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InternationalJournalofPlasticity,2022.150:p.103201,https://doi.org/10.1016/j.ijplas.2021.103201.
1
Temperature effect on tensile behavior of an interstitial high
entropy alloy: Crystal plasticity modeling
Xu Zhang a,*,1, Xiaochong Lu a,b,1, Jianfeng Zhao c, Qianhua Kan a, Zhiming Li d, Guozheng Kang a,*
a Applied Mechanics and Structure Safety Key Laboratory of Sichuan Province, School of Mechanics and
Aerospace Engineering, Southwest Jiaotong University, Chengdu, 610031, China
b School of Aeronautics and Astronautics, Sichuan University, Chengdu 610065, China
c Institute of Systems Engineering, China Academy of Engineering Physics (CAEP), Mianyang, 621999, China
d School of Materials Science and Engineering, Central South University, Changsha, 410083, China
* Corresponding authors: xzhang@swjtu.edu.cn (Xu Zhang)；guozhengkang@swjtu.edu.cn (Guozheng Kang)
Homepages: https://faculty.swjtu.edu.cn/xu_zhang/; https://faculty.swjtu.edu.cn/kangguozheng
1 These authors contributed equally to this work.
Graphical abstract
Highlights
1. The crystal plasticity model well describes the tensile behavior of the interstitial
high entropy alloy at different temperatures.
2. The temperature effects on the yield stress and strain hardening ability are further
revealed by constitutive modeling.
3. The strength-ductility map is predicted to explore performance optimization by
adjusting the temperature and grain size.
InternationalJournalofPlasticity,2022.150:p.103201,https://doi.org/10.1016/j.ijplas.2021.103201.
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Abstract
Previous tensile tests showed that the typical coarse-grained (CG) interstitial high
entropy alloy (iHEA) with the nominal composition Fe49.5Mn30Co10Cr10C0.5 (at.%) has
higher yield stress and better strain hardening at cryogenic conditions compared to that
at room temperature. However, the yield stress of the fine-grained (FG) iHEA is little
influenced by decreasing temperature, while the strain hardening is significantly
enhanced. The fundamental reasons for these observations need to be further
investigated. Thus, a micromechanism-based crystal plasticity model is developed to
investigate the temperature effect on the tensile behavior of the iHEA. A
thermodynamic model is established to calculate the stacking fault energy of the iHEA
at different temperatures. The developed constitutive model is verified by comparing
the simulated the stress-strain curves and martensite volume fraction of the CG and FG
iHEAs at different temperatures with the corresponding experimental results. Moreover,
the contributions of different strengthening mechanisms, such as dislocations, grain
boundaries, nano-precipitation, and lattice friction, are quantified to reveal the effect of
temperature on the iHEA’s yield stress. Furthermore, the grain size effects on
deformation twinning and martensite phase transformation are considered. The
developed constitutive model is further applied to predict the stress-strain curves of the
iHEA with various grain sizes at different temperatures. The present study thus provides
a useful modeling tool for understanding the underlying mechanisms of the strength
and plasticity in the iHEA, paving a way for optimizing the mechanical properties of
advanced alloys at various temperatures.
Keywords: High entropy alloy; crystal plasticity finite element method; temperature
effect; strength and ductility
1. Introduction
During service, alloys are inevitably subject to the environment with extremely low
temperatures, such as the metallic devices in submarines, satellites, Mars rovers, and
superconducting magnets, etc. The temperature significantly affects alloys’ deformation
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behavior (yielding, strain hardening and fracture) (Li et al., 2021; Wu et al., 2021). Most
conventional alloys lose a large amount of plasticity at low temperatures (Benac et al.,
2016). Such ductile-to-brittle transition of fracture at low temperatures potentially leads
to serious accidents. For example, the frigid seawater made the steel abnormally brittle
and less impact-resistant, causing the big hole and rapid sinking of the Titanic ship.
Thus, investigating alloys’ deformation behavior and mechanism at different
temperatures (especially low temperatures) is very important to evaluate their service
safety (Li et al., 2020; Zhang et al., 2020). It is also a burning issue to break the strength-
ductility trade-off in alloys at cryogenic temperatures. In recent years, high entropy
alloys (HEAs) first named by Yeh et al. (2004) have drawn much attention due to their
excellent mechanical properties (Diao et al., 2017; George et al., 2019; Ye et al., 2016;
Zhang et al., 2018). The most obvious difference between HEAs and conventional
alloys is that HEAs consist of multi-principal elements. Therefore, high mixing entropy
can facilitate the formation of stable solid solutions with single-phase or dual-phase
microstructures (George et al., 2020). Such design concept enables HEAs to achieve
satisfactory performances by composing different elements and adjusting the atomic
proportions over a wide range (Gao et al., 2016; Li and Raabe, 2017).
As a benchmark, the equiatomic CrMnFeCoNi developed by Cantor et al. (2004) has
been extensively studied at cryogenic temperatures (Gali and George, 2013; Gludovatz
et al., 2014; Laplanche et al., 2016). The tensile tests showed that the yield stress of the
CrMnFeCoNi HEA doubles and the ductility increases to >60% at 77 K compared to
that at room temperature, exhibiting a good combination of high strength and
considerable fracture toughness in the cryogenic environment (Gludovatz et al., 2014).
The strain hardening ability was also enhanced at 77 K, caused by the more extensive
generation of twin laminates during deformation at cryogenic temperature compared to
that at room temperature (Laplanche et al., 2016). The nano-twins led to the “dynamic
Hall-Petch effect”, improving strain hardening and ductility (De Cooman et al., 2018).
Such simultaneous increases of strength and ductility at cryogenic temperatures were
also observed in quaternary FeNiCoCr, FeNiCoMn, NiCoCrMn HEAs (Wu et al., 2014).
Wang et al. (2018) demonstrated that the decrease of stacking fault energy (SFE)
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improved the probability of forming stacking faults (SFs) and deformation twins in
FeNiCoCr HEA, the dislocation-twin and twin-twin intersections enhanced the
strength-ductility synergy at cryogenic temperatures. Moreover, Naeem et al. (2020)
revealed that the CoCrFeMnNi HEA had extraordinarily high ultimate strength of ~2.5
GPa and high ductility of ~62 % at 15 K. They revealed that the extreme strain
hardening ability was attributed to the cooperation of different plastic mechanisms,
such as dislocation slip, SFs and deformation twinning. Similarly, Liu et al. (2019)
revealed that deformation twinning together with phase transformation at extremely
low temperature (≤ 20 K) produced an adequate strain hardening ability of CoCrFeNi
HEA. The above studies indicated that HEAs have better cryogenic mechanical
properties than conventional alloys. Thus, HEAs have a great potential for engineering
applications in low-temperature environments (Miracle et al., 2014).
Based on equiatomic HEAs, non-equiatomic HEAs with metastable phase have
appeared and greatly broadened the variety of HEAs (Huang et al., 2017; Huang et al.,
2018; Li and Raabe, 2017; Pradeep et al., 2015). Besides, interstitial elements have
been added into metastable HEAs to improve their mechanical properties further. Li et
al. (2017) reported the Fe49.5Mn30Co10Cr10C0.5 (at.%) interstitial HEA (iHEA) with joint
twinning and transformation induced plasticity (TWIP and TRIP) effects. The initial
face-centered cubic (FCC) structure of the iHEA partially transforms to the hexagonal
closed-packed (HCP) structure during plastic deformation. Thus, the iHEA has better
strain hardening and cyclic hardening abilities than the equiatomic HEAs and
conventional alloys (Li et al., 2017; Lu et al., 2020). Moreover, incorporating carbon
atoms generated the interstitial solid solution strengthening and carbide precipitation
strengthening in the iHEA. The activations of multiple strengthening and plastic
mechanisms bring the iHEA an excellent balance of strength and ductility at room
temperature. In addition, Wang et al. (2019) investigated the cryogenic tensile
deformation of the coarse-grained (CG, ~100 m) and fine-grained (FG, ~6 m) iHEAs
at 77 K. For the CG iHEA, both the yield stress and strain hardening ability at 77 K
were improved compared to that at room temperature. The microscopic
characterizations showed that plentiful martensite phases were generated and almost no
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twin occurred at 77 K (Wang et al., 2019), indicating that the TRIP effect is enhanced
and the TWIP effect is inhibited at low temperature due to the decreased SFE. For the
FG iHEA, the strain hardening ability was also improved by the strong TRIP effect at
77 K, but it is abnormal that the yield stress is almost unchanged with decreasing the
temperature (Wang et al., 2019). Besides, both the fracture strains of the CG and FG
iHEA at 77 K were less than those at 293 K. These phenomena could not be fully
explained by the experimental investigations, which should be further revealed by
theoretical analysis.
A feasible constitutive model can describe the deformation behaviors of materials
and help reveal the underlying mechanisms. For example, constitutive modeling has
been applied to the compressive deformation of the CoCrFeMnNi HEA (Jang et al.,
2017), and the tensile deformation of the AlCoCrFeNi2.1 (at.%) HEA (Wang et al., 2021).
