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Phase field modeling of coupled crystal plasticity and deformation twinning in polycrystals with monolithic and splitting solvers

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Phase field modeling of coupled crystal plasticity and deformation 1 twinning in polycrystals with monolithic and splitting solvers 2 Ran Ma · WaiChing Sun 3 4 Abstract For some polycrystalline materials such as austenitic stainless steel, magnesium, TATB, and HMX, 6 twinning is a crucial deformation mechanism when the dislocation slip alone is not enough to accommodate 7 the applied strain. To predict this coupling effect between crystal plasticity and deformation twinning, 8 we introduce a mathematical model and the corresponding monolithic and operator splitting solver that 9 couples the crystal plasticity material model with a phase field twining model such that the twinning 10 nucleation and propagation can be captured via an implicit function. While a phase-field order parameter 11 is introduced to quantify the twinning induced shear strain and corresponding crystal reorientation, the 12 evolution of the order parameter is driven by the resolved shear stress on the twinning system. To avoid 13 introducing an additional set of slip systems for dislocation slip within the twinning region, we introduce a 14 Lie algebra averaging technique to determine the Schmid tensor throughout the twinning transformation. 15 Three different numerical schemes are proposed to solve the coupled problem, including a monolithic 16 scheme, an alternating minimization scheme, and an operator splitting scheme. Three numerical examples 17 are utilized to demonstrate the capability of the proposed model, as well as the accuracy and computational 18 cost of the solvers. 19
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International Journal for Numerical Methods in Engineering manuscript No.
(will be inserted by the editor)
Phase field modeling of coupled crystal plasticity and deformation1
twinning in polycrystals with monolithic and splitting solvers2
Ran Ma ·WaiChing Sun3
4
Received: October 27, 2020/ Accepted: date5
Abstract
For some polycrystalline materials such as austenitic stainless steel, magnesium, TATB, and HMX,
6
twinning is a crucial deformation mechanism when the dislocation slip alone is not enough to accommodate
7
the applied strain. To predict this coupling effect between crystal plasticity and deformation twinning,
8
we introduce a mathematical model and the corresponding monolithic and operator splitting solver that
9
couples the crystal plasticity material model with a phase field twining model such that the twinning
10
nucleation and propagation can be captured via an implicit function. While a phase-field order parameter
11
is introduced to quantify the twinning induced shear strain and corresponding crystal reorientation, the
12
evolution of the order parameter is driven by the resolved shear stress on the twinning system. To avoid
13
introducing an additional set of slip systems for dislocation slip within the twinning region, we introduce a
14
Lie algebra averaging technique to determine the Schmid tensor throughout the twinning transformation.
15
Three different numerical schemes are proposed to solve the coupled problem, including a monolithic
16
scheme, an alternating minimization scheme, and an operator splitting scheme. Three numerical examples
17
are utilized to demonstrate the capability of the proposed model, as well as the accuracy and computational
18
cost of the solvers.19
Keywords Crystal plasticity; deformation twinning; phase field; operator split; Lie algebra20
1 Introduction21
Twinning is a common deformation mechanism usually observed in metallic materials (e.g. TWIP steels [
1
]
22
and magnesium [
2
]) with either low stacking fault energy (SFE) or insufficient slip systems to accommodate
23
the plastic strain. Figure 1shows a schematic illustration of the difference between dislocation slip and
24
deformation twinning. Compared with the initial configuration in Figure 1(a), pure dislocation slip has
25
no influence on the crystal orientation, as shown in Figure 1(b). Figure 1(c) shows the orientation relation
26
between the twinning region and the parent region separated by a low-angle boundary. Unlike dislocation
27
slip mechanism where plastic strain is generated by full dislocations sliding along the slip plane without
28
crystal reorientation, the shear strain associated with twinning is polar in nature where the crystallographic
29
directions of the twinning region and the parent region are symmetric with respect to the twinning habit
30
plane.31
Deformation twinning is so crucial in enhancing the strength and formability of certain materials that
32
efficient numerical prediction is required to design such materials with desired properties [
3
,
4
]. For example,
33
deformation twinning can increase the tensile elongation of AZ31 [
2
]. Also, an efficient twinning model
34
is important in explaining many twinning-related material behaviors observed in experiments, including
35
Corresponding author: WaiChing Sun
Associate Professor, Department of Civil Engineering and Engineering Mechanics, Columbia University , 614 SW Mudd, Mail
Code: 4709, New York, NY 10027 Tel.: 212-854-3143, Fax: 212-854-6267, E-mail: wsun@columbia.edu
2 Ran Ma, WaiChing Sun
(a) (b) (c)
Twin Plane
(mirror plane)
Slip Plane
Fig. 1: Schematic illustration of dislocation slip and deformation twinning. (a) Original single crystal before
loading; (b) Dislocation slip; (c) Deformation twinning.
atomic rearrangement and tension-compression asymmetry [
2
], stress concentration and microcrack initi-
36
ation [
5
]. This paper focuses on developing a mathematical model and efficient solution schemes for the
37
coupled crystal plasticity and deformation twinning problem.38
1.1 Models for deformation twinning39
Traditionally, the coupling effect between crystal plasticity and twinning is modeled through a volume
40
fraction approach such that the twinning fraction is treated as an internal variable [
6
,
7
]. This approach is
41
computationally convenient and efficient, but the morphology of the twinning region, which is usually a
42
thin layer stretching across the grain, is not captured explicitly.43
There is an increasing number of studies focusing on capturing the twinning morphology explicitly.
