], and KhanHuangLiang model [2932]. Thesemodels have improved the constitutive descriptions of thedynamic plasticity of metals and have described the strain,strain-rate, and temperature relations formetals in large strainand high strain-rate regimes. The above models have allplayed important roles in the investigation of thermo-elasto-plastic deformations for polycrystalline metals. However,none of them have accounted for the thermal expansion inthe deformation histories, which limits their applications tostructural calculation with some boundary constraints. At thesame time, we also need concise descriptions to reflect thetemperature effects on the yield stress and hardening behav-iour.

In this study,we propose a thermo-elasto-plasticity consti-tutive theory to describe behaviors of polycrystalline metalsat different temperatures. First of all, a new decomposi-tion of the deformation gradient is presented, and then thethermo-elasticity constitutive equation for single crystals isestablished. This is followed by obtaining the polycrystalelastic constants from single crystal elastic constants throughintegral transformation. The next focus is a macroscale plas-tic constitutive equation implementation to obtain the plasticdeformation of the polycrystal. In the calculation process,a simple exponential relationship is proposed in order todescribe the decrease of the yield stress with temperatureincreases, so that a law change of the material behaviour caneasily be obtained. Lastly, the comparisons between the cal-culated and experimental results are presented.

2 Thermo-elasto-plasticity constitutive relationshipfor single crystals

2.1 Decomposition of the deformation gradient

In this paper, a new decomposition of deformation gradientis proposed to describe the thermo-elasto-plasticity deforma-tion behavior, which is different from the kinematical theory[3335]. As shown in Fig. 1, the whole deformation processis decomposed into four parts: the initial configuration at theundeformed state of 0 K (Fig. 1a), the first intermediate con-figuration after free thermal expansions at T K (Fig. 1b), thesecond intermediate configuration after elastic deformationat T K (Fig. 1c), and the current configuration after plasticdeformation at T K (Fig. 1d).

The total deformation gradient is decomposed as

F = FpFeF, (1)

where Fe is the elastic deformation gradient, Fp is the plas-tic deformation gradient, and F is the thermal deformationgradient due to the free thermal expansion.

The thermal strain tensor E, elastic strain tensor Ee, andplastic strain tensor Ep take the respective forms as

Fig. 1 Decomposition of deformation configuration. a Initial con-figuration. b First intermediate configuration. c Second intermediateconfiguration. d Current configuration

E = 12

(FTF I

), (2a)

Ee = 12

(FeTFe I

), (2b)

Ep = 12

(FpTFp I

). (2c)

Therefore, the total strain tensor is expressed as

E = 12

[(FTFeTFpTFpFeF

) I

]

= 12

(FTFeTFeF I

)+ FTFeTEpFeF

= E + FTEeF + FTFeTEpFeF. (3)Based on the polar decomposition of the tensor, the defor-

mation gradients F and Fe are written respectively as

F = RU, (4)Fe = ReUe, (5)where R and Re are the rotation tensors, and U and Ueare the stretch tensors.

Assuming that R = I , Re = I , the total strain tensor isexpressed as

E = E + UEeU + UUeEpUeU. (6)Based on Eqs. (2a), (2b), (4) and (5), we can obtain

E = 12

[(U

)2 I], (7a)

Ee = 12

[(Ue

)2 I]. (7b)

The Taylor expansions of U and Ue are

U = (I + 2E)1/2 = I + E 12

(E

)2 + , (8a)

Ue = (I + 2Ee)1/2 = I + Ee 12

(Ee

)2 + . (8b)

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340 C. Chen et al.

If the thermal strain tensor E and elastic strain tensor Eeare small, we can obtain

U = I + E, (9a)Ue = I + Ee, (9b)

and

U = I, (10a)Ue = I . (10b)

Then, the total strain tensor takes

E = E + Ee + Ep. (11)

Equation (11) is a new strain tensor expression of the elas-tic and plastic deformation at the finite temperature, and itextends the kinematical theory of the elastic-plastic defor-mation of the crystal.

2.2 Thermal strain

When an undeformed body is heated up from temperature T0to T , the thermal strain T is given by [36]

T =T

T0

dT , (12)

where T0 is the reference temperature. is the coeffi-cient of thermal expansion, which can be obtained from theexperimental results [37] and also can be calculated by thetheoretical method [38].