These studies referred to the constitutive model of conventional alloys, such as TWIP
steel (Bouaziz et al., 2011) and TRIP steel (Wong et al., 2016), but the temperature
effect was not considered. Moreover, Patnamsetty et al. (2020) developed a constitutive
model using hyperbolic-sinusoidal functions to describe the hot deformation behavior
(1023~1423 K) of the CoCrFeMnNi HEA. This model took account of the temperature
effect but was too phenomenological. It is certain that macroscopic phenomenological
models can describe the deformation behavior of materials, and the related calculation
can be more time-saving compared to that by physics-based constitutive models.
However, the intrinsic effect of plastic deformation mechanisms cannot be reflected by
these phenomenological models. This shortage is adverse to the further understanding
of the relationship between the microstructures and mechanical behavior, and restricts
the performance optimization based on microstructure adjustments. Among various
constitutive models, the crystal plasticity model has an obvious advantage in relating
the macroscopic behaviors of polycrystalline materials with their single-crystal
deformation responses and microstructural evolutions (Roters et al., 2010). Many
efforts have been paid to develop the temperature-dependent crystal plasticity models
for different materials with multiple plastic mechanisms (Connolly et al., 2020;
Ghorbanpour et al., 2020; Li et al., 2019; Liang et al., 2019; Liu et al., 2017; Yang et
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6
al., 2020), which can help introduce the temperature effect in the crystal plasticity
model of the iHEA.
The present investigation is undertaken with three major objectives using crystal
plasticity modeling: (1) describing the tensile deformation behaviors of the CG and FG
iHEA at different temperatures; (2) explaining the intrinsic reasons for the temperature
insensitivity of the FG iHEA’s yield stress, the enhanced martensite phase
transformation and the decrease of ductility at cryogenic temperature; (3) predicting the
strength and ductility of the iHEA with different grain sizes at different temperatures.
This paper is structured as follows: first, the crystal plasticity modeling and simulation
methods of the iHEA are expounded (Section 2); next, the finite element simulation
results are presented to verify the accuracy of the constitutive model (Section 3); next,
the cryogenic mechanical properties of the iHEA are further discussed based on the
analysis of the simulations results, and the possible model-guided strength-ductility
optimization is explored (Section 4); finally, some conclusions obtained from this study
are given (Section 5). The flowchart of the article structure is shown in Fig. 1.
Fig. 1 The article structure.
InternationalJournalofPlasticity,2022.150:p.103201,https://doi.org/10.1016/j.ijplas.2021.103201.
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2. Modeling and simulation methods
2.1 Thermodynamic calculation of stacking fault energy
Nucleations of twin and martensite in the iHEA depend on the same dislocation
reaction (Idrissi et al., 2010), namely the dissociation of a perfect dislocation on the
{111} plane into two Shockley partial dislocations (Hirth et al., 1983)
01 1 10 1 3 112
22 6
aa a
〈〉〈〉 〈〉
, (1)
where 0.362 nmais the lattice constant for the iHEA (Lu et al., 2020). These two
Shockley partials are connected by an SF (He et al., 2020). Experimental studies
showed that the formations of twin and
-martensite are related to the overlapping nano-
sized SFs (Fujita and Ueda, 1972). The separation distance of the two Shockley partials
determines the value of SFE. With the variation of deformation temperature, SFE also
changes, which influences the activations of TWIP and TRIP effects. Therefore,
studying the evolution of SFE with temperature should be done prior to modeling the
plastic deformation mechanisms of the iHEA.
Based on the fact that a single SF can be regarded as a two-layer HCP structure
inserted in the FCC matrix, the SFE in FCC lattice can be thermodynamically calculated
by (Olson and Cohen, 1976):
/
sf 2
A
41
22,
3
GaN
, (2)
where
is the molar surface density along {111} planes; 23 1
A6.022 10 molN
is the Avogadro’s constant; /
is the interfacial energy per unit area of the boundary
between
-austenite and
-martensite phases. Olson and Cohen (1976) proposed the
possible temperature dependence of interfacial energy. They considered that /
was
between 5 ~ 15 mJ/m2 when the temperature increases from 0 to 500 K. Saeed-Akbari
et al. (2009) demonstrated that /
has an inverse linear relationship with the
temperature:
//
0sf
kT
, (3)
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where /
0
is the interfacial energy at 0 K; T is the absolute temperature; sf
k is the
change in interfacial energy per temperature.
The term G
in Eq. (2) is the change in molar Gibbs free energy during
phase transformation, which is calculated by (Curtze and Kuokkala, 2010; Lee
et al., 2019):
ch mg
ΔΔΔGGG
, (4)
where ch
G
is the thermochemical molar free energy difference between
and
phases (Xiong et al., 2014); mg
G
is the magnetic contribution to the change in the
Gibbs free energy due to the Néel transition between
and
phases (Curtze et al., 2011).
The processes of calculating ch
G
a n d mg
G
are summarized in Fig. 2. The
parameters used in this thermodynamic calculating approach are listed in Table 1.
Fig. 2 Flow chart showing a modified thermodynamic approach for calculating the
InternationalJournalofPlasticity,2022.150:p.103201,https://doi.org/10.1016/j.ijplas.2021.103201.
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stacking fault energy of the iHEA.
Table 1 Thermodynamic parameters and functions for calculating the stacking fault
energy of the iHEA.
Symbol Function Reference
Fe J/molG
2243.38 4.309T
(Allain et al., 2004)
Mn J/molG
1000 1.123T (Allain et al., 2004)
Co J/molG
427.59 0.615T (Achmad et al., 2017)
Cr J/molG
2284 0.163T (Achmad et al., 2017)
CJ/molG
62 83
4.88 0.01ln 135400 /
3.3 10 / 9 10 /
TT T
TT
(Wan et al., 2001)
FeMn J/mol
Fe Mn
2873 717
x
x
(Dumay et al., 2008)
FeCo J/mol
2
Fe Co
13968.75 3528.8 xx
(Achmad et al., 2017)
FeCr J/mol
2095(Yakubtsov et al., 1999)
FeC J/mol
42500(Allain et al., 2004)
MnCo J/mol
Mn Co
2756 1657
x
x
(Achmad et al., 2017)
MnCr J/mol
19088 17.5423T (Dinsdale, 1991)
MnC J/mol
26910(Xiong et al., 2014)
CoCr J/mol
Co Cr
7341.73 7.93
4621.59 7.32
Tx x
T
(Achmad et al., 2017)
CoC J/mol
Unavailable
CrC J/mol
11977 6.8194T (Lee, 1992)
Fe Mn Co
Cr F CeMn
0
4
0.62 1.35
0.8 0.
7
6
.
4
xx
x
x
x
xx
(Dumay et al., 2008)
modified
Mn C
0.62 4
x
x
(Dumay et al., 2008)
InternationalJournalofPlasticity,2022.150:p.103201,https://doi.org/10.1016/j.ijplas.2021.103201.
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Neel KT
Mn Cr
Co C
251.71 681 272
1396 1740
x
x
xx
(Hillert and Jarl, 1978)
modified
Neel KT
Mn
580
x
(Allain et al., 2004)
/2
0mJ/m
15.05This work
2
sf mJ/m /Kk-0.02662This work
J/K/molR8.314472(Mohr et al., 2008)
p 0.28(Li and Hsu, 1997)
The SFEs of the iHEA at different temperatures (50~500 K) are shown in Fig. 3 (a).
It can be seen that the SFE increases linearly with increasing the temperature when the
temperature is > 300 K. However, when the temperature is < 300 K, the SFE has a
nonlinear relationship with the temperature. The change of SFE during 50~300 K is
smaller than that during 300~550 K. Similar tendencies are also observed in Fig. 3 (b),
which gives the Gibbs free energy’s change G
and the respective contribution
from chemical and magnetic parts. At cryogenic temperatures, ch
G
is negative
while mg
G
is quite large. With increasing the temperature, mg
G
is gradually
close to zero and ch
G
becomes the predominant part of the change in Gibbs free
energy.
The free energy difference between the austenite and martensite phase G
determines the stability of the initial austenite phase. With decreasing G
, t h e
metastability of the FCC phase is increased; thus, the phase transformation is more
likely to be activated at low temperatures. For the FCC alloys, it is accepted that
martensite transformation is activated in the low range of SFE (< 20 mJ/m2),
deformation twin can be observed in the medium range of SFE (20 ~ 40 mJ/m2), and
dislocation slip dominates the plastic deformation in the high range of SFE (> 40 mJ/m2)
(Allain et al., 2004; De Cooman et al., 2018). From Fig. 3 (a), the SFE of the iHEA is
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11
close to 20 mJ/m2 at room temperature (293 K). At this value of SFE, the transition
between TRIP and TWIP occurs. Therefore, the iHEA can have joint TWIP and TRIP
effects during room-temperature deformation (Li et al., 2017).
0 50 100 150 200 250 300 350 400 450 500 550
0
10
20
30
40
50
Dislocation slip + TRIP
Dislocation slip + TWIP
Stacking fault energy (mJ/m
2
)
Temperature (K)
(a)
Dislocation slip
0 50 100 150 200 250 300 350 400 450 500 550
-800
-600
-400
-200
0
200
400
600
800
Gibbs free energy change (J/mol)
Temperature (K)
G
→
(total energy)
G
→
ch
(chemical part)
G
→
mg
(magnetic part)
(b)
Fig. 3 (a) Theoretically predicted stacking fault energy as a function of temperature for
the iHEA based on the thermodynamic approach; (b) Variation of Gibbs free energy
change G
with temperature, including chemical and magnetic parts of G
.