44
A physics-based twin nucleation [
8
] and propagation [
9
] model is proposed, where twin propagation is
45
achieved by checking the status of the neighboring element around the twin nucleus at the end of each
46
time step. Since the twinning region is formed by the element sets, this approach may require specific
47
treatment to circumvent the mesh bias [
10
,
11
]. Other numerical methods, for example adaptive mesh [
12
]
48
and embedded weak discontinuity [
13
,
14
], are also applied to explicitly model the evolution of deformation
49
twinning and its interaction with surrounding grains.50
To handle the evolution of the deformation twinning, an alternative approach is to represent the location
51
of the twinning region via an implicit function in a phase-field framework. Similar to the relation between52
damage mechanics and phase-field fracture [
15
,
16
], the twinning phase-field method can be also viewed
53
as an extension of the volume fraction approach. Starting from the energy pathway for the formation of
54
stacking fault, a twinning free energy is proposed for multi-phase-field method and the coupling with linear
55
elasticity is investigated [
17
]. Later, the twinning phase field is coupled with finite strain elasticity [
18
] and
56
fracture phase-field [
19
], and the coupled problem is solved by a nonlinear conjugate gradient method. It is
57
also proved that the coupled free energy may not be convex, such that multiple local minima may exist and
58
the initial guess is crucial in predicting the phase-field evolution. Based on these preliminary studies, the
59
twinning phase field is also coupled with crystal plasticity for single crystal [20] and polycrystal [21].60
In these aforementioned models, an additional set of slip systems is introduced to characterize the dislo-
61
cation slip within the twinning region, which may increase the computational cost during the stress update
62
algorithm. Furthermore, the mechanical residual and the phase-field residual are minimized alternatively
63
by fixing the other field at each iteration. The iterative alternation between the mechanical and phase-field
64
steps may significantly increase the computational cost.65
1.2 Solution techniques for phase field models: alternating minimization and operator splitting66
The coupled mechanics and phase field problems, for example the deformation-twinning phase-field prob-
67
lem [
21
] and the phase field fracture problem [
22
,
23
,
24
,
25
,
26
,
27
], are usually solved in an alternating
68
Solvers for crystal plasticity/twinning phase field models 3
minimization way. The displacement field and the phase field are update alternatively with fixing the other
69
field until both residuals vanish eventually. Through this approach, the coupled non-convex energy func-
70
tional is minimized iteratively by alternatively minimizing two convex energy functionals. This approach
71
may increase the computational cost since many iterations are usually needed before the final convergence,
72
and the crack propagation progress are delayed compared with corresponding monolithic approach [28].73
The operator splitting methods are efficient numerical techniques for solving some types of partial
74
differential equations (PDE) [
29
,
30
]. The key idea is to approximate a complex differential operator by
75
splitting it into the sum of sub-operators and solve the sub-problems sequentially to approximate the
76
original solution, often at the expense of increasing error.77
To reduce the computational cost of the phase-field method, the operator splitting method has been
78
applied for both molecular beam epitaxy (MBE) equation and Cahn-Hilliard (CH) equation [
31
], and also
79
the binary phase-field crystal model [
32
]. The nonlinear part of the phase-field operator is reduced to a
80
local ODE solved by the third-order Runge-Kutta method, and the linear part is solved efficiently with
81
the spectral method. The operator splitting method is also applied to solve the phase-field crystal model
82
with the FFT based spectral method [
33
], and the numerical accuracy and stability have been analyzed
83
and discussed [
34
]. Nevertheless, this operator splitting method has not been applied for the coupled
84
problems that involve crystal plasticity and phase field. One of the contributions of this paper is to assess
85
the feasibility of the operator splitting approach and compare it with the alternating minimization method
86
and the monolithic solver.87
1.3 Outlines, major contributions and notations88
In this paper, we extend the previous studies of the twinning phase field coupled with elasticity [
18
], single
89
crystal plasticity [
20
], and polycrystal plasticity [
21
]. More specifically, Lie algebra is utilized to interpolate
90
the Schmid tensor within the phase field interfacial region to avoid introducing an additional set of slip
91
systems for the dislocation slip within the twinning region. Inspired by the operator splitting method
92
for pure phase field problem [
31
], an operator splitting method is also proposed for the coupled crystal
93
plasticity and twinning phase field problem, such that the stress update and the ODE part of the phase field
94
governing equation are solved simultaneously at each Gauss point, while the linear PDE part of the phase
95
field governing equation is solved separately. As a comparison, the monolithic method and alternating
96
minimization method are also derived. Three numerical examples, including one single crystal example
97
and two polycrystal examples, are performed to demonstrate the capability of the proposed model, as well
98
as their accuracy and computational cost.99
This paper will proceed as follows. Section 2presents the strong form of the coupled mechanics
100
and twinning phase field equations with thermodynamic consistency. Section 3presents the constitutive
101
relation and stress update algorithm for small strain crystal plasticity, including the shear strain and crystal
102
reorientation due to twinning. In Section 4, the monolithic method, the alternating minimization method,
103
and the operator splitting method are introduced for solving the coupled crystal plasticity and twinning
104
phase field problem. Section 5discusses the numerical performance of the three solution scheme with three
105
numerical examples. Section 6summarizes the major results and concluding remarks.106
As for notations and symbols, bold-faced letters denote tensors (including vectors which are rank-one
107
tensors); the symbol ’
·
’ denotes a single contraction of adjacent indices of two tensors (e.g.
a·b=aibi
108
or
c·d=cij djk
); the symbol ‘:’ denotes a double contraction of adjacent indices of tensor of rank two or
109
higher (e.g.
C:εe
=
Cijk l εe
kl
); the symbol ‘
’ denotes a juxtaposition of two vectors (e.g.
ab=aibj
) or two
110
symmetric second order tensors (e.g.
(αβ)ijkl =αij βkl
). The operators
X
and
x
denote the gradient
111
operation with respect to the initial undeformed configuration and the current deformed configuration,
112
respectively. For infinitesimal strain theory, they are equivalent, i.e.
X=x
. The symmetric operator
sym113
projects a second order tensor to its symmetric part, i.e.
sym(A) = (A+AT)/
2. The dot over a quantity
114
denotes its first order material time derivative, i.e.
˙
a=da/dt
. Similarly, double dot over a quantity denotes
115
its second order material time derivative.116
4 Ran Ma, WaiChing Sun
2 Governing equations117
In this section, the balance laws for linear momentum and microforce, which constitute the governing partial
118
differential equations for this multi-physics problem, are first introduced. Then, the thermodynamic laws
119
are discussed, which set constraints to admissible constitutive relations. Lastly, a specific form of Helmholtz
120
free energy is defined, such that the thermodynamic forces work-conjugate to the field variables are defined
121
with thermodynamic consistency.122
2.1 Balance of linear momentum and microforce123
Consider a deformable body occupying an open bounded domain
in the Euclidean space deforming
124
within a time interval
I= [
0,
t]
. The deformable body
has a piece-wisely smooth boundary
Γ
such that
125
the closure of the open set is
¯
=Γ
. A displacement field
u:×IRd
and a twinning phase field
126
ηt:×IR
are introduced to describe the deformation and twinning transformation, where
d
2 is the
127
spatial dimension. Displacement constraint
˜u
is imposed on the Dirichlet boundary
Γu
, and surface traction
128
t
is imposed on the von Neumann boundary
Γt
, with
ΓuΓt=Γ
and
ΓuΓt=
. Furthermore, we do
129
not impose Dirichlet boundary condition for the phase field and the von Neumann boundary condition is
130
trivial (cf. Eq. (2)).131
Assume that the deformation is geometrically linear such that the infinitesimal strain
ε=sym(xu)
is
132
an appropriate strain measure. For an arbitrary point
p
, the balance law of linear momentum is written
133
as:134
x·σ+b=ρ¨u, in
u=˜u, on Γu
σ·n=t, on Γt,
(1)
where
σ
is the Cauchy stress and
b
is the body force. A non-conserved phase field variable
ηt[
0, 1
]135
is defined to describe the twinning evolution within the domain
, where
ηt=
1 denotes the twinning
136
region,
ηt=
0 denotes the parent region, and
ηt(
0, 1
)
denotes the interfacial region. Define
πt
as the
137
thermodynamic force work-conjugate to the twinning phase field
ηt
, and define
ξt
as the thermodynamic
138
force work-conjugate to the gradient of the phase field
xηt
, the micro-force balance equation for the
139
twinning phase field is written as:140
(x·ξt+πt=˙
ηt
Mt, in
xηt·n=0, on Γ,(2)
where
Mt
is a mobility parameter. A monotonic interpolation function
ϕ(ηt)
is used to represent the local
141
twinning volume fraction:142
ϕ(ηt)=αη2
t+2(2α)η3
t+(α3)η4
t, (3)
where
α(
0, 6
)
is a scalar constant. As shown in Figure 2, the value of
α
has a minor influence on the
143
simulated twinning configuration [
18
]. An important property of the interpolation function is that its first-
144
order derivative vanishes at
ηt=
0 and
ηt=
1, such that the material properties within the parent region and
145
the twinning region remains stable. Also, when the scalar constant
α
equals to 3, the interpolation function
146
is anti-symmetric such that
ϕ(
1
ηt) =
1
ϕ(ηt)
, which is more suitable for modeling flat interfaces [
18
].