For the metal material, the thermal strain tensor E is as

E =

T 0 00 T 00 0 T

. (13)

The calculations for lattice constant r (0)(T ) at temperatureT were given by Jiang [39] as follows:

r (0)(T ) = r (0)(T0)1 +

T

T0

dT

. (14)

2.3 Thermo-elasticity constitutive equation for singlecrystals

The second PiolaKirchhoff stress is expressed as

S = WEe

= 1V

[Utot (Ee)

Ee

], (15)

where V is the volume at the first intermediate configurationas shown in Fig. 1b, Utot is the total potential energy of thesystem.

The rate of the second PiolaKirchhoff stress takes

S = 1V

[U 2tot (E

e)

EeEe

]: Ee

= 1V

[U 2tot (E

e)

EeEe

]: (E Ep E). (16)

Equation (16) can be written as

S = Csig : Ee, (17)

where Csig is the thermo-elastic stiffness tensor for singlecrystals. And the change of lattice constant with temperatureis considered in the calculation of total potential energyUtot.So the stiffness Csig changes with temperature.

The constitutive equation (16) is established by the rate ofthe second PiolaKirchhoff stress and the rate of the Greenstrain based on the new deformation gradient decomposi-tion. Since the decomposition is obtained under the conditionthat the elastic and thermal strain is small (Eqs. (910)), theconstitutive equation can also be applied under this condi-tion. Although the plastic strain of metal material alwaysexceeds small deformation range, the decomposition is avail-able because the elastic and thermal strain is small enough.

3 Thermo-elasto-plasticity constitutive relationshipfor polycrystal

3.1 Thermo-elastic constants for polycrystal

Polycrystalline material can be considered an aggregate ofrandomly oriented single crystals. The orientation of a crys-tallite in a polycrystalline sample is specified by means ofEuler angles (, , ) [40]. The component of the thermo-elastic stiffness tensor for the polycrystalline Cpoli jkl can beobtained by

Cpoli jkl = 0

20

20

(Rim)1 (R jn

)1 (Rkp

)1 (Rlq

)1

Csigmnpq f (, , ) sin ddd, (18)

where Csigmnpq is the component of thermo-elastic stiffnesstensor for the single crystals, which can be obtained by Eq.(17), R is the rotation tensor described by the Euler angles,and its expression is given by Reo [40], Ri j is the componentof R, and f (, , ) is the crystalline orientation distribu-tion function in the polycrystalline sample. This satisfies

0

20

20

f (, , ) sin ddd = 1. (19)

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Then, assuming that crystalline orientations satisfy theuniform distribution, f (, , ) would be constant as:

f (, , ) = 182

. (20)

The component of the thermo-elastic stiffness tensor forthe polycrystal is written as

Cpoli jkl

= 182

0

20

20

(Rim)1 (R jn

)1 (Rkp

)1 (Rlq

)1

Csigmnpq sin ddd. (21)

For the single crystals of cubicmetals, there are three inde-pendent elastic constants:Csig1111,C

sig1122,C

sig1212, and the elastic

constants of polycrystalline cubic materials are calculated as

Cpol1111 = 0.6Csig1111 + 0.4(Csig1122 + 2Csig1212), (22a)Cpol1122 = 0.2Csig1111 + 0.8Csig1122 0.4Csig1212, (22b)Cpol1212 = 0.2Csig1111 0.2Csig1212 + 0.6Csig1212. (22c)

While there are five independent elastic constants for the sin-gle crystals of hexagonal metals: Csig1111, C

sig3333, C

sig1122, C

sig1133,

Csig1212, Csig1313. The elastic constants of polycrystalline hexag-

onal materials are calculated as:

Cpol1111 = 0.533Csig1111 + 0.2Csig3333+ 0.266

(Csig1133 + 2.0Csig1313

), (23a)

Cpol3333 = 0.533Csig1111 + 0.2Csig3333+ 0.266

(Csig1133 + 2.0Csig1313

), (23b)

Cpol1122 = 0.0667(Csig1111 + Csig3333

)+ 0.333Csig1122

+ 0.533Csig1133 0.2668Csig1313, (23c)Cpol1133 = 0.0667

(Csig1111 + Csig3333

)+ 0.333Csig1122

+ 0.533Csig1133 0.2668Csig1313, (23d)Cpol1313 = 0.233c1111 + 0.0667Csig3333 0.1667Csig1122

0.133Csig1133 + 0.4Csig1313, (23e)Cpol1212 =

1

2

(Cpol1111 Cpol1122

)

= 0.233Csig1111 + 0.0667Csig3333 0.1667Csig1122 0.133Csig1133 + 0.4Csig1313. (23f)

From Eqs. (23a)(23f), it can be found that though the sin-gle crystals of hexagonalmetals are anisotropic, there also arethree independent elastic constants of polycrystalline hexag-onal materials. This result is due to the assumption that the

crystalline orientations satisfy uniform distribution in thepolycrystalline sample.

