-
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
123
http://crossmark.crossref.org/dialog/?doi=10.1007/s10409-015-0462-1&domain=pdf
-
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)
123
-
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)
123
-
A new thermo-elasto-plasticity constitutive theory for
polycrystalline metals 341
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)
123
-
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
123
-
A new thermo-elasto-plasticity constitutive theory for
polycrystalline metals 343
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
123
-
A new thermo-elasto-plasticity constitutive theory for
polycrystalline metals 345
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.
References
1. Chen, S.R., Kocks, U.: High-Temperature Plasticity in
CopperPolycrystals. LosAlamosNational Laboratory, LosAlamos
(1991)
2. Nemat-Nasser, S., Li, Y.: Flow stress of fcc polycrystals
with appli-cation to OFHC Cu. Acta Mater. 46, 565577 (1998)
123
-
A new thermo-elasto-plasticity constitutive theory for
polycrystalline metals 347
3. Ono, N., Nowak, R., Miura, S.: Effect of deformation
temperatureon HallPetch relationship registered for polycrystalline
magne-sium. Mater. Lett. 58, 3943 (2004)
4. Lennon, A., Ramesh, K.: The influence of crystal structure on
thedynamic behavior of materials at high temperatures. Int. J.
Plast.20, 269290 (2004)
5. Vagarali, S.S., Langdon, T.G.: Deformation mechanisms in
hcpmetals at elevated temperaturesI. Creep behavior of
magnesium.Acta Metall. 29, 19691982 (1981)
6. Roberts, W., Bergstrm, Y.: The stressstrain behaviour of
singlecrystals and polycrystals of face-centered cubicmetalsa new
dis-location treatment. Acta Metall. 21, 457469 (1973)
7. Viguier, B., Kruml, T., Martin, J.L.: Loss of strength in
Ni3Al atelevated temperatures. Philos. Mag. 86, 40094021 (2006)
8. Prasad, Y.V.R.K., Rao, K.P.: Kinetics of high-temperature
defor-mation of polycrystalline OFHC copper and the role of
dislocationcore diffusion. Philos. Mag. 84, 30393050 (2004)
9. Taylor, G.I.: Plastic strain in metals. J. Inst. Met. 62,
307324(1938)
10. Hill, R.: Continuummicro-mechanics of elastoplastic
polycrystals.J. Mech. Phys. Solids 13, 89101 (1965)
11. Roters, F., Eisenlohr, P., Hantcherli, L., et al.: Overview
of consti-tutive laws, kinematics, homogenization and multiscale
methodsin crystal plasticity finite-element modeling: Theory,
experiments,applications. Acta Mater. 58, 11521211 (2010)
12. Raabe, D., Mao, W.: Experimental investigation and
simulation ofthe texture evolution during rolling deformation of an
intermetallicFe-28 at.% A12 at.% Cr polycrystal at elevated
temperatures.Philos. Mag. A 71, 805813 (1995)
13. Kocks, U.: The relation between polycrystal deformation
andsingle-crystal deformation. Metall. Mater. Trans. 1,
11211143(1970)
14. Balasubramanian, S., Anand, L.: Elasto-viscoplastic
constitutiveequations for polycrystalline fcc materials at low
homologous tem-peratures. J. Mech. Phys. Solids 50, 101126
(2002)
15. Hutchinson, J.: Bounds and self-consistent estimates for
creep ofpolycrystalline materials. Proc. R. Soc. Lond. A 348,
101127(1976)
16. Landis, C.M., McMeeking, R.M.: A self-consistent
constitutivemodel for switching in polycrystalline barium titanate.
Ferro-electrics 255, 1334 (2001)
17. Wang, H., Raeisinia, B., Wu, P., et al.: Evaluation of
self-consistentpolycrystal plasticity models for magnesium alloy
AZ31B sheet.Int. J. Solids Struct. 47, 29052917 (2010)
18. Agnew, S.R., Duygulu, .: Plastic anisotropy and the role of
non-basal slip in magnesium alloy AZ31B. Int. J. Plast. 21,
11611193(2005)
19. Askari, H., Young, J.P., Field, D.P., et al.: Prediction of
flow stressand textures of AZ31 magnesium alloy at elevated
temperature.Philos. Mag. 94, 33533367 (2014)
20. Turner, P.A., Tom, C.N., Christodoulou, N., et al.: A
self-consistent model for polycrystals undergoing simultaneous
irra-diation and thermal creep. Philos. Mag. A 79, 25052524
(1999)
21. Beyerlein, I., Tom, C.: A dislocation-based constitutive law
forpure Zr including temperature effects. Int. J. Plast. 24,
867895(2008)
22. Bower, A.F., Wininger, E.: A two-dimensional finite
elementmethod for simulating the constitutive response and
microstruc-ture of polycrystals during high temperature plastic
deformation.J. Mech. Phys. Solids 52, 12891317 (2004)
23. Agarwal, S., Briant, C.L., Krajewski, P.E., et al.:
Experimental val-idation of two-dimensional finite element method
for simulatingconstitutive response of polycrystals during high
temperature plas-tic deformation. J. Mater. Eng. Perform. 16,
170178 (2007)
24. Ma, A., Roters, F.: A constitutive model for fcc single
crystalsbased on dislocation densities and its application to
uniaxial com-
pression of aluminium single crystals. Acta Mater. 52,
36033612(2004)
25. Zamiri, A., Bieler, T., Pourboghrat, F.: Anisotropic crystal
plastic-ity finite element modeling of the effect of crystal
orientation andsolder joint geometry on deformation after
temperature change. J.Electron. Mater. 38, 231240 (2009)
26. Staroselsky, A., Anand, L.: A constitutive model for hcp
materialsdeforming by slip and twinning: Application to magnesium
alloyAZ31B. Int. J. Plast. 19, 18431864 (2003)
27. Johnson, G. R., Cook W. H.: A constitutive model and data
formetals subjected to large strains, high strain rates and high
tem-peratures. In: Proceedings of the 7th International Symposium
onBallistics, The Hague (1983)
28. Zerilli, F.J., Armstrong, R.W.:
Dislocationc-mechanicsc-basedconstitutive relations for material
dynamics calculations. J. Appl.Phys. 61, 18161825 (1987)
29. Khan, A.S., Huang, S.: Experimental and theoretical study
ofmechanical behavior of 1100 aluminum in the strain rate range105
104 S1. Int. J. Plast. 8, 397424 (1992)
30. Khan, A.S., Liang, R.: Behaviors of three BCC metal over a
widerange of strain rates and temperatures: experiments and
modeling.Int. J. Plast. 15, 10891109 (1999)
31. Khan, A.S., Yu, S., Liu, H.: Deformation induced
anisotropicresponses of Ti6Al4V alloy part II: A strain rate and
temper-ature dependent anisotropic yield criterion. Int. J. Plast.