147
Therefore,
α=
3 is prescribed throughout this paper. Unlike the treatment used in phase field fracture
148
problem where the phase field order parameter is monotonically increasing (cf. [
35
]), detwinning is allowed
149
in our model where the order parameter is allowed to decrease upon unloading [36,17].150
2.2 Balance of energy and dissipation inequality151
Under the assumption that the deformation is infinitesimal, the total strain
ε
is additively decomposed into
152
the elastic strain εe, the plastic strain εp, and the shear strain due to deformation twinning εt:153
ε=εe+εp+εt. (4)
Solvers for crystal plasticity/twinning phase field models 5
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Fig. 2: Interpolation function in Equation (3) [18].
From the continuum mechanics point of view, twinning deformation occurs in the form of shear strain154
on the twinning system. As a result, the original crystal plasticity slip systems and those after the twinning
155
deformation are symmetric with respect to the twinning plane. The shear strain due to deformation twinning
156
εtis written as an explicit function of the twinning phase field ηt:157
εt=γ0ϕ(ηt)Pt,Pt=1
2(stmt+mtst), (5)
with
γ0
being the magnitude of the shear strain,
mt
representing the normal direction of the twinning habit
158
plane, and
st
representing the shear direction of the twinning habit plane Here,
Pt
represents the second
159
order Schmid tensor of the twinning system.160
The first law of thermodynamics requires that the internal energy rate
˙
e
equals to the energy input rate,
161
including the mechanical work and the work due to the microforces. Therefore, the local form of the energy
162
balance law is written as:163
˙
e=σ: ˙ε+ξt· ∇x˙
ηtπt˙
ηt. (6)
Under the local thermodynamic equilibrium assumption and isothermal condition, the second law of
164
thermodynamics (Clausius-Duhem inequality) requires that the internal energy rate is larger than or equal
165
to the free energy rate:166
Dint =˙
e˙
ψ0, (7)
where
ψ
represents the Helmholtz free energy. A general form of the Helmholtz free energy
ψ
, which
167
is a function of all the independent state variables, is related to the internal energy
e
through Legendre
168
transformtion as:169
ψ=ψ(εe,ηt,xηt)=eθs, (8)
where
θ
is the absolute temperature and
s
is the entropy. Note that the free energy
ψ
is assumed to be
170
independent of the crystal plasticity internal variables [37]. Substituting equation (6) and equation (8) into171
equation (7), the dissipation inequality is written as172
σ∂ψ
εe: ˙εe+σ: ˙εp+σ: ˙εtπt˙
ηt∂ψ
∂ηt
˙
ηt+ξt∂ψ
xηt· ∇x˙
ηt0. (9)
With this general form of dissipation inequality, the traditional Coleman-Noll arguments is written as:173
σ=∂ψ
εe. (10)
6 Ran Ma, WaiChing Sun
Furthermore, define the resolved shear stress on the twinning system as:174
τt=1
2σ:(stmt+mtst), (11)
which acts as the driving force for the twinning phase field evolution. Then, the microforces
πt
and
ξt
take
175
the form:176
πt=γ0τtϕ0(ηt)∂ψ
∂ηt,ξt=∂ψ
xηt. (12)
2.3 A specific free energy for deformation twinning177
A specific form of free energy is chosen which splits the total free energy into the elastic free energy
ψe
and
178
the twinning free energy ψtas:179
ψ=ψe(εe,ηt) + ψt(ηt,xηt). (13)
Note that the free energy
ψ
is assumed to be independent of the crystal plasticity internal variables so that
180
the plastic stored energy vanishes [37].181
Anisotropic linear elasticity is assumed throughout this paper, even though nonlinear elasticity can also
182
be applied in the proposed framework. The phase-field dependent elastic stiffness tensor C(ηt)reads [18]:183
C(ηt) = C(0) + [C(1)C(0)]ϕ(ηt),Cijk l (1) = Qim(1)Qjn(1)Qk p(1)Ql q(1)Cmnpq (0), (14)
where
Q
, as elaborated further in Section 3, is a rotation tensor that replicates the distortion due to the
184
deformation twinning. Here,
C(
0
)
is the elastic stiffness tensor before twinning which is determined by the
185
initial crystal orientation, and
C(
1
)
is the elastic stiffness tensor when the twinning phase field
ηt=
1. The
186
actual elastic stiffness tensor
C(ηt)
is then computed through the algebraic interpolation by the interpolation
187
function ϕ(ηt).188
A quadratic form of elastic free energy
ψe
and a double-well potential for twinning free energy are used
189
ψt[18,17]:190
(ψe=1
2εe:C(ηt):εe
ψt=Atη2
t(1ηt)2+kt:(xηt⊗ ∇xηt),(15)
where for isotropic twinning phase field, the parameters
At=
12
Gt/lt
and
kt=
3
GtltI/
4 represent the
191
equilibrium energy per unit area and per unit thickness, with
Gt
representing the twinning boundary energy
192
per unit volume,
lt
representing the equilibrium boundary thickness of the interfacial region, and
I
being
193
the second order identity tensor. Substituting the specific form of the free energy
(15)
into equation
(12)
, the
194
microforces πtand ξtare derived as:195
πt=γ0τtϕ0(ηt)2Atηt13ηt+2η2
t1
2ϕ0(ηt)εe:[C(1)C(0)] :εe, (16)
196
ξt=∂ψ
xηt
=2kt· ∇xηt. (17)
Then, the microforce balance equation (2) has the following specific form:197
˙
ηt
Mt
=γ0τtϕ0(ηt)2Atηt13ηt+2η2
t¯
W+2x·kt· ∇xηt, (18)
where the partial derivative of the elastic free energy
ψe
with respect to the twinning phase field is introduced
198
as ¯
Wfor illustration convenience:199
¯
W(εe,ηt)=1
2ϕ0(ηt)εe:[C(1)C(0)] :εe. (19)
Note that instead of the double-well potential in equation
(15)
, other forms of Helmholtz free energy
200
may also be feasible [21].201
Solvers for crystal plasticity/twinning phase field models 7
3 Constitutive law202
In this section, the small strain crystal plasticity model with phase-field dependent plastic flow is first
203
introduced. Instead of introducing an additional set of slip systems to characterize the dislocation slip after
204
twinning, a Lie algebra based averaging is used such that the rank of the Schmid tensor is maintained
205
and the resultant plastic flow remains consistent with a slip deformation mode. Next, the stress update
206
algorithm is introduced with consistent linearization. A simplified stochastic twinning nucleation model is