Based on the above derivation, we are able to determinethe rate of the second PiolaKirchhoff stress of the polycrys-talline material as

S = Cpol : Eepol, (24)

where Eepol

is the rate of the Green strain for the polycrys-talline material.

3.2 Plastic constitutive equation for polycrystal

In order to determine the relationship between the plasticstrain, and the stress at a given temperature, the power lawof the macroscopic uniaxial strainstress curve is adopted,and has been effectively adopted by the theoretical model[27,31,32,41]:

={cm, ysE , < ys

, (25)

where c and m are parameters, E is secant modulus of elas-ticity, and ys is yield stress, which can be obtained by

ys = cmys = E ys, (26)

where ys is yield strain, which remains unchanged for thesame material; then Eq. (25) can be written as follows:

ys=

(

ys

)m, ys

ys, < ys

. (27)

In Eq. (27), when the yield stress ys and parameterm aredetermined at a given temperature, the stressstrain curvecan then be obtained.

With consideration to the yield stress always changingwith temperature, and in order to describe the temperatureeffects on the yield stress, we proposed the exponential curvebased on the previous experiment and theoretical investiga-tions as:

ys = 0yseT, (28)

where, T = TT0 1, and T0 is the reference temperature, 0ysis the yield stress at reference temperature, and is parameterwhich reflects the change of yield stress with temperature.

When, ys (T), the stress is obtained by:

=(

ys

)m 0yse

T . (29)

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342 C. Chen et al.

Fig. 2 Thermal strain of copper

4 Calculation results

In this study, the uniaxial stressstrain curves of FCC poly-crystalline copper and HCP polycrystalline magnesium atvarious temperatures are calculated based on the presentmodel, and the calculation results are compared with theexperiment results [3,5,6].

4.1 FCC polycrystalline copper

The experimental results for the quasi-static uniaxial stressstrain curves of polycrystalline copper from 77 to 764Kwereobtained by Roberts and Bergstrm [6].

4.1.1 Thermal strain and lattice constants for copper

The thermal strain and lattice constants at different tempera-tures are calculated based on Eqs. (12) and (14), respectively,and the thermal expansion in Eqs. (12) and (14) is obtainedfrom the experimental results [37]. Figures 2 and 3 show thethermal strain and the lattice constant versus temperature forcopper. The thermal strain at room temperature is set as zero.

4.1.2 Elastic constants for copper

For copper, the EAM potential proposed by Mishin [42] isadopted to calculate the potential energy Utot in Eq. (16).The change of lattice constant with temperature is consid-ered in the calculation of thermo-elastic stiffness tensor [Eq.(17)], and the elastic constants of the polycrystalline copperat different temperatures can be easily obtained (Fig. 4).

4.1.3 Determination of calculated parameters for copper

For copper, the reference temperature is 293 K and yieldstrain ys is 0.2 %. Based on Eq. (29), only three parameters

Fig. 3 Lattice constants of copper

Ela

stic

con

stan

ts/G

Pa

Temperature/K

100 200 300 400 500 600 700 800-50

0

50

100

150

200

250

CopperC11 C22 C12

Fig. 4 Elastic constants of copper

are required: 0ys, , and m. The yield stress 0ys and para-

meter m are determined firstly by the stressstrain curve at293 K. Then, from another stressstrain curve at a differenttemperature, the parameter can be obtained. Figure 5 is thecalculated curve of the yield stress versus the temperature forcopper, which agrees well with the experimental results. Andthe calculated parameters for copper are shown in Table 1.

4.1.4 Calculated results for copper

Figure 6a6c is the comparison of the uniaxial stressstraincurves for copper between the simulation and the experi-mental results at different temperatures (77764 K). It canbe found that the present model can potentially describe thebehavior of the FCC polycrystalline copper effectively.

4.2 HCP polycrystalline magnesium

The experimental results of the magnesium come from dataof two different studies: low and medium temperatures [3]:77523 K; high temperatures [5]: 675870 K. Therefore, the

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0

ys

0

ysys

Fig. 5 Comparison of yield stress for copper between the calculatedcurve and experimental data at different temperatures

Table 1 Calculated parameters for copper

T (K) 0ys (MPa) m

77764 16.5 0.387 0.67

reference temperature is 293 K for low and medium temper-atures, and 675 K for high temperatures, and the yield strainys is 0.05 %.