2.2 Crystal plasticity model
2.2.1 Finite deformation framework
With the introduction of intermediate configuration (Lee, 1969), the multiplicative
decomposition of total deformation gradient tensor F can be expressed by (Asaro and
Rice, 1977; Kröner, 1959)
ep
FFF, (5)
where e
F
is the elastic deformation gradient tensor describing the rotation and stretch
of the lattice structure from the intermediate configuration to the current configuration.
Following the elastic stress-strain relation, the second Piola-Kirchhoff stress tensor S
in the intermediate configuration can be expressed in terms of e
F as
T
ee
1
2
SFFI： ， (6)
where is the fourth-order anisotropic elasticity tensor, and I denotes the second-
order identify tensor. For the FCC lattice, the elasticity tensor depends on three
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independent elastic constants 11
C, 12
C, and 44
C. For the HCP lattice, the matrix of
consists of 11
C, 12
C, 13
C, 33
C, and 44
C.
In Eq. (5),
p
F is the plastic deformation gradient tensor describing the
irrecoverable deformation, which maps the material point in the reference configuration
to the intermediate configuration. The evolution rate of
p
F follows (Asaro and
Needleman, 1985)
p
pp
FLF
, (7)
where
p
L is the plastic velocity gradient tensor in the intermediate configuration.
Owing to that multiple plastic deformation mechanisms can be activated in the iHEA,
the total plastic deformation gradient tensor should consider the contributions of
dislocation slip, deformation twinning, and martensite phase transformation (Kalidindi,
1998; Lu et al., 2020; Madivala et al., 2018; Wong et al., 2016)
stwtr
ptwtr
111
1
NN N
ff
Lnsnsns
, (8)
where
tw
tw
1
N
f
f
and
tr
tr
1
N
f
f
(
f
and
f
are the volume fractions in the
twin system
and the transformation system
are the total volume fractions of the
twinned region and martensite phase, respectively; stwtr
12NN N
are the
numbers of slip/twinning/transformation systems in FCC lattice; n and s are the two
orthogonal unit vectors representing the normal and shear directions of each
slip/twin/transformation system, the symbol
represents the dyadic product of two
vectors;
,
, and
are the shear strain rates provided by the corresponding
plastic mechanisms. It should be mentioned that the dislocation slip in twins and
martensites is neglected in this model, owing to that the twin fraction is relatively low
in the iHEA (Wang et al., 2019), and the martensite has relatively high slip resistance
and acts as a hard phase (Liang et al., 2018). The calculations of
,
, and
are
described as follows.
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2.2.2 Kinematics of dislocation slip
The slip shear rate of the slip system
based dislocation glide is expressed by a
modified Orowan equation (Orowan, 1934; Wong et al., 2016)
pGB
s
0
Bpref
exp 1 sign
q
p
Q
bv bv kT
, (9)
where is the McCauley’s bracket;
is the current dislocation density of the
slip system
; 0.256 nmb
is the magnitude of the Burgers vector of perfect
dislocations in the iHEA (Lu et al., 2020); 4
0110 m/sv
is the reference dislocation
slip velocity (Kocks and Mecking, 2003); 19
s3.5 10 JQ
is the activation energy
for dislocation slip (Wong et al., 2016); 23
B1.38 10 J/Kk
is the Boltzmann
constant (Callister Jr and Rethwisch, 2020); p and q are the material constants related
to the obstacle profile;
is the resolved shear stress on the slip system
, which is
the driving force for dislocation slip and calculated by Schmid law (Mark et al., 1923a,
b)
p
:
Mn s
. (10)
where
p
M is the Mandel stress defined from the second Piola-Kirchhoff stress tensor
S as T
p
ee
MFFS
(Roters et al., 2019).
In Eq. (9),
p
, GB
,
p
re
and f
are respectively the slip resistances caused by
the forest dislocations, the grain boundaries, the carbide precipitations, and the lattice
friction. The expression of each term is described below.
(1) The glide of mobile dislocations can be impeded by the immobile forest
dislocations penetrating the current slip plane. The dislocation interactions lead to either
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14
jogs or Lomer locks (Hull and Bacon, 2011). According to the classical strengthening
theory proposed by Taylor (1934), the dislocation passing stress
p
is expressed by
p
1
s
N
Gb
, (11)
where 76 GPaG is the shear modulus of the iHEA (Zhu et al., 2019);
characterizes the interaction between the dislocations on the slip systems
and
,
including self-hardening
0.122
, coplanar
0.122
, collinear
0.625
, orthogonal
0.07
, glissile
0.137
, and sessile
0.122
types (Kubin et al., 2008). The initial dislocation density 0
determines the initial value of
p
and contributes to the yield stress. During the plastic
deformation,
p
improves with increasing the dislocation density, which contributes
to the strain hardening. The evolution rate of dislocation density can be calculated by
(Mecking and Kocks, 1981)
1/
0
s0
1n
k
b
, (12)
where the two terms in the square brackets describe the multiplication and annihilation
of dislocations; 0
k
is a material constant controlling the dislocation annihilation rate;
0
is the reference shear rate; n characterizes the strain rate sensitivity of annihilation
process, and is inversely proportional to the absolute temperature as r/nTT (Baik
et al., 2003; Yu et al., 2018), where r
T is the reference temperature; s
represents
the mean free path (MFP) of mobile dislocations and is calculated by (Lu et al., 2020)
11 1
sgrains tw tw tr tr
Dislocation
GB Twin Martensite
stw tr
111 1 1
11
NN
Nff
di t f tf
, (13)
where grain
d is the grain size; s
i is a fitting parameter controlling the space between
the forest dislocations;
describes the interaction between the slip system
and the
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twin system
, containing coplanar and non-coplanar types; similarly,
describes
the interaction between the slip system
and the transformation system
; tw
t and
tr
t are the average thicknesses of twin and martensite lamellas, respectively. From Eq.
(13), it can be seen that the dislocation MFP has the combined impact of the grain
boundaries (GBs), forest dislocations, twin boundaries (TBs), and martensite interfaces.
(2) The yield stresses of metals are dependent on the grain size, which is well-known
as the Hall-Petch relationship (Hall, 1951; Petch, 1953). The GBs provide strong slip
resistance for dislocations, especially in nano-grained materials. In this study, a
temperature-dependent GB
is introduced to reflect such grain size effect, which is
calculated by
HP
GB
grain
kT
d
, (14)
where
HP
kT
is the temperature-dependent Hall-Petch coefficient and expressed in
a linear form Sun et al. (2019)
HP
kATB
, (15)
where A and B are fitting parameters.
(3) Through microstructure characterizations, Li et al. (2017) and Wang et al. (2019)
revealed that the nano-sized precipitations M23C6 exist in the FG iHEA matrix but not
in the CG iHEA. Therefore, precipitation strengthening should be additionally
considered for the FG iHEA. According to the Orowan bypass mechanism (Gottstein,
2013), when the dislocations meet the carbide barrier, they bow out between the
particles and pass through them (Chatterjee et al., 2021; Fomin et al., 2021). The critical
stress
p
re
needed to make a dislocation fully bow out is calculated by (Gottstein, 2013)
p
re
pre
pre
Gb f
r
, (16)
where
p
re
f
is the volume fraction of carbide particles;
p
re
r is the particle radius.
(4) Owing to the radius difference between the C atom and other metal atoms, the
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introduction of C leads to great lattice distortion in the iHEA. The lattice friction
provides a considerable contribution to the slip resistance of dislocations. As
demonstrated by Wu et al. (2014), the lattice friction f
has an inverse relationship
with the temperature
0
ff0
2
exp T
b
, (17)
where 0
is the dislocation width at 0 K;
is a small positive material constant; f0
is the lattice friction at 0 K, which can be expressed by (Wu et al., 2014)
0
f0
2
2exp
1
G
vb
, (18)
where v is the Poisson’s ratio.
2.2.3 Kinematics of deformation twinning
The shear rate of each twin system
is given by
tw
f
, (19)
where tw 2/2
is the characteristic shear strain induced by the deformation
twinning process;
f
is the evolution rate of twin volume fraction as
tw tr tw
1
f
ffVN
, (20)
where V
is the volume of a new twin and calculated by (Steinmetz et al., 2013)
2
tw tw
4
Vt
, (21)
where tw
t is the average twin thickness, tw
is the MFP for twin growth, which
should consider the effects of GBs and TBs as
tw
1
tw grain tw tw
11 1
1
Nf
dtf
, (22)
where
characterizes the interaction between the twin systems
and
,
including coplanar and non-coplanar types.
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The term tw
N
in Eq. (20) is the twin nucleation rate and given by
tw 0 tw ncs
NNpp
, (23)
where 0
N
is the number density of the potential twin nuclei per unit time, tw
p
is the
probability that a twin nucleus bows out to from a twin lamella as
tw
tw
ˆ
exp
C
p
, (24)
where C is a fitting parameter determining the sharpness of the transition from the non-
twinned region to the twinned one (Steinmetz et al., 2013); The resolved shear stress
(driving force)
in the twin system
is calculated by
p:
Mn s
. (25)
The critical stress tw
ˆ
for deformation twinning is calculated according to a
modified Mahajan-Chin nucleation model (Mahajan and Chin, 1973)
p
grain
sf tw
tw
ptw p ref
3
ˆexp
33
Gb d
E
bL b d
, (26)
where p/ 3 0.147 nmbb is the magnitude of the Burgers vector of partial
dislocations. tw
L
is the length of the sessile partials forming a new twin nucleus. The
first term in Eq. (26) represents the stress needed to overcome the attractive force
between the partials, because the leading Shockley partial should be separated from the
trailing one to form a twin embryo (Mahajan and Chin, 1973; Steinmetz et al., 2013).
The second term in Eq. (26) describes that the mobile partials should form a semicircle
between the two pinning points to promote twin growth (Wong et al., 2016). Moreover,
grain size truly has an influence on the nucleation of twins and martensites (Gutierrez-
Urrutia et al., 2010; Sun et al., 2018; Zhu et al., 2013). The effect of grain size on the
critical driving force of deformation twinning and martensite phase transformation is
usually described as an exponential form (Lee and Choi, 2000; Takaki et al., 1993), as
expressed by the third term in Eq. (26). tw
E
is a fitting parameter representing the
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18
characteristic energy barrier for deformation twinning, and ref
d is the reference grain
size.
Moreover, the possibility that dislocation cross-slip occurring should be excluded in
the calculation of nucleation rate, as the term ncs
p
in Eq. (23) described. Sufficient
local stress and dislocations are needed to nucleate twin and martensite. However,
cross-slip can decrease the number of piled-up dislocations and relief the local stress
concentration (Steinmetz et al., 2013). The probability
ncs
p
of no cross-slip occurring
is expressed by (Groh et al., 2009; Nix et al., 1985)
cs
ncs r
1exp ,
B
V
pkT
, (27)
where 29 3
cs 1.67 10 mV
is the activation volume for dislocation cross-slip (Wong
et al., 2016); r
is the stress needed to bring two partial dislocations to form the twin
nucleus without the help of externally applied stress (Mahajan and Chin, 1973)
r
0c 0
cos / 3
22
Gb
Gb
xx x
, (28)
where 9
c110 mx
is the extreme critical distance between two partial dislocations
to form a twin nucleus (Steinmetz et al., 2013), 0
x
is the equilibrium separation of two
partials. For the FCC lattice, 0
x
is expressed by (Gottstein, 2013)
2
0
sf
2
81
Gb v
xv
. (29)
2.2.4 Kinematics of martensite phase transformation
Similar to the deformation twinning, the shear rate for each transformation system
is given by
tr
f
, (30)
where tr 2/2
is the characteristic shear strain induced by the phase
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transformation process;
f
is the evolution rate of martensite volume fraction as
(Wong et al., 2016)
tw tr tr
1
f
ffVN
, (31)
where V
is the volume of a new martensite, which is calculated by
2
tr tr
4
Vt
, (32)
where tr
t is the average martensite thickness, tr
is the MFP for martensite growth,
which is blocked by the GBs and existing martensites as
tr
1
tr grain tr tr
11 1
1
Nf
dtf
, (33)
where
characterizes the interaction between the transformation system
and
, including coplanar and non-coplanar types.
The term tr
N
in Eq. (31) is the martensite nucleation rate in the transformation
system
, which is given by
tr 0 tr ncs
NNPP
, (34)
where tr
p is the probability that a nucleus bows out to form a martensite as
tr
tr
ˆ
exp
D
p
, (35)
where D is a fitting parameter determining the sharpness of the transition from the
austenite phase to the martensite phase (Wong et al., 2016);
is the resolved shear
stress (driving stress) in the phase transformation system
and calculated by
p:
Mn s
. (36)
According to the model originally proposed by Mahajan et al. (1977), the critical
stress tr
ˆ
for martensite phase transformation is calculated by (Takaki et al., 1993;
Wong et al., 2016)
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20
/
pgrain
tr
tr
ptr p p ref
3
2
ˆexp
333
Gb d
E
hG
bL b b d
, (37)
where tr
L is the length of the sessile partials forming a new martensite nucleus; tr
E
is a fitting parameter representing the characteristic energy barrier for martensite phase
transformation; h is the thickness of the martensite nucleus. Owing to that the lattice
constant of the iHEA is 0.362 nm, the interplanar spacing of the {111} plane is
calculated by 222
{111} / 1 1 1 0.21 nmda (Wong et al., 2016). Thus, the
thickness {111}
51.05nmhd .
The FCC and HCP lattices are both close-packed with a 74% atomic packing factor
(26% void space in unit cell), 12 nearest neighbors, and the same interstitial sites
(Gottstein, 2013). Owing to the similar packing density between the HCP martensite
and FCC matrix, the phase transformation only accompanies a slight volume change
(Moyer and Ansell, 1975). Such a volume change plays a weak role in the total plastic
deformation (Bhadeshia, 1997; Zaera et al., 2012). Thus, the volume change associated
with the FCC to HCP transformation is not considered in this crystal plasticity
constitutive model. The numerical integration procedure of the constitutive equations
described above and the updating scheme of the microstructure internal variables are
elaborated in Appendix A and B.
2.3 Finite element model and boundary conditions
A polycrystalline representative volume element (RVE) constructed by the Voronoi
tessellation was used in the CPFEM simulation. Such RVE contains 50 grains with
random orientations, as shown in Fig. 4 (a). The generating and meshing processes were
performed by the open-source software Neper (Quey et al., 2011). The validation of
cubic polycrystalline RVE showed that the simulated stress-strain curves were nearly
the same for the models with 64, 100, 216 and 512 grains (Zhang et al., 2015). Thus,
50 grains were enough to simulate the macroscopical homogenized deformation
response of the polycrystalline alloy. The tensile simulations were conducted in the
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21
commercial finite element software Abaqus 6.14/CAE (Dassault Systemes Simulia
Corporation). The developed crystal plasticity model described in Section 2.2 was
implemented into finite element method through the Düsseldorf Advanced Material
Simulation Kit (DAMASK) (Roters et al., 2019; Roters et al., 2012)
Fig. 4 (a) Polycrystalline cubic representative volume element (RVE) contains 50
grains with random orientations. (b) Schematic of the boundary conditions applied to
the RVE.
For simulating the uniaxial tensile deformation, appropriate boundary conditions
should be applied to the RVE. For simplification, the displacement degrees of freedom
along the X-axis, Y-axis and Z-axis are expressed by X
U, Y
U, and
Z
U. As shown in
Fig. 4 (b), the displacive load along the Z-axis is applied to point A chosen as the loading
point.
Z
Uof point A is coupled with those of all other nodes on face EHDA. Thus, the
overall engineering strain-stress curve of the RVE can be obtained by extracting the
reaction force and displacement of point A. Moreover, to prevent rigid body
displacements during tensile deformation, 0
Z
U
is set for face FGCB as the fixed
surface, X0U and Y0U are applied to line BF and BC, respectively. The tensile
strain rate in the simulations is 0.001 s−1, the same as that used in the experiment.
Temperature is specified for each element as a predefined field in simulation. During
deformation, the temperature keeps constant, and no thermal diffusion is considered.
Generally, the simulation results of the polycrystalline RVE display a significant
mesh sensitivity (Lim et al., 2019), a finer mesh results in better accuracy. However,
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22
overmuch elements will lead to unnecessary computing costs. Therefore, the influence
of element number on the simulation results should be investigated to choose an
appropriate mesh. As shown in Fig. 5, the uniaxial tensile deformations of six models
with different numbers of C3D8 (linear interpolation, 2×2×2 Gauss points) elements
were simulated, and all other conditions of simulations were the same. The engineering
stress-strain curves show that the difference becomes smaller and smaller with
increasing the element number. From 3375 to 8000 elements, the stress-strain curves
are nearly overlapped, and the variation of the maximum stress is within 8 MPa. Thus,
for obtaining a balance of calculation cost and accuracy, 3375 (15 × 15 × 15) elements
are used to mesh the polycrystalline model.
0 1020304050
0
150
300
450
600
750
900
Engineering stress (MPa)
Engineering strain (%)
216 elements
512 elements
1000 elements
3375 elements
5832 elements
8000 elements
(b)
Fig. 5 (a) Morphologies and (b) simulated engineering stress-strain curves of the
polycrystalline finite element models with different mesh numbers.
2.4 Parameters of the constitutive model
SFE is the key parameter to describe the temperature effect on the activation of
plastic mechanisms. Based on the thermodynamic approach described in Section 2.1,
the SFEs of the iHEA at 77 K and 293 K can be calculated, as shown in Table 2. In
addition, the Gibbs free energy change G
and interfacial energy /
between
the FCC and HCP phases are also given, which are essential to calculate the critical
stress for martensite phase transformation. The calculated iHEA’s SFE at room
temperature is about 18.04 mJ/m2, close to the value estimated by Su et al. (2019) as 18
mJ/m2.
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Table 2 Thermodynamically calculated stacking fault energy, Gibbs energy change, and
interfacial energy of the iHEA at 77 K and 293 K.
Temperature (K) sf
(mJ/m
2) G
(J/mol) /
(mJ/m2)
293 18.04 60.44 7.25
77 13.38 215.58 13
Moreover, the parameters relevant to the strengthening mechanisms are listed in
Table 3. These parameters were obtained from the existing literature or by fitting the
experimental value of yield stress (elasticity limit). The shear modulus is adopted from
the previous study of iHEA (Zhu et al., 2019). It has been indicated that the shear moduli
of the alloys have negative linear dependencies with temperature (Juuti et al., 2018; Wu
et al., 2014). However, Haglund et al. (2015) found that the temperature dependence of
the shear modulus of the FCC HEA was weaker than that of pure metal; thus, the
uncharacteristic temperature dependence of the shear modulus is not considered here.
The initial dislocation density of the as-homogenized CG iHEA is assumed to be
12 2
210 m
. By contrast, very few dislocations are observed in the thermo-
mechanically treated FG iHEA (Wang et al., 2019). Thus, the initial dislocation density
in the FG iHEA is assumed to be 10 2
110 m
. According to the experimental yield
stress (Wang et al., 2019), the Hall-Petch slopes are fitted as 0.5
918.01MPa μm
at
293 K and 0.5
629.74MPa μm
at 77 K. By dividing the Taylor factor 3.06M, the
Hall-Petch coefficients HP
k in the slip system level are 0.5
300.0033 MPa μm
at
293 K and 0.5
205.7974 MPa μm
at 77 K. Therefore, the fitting parameters A and B
in Eq. (15) can be determined. The average radius and volume fraction of carbide
precipitations are estimated from the characterization results by Wang et al. (2019) and
Li et al. (2017). Finally, Ohsawa et al. (1994) proposed that a reasonable approximation
of lattice friction could be obtained if 0
/b
ranged from about 0.4 to 2 (i.e.
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24
0
0.4 2bb
). For a better fitting effect, the dislocation width is set to be 1.134b, and
0.00195
.
Table 3 Parameters for the strengthening mechanisms of the iHEA.
Description Symbol Value
Initial dislocation density 0
CG: 2 × 1012 m-2
FG: 1 × 1010 m-2
Fitting parameters in the Hall-Petch
coefficient
A 0.5 1
0.436138 MPa μm K
B 0.5
MP172.21 a μm48
Precipitation volume fraction
p
re
f
1.5%
Precipitation radius
p
re
r 50 nm
Dislocation width in Burgers vector 0
/b
1.134
Fitting parameter in lattice friction
0.00195
The parameters related to plastic mechanisms are listed in Table 4. These parameters
were obtained from the existing literature or by fitting the experimental results,
including stress-strain curves and evolution of martensite volume fractions. The
exponents p/q in glide velocity were adopted from the previous study (Lu et al., 2020).
The exponents C and D in the twin and martensite formation probabilities were adopted
from the literature (Wong et al., 2016). The average thicknesses of twins and
martensites were estimated according to the microscopic characterizations (Wang et al.,
2019). The lengths of twin and martensite nuclei were determined from the previous
studies (Lu et al., 2020; Madivala et al., 2018; Steinmetz et al., 2013; Wong et al., 2016).
The reference shear rate was adopted from the literature (Baik et al., 2003). The
parameters s
i , 0
k and r
T were obtained by fitting the experimental stress-strain
curves. The parameters tw
E and tr
E are determined by fitting the start of
deformation twinning and martensite phase transformation. Finally, the interaction
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25
coefficients
and
were obtained by fitting the experimental martensite
volume fractions, and the interaction coefficients
and
for twi ns are the same
as that of martensites.
Table 4 Parameters for the plastic mechanisms of the iHEA.
Description Symbol Value
Exponent in glide velocity p/q 1.0/1.15
Parameter controlling forest dislocation
mean spacing s
i 12
Parameter controlling the dislocation
annihilation rate 0
k 4.8
Reference temperature r
T 1800 K
Reference shear rate 0
1.0 s-1
Average thickness of twins tw
t 50 nm
Length of twin nuclei tw
L 128 nm
Characteristic energy barrier for deformation
twinning tw
E 0.01 J/m2
Reference grain size ref
d 18.55 m
Twinning transition profile width exponent C 5.0
Average thickness of martensites tr
t 200 nm
Length of martensite nuclei tr
L 307.2 nm
Characteristic energy barrier for martensite
phase transformation tr
E 0.023 J/m2
Martensite transition profile width exponent D 8.0
Interaction coefficient between slip system
and twin/transformation system
,
Coplanar: 0.0
Non-coplanar: 0.5
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Interaction coefficient between two different
twin/transformation systems
,
Coplanar: 0.0
Non-coplanar: 5.0
Furthermore, the anisotropic elastic constants for the austenite and martensite phases
are given in Table 5. These constants form the elastic matrixes according to the lattice
structures, which are used in the calculation of Eq. (6). It should be noted that the
single-crystal elastic constants for the HCP phase are derived from the elastic constants
for the FCC phase, through the approach proposed by Fuller and Weston (1974).
Table 5 Single-crystal elastic constants for the austenite and martensite phases used in
the simulations of the iHEA.
Lattice Elastic constants (GPa) Elastic matrix
Austenite
(FCC phase)
A
11 247.71C
A
12 122.04C
A
44 84.21C
AAA
11 12 12
AA
11 12
A
11
A
44
A
44
A
44
000
000
000
00
0
CCC
CC
C
C
C
C
Martensite
(HCP phase)
M
11 267.634C
M
12 116.366C
M
13 107.79C
M
33 276.21C
M
44 68.643C
MMM
11 12 13
MM
11 13
M
33
M
44
M
44
M
44
000
000
000
00
0
CCC
CC
C
C
C
C
3. Simulation results
The feasibility of the developed crystal plasticity constitutive model was verified by
simulating the macroscopic deformation responses and microstructure evolutions
simultaneously. According to the experimental conditions (Wang et al., 2019), the
tensile simulations contained four cases: (1) the CG iHEA at 293 K; (2) the CG iHEA
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27
at 77 K; (3) the FG iHEA at 293 K; (4) the FG iHEA at 77 K. The constitutive
parameters listed in Table 3 and Table 4 are the same for these four cases. However,
grain size
grain
d and the temperature-dependent energies
/
sf ,,G
listed
in Table 2 are different for different cases, which correspond to the simulated objects.
The engineering stress-strain curves and true strain hardening curves are shown in Fig.
6. The hardening rate curves are calculated from the corresponding stress-strain curves.
It can be seen that the simulation results can match the experimental data well,
indicating that the developed model can effectively describe the effects of temperature
and grain size on the macroscopic tensile behavior of iHEA.
0 1020304050
0
300
600
900
1200
1500
Exp. Sim.
CG 293 K
CG 77 K
FG 293 K
FG 77 K
Engineering stress (MPa)
Engineering strain (%)
(a)
0 10203040
0.0
1.5
3.0
4.5
6.0
7.5
9.0
Hardening rate (GPa)
True strain (%)
Exp. Sim.
CG 293 K
CG 77 K
FG 293 K
FG 77 K
(b)
Fig. 6 Simulation results of CG (100 m) and FG (6 m) iHEAs at 77 K and 293 K: (a)
engineering strain-stress curves; (b) strain hardening rates. The experimental data were
adopted from Wang et al. (2019).
Moreover, the evolutions of twin and martensite volume fractions were extracted
from the simulation results, as shown in Fig. 7 (a). The experimental data are taken
from Li et al. (2017) and Wang et al. (2019). The comparisons between the simulated
and experimental martensite volume fractions indicate that the developed constitutive
model can also describe the microstructural evolution well. From Fig. 7 (a), it can be
seen that the strain corresponding to the start of martensite phase transformation in
simulation is very close to that in the experiment, demonstrating that the critical stress
of martensite nucleation is predicted accurately. Furthermore, at 77 K, the martensite
volume fractions of the CG and FG iHEAs are similar, and much higher than the twin
volume fractions (almost zero); these phenomena are also consistent with the
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experimental observation (Wang et al., 2019). The activations of deformation twinning
and martensite phase transformation depend on their critical stresses influenced by the
SFE and grain size. The underlying reason for the differences in plastic mechanisms at
various temperatures and their effects on the mechanical behavior of the iHEA will be
discussed in Section 4. Fig. 7 (b) shows the dislocation density evolutions of the four
simulated cases. It can be seen that the increasing rates of dislocation densities are
similar for the four cases at 0~5% strain. However, the dislocation densities at 77 K
increase faster than those at 293 K with increasing the strain. The better ability of
dislocation storage at 77 K provides continuous strain hardening of the iHEA, as shown
in Fig. 6 (b).
0 1020304050
0
20
40
60
80
Volume fraction (%)
Engineering strain (%)
f
f
f
twin
CG 293 K
CG 77 K
FG 293 K
FG 77 K
(a)
Experiment: symbols
Simulation: lines
0 1020304050
0
1
2
3
4
5
6
Dislocation density (10
15
m
-2
)
Engineering strain (%)
CG 293 K
CG 77 K
FG 293 K
FG 77 K
(b)
Fig. 7 Simulation results of CG (100 m) and FG (6 m) iHEAs at 77 K and 293 K: (a)
martensite and twin volume fractions versus engineering strain; (b) dislocation density
versus engineering strain. The experimental data were adopted from Wang et al. (2019).
Visualizing the field and internal variables at different loading times is an advantage
of the CPFEM simulation. Fig. 8 shows the contours of the strain, stress, and martensite
volume fraction of the CG 77 K simulation case. It can be seen that the contours exhibit
obvious inhomogeneity owing to the differences in the grain orientations, and the
inhomogeneity is enhanced with increasing the strain. Moreover, in the locations with
high stress, the corresponding martensite volume fractions are also higher than other
places. It indicates that stress localization provides enough driving force and promotes
the martensite phase transformation.
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Fig. 8 Contours of (a) Mises stress; (b) component along Z-axis of the logarithmic (true)
strain tensor; (c) martensite fraction of the CG 77 K simulation case.
4. Discussion
The comparisons between the simulated and experimental results validate the
feasibility of the developed crystal plasticity constitutive model. Moreover, some
details should be discussed to further reveal the intrinsic mechanisms for temperature
and grain size effects on the tensile deformation of iHEA.
With the temperature decreasing from 293 K to 77 K in the experiments (Wang et al.,
2019), the change of the FG (6 m) iHEA’s yield stress is y-FG 32.3 MPa
, which
is much lower than the change of the CG (100 m) iHEA’s yield stress
y-CG 105.7 MPa
. Since multiple strengthening mechanisms were jointly activated
in the iHEA, the contribution of each mechanism to the overall yield stress was
estimated at different temperatures, as shown in Fig. 9 (a). It should be noticed that the
slip resistances were converted from slip system level to macroscopic level by Taylor
factor, namely
3.06MM
. Besides, the contribution of forest dislocation
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hardening was calculated by MGb
, and the Taylor coefficient
was set to
be 0.4, which usually changes from 0.2 to 0.5 (Zhang et al., 2019).
From Fig. 9 (a), both the GB stress and lattice friction exhibit significant temperature
dependences for the same grain size, but the change trends with temperature are
opposite. Thus, the contributions of these two strengthening mechanisms complement
each other. For example, when the temperature decreases from 293 K to 77 K, the
reduction of GB strengthening is much lower than the improvement of lattice friction,
leading to the obvious increase of the CG iHEA’s yield stress. However, GB
strengthening plays a more important role in the FG iHEA than the CG iHEA. With
decreasing the temperature, The reduction of GB strengthening is comparable to the
improvement of lattice friction, resulting in a slight change of the FG iHEA’s yield
stress. The quantification of each strengthening mechanism helps us further understand
the different temperature dependencies of yield stresses in the FG and CG iHEAs.
0
160
320
480
640
800
Experimental value
Yield stress
Yield stress (MPa)
CG, 293 K CG, 77 K FG, 293 K FG, 77 K
Theoretical contribution
Grain boundary
Lattice friction
Forest dislocation
Precipitation
(a)
0
100
200
300
400
500
140.03
308.44
197.03
318.07
102.52
292.14
159.53
CG, 293 K CG, 77 K FG, 293 K FG, 77 K
Critical stress (MPa)
Deformation twin
Martensitic phase transformation
(b)
301.76
Fig. 9 (a) Theoretical estimation of each strengthening mechanism’s contribution to the
yield stresses of CG and FG iHEAs at 293 K and 77 K. (b) The critical shear stresses
for the activation of deformation twinning and martensite phase transformation in the
CG and FG iHEAs at 77 K and 293 K.
Furthermore, Fig. 7 (a) shows that the temperature effect on the evolutions of twins
and martensites volume fractions are opposite. For example, the FG iHEA at 293 K has
the lowest martensite volume fraction but the highest twin volume fraction at the strain
of 50%. Wong et al. (2016) and Madivala et al. (2018) also demonstrated that with the
increase of SFE, martensite phase transformation is suppressed, but deformation
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31
twinning is promoted. The competition between these two plastic mechanisms depends
on the changes of SFE, Gibbs free energy and grain size. The critical stresses for
deformation twinning and martensite phase transformation are calculated and given in
Fig. 9 (b). It can be seen that the critical stresses for deformation twinning are around
300 MPa for all four cases. However, the critical stress for martensite phase
transformation changes significantly at different grain sizes and temperatures.
Therefore, martensite phase transformation is easier to activate at cryogenic
temperature, accommodating the plastic deformations and relieving the stress
localization. At this time, deformation twinning is difficult to activate, leading to almost
no twin being observed in the iHEA at 77 K (Wang et al., 2019).
From Fig. 7 (a), it can also be seen that the volume fraction of martensites is much
larger than that of twins in the simulations at 77 K. Therefore, martensite plays a
dominant role in the plastic deformation of the iHEA, in addition to dislocation slip.
According to Eq. (13), the martensites generated during deformation can gradually
reduce the dislocation MFP. With decreasing the MFP, the multiplication rate of
dislocations increases, as described in Eq. (12). According to Eq. (11), the current
dislocation density determines the dislocation passing stress and the strain hardening
ability of the iHEA. Thus, the enhanced martensite phase transformation at 77 K
promotes the quick improvement of dislocation density, resulting in the higher strain
hardening rate of the iHEA at cryogenic temperature, as shown in Fig. 6 (b) and Fig. 7
(b).
As discussed above, the enhancement of martensite phase transformation at low
temperatures is related to the decrease of critical stress caused by the reduced SFE.
Besides, from Fig. 6 (a), it can be seen that the stress level is improved at 77 K for the
same grain size. According to Eq. (35), the nucleation probability of martensite is
higher when the shear stress (driving force) is higher, which also promotes the
activation of martensite phase transformation. Therefore, The enhanced phase
transformation at 77 K depends on two factors: (1) the decreased critical stress for
martensite nucleation; (2) the increased shear stress on the transformation systems.
Moreover, according to Eq. (26) and Eq. (37), the decrease of grain size can also
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32
increase the critical stresses for deformation twinning and martensite phase
transformation, as shown in Fig. 9 (b). However, the stress level of the FG iHEA is also
higher than that of the CG iHEA at the same temperature, which provides a higher
driving force and benefits the generation of twins and martensites. Therefore, the
dislocation density evolutions of the CG and FG iHEA are similar at the same
temperature, as shown in Fig. 7 (b). As a result, grain refinement has a weak influence
on the strain hardening ability of the iHEA. Such a mechanism helps the FG iHEA
contain the considerable strain hardening and achieve a strength-ductility balance
during grain refinement (Li et al., 2017; Zhang et al., 2021).
For showing the TRIP effect on the strain hardening more directly, the comparative
simulations with and without the TRIP effect were conducted for the CG and FG iHEA
at 77 K. The simulation results are given in Fig. 10. It can be seen that the activation of
the TRIP significantly increases the stress level when the engineering strain is larger
than 15%, namely the improvement of strain hardening ability. From Fig. 10 (b), if the
TRIP effect is considered, the hardening rates first become lower than the cases with
the TRIP effect (2.5%~8.5% true strain), and then increase beyond the cases with TRIP
(8.5%~40% true strain). These phenomena are attributed to the fact that martensite
phase transformation initially provides the plasticity and accommodates the local
deformations, leading to the quick drop in the hardening rate curve. With the
accumulation of martensite, the dislocation MFP reduces continuously and accelerates
the dislocation multiplication, resulting in the up-turn in the hardening rate curve. The
above analysis further reveals the effect of martensite phase transformation on the
deformation behavior of the iHEA.
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33
0 1020304050
0
300
600
900
1200
1500
Engineering stress (MPa)
Engineering strain (%)
CG 77K (with TRIP)
FG 77K (with TRIP)
CG 77K (without TRIP)
FG 77K (without TRIP)
(a)
0 10203040
2
3
4
5
6
CG 77K (with TRIP)
FG 77K (with TRIP)
CG 77K (without TRIP)
FG 77K (without TRIP)
Hardening rate (GPa)
True strain (%)
(b)
Fig. 10 (a) Engineering stress-strain curves and (b) true strain hardening curves of the
CG and FG iHEA with and without martensite phase transformation at 77 K.
The contours of the martensite volume fraction of the four simulation cases are
shown in Fig. 11. It can be seen that the distributions of martensite in the polycrystalline
RVE are very inhomogeneous, owing to different grain orientations. At 77 K, the local
volume fractions of martensite can exceed 80% (regions in red color) in the CG and FG
iHEAs, which are higher than the average values at the strain of 50% (CG: 79% and
FG: 75%) at the engineering strain of 50%, as shown in Fig. 7 (a). The various grain
orientations lead to different shear stress on the transformation system of different
grains. Thus, the activation time of martensite phase transformation is also different.
Wei et al. (2019) demonstrated that the asynchronously generated martensites caused
the nucleation of GB damage and the formation of various martensite variants.
Furthermore, the martensite variants restrain slip transfer between adjacent grains,
accelerating the GB damage evolution. At 77 K, the enhanced martensite phase
transformation induces microvoids in the GBs of the iHEA. The improved stress level
promotes the growth of microvoids to crack, resulting in the final fracture. Therefore,
the ductility of the iHEA with the same grain size is lower at 77 K than that at 293 K
(Wang et al., 2019). The constitutive model coupled damage has a better capability in
describing the ductile failure. Some microstructure-based crystal plasticity models with
damage mechanism have been established for FCC or HCP metal materials (Frodal et
al., 2021; Kim and Yoon, 2015; Liu et al., 2021; Park et al., 2021), and the effects of
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34
dislocation slip and twinning can also be considered. These studies provide valuable
references for the modeling of the iHEA. In the future, damage evolution will be
considered in the developed crystal plasticity model to describe the fracture behavior
of the iHEA better.
Fig. 11 Contours of the martensite volume fractions at the strain of 50 % in the CG and
FG iHEAs at 77 K and 293 K.
Based on the verifications of the developed constitutive model, some tensile
simulations at different temperatures for the iHEAs with different grain sizes were
conducted. The predicted deformation behavior can help us explore the potential for
optimizing the iHEA’s performance. Table 6 shows the SFE, Gibbs free energy change
and interfacial energy of iHEAs at 100 K, 250 K, and 400 K, calculated by the
thermodynamic approach described in Section 2.1. The constitutive parameters, finite
element model, and boundary conditions are the same as those described in Section 2.3
and 2.4. It should be noted that precipitation strengthening is considered in all these
simulations because all the grain sizes are smaller than that of the as-homogenized CG
iHEA (100 m). Wang et al. (2019) revealed that the thermo-mechanical grain
refinement could induce carbide precipitations in the iHEA.
Table 6 Thermodynamically calculated stacking fault energy, Gibbs energy change, and
interfacial energy of the iHEAs at 100 K, 250 K, and 400 K.
Grain size Symbol T = 100 K T = 250 K T = 400 K
0.5 m, 5 m,
50 m
sf
(mJ/m2) 13.73 14.36 30.01
G
(J/mol) 186.83 34.55 373.12
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/
(mJ/m2) 12.4 8.2 4.0
The Considère criterion d/d
is used here to determine the uniform
elongation; namely, the true strain corresponding to the intersection of the stress-strain
curve and strain hardening curve, as shown in Fig. 12 (a), which gives the simulation
results of the iHEAs with the grain sizes of 0.5 m, 5 m, and 50 m at 100 K.
Moreover, it can also be seen that the up-turn in strain hardening curves becomes more
obvious with increasing the grain size. Fig. 12 (b) gives the simulated strength-ductility
map for the iHEAs with various grain sizes at different temperatures. It can be seen that
the trade-off between yield stress and uniform elongation caused by grain refinement
still occurs in the iHEA at 400 K. However, the extent of the trade-off gradually
becomes weaker with decreasing the temperature. The shape of strength-ductility curve
change from concave to linear, and then convex type, as shown by the blue, green, and
red dashed lines in Fig. 12 (b). It means that the balance between strength and ductility
during grain refinement can be optimized by decreasing the temperature in the iHEA.
Therefore, controlling the grain size of the iHEA at low temperatures can obtain a better
synergy of stress and ductility.
0 102030405060
0
1
2
3
4
5
6
True stress/hardening rate (GPa)
True strain (%)
d
d
0.5
m
5
m
50
m
(a)
0 300 600 900 1200 1500
10
20
30
40
50
60
70
Uniform elongation (%)
Yield stress (MPa)
Simulation
0.5
m
5
m
50
m
Red: 100 K
Green: 250 K
Blue: 400 K
(b)
Fig. 12 (a) True stress-strain and strain hardening curves of the iHEAs with the grain
sizes of 0.5 m, 5 m, and 50 m at 100 K. (b) Experimental and simulated strength-
ductility maps of iHEAs with different grain sizes and at different temperatures.
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5. Conclusions
In this paper, a crystal plasticity constitutive model considering the effects of
temperature and grain size is developed and implemented in the finite element method
through DAMASK. The simulated stress-strain responses and microstructural
evolutions reproduce the experimental data well, indicating the reliability of the
developed model. Furthermore, the analysis of the simulation results and discussion
help further understand the temperature effect on the tensile deformation behavior of
the CG and FG iHEA. Finally, the crystal plasticity model is applied to predict the
temperature-dependent strength-ductility map of the iHEAs with different grain sizes.
From this study, some conclusions can be summarized as follows:
(1) The contributions of various strengthening mechanisms to the yield stress at
different temperatures were quantified, revealing that the change tendencies of GB
strengthening and lattice friction with temperature are opposite in the iHEA. The
counterbalance of their change results in that the yield stress of FG iHEA has a weak
temperature dependence.
(2) There are two reasons for the enhanced martensite phase transformation at low
temperatures in the iHEA: first, the decreasing temperature leads to the decreasing
SFE and critical stress for martensite nucleation; second, the higher stress level at
lower temperature promotes the martensite nucleation and evolution.
(3) From simulations, the TRIP effect on the deformation behavior is exhibited directly,
especially the contribution to the strain hardening ability of the iHEA. The contours
show that martensites distribute inhomogeneously in the polycrystal RVE, owing to
the different grain orientations. The damage induced by martensite phase
transformation is speculated to be the reason for ductility loss of the iHEA at 77 K.
(4) The strength-ductility maps of the iHEAs with various grain sizes are predicted at
different temperatures. The trade-off between strength and ductility caused by grain
refinement still occurs in the iHEA at 400 K. With decreasing the temperature, the
curve shape changes from concave to linear, and then convex type, reflecting a
better synergy of strength and ductility.
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37
This study provides an effective approach to describe and predict the temperature-
and grain size-dependent deformation behavior of iHEA. It also offers a strategy for
achieving a superior strength-ductility combination of the iHEA in engineering
applications. In further work, damage mechanisms will be considered in this crystal
plasticity model to describe the fracture behavior of the iHEA.
Acknowledgments
This work was supported by the National Natural Science Foundation of China
(Grant No. 11872321, 12192214, 11672251), State Key Lab of Advanced Metals and
Materials (Grant No. 2019-Z07).
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38
Appendix A. Numerical solution of the equilibrium equations
Considering a deformation map
0t
:
xx y
BB
maps the material point x
in the reference configuration to y in the current (deformed) configuration, the total
deformation gradient tensor is expressed by /
Fx
. Further introducing an
intermediate configuration (unstressed state) between the reference and current
configurations, the total deformation gradient tensor F can be multiplicatively split into
the elastic part e
F and the plastic part
p
F, as described by ep
FFF (namely Eq. (5)).
Three stress measures in the finite deformation framework are introduced: the first
Piola-Kirchhoff stress tensor P, the second Piola-Kirchhoff stress tensor S, and the
Cauchy stress tensor The conversions of these three stress measures follow the
formulas as
T
det
PFσF
, (A.1)
1T
ee e
det
SFFσF
, (A.2)
where
det F denotes the determinant of the deformation gradient tensor.
The mechanical equilibrium in terms of linear momentum can be expressed by
00
Div in
Py
B
, (A.3)
tt
div in
σy
B
, (A.4)
where
is the mass density. For static equilibrium, y
should be zero. Thus, the
balance equation of linear momentum is
Div 0
P. (A.5)
Assuming the deformation is an isothermal and quasi-static process without the
supplement of external heat, the thermodynamic equilibrium is expressed by
:
PF
, (A.6)
where
is the dissipation energy density,
is the free energy density. For satisfying
the thermodynamic consistency, the dissipation energy density should be 0
. The
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39
demonstration of this inequality is given in Appendix B.
In the numerical implementation of the developed constitutive model, the stress
equilibrium solution is conducted in the intermediate configuration. The work
conjugacy is established between the second Piola-Kirchhoff stress tensor S and the
Green strain tensor E. The elastic strain tensor is expressed by
T
eee
1
2
EFFI
.
According to the generalized Hooke’s law, the second Piola-Kirchhoff stress tensor S
can be described by
T
eee
1
:2
SE FFI： (namely Eq. (6)).
For describing the rate of deformation gradient tensor, the velocity gradient tensor is
defined as
11 11
ee e pp e
LFF FF FFF F
. (A.7)
The plastic velocity gradient tensor in intermediate configuration is
1
p
pp
LFF
, (A.8)
which can be rewritten in
p
pp
FLF
(namely Eq. (7)).
The whole deformation process is divided into a series of small loading steps. Each
step has a time increment t . At the start of each step
0
t , the change of the
deformation gradient tensor F in this step should be given. At the final of each step
0
tt t, the deformation-resulted stress response S and internal variables will be
updated until their integrations achieve convergence. In the fully implicit integration
scheme, Eq. (7) can be expressed as
pp0
ppp
tt tt
t
FF
FLF
. (A.9)
Integrating Eq. (A.9) results in
1
p
pp0
tttt
FILF
, (A.10)
11
pp0 p
tttt
FFIL
, (A.11)
TTT
p
pp0
tttt
FILF
. (A.12)
Eq. (5) can be written in
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40
1
ep
FFF
. (A.13)
Substituting Eq. (A.11) into Eq. (A.13), the elastic deformation gradient tensor at
time t is obtained
1
ep0p
ttt tt
FFF IL
. (A.14)
Substituting Eq. (A.14) into Eq. (6), the second Piola-Kirchhoff stress tensor can be
expressed by
T
T
ee
TTT 1
pp0 p0 p
1:
2
1:
2
ttt
tt t tt t tt
BA
A
SFFI
IL F FFF IL I
, (A.15)
where
1
p
0
t
F and
tF are known for this time increment,
tS and
ptL are
unknown. According to Eq. (8), (9), (19) and (30), the plastic velocity gradient tensor
ptL depends on the stress tensor
tS and microstructure variables
tm:
p,tgt t
LSm
, (A.16)
where the matrix m contains the microstructure internal variables about dislocation
density, twin volume fraction, martensite volume fraction, etc. From Eq. (12), (20) and
(31), the evolutions of microstructure internal variables are expressed in rate form,
depending on the current states of stress and microstructures:
,tht t
mSm
. (A.17)
Combining Eqs. (A.10), (A.13) (A.15), (A.16) and (A.17), the stress state can be
calculated from the given
tF, and the microstructure state is also updated at the end
of the time step 0
tt. Each time increment t
is divided into several iterative steps,
and the Newton-Raphson iteration method is used here. Considering that it is difficult
to solve
tSand
tmtogether, the two-level (stress level and microstructure level)
iteration scheme proposed by Kalidindi (1998) is adopted in this study.
(1) Stress level
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41
The first level of iterations solves for
ptL a n d
tS with
tm fixed; thus,
tmis known in this level. When the stress response of the material is close to the
yield strength, a small fluctuation of S can lead to a big change in
p
L . Therefore,
choosing S as the iterative object will result in numerical instability. For a better
convergence,
p
L is chosen as the target in iterations, and then S is calculated by the
converged value of
p
L.
The predicted plastic velocity gradient tensor after n iterative steps is denoted by
p
,n
L
. According to Eq. (A.15) and Eq. (A.16), the resulting value of
p
L from the
constitutive relationship is described by
p, p,nn
g
LSL
. (A.18)
The convergency value for
p
L is achieved using a predictor-corrector scheme. At
the nth step of iterations, the residual of
p
L is defined as
p, p, p, p,nnnn n
g
RL L L SL
. (A.19)
Thus, the Newton-Raphson iterative equation is
1
p, 1 p,
p,
:
n
nn n
n
R
LL R
L
. (A.20)
The part
p
,
/
nn
RL
contains the partial derivative
p
,
/
nn
SL
, which can be
calculated by
TT
T
p, p, p, p,
:/2
1:
2
n
nn n n
ABA I
SAA
BA A B
LL L L
. (A.21)
The iterative loop is continued until a self-consistent solution is obtained. The
Newton-Raphson method is regarded as converged when the residual decreases below
a reasonable error
, which is given by
ar p r p
22
max , ,
LL
, (A.22)
where
a
and r
are the absolute and relative errors, 2
denotes the 2-norm of a
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42
matrix. The stress tensor S is calculated when the plastic velocity gradient tensor
p
L
is converged. The integration algorithm at the stress level with a fixed internal material
state can be summarized by Algorithm 1 (Roters et al., 2019).
Algorithm 1 Self-consistent integration of kinematic variables at the stress level.
(2) Microstructure level
The second level of iterations updates the microstructure state
tm based on the
calculated second Piola-Kirchhoff stress tensor
tS . Thus,
tS is known at this
level. Newton-Raphson method is also used here. Similar to Eq. (A.19), the
0,
nn nn
tt
rmm mSm
. (A.23)
Same as Algorithm 1, if all components of n
r drop below the given error, the
iterations at the microstructure level are considered converged, and the microstructure
internal variables are updated.
In addition, the new crystal orientations should also be updated at the end of each
time step. In the crystal plasticity model, the initial orientation at time 0
t can be
described by the two orthogonal unit vectors
0
tn and
0
ts , which denote the
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43
normal and slip directions of a specific slip system. The updated orientation at time t is
expressed by
T
e0
ttt
nFn
, (A.24)
e0
tttsFs
. (A.25)
Moreover, for the implicit solver of Abaqus used in this study, the tangent stiffness
of stress with respect to the deformation gradient is needed to solve the mechanical
equilibrium. In the DAMASK framework, the stress tangent d/dPF
is analytically
derived (Roters et al., 2019). The first Piola-Kirchhoff stress tensor P can be converted
by the second Piola-Kirchhoff stress tensor S as
T1T
ep p p
P F SF FF SF . (A.26)
Thus, the stress tangent can be expressed by
p
1
p
T
p
11
p
1T T
pp
pp
T
d
dd
::
ddd
d
d
::
dd
F
PF
IF
FFF
F
S
FF FF S
FF
FSF SF
FI
, (A.27)
where denotes the tensor product of two second-order tensors (
ijkl ij kl
TA
B
such that ijkl kl im mn nj
TC ACB), and
d/d
IIFF .
The total derivative of Eq. (A.11) yields
1
1
p0 p
pp
1
p0
d
dd
d
::
dd dd
tt
FI L
FL
S
F
FF SF
I, (A.28)
where
11
p
0p0
t
FF .
Taking total derivatives of Eq. (A.12) gives T
p
d/d
FF
as
T
pp
T
p0
dd
d
::
ddd
t
FL
S
F
FSF
I. (A.29)
Using the chain-rule, d/dSF
can be calculated by
e
e
d
dd
:
ddd
F
SS
FFF
. (A.30)
The term e
d/dSF
is obtained by taking the total derivative of Eq. (6)
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44
T
T
ee
ee
ee
T
e
ee
dd
d1 1
::
d2d d 2
FF I
FF
SFF I
FFF
. (A.31)
Taking the total derivative of Eq. (A.13) gives
e
d/dFF
1
p
e1
p
1
p
d
d::
ddd
d
F
FF
FF
FI
FF FI
. (A.32)
Combining Eqs. (A.31) and (A.32), the term d/dSF
is solved. The stress tangent
d/dPF
can be calculated by substituting Eq. (A.28), Eq. (A.29) and Eq. (A.30) into
Eq. (A.27).
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45
Appendix B. Demonstration of the thermodynamic consistency
The Helmholtz free energy
per unit volume can be decomposed into elastic and
plastic parts as
ep
, (B.1)
where the elastic strain energy density can be expressed by
eee
1::
2
EE. (B.2)
The plastic strain energy density can be given in rate form as
s
ptwtrc c c
111
tw tr
1
NN
N
ff
. (B.3)
where c
, c
, a n d c
are the resistant force for dislocation slip, deformation
twinning and martensite phase transformation, respectively. Thus, the rate form of
is given by
s
ep
e e tw tr c c tw tw c tr tr
11 1
Elastic
Dislocation Twin Martensite
tw tr
:: 1
NN
N
f
fff
EE
. (B.4)
The second law of thermodynamics should be satisfied, which can be expressed by
the Clausius-Duhem dissipative inequality (Acharya and Shawki, 1996)
:0
PF
. (B.5)
Using the continuum kinematic relations, the term :PF
can be written as (Dhala
et al., 2019)
T
eee p
:: :PF SE FFS L
. (B.6)
Thus, the dissipative inequality can be written in
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46
s
s
T
eee p
e e tw tr c c tw tw c tr tr
11 1
eee
T
ee p tw tr c ctwtw ctrtr
11 1
eee
tw tr
tw tr
:
::
:: 1
:::
:1
:::
NN
N
NN
N
ff f f
ff f f
PF
SE FFS L
EE
SE E E
FFS L
SE E E
F
stwtr
s
ss
T
e e tw tr
111
tw tr c c tw tw c tr tr
11 1
eee
e e tw tr tw tr c
11
ee
tw tr
:1
1
:::
:1 1
NN N
NN
N
NN
ff
ff f f
ff ff
FS n s n s n s
SE E E
FFS n s
FFS
tw
tr
s
tw
t
ctwtw
11
ee ctrtr
11
ee
tw tr e e c
1
ee c
1
ee c
1
tw
tr
:
:
:: (I)
1: (II)
:(III)
:
N
N
N
N
N
N
N
f
f
ff
ns
FFS n s
SEE
FFS n s
FFS n s
FFS n s
r
(IV)
(B.7)
Owing to e
:SE, the term (I) is equal to zero. The term T
ee
FFS
is the Mandel
stress
p
M defined in Eq. (10); thus, the term
T
ee :
FFS n s means the driving
forces for dislocation slip, deformation twinning, and martensite phase transformation.
In this constitutive model, the activation criteria of plastic mechanisms are always that
the driving force should exceed the resistant force, as shown in Eq. (9), (24) and (35).
These criteria result in the terms (II~IV) always equal or greater than zero. Thus, the
dissipative inequality :0
PF
and the thermodynamic consistency of the model
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are fulfilled.
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References
Acharya, A., Shawki, T.G., 1996. The Clausius-Duhem Inequality and the structure of rate-independent
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