207
incorporated to predict the twinning nucleation.208
3.1 Small strain crystal plasticity209
Based on the Coleman-Noll argument
(10)
and the specific form of free energy
(15)
, the Cauchy stress is
210
resolved as211
σ=∂ψe(εe,ηt)
εe=C(ηt):εe. (20)
Here, the elastic stiffness tensor C(ηt)is defined in equation (14).212
For crystal plasticity without twinning, the Schmid tensor
P(α)
is usually defined by the normal direction
213
m(α)and shear direction s(α)of slip system (α):214
P(α)=1
2s(α)m(α)+m(α)s(α). (21)
As a result of the deformation twinning, the orientations of the slip systems within the twinning region
215
are symmetric to those within the parent region with respect to the twinning habit plane. In the volume
216
fraction approach [
7
] and other twinning phase field approach [
21
], an additional set of slip systems is
217
introduced to characterize the dislocation slip within the twinning region:218
˙εp=[1ϕ(ηt)]
α
˙
γ(α)
sP(α)+ϕ(ηt)
α
˙
γ(α)
stw Q(1)P(α)QT(1) + ˙
γtPt, (22)
where
˙
γ(α)
s
is shear strain of the slip system without twinning,
˙
γ(α)
stw
is shear strain of the rotated slip system
219
after twinning, and ˙
γ(α)
tis the twinning shear strain.220
Unlike other models where algebraic interpolation is used for the Schmid tensor within the phase field
221
interfacial region, we use the Lie algebra to interpolate the Schmid tensor within the interfacial region, such
222
that the rank of the Schmid tensor is maintained and there is no need to introduce an additional set of slip
223
systems.224
The phase-field dependent Schmid tensor
P(α)
of slip system
(α)
is constructed through the regular
225
Schmid tensor P(α)as:226
P(α)(ηt)=Q(ηt)P(α)QT(ηt), (23)
where the re-orientation tensor associated with twinning
Q(ηt)SO(
3
)
is defined through the Lie-algebra
227
based interpolation [38,39,11]:228
Q(ηt)=exp [±πϕ (ηt)m×],m×·umt×uuR3. (24)
The normal direction of the twinning habit plane
mt
is the axial vector of the second order skew symmetric
229
tensor
m×so(
3
)
where so(3) is the Lie algebra corresponding to the special orthogonal group of degree 3
230
denoted as SO(3) in the literature. For completeness, the definitions of SO(3) and so(3) are given below (cf.
231
[38,11,39]):232
SO(3) = {AGL(3)|AAT=Iand det A=1}; (25)
233
so(3) = {Agl(3)|A=AT}; (26)
8 Ran Ma, WaiChing Sun
where GL(3) is the set of 3-by-3 invertible tensor and gl(3) is the corresponding Lie algebra of GL(3), which
234
is the set for 3-by-3 tensor. When the twinning phase field equals to one, the reorientation matrix represents
235
the rotation operation 180around the axial vector mt:236
Q(1)=2mtmtI. (27)
The sign in Eq.
(24)
is selected according to the continuity condition from the field in the neighborhood.
237
Note that the phase-field dependent Schmid tensor
P(α)
should have the same rank as the regular Schmid
238
tensor P(α)and remains free of volumetric mode, i.e.239
tr P(α)=0. (28)
Several conventions on the Lie algebra interpolation are prescribed to ensure consistent and reliable
240
numerical results. First, it is required that the normal direction of the twinning habit plane
mt
has the
241
same positive direction within each grain. Also, only positive sign in equation
(24)
is used to calculate the
242
intermediate rotation tensor Q(ηt)to avoid ambiguity.243
Similar to the typical small strain crystal plasticity models [
40
], the total plastic strain
εp
is the summation
244
of shear strain γ(α)on each slip system:245
˙εp=
α
˙
γ(α)P(α). (29)
The Voce type crystal plasticity model is used, where the flow rule is written as:246
˙
γ(α)=˙
γ0
τ(α)
g
m
sign τ(α), (30)
and the hardening law is written as:247
g=τy+τw,˙
τw=h
β˙
γ(β). (31)
Here,
˙
γ0
is the referential shear rate,
τy
is the initial hardening variable,
m
is the exponential coefficient, and
248
his the hardening rate.249
3.2 Stress update algorithm and consistent tangent stiffness250
This section presents the details for the stress update algorithm and consistent linearization for the crys-
251
tal plasticity model described in Section 3.1 based on a general stress update algorithm for both rate-
252
independent and rate-dependent crystal plasticity [
40
]. The original algorithm is adjusted to account for
253
twinning shear strain and phase field dependent crystal orientation.254
The numerical time integration of the continuous crystal plasticity model is based on the implicit
255
backward Euler integration over time interval
[tn
,
tn+1]
. Suppose that all the state variables at
t=tn
are
256
known, the backward Euler integration of the evolution equations (29)–(31) reads257
εp
n+1=εp
n+
α
γ(α)P(α)
n+1, (32a)
258
γ(α)=t˙
γ0
τ(α)
n+1
gn+1
m
sign τ(α)
n+1, (32b)
259
gn+1=gn+
β
hγ(β), (32c)
where
γ(α)
is the incremental shear strain of slip system
(α)
. Here, the variables with subscript
n+
1 and
n260
denotes their values evaluated at time tn+1and tn, respectively.261
Solvers for crystal plasticity/twinning phase field models 9
Suppose that the displacement and twinning phase field are known at
tn+1
, the total strain
εn+1
, the
262
twinning deformation
εt
n+1
, and the Schmid tensor
P(α)
n+1
are frozen throughout the stress update process.
263
Then, the elastic trial strain is defined as264
εetr =εn+1εp
nεt
n+1, (33)
which is also frozen throughout the stress update. The resolved shear stress
τ(α)
n+1
on the slip system
(α)265
follows the Schmid law as:266
τ(α)
n+1=P(α)
n+1:σn+1=P(α)
n+1:Cn+1:εetr
β
γ(β)P(α)
n+1:Cn+1:P(β)
n+1, (34)
where
Cn+1
is the elastic stiffness tensor defined in equation
(14)
depending solely on the twinning phase
267
field. Combining Equation
(32b)
and Equation
(34)
, the following set of nonlinear equations holds with
268
γ(β)being the set of unknown variables269
r(α)=τ(α)
n+1
msign τ(α)
n+11
˙
γ0t gn+
β
hγ(β)!m
γ(α)=0, (35)
This set of nonlinear equations is solved by Newton-Raphson method, and the consistent linearization is
270
expressed as:271
Dαβ =r(α)
γ(β)=mτ(α)
n+1
m1P(α)
n+1:Cn+1:P(β)
n+1
1
˙
γ0thmgm1
n+1γ(α)sign γ(β)1
˙
γ0tgm
n+1δαβ.