Based on previous investigations regarding magnesium[43], it can be determined that the low symmetry of thecrystallographic structure, as well as the twinning behavior,would make the plastic deformation of the HCP polycrys-talline more complicated. In comparing the two groups ofexperimental results, we found that the temperature effectson the plastic behavior at lower temperatures are more dif-ficult to describe. Therefore, we adopted a special handlingin the calculation of the magnesium at the low and mediumtemperatures.

4.2.1 Thermal strain and lattice constants for magnesium

The thermal strain and lattice constants for magnesium areobtained by the same method as copper, and the results areshown in the Figs. 7 and 8.

4.2.2 Elastic constants for magnesium

For magnesium, the EAM potential proposed by Zhou [44]is adopted to calculate the potential energy. The elastic con-stants at different temperatures for magnesium are as shownin Fig. 9.

4.2.3 Determination of calculated parametersfor magnesium at high temperature

Similar to copper, the calculated parameters for magnesiumat high temperatures is determined by two experimental

77K-196K

a

b

c

298K-479K

568K-764K

Copper

Copper

Copper

E

C

E

C

E

C

Fig. 6 Comparisons of uniaxial stressstrain curves for copperbetween the simulation and experiment results at different tempera-tures. a 77196 K. b 298479 K. c 568764 K

stressstrain curves [5]. The calculated curve of the yieldstress versus temperature for magnesium at high tempera-

123

344 C. Chen et al.

Magnesium

Fig. 7 Thermal strain of magnesium

Magnesium

Fig. 8 Lattice constants of magnesium

ture is as shown in Fig. 10. The calculated parameters areillustrated in Table 2.

4.2.4 Calculated results for magnesium at high temperature

Figure 11a11b shows the comparison of the uniaxial stressstrain curves of the simulation and experiment results atdifferent temperatures for copper (675870 K).

4.2.5 Determination of calculated parameters formagnesium at low and medium temperatures

From the stressstrain curves of the experiment [3], it can bedetermined that the plastic deformation of the HCP polycrys-talline magnesium at low and medium temperatures appearsto be more complicated than the above calculated results.If the parameter m remains unchanged at different tempera-tures, we are unable to obtain the correct calculated results.In order to describe accurately the temperature effects onthe hardening behavior for magnesium at low and medium

Magnesium

Fig. 9 Elastic constants of magnesium

Magnesium

F

E

Fig. 10 Comparison of yield stress for magnesium between the calcu-lated curve and experimental data at high temperatures

Table 2 Calculated parameters for magnesium at high temperature

T (K) 0ys (MPa) m

675870 6.5 4.28 0.07

temperatures, we adopted a bilinear function to describe thechange of parameter m with the various temperatures as

m = m0(1 + aT + bT 2), (30)

where m0 is the value of m at reference temperature, and aand b are parameters.

Then, the yield stress 0ys andm0 are first determinedby thestressstrain curve at the reference temperatures. The para-meters a and b can be obtained by another two stressstraincurves at different temperatures. Meanwhile, the parameter can be obtained. Figure 12 is the comparison of the calcu-lated curve for m with the experimental results at differenttemperatures. Figure 13 is the comparison of the calculated

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a

b

Magnesium

Magnesium

E

C

E

C

Fig. 11 Comparisons of uniaxial stressstrain curves for magnesiumbetween the simulation and experiment results at high temperature.a 675773 K. b 813870 K

curve for yield stress ys with the experimental results. Itcan be seen that the calculated curves agree well with theexperimental results.

The calculated parameters for magnesium at low andmedium temperatures are as shown in Table 3.

4.2.6 Calculated results for magnesium at low and mediumtemperatures

Figure 14a, 14b is the comparison of the uniaxial stressstrain curves for the magnesium between the calculation andthe experimental results at low and medium temperatures(77523 K). It can be determined that the present model canpotentially describe the behavior of the HCP polycrystallinemagnesium efficiently.

MagnesiumEC

Fig. 12 Comparison of parameter m for magnesium between the cal-culated curve and experimental data at low and medium temperatures

MagnesiumE

C

Fig. 13 Comparison of yield stress for magnesium between the cal-culated curve and experimental data at low and medium temperatures

5 Discussion

When compared with the kinematical theory described byAsaro [33,34], whose deformation gradient is written asF = FeFp, the new decomposition of the total deforma-tion gradient is effective when the constitutive equation isapplied to the thermal strain. Moreover, the new decompo-sition equation F = FpFeF in this study is able to obtainthe simple strain tensor in Eq. (11), which is more favorablethan the other decomposition expressions. Then, based on thenew decomposition, we established the constitutive equationfor crystals. However, the new decomposition is establishedunder the condition that the elastic and thermal strain is smallenough, so the new constitutive equation is also applicativein this condition.