38, 1426(2012)
32. Liang, R., Khan, A.S.: A critical review of experimental
results andconstitutive models for BCC and FCC metals over a wide
range ofstrain rates and temperatures. Int. J. Plast. 15, 963980
(1999)
33. Asaro, R.J.: Crystal plasticity. J. Appl. Mech. 50, 921934
(1983)34. Asaro, R.J., Rice, J.R.: Strain localization in ductile
single crystals.
J. Mech. Phys. Solids 25, 309338 (1977)35. Hill, R.: Generalized
constitutive relations for incremental defor-
mation of metal crystals by multislip. J. Mech. Phys. Solids
14,95102 (1966)
36. Liu, X.L., Tang, Q.H., Wang, T.C.: A continuum thermal
stresstheory for crystals based on interatomic potentials. Sci.
China Phys.Mech. Astron. 57, 110 (2014)
37. Nix, F.C., MacNair, D.: The thermal expansion of pure
metals:Copper, gold, aluminum, nickel, and iron. Phys. Rev. 60,
597605(1941)
38. Tang,Q.,Wang, T., Shang, B., et al. Thermodynamic properties
andconstitutive relations of crystals at finite temperature. Sci.
ChinaPhys. Mech. Astron. 55, 918926 (2012)
39. Jiang,H.,Huang,Y.,Hwang,K.C.:Afinite-temperature
continuumtheory based on interatomic potentials. J. Eng.Mater.
Technol. 127,408416 (2005)
40. Roe, R.J.: Description of crystallite orientation in
polycrystallinematerials. III. General solution to pole figure
inversion. J. Appl.Phys. 36, 20242031 (1965)
41. Khan, A.S., Yu, S.: Deformation induced anisotropic
responsesof Ti6Al4V alloy. Part I: Experiments. Int. J. Plast. 38,
113(2012)
42. Mishin, Y., Mehl, M., Papaconstantopoulos, D., et al.
Structuralstability and lattice defects in copper: Ab initio,
tight-binding, andembedded-atom calculations. Phys. Rev. B 63,
224106 (2001)
43. Barnett, M., Keshavarz, Z., Beer, A., et al.: Influence of
grain sizeon the compressive deformation of wrought Mg-3Al-1Zn.
ActaMater. 52, 50935103 (2004)
44. Zhou, X.W., Johnson, R.A., Wadley, H.N.G.:
Misfit-energy-increasing dislocations in vapor-deposited CoFe/NiFe
multilayers.Phys. Rev. B 69, 144113 (2004)
45. Agnew, S.R., Brown, D.W., Tom, C.N.: Validating a
polycrystalmodel for the elastoplastic response of magnesium alloy
AZ31using in situ neutron diffraction. Acta Mater. 54,
48414852(2006)
123
-
348 C. Chen et al.
46. Gehrmann, R., Frommert, M.M., Gottstein, G.: Texture effects
onplastic deformation of magnesium. Mater. Sci. Eng. A 395, 338349
(2005)
47. Yoo, M.H., Lee, J.K.: Deformation twinning in h.c.p. metals
andalloys. Philos. Mag. A 63, 9871000 (1991)
48. Matsunaga, T., Kameyama, T., Ueda, S., et al. Grain boundary
slid-ing during ambient-temperature creep in hexagonal
close-packedmetals. Philos. Mag. 90, 40414054 (2010)
49. Liu, Y.,Wei, Y.: A polycrystal based numerical investigation
on thetemperature dependence of slip resistance and texture
evolution inmagnesium alloy AZ31B. Int. J. Plast. 55, 8093
(2014)
50. Knezevic,M.,McCabe, R.J., Tom, C.N., et
al.:Modelingmechan-ical response and texture evolution of -uranium
as a function ofstrain rate and temperature using polycrystal
plasticity. Int. J. Plast.43, 7084 (2013)
123
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