(36)
The consistent tangent stiffness for the crystal plasticity model is written as:272
Ceq =Cn+1
α
β
mτ(β)
n+1
m1D1
αβ Cn+1:P(α)
n+1P(β)
n+1:Cn+1. (37)
3.3 Twinning nucleation model273
In general, the twinning free energy functional is not convex, and the local minimizer of the potential for
274
the coupled problem may not be unique. This implies that different solutions may become admissible and
275
could be sought using different initial guesses [
18
]. For example,
ηt(x
,
t)
0 is a local minimum of the
276
twinning free energy functional satisfying equation
(18)
, but the twinning behavior is not well characterized.
277
To avoid this issue, a coupling term between mechanics and phase field has been introduced in [
17
], but
278
this coupling term is not derived from the thermodynamic theory. Here, we pursue a different approach
279
where a twinning nucleation model within the grain boundary region is introduced to initiate the twinning
280
propagation, which is also physically consistent with the fact that the twinning partial dislocations usually
281
stem from grain boundary defects under external stress.282
A stochastic twinning nucleation model [
21
] is employed here by assuming that the twin nucleus is
283
energetically stable and the nucleation time is negligible. In this model, a random threshold stress
τnu
284
following Poisson distribution is assigned to each integration point within the grain boundary region:285
τnu =τ0
nu ln (1Y)1/ζ, (38)
where
Y[
0, 1
]
is a uniform random variable,
ζ
is a material parameter indicating the fluctuation of the
286
nucleation threshold along grain boundary, and
τ0
nu
is a material parameter indicating the average threshold
287
10 Ran Ma, WaiChing Sun
value. When the resolved shear stress
τt
on the twinning system is larger than the nucleation threshold
288
stress
τnu
, a penalty term enforcing
ηt=
1 is introduced in equation
(18)
which serves as a twinning nucleus
289
˙
ηt
Mt
=γ0τtϕ0(ηt)2Atηt13ηt+2η2
t¯
W+2x·kt· ∇xηt
+βp(ηt1)sign max
τ[0,t]τtτnu +1,
(39)
where βpis a large number to enforce the initial twin nucleus.290
4 Finite element discretization291
In this section, three numerical schemes, including a monolithic scheme, an alternating minimization scheme,
292
and an operator splitting scheme, are derived to solve the coupled crystal plasticity and twinning phase field
293
problem. The operator splitting scheme is proposed to reduce the computational cost of the coupled problem,
294
where the phase field equation is split into an ODE part and a PDE part which are solved sequentially. Also,
295
the operator splitting scheme is more flexible for dynamic problems, where the mechanical part can be
296
solved either explicitly or implicitly. All three numerical schemes are spatially discretized by finite element
297
method using the open-source finite element library deal.II [41,42].298
4.1 Solution scheme: monolithic299
The displacement field
u
is first discretized in time to incorporate the inertia effect. The Newmark method is
300
used to discretize the displacement field in time from t=tnto t=tn+1:301
¨un+1=1
βt2un+1+11
2β¨un1
βt˙un1
βt2un1
βt2un+1+˜an+1, (40)
˙un+1=γ
βtun+1+1γ
β˙un+γt11
2β¨unγ
βtunγ
βtun+1+˜vn+1. (41)
Here,
un
,
˙un
, and
¨un
stands for the displacement, velocity, and acceleration at
t=tn
, while
un+1
,
˙un+1
, and
302
¨un+1
stands for the displacement, velocity, and acceleration at
t=tn+1
. The numerical parameters
β
and
γ303
control the stability of the time integration. Similarly, the material time derivative of the twinning phase
304
field ˙
ηtis approximated by first order finite difference method305
˙
ηt,n+1=ηt,n+1ηt,n
t. (42)
Two weight functions are introduced to derive the weak form of the governing equations, including
306
the virtual displacement field
δu∈ {ϕ|ϕ[H1()]dim
,
ϕ(Γu) =
0
}
and the virtual phase field
δηtH1()
.
307
The weak form for the balance of linear momentum is derived through the inner product between the
308
virtual displacement δuand equation (1)309
Ru=Z(δu·bρδu·¨u− ∇xδu:σ)d+Zδu·tdS=0, (43)
where the acceleration
¨u
is discretized in time by Newmark method in equation
(40)
. Similarly, the weak
310
form for the balance of microforce is derived through the inner product between the virtual phase field
δηt
311
and equation (18)312
Rt=Z"γ0τtδηtϕ0(ηt)˙
ηtδηt
Mt
2Atδηtηt13ηt+2η2
t¯
W2xδηt·kt· ∇xηt#d=0, (44)
where the phase field rate ˙
ηtis discretized in time by implicit backward Euler method.313
Solvers for crystal plasticity/twinning phase field models 11
Directional derivatives of the weak form are derived in order to solve the coupled problems by Newton-
314
Raphson method. Within each time step, the displacement field and the phase field are simultaneously
315
updated until both the mechanical residual
kRukL2
and the phase field residual
kRtkL2
satisfy the conver-
316
gence criteria:317
M+Kuu Kut
Ktu Ktt u
ηt=Ru
Rt. (45)
Here,
M
is the mass matrix normalized by
βt2
, and
Kuu
,
Kut
,
Ktu
, and
Ktt
are the consistent linearization
318
of the residuals as discussed in detail in Section 4.2. Note that the monolithic scheme is less flexible than
319
the alternating minimization scheme and the operator splitting scheme in that the dynamic momentum
320
equation can only be solved implicitly. Also, the assembled stiffness matrix is symmetric since the crystal
321
plasticity model is associative and all the coupling terms are derived from the Helmholtz free energy.322
4.2 Consistent linearization for the monolithic solution scheme323
This section presents the consistent linearization for the global Newton iterations for the monolithic scheme
324
in Section 4.1. Note that the same linearization is also applied for the alternating minimization scheme in
325
Section 4.3.326
The following partial derivatives are necessary to derive the coupled stiffness Kuu,Kut ,Ktu,Ktt.327
r(α)
ε=0∂γ(α)
ε=
β
D1
αβ mτ(β)
m1P(β):C. (46)
r(α)
∂ηt
=0∂γ(α)
∂ηt
=
β
D1
αβ mτ(β)
m1P(β)
∂ηt:C:εetr +P(β):C:εetr
∂ηt
+ϕ0(ηt)P(β):[C(1)C(0)] :εetr
c
γ(c)P(β)
∂ηt:C:P(c)
c
γ(c)P(β):C:P(c)
∂ηt
ϕ0(ηt)
c
γ(c)P(β):[C(1)C(0)] :P(c).
(47)
σ
ε=Ceq. (48)
εp
∂ηt
=
α∂γ(α)
∂ηt
Pα+γ(α)P(α)
∂ηt,P(α)
∂ηt
=πϕ0(ηt)m×P(α)+P(α)mT
×. (49)
εt
∂ηt
=1
2γ0ϕ0(ηt)(stmt+mtst). (50)
σ
∂ηt
=ϕ0(ηt)[C(1)C(0)] :εeC:εp
∂ηt
+εt
∂ηt. (51)
∂τt
ε=Pt:Ceq. (52)
∂τt
∂ηt
=Pt:σ
∂ηt. (53)
The full mass matrix normalized by βt2for the implicit dynamic problem is written as:328
δuIMI J uJ=Zρ
βt2δu·ud, (54)
12 Ran Ma, WaiChing Sun
where the quantities with superscript
I
or
J
refers to the nodal value corresponding to the basis function. A
329
standard Galerkin method is used such that the trial and the interpolation are spanned by the same set of
330
basis function. Finally, the coupled stiffness matrix between the displacement field and the twinning phase
331
field is derived as the following. Here, the operator
D×
refers to the directional derivative with respect to
×
,
332
and the operator (·,·)refers to inner product over the domain.333
δuIKIJ
uuuJ=DuRu,u=Zxδu:σ
ε:xud. (55)
δuIKIJ
utηJ
t=DηtRu,ηt=Zxδu:σ
∂ηt
ηtd. (56)
δηI
tKIJ
tuuJ=DuRt,u=Zϕ0(ηt)εe:[C(1)C(0)] :"xu
α ∂γ(α)
ε:xu!P(α)#
γ0ϕ0(ηt)δηt∂τt
ε:xud.
(57)
δηI
tKIJ
tt ηJ
t=DηtRt,ηt=Zγ0∂τt
∂ηt
ϕ0(ηt) + τtϕ00(ηt)δηtηt
+1
Mttδηtηt+2At16ηt+6η2
tδηtηt
+¯
W
∂ηt
δηtηt+¯
W
εe:εe
∂ηt
δηtηt
+2xδηt·kt· ∇xηtd.
(58)
4.3 Solution scheme: alternating minimization334
The monolithic scheme for multiphysics often leads to saddle point problems that often exhibit higher
335
condition number and in many cases when equal-order discretization is employed, may lead to stability
336
issue [
43
,
44
,
45
,
46
]. Furthermore, the derivation and implementation of the consistent linearization of
337
the coupling stiffness can be time consuming [
47
]. An alternative is to solve the coupling problem via
338
the alternating minimization strategy [
48
,
49
]. The key idea is that, within each time step, the mechanical
339
equation
(43)
is first solved with fixing the phase field parameter
ηt
, and then the phase field equation
(44)
is
340
solved with fixing the displacement
u
. This process is repeated within each time step until both mechanical
341
residual
Ru
and phase field residual
Rt
becomes sufficiently low. A similar approach is widely used for
342
solving phase field fracture problems, which is more robust than the corresponding monolithic scheme [
35
].
343
The convergence criteria for the alternating minimization strategy is demonstrated as follows. Within
344
each time step, the displacement residual
kRukL2
and the phase field residual
kRtkL2
are minimized
345
alternatively by Newton-Raphson method. Within each alternation between the displacement residual and
346
the phase field residual, the initial
L2
norm of the displacement residual
kR0
ukL2
and the phase field residual
347
kR0
tkL2serves as the stopping criteria of the alternating minimization process348
kR0
ukL2<tol, kR0
tkL2<tol. (59)
The displacement residual and the phase field residual are both minimized simultaneously at the end of
349
the alternating minimization process. Algorithm 1shows the pseudocode for the alternating minimization
350
algorithm.351
The alternating minimization strategy is often considered to be more robust than the monolithic scheme
352
in the literature [
50
,
51
,
22
]. This is attributed to the fact that the alternating minimization strategy may
353
approximate a complex saddle-point problem originated from a non-convex coupled free energy with two
354
alternating sub-problems that minimize two convex free energies (one correspond to each sub-problem)
355
in an alternative manner [
51
]. In each incremental time step, one set of the solution is frozen while the
356
Solvers for crystal plasticity/twinning phase field models 13
other one is updated by solving the decoupled sub-problems. Therefore, this strategy avoids many of the
357
common issues of the monolithic scheme such as ill-conditioned tangent, different signs of eigenvalues and
358
the resultant non-positive definite tangential matrix.359
However, the computational cost of the alternative minimization scheme could be higher (unless the
360
number of degree of freedom is very high) and a smaller time step is usually required to maintain the proper
361
coupling behaviors. Furthermore, the alternating minimization scheme is more flexible in the sense that the
362
momentum equilibrium equation can be solved both implicitly and explicitly and different pre-conditioners
363
can be employed to each sub-problem if needed.364
1Data: un,ηt,n
2Result: un+1,ηt,n+1
3i=0 ;
4u(0)=un;
5η(0)
t=ηt,n;
6while kR0
ukL2>tol. or kR0
tkL2>tol. do
7solve Ruu(i+1),η(i)
t=0 for u(i+1);
8solve Rtu(i+1),η(i+1)
t=0 for η(i+1)
t;
9i=i+1 ;
10 end
11 un+1=u(i+1);
12 ηt,n+1=η(i+1)
t;
Algorithm 1:
Algorithm for alternating minimization. The superscript
(i+
1
)
and
(i)
refers to the
number of alternating minimization iterations. The subscript
(n+
1
)
and
(n)
refers to the time step.
4.4 Solution scheme: operator splitting method365
Consider that the monolithic scheme is less robust and the alternating minimization scheme is less efficient,
366
an operator splitting approach is proposed here to solve the coupled crystal plasticity and twinning phase
367
field problem.368
The basic idea of the operator splitting method is to split the partial differential operator into a sequence
369
of sub-problems, usually representing different physical processes, and solve these sub-problems sequen-
370
tially with different numerical methods. In the coupled deformation twinning problem, the phase field
371
operator as defined in Equation (18) is split into a nonlinear ODE part and a linear PDE part:372
˙
ηt
Mt
=γ0τtϕ0(ηt)2Atηt13ηt+2η2
t¯
W, (60a)
373
˙
ηt
Mt
=2x·kt· ∇xηt. (60b)
Let
SO(t)
and
SP(t)
denote the exact solution operators for the nonlinear ODE problem
(60a)
and
374
the linear PDE problem (60b), which project the phase field from time tto t+twhile satisfying equation375
(60a)
and
(60b)
, respectively. Suppose that the state variables at time
tn
are known, the phase field at time
376
t=tn+1satisfying equation (18) is aproximated through the Lie splitting method:377
˜
ηt=SO(t)ηt,n,ηt,n+1=SP(t)˜
ηt,ηt,n+1=SP(t)SO(t)ηt,n, (61)
where the intermediate phase field
˜
ηt
is introduced for the operator splitting. A similar approach is also
378
applied to solve a pure phase field problem for molecular beam epitaxy (MBE) equation and Cahn-Hilliard
379
14 Ran Ma, WaiChing Sun
(CH) equation [
31
], but no mechanics field is coupled with the phase field. Note that Lie splitting method
380
is used here, and Strang splitting method can be used to increase the order of accuracy regarding time
381
integration.382
In the actual implementation, the exact solution operators
SO(t)
and
SP(t)
are approximated by
383
numerical solutions. The time integrations of equations
(60a)
and
(60b)
are performed with backward Euler
384
method, and a weight function δηtH1()is introduced to derive the weak form of equation (60b):385
˜
Rloc =γ0τt(εn+1,˜
ηt)ϕ0(˜
ηt)˜
ηtηt,n
Mtt2At˜
ηt13˜
ηt+2˜
η2
t¯
W=0, (62a)
386
˜
Rt=Z(ηt,n+1˜
ηt)δηt
Mtt2xδηt·kt· ∇xηt,n+1d=0, (62b)
where ˜
ηtis the intermediate phase field.387
Algorithm 2shows the pseudocode for this operator splitting scheme. Within each time step, the
388
displacement field and the intermediate phase field
˜
ηt
are first solved simultaneously. The intermediate
389
phase field is solved together with the crystal plasticity stress update algorithm at each Gauss point, and the
390
discretized intermediate phase field is only stored at each Gauss point. Then, the linear PDE equation is
391
solved to update the phase field.392
Similar to the alternating minimization scheme, the linear momentum equation can be solved both
393
implicitly and explicitly. After the linear momentum equation successfully converges, the phase field at
394
t=tn+1is updated by solving the linear equation (60b).395
1Data: un,ηt,n
2Result: un+1,ηt,n+1
3solve Ru(un+1,˜
ηt)=0
˜
Rloc (un+1,˜
ηt)=0for un+1and ˜
ηt;
4solve ˜
Rt(ηt,n+1,˜
ηt)=0 for ηt,n+1;
Algorithm 2: Algorithm for operator splitting method.
The local stress update algorithm at each Gauss point for the coupled crystal plasticity and twinning
396
phase field is shown in Algorithm 3. For the sake of implementation convenience, two Newton iterations
397
are used to solve the coupled problem. The outer iteration solves the local twinning evolution problem in
398
equation
(62a)
, while the inner iteration performs the regular stress update for crystal plasticity as shown in
399
Section 3.2. Observe that for the local phase field ODE problem in equation (62a),400
˜
Rloc(0) = ηt,n
Mtt0, ˜
Rloc(1) = 1ηt,n
Mtt0, (63)
therefore, the existence of the solution within the closed set
[
0, 1
]
is guaranteed. Equation
(62a)
is highly
401
nonlinear and not monotonic within the
[
0, 1
]
region, and the bisection method is used to provide the initial
402
guess for the Newton iteration. Note that it is possible to consider the local phase field
˜
ηt
as an internal
403
variable and solve the coupled problem with one Newton iteration.404
Given the required partial derivatives derived in Section 4.2, the consistent tangent stiffness for the
405
coupled crystal plasticity and twinning phase field problem is derived as:406
d˜
ηt
dε=γ0ϕ0(˜
ηt)∂τt
ε¯
W
εe:εe
ε
γ0ϕ0(˜
ηt)∂τt
˜
ηt+γ0τtϕ00(˜
ηt)2At16˜
ηt+6˜
η2
t¯
W
∂ηt¯
W
εe:εe
∂ηt1
Mtt
, (64)
Ceq
II =Ceq +σ
˜
ηt
d˜
ηt
dε. (65)
For the operator splitting scheme, two Newton iterations are nested to perform the stress update.
407
The convergence criteria for these two Newton iterations must be carefully determined. Otherwise, the
408
error of the final results might be increased as shown in the results section. Another possibility is that the
409
intermediate phase field is treated as an internal variable so that the two Newton iterations are merged.410
Solvers for crystal plasticity/twinning phase field models 15
1Data: εn+1,εn,gn,ηt,n
2Result: σn+1,Ceq
I I ,gn+1,˜
ηt
3˜
ηt=ηt,n;
4while k˜
Rlock>tol. do
5˜
ηt=d˜
Rloc/d ˜
ηt1˜
Rloc ;
6˜
ηt=˜
ηt+˜
ηt;
7Crystal plasticity stress update in Section 3.2 ;
8Compute ˜
Rloc ;
9end
Algorithm 3: Stress update algorithm for operator splitting method.
Table 1: Material properties for crystal plasticity
Parameters Description Value Unit
KBulk modulus 50.0 GPa
νPoisson’s ratio 0.3 -
τyCritical resolved shear stress 10.0 MPa
hHardening parameter 20.0 MPa
˙
γ0Reference shear strain rate 1.0 ×1012 s1
mRate sensitivity exponent 5.0 -
Remark 1.The computational cost of the three numerical schemes is estimated by the linear solver. Suppose
411
that direct solver is used for all three numerical schemes and there are four dofs at each node including three
412
displacement dofs and one phase field dof. The computational cost of the monolithic scheme is 64
O(N3)413
where
N
is the total number of nodes. The computational cost of the operator splitting scheme is 27
O(N3)414
for the linear momentum equation and
O(N3)
for the phase field equation. For the alternating minimization
415
scheme, the computational cost is difficult to estimate since the total number of iterations within each step is
416
not determined. But the alternating minimization scheme is less efficient in general than the monolithic
417
scheme.418
5 Numerical examples419
This section presents three numerical examples, including one example for a single crystal shear test and two
420
examples for the polycrystal shear test, to demonstrate the accuracy and capability of the three numerical
421
schemes for solving the coupled crystal plasticity and twinning phase field problems. More emphasis is put
422
on the homogenized stress-strain relation, the twinning phase field distribution, and the equivalent plastic
423
strain distribution.424
The material parameters for the crystal plasticity material model and the twinning phase field are listed
425
in Table 1and Table 2, respectively. The material parameters
Mt
and
kt
depend on the interfacial energy
Gt
426
and the length parameter
lt
as shown in equation
(15)
. The single crystal is assumed to be a face-centered
427
cubic (FCC) with 12 slip systems. The default material parameters are used throughout the numerical
428
examples unless otherwise specified.429
5.1 Single crystal with isotropic elasticity430
In this example, the nucleation and evolution of deformation twinning in single crystal is simulated in
431
a two-dimensional domain in plane strain condition under the assumption that the elastic response is
432
isotropic. Figure 3(a) shows the initial setup and boundary condition of the single crystal shear test. The
433
square domain represents a single crystal with 12 face-centered cubic (FCC) slip systems. The normal
434
direction
mt
of the twinning habit plane is
(
0, 1, 0
)
, and the shear direction
st
of the twinning system is
435
16 Ran Ma, WaiChing Sun
Table 2: Material properties for twinning phase field
Parameters Description Value Unit
GtInterfacial energy 5.85 ×101J m2
ltLength scale 5.0 ×109m
MtMobility parameter 1000.0 J1s1
αInterpolation parameter 3.0 -
τt
0Nucleation threshold stress 10.0 MPa
γ0Twin shear 0.1 -
(
1, 0, 0
)
. The initial Euler angles are (
0
,
0
,
0
) in Bunge notation. This square domain is discretized into
436
128
×
128 structured finite elements. Simple shear boundary condition is applied on the top surface, while
437
the bottom surface is fixed. Within the nucleation region in the middle of the square domain, the nucleation
438
threshold τnu equals to 10 MPa, while τnu outside the nucleation region.439
Twin nucleation
48 nm
6 nm
ux= 2.4t(nm)
(a) numerical setup
0 1 2 3 4 5
shear strain 10-3
-100
0
100
200
300
400
500
shear stress (MPa)
monolithic
AM
OP
(b) stress strain ralation
Fig. 3: The numerical setup of the single crystal shear test and homogenized shear-stress shear-strain
response. The Euler angles are (
0
,
0
,
0
) in Bunge notation. In the legend of Figure (b), AM stands for
alternating minimization, and OS stands for operator splitting.
The stress-strain relation for the simple shear test is shown in Figure 3(b). The shear strain is calculated
440
by the total displacement of the top surface divided by the edge length, and the shear stress is homogenized
441
within the square domain. At the beginning of the loading, the shear stress drops due to twin nucleation.
442
Then, the shear stress increases monotonically until the twinning region propagates through the domain.
443
After the twinning region propagates through the domain, the shear stress remains constant when the
444
twinning region propagates along the thickness direction. It is observed that the stress-strain relations of the
445
monolithic scheme and the alternating minimization scheme coincide with each other, while higher stiffness
446
is observed in the operator splitting scheme. Also, the twinning propagation occurs earlier in the operator
447
splitting scheme than the other two numerical schemes.448
Figure 4shows twinning phase field distribution for different numerical schemes. It is observed that the
449
phase field distribution for the three numerical schemes are comparable in general, but the interfacial region
450
for the operator splitting scheme is smaller than the other two numerical schemes.451
Figure 5shows the equivalent plastic strain distribution obtained from different numerical schemes.
452
Again, the distribution patterns of the equivalent plastic strain are comparable among the three numerical
453
schemes in general, but the operator splitting scheme over-predicts the plastic strain.454
Solvers for crystal plasticity/twinning phase field models 17
(a) monolithic scheme (b) alternating minimization (c) operator splitting
Fig. 4: Phase field distribution for different numerical schemes.
(a) monolithic scheme (b) alternating minimization (c) operator splitting
Fig. 5: Equivalent plastic strain distribution for different numerical schemes.
In general, the phase field propagation progress of the operator splitting method is faster than the other
455
two numerical schemes, and the resultant equivalent plastic strain is also higher. One possible reason is
456
that, for the operator splitting scheme, two Newton iterations are used to perform the stress update at each
457
Gauss point. The nested Newton iteration may introduce error accumulation. Another possible reason is
458
that the Lie splitting is used for the operator splitting scheme, which is only first order accurate in time.
459
Other splitting methods, such as Strang splitting method [
30
], might improve the accuracy. More detailed
460
analysis on the performance of the variety of choices and setups for the splitting methods will be considered
461
in the future but is out of the scope of this study.462
5.2 Polycrystal with isotropic elasticity463
The twinning nucleation and evolution is then simulated in a 2D shear test performed on a polycrystalline
464
domain. The material parameters remain the same as the single crystal shear test, except that the interfacial
465
energy
Gt=
0.5
×
10
1J m2
and the length scale
lt=
2.4
×
10
9m
such that twinning propagation is
466
observed within small strain region. The normal direction
mt
of the twinning habit plane is
(
0.5, 0.866, 0
)
,
467
and the shear direction
st
of the twinning system is
(
0.866,
0.5, 0
)
. Compared with the single crystal
468
shear test where the twinning habit plane is aligned with the crystal axis, the twinning habit plane in the
469
polycrystal test is not aligned with any crystal axis such that the crystal orientations of the parent region
470
and the twinning region are not equivalent.471
Figure 6(a) shows the initial setup and boundary condition for the polycrystal shear test. Compared with
472
the initial setup of the single crystal test, the only differences are the crystal orientation and the nucleation
473
region size. A total number of 10 grains are included in the RVE, with their
[
001
]
axis aligned along the
z
axis.
474
The grain structure of the 2D domain is produced by the open source software Neper [
52
]. A single twinning
475
18 Ran Ma, WaiChing Sun
Twin nucleation
48 nm
2 nm
ux= 2.4t(nm)
x
y
w.r.t. xaxis
(a) numerical setup
0 0.005 0.01 0.015 0.02 0.025 0.03
shear strain
0
100
200
300
400
500
600
shear stress (MPa)
monolithic
alternating minimization
operator splitting
(b) stress strain ralation
Fig. 6: The numerical setup of a polycrystal RVE shear test with isotropic elasticity and corresponding
homogenized shear-stress shear-strain response.
nucleation region with fixed nucleation stress is set in the middle of the domain instead of assigning random
476
nucleation stress following Poisson distribution to the grain boundary elements, so that the three numerical
477
simulations are comparable to each other.478
Figure 6(b) shows the stress-strain response of the shear test. Again, the shear strain and the shear stress
479
are calculated in the same manner as the single crystal test. Similar stress-strain response is also observed in
480
the polycrystalline test. The monolithic scheme and the alternating minimization scheme coincide with each
481
other, while twinning propagation occurs earlier in the operator splitting scheme.482
Figure 7shows the twinning phase field distribution of different numerical schemes, and Figure 8
483
shows corresponding orientation distribution. It is observed that all three schemes produce similar twinning
484
propagation pattern, which is determined by the grain orientation and the stress distribution. The phase
485
field propagation progress of the operator splitting scheme falls behind that of the other two numerical
486
schemes. The benefit of using Lie algebra to interpolate the crystal orientation between the twinning region
487
and the parent region is shown in Figure 8, where the orientation transition from the parent region to the
488
twinning region is smooth. Note that in the current simulation, although the
[
001
]
axis of all the grains is
489
aligned with the
z
axis, the
[
001
]
axis of the twinning interfacial region is not aligned with the
z
axis and the
490
twinning shear strain is out of the
xy
plane. This is due to the Lie algebra interpolation, which guarantees
491
that any interpolation between two rotation tensors in
SO(
3
)
group remains in the
SO(
3
)
[
11
,
39
,
38
]. As a
492
result, the crystal orientation is rotated smoothly from the parent region to the twinning region.493
Figure 9shows the equivalent plastic strain within the polycrystalline domain from all the numerical
494
schemes. Similar as the single crystal test, the monolithic scheme and the alternating minimization scheme
495
produce the same equivalent plastic strain, while the operator splitting scheme produces higher equivalent
496
plastic strain.497
Remark 2.In the alternating minimization scheme, the iteration between the mechanical part and the phase
498
field part of the problem is necessary. Otherwise, the twinning propagation will be significantly delayed, and
499
the twinning morphology is not correctly predicted. Corresponding results are not provided for simplicity.
500
5.3 Polycrystal with anisotropic elasticity501
In the third numerical example, the initial setup is exactly the same as the previous polycrystal test with
502
isotropic elasticity, except that the elastic stiffness tensor
C(
0
)
possesses cubic symmetry instead of being
503
Solvers for crystal plasticity/twinning phase field models 19
(a) monolithic scheme (b) alternating minimization (c) operator splitting
Fig. 7: Phase field distribution for different numerical schemes with isotropic elasticity.
(a) monolithic scheme (b) alternating minimization (c) operator splitting
Fig. 8: Crystal reorientation of different numerical schemes with isotropic elasticity. The inverse pole figures
are with respect to the xaxis as shown in Figure 6.
(a) monolithic scheme (b) alternating minimization (c) operator splitting
Fig. 9: Equivalent plastic strain distribution of different numerical schemes with