The plastic deformation and the decrease of yield stresswith temperature are reflected by simple power and expo-nential relationships respectively. The calculation is more

123

346 C. Chen et al.

Table 3 Calculated parameters formagnesium at low andmedium tem-peratures

T (K) 0ys (MPa)

77523 14.5 0.178

m0 a b

0.5 0.57 0.58

a

b

C

MagnesiumExperiment

C

MagnesiumExperiment

Fig. 14 Comparisons of uniaxial stressstrain curves for magnesiumbetween the simulation and experimental results at low and mediumtemperatures. a 77293 K. b 423523 K

concise, and the parameters can be determined by onlytwo or three uniaxial stressstrain curves at different tem-peratures. Therefore, it is easy to describe the temperatureeffects on the thermo-elasto-plasticity behaviors of the mate-rials.

By comparing the calculations between the FCC poly-crystalline copper and the HCP polycrystalline magnesium,it can be found that the material behaviors of the magnesiumat low and medium temperature is the more complex. Theparameterm drops off for the magnesium at low andmedium

temperatures, which is described by a bilinear function in Eq.(30). Based on many previous theories investigated in the lit-erature [4548], these phenomena can be explained by thedifferences of the crystallographic structure, the interactionamong grains, and the plastic mechanism between the FCC(or BCC) and the HCP polycrystalline metals. First of all, theFCC (or BCC) polycrystalline metals keep the symmetry ofthe crystallographic structure, and their slip systems for thedislocation motion have roughly equal resistance. Therefore,the plastic deformation in most FCC materials is dominatedby crystallographic slip. However, due to the low symmetryof the crystallographic structure of the HCP polycrystallinemetals, one must carefully take into account the details ofthe stress/strain variations from the grain, and the interac-tion between the crystals and grains. Also, different types ofslip systems exist, and both slip and twinning contribute tothe plastic deformation in the HCP crystals [17,49,50]. Asthe temperature increases, the plastic deformation transitsfrom being twinning dominated to a combination of slip andtwinning [49] or slip dominated [43]. All the above factorsshow that the simulation of the plastic behaviors at differ-ent temperatures for the HCP is more difficult than for theFCC. In this paper, we provide a concisemethod to reflect thedifferent of macroscopic behavior of the FCC and the HCPpolycrystalline materials.

6 Conclusion

In this study, a new thermo-elasto-plasticity constitutive the-ory is proposed to investigate the behaviors of polycrystallinemetals. First of all, the present new decomposition of thedeformation gradient is effective when applied to the ther-mal strain, and the calculation of the total strain is simplerandmore explicit. Then, the constitutive equations of a singlecrystal and polycrystal are established, and we provide a newmethod to describe the temperature effects on the yield stress,as well as the hardening behavior of the FCC and HCP poly-crystalline. Lastly, the comparisons between the calculationsand the experimental results show that the present model canpotentially accurately reflect the behavior of the polycrys-talline metals at different temperatures with a concise andclear calculation process.

Acknowledgments This work is supported by the National Nat-ural Science Foundation of China (Grants 11021262, 11172303,11132011) and National Basic Research Program of China through2012CB937500.

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A new thermo-elasto-plasticity constitutive theory for polycrystalline metalsAbstract1 Introduction2 Thermo-elasto-plasticity constitutive relationship for single crystals2.1 Decomposition of the deformation gradient2.2 Thermal strain2.3 Thermo-elasticity constitutive equation for single crystals3 Thermo-elasto-plasticity constitutive relationship for polycrystal3.1 Thermo-elastic constants for polycrystal3.2 Plastic constitutive equation for polycrystal4 Calculation results4.1 FCC polycrystalline copper4.1.1 Thermal strain and lattice constants for copper4.1.2 Elastic constants for copper4.1.3 Determination of calculated parameters for copper4.1.4 Calculated results for copper4.2 HCP polycrystalline magnesium4.2.1 Thermal strain and lattice constants for magnesium4.2.2 Elastic constants for magnesium4.2.3 Determination of calculated parameters for magnesium at high temperature4.2.4 Calculated results for magnesium at high temperature4.2.5 Determination of calculated parameters for magnesium at low and medium temperatures4.2.6 Calculated results for magnesium at low and medium temperatures5 Discussion6 ConclusionAcknowledgmentsReferences

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A New Thermo-elasto-plasticity Constitutive Theory for Polycrystalline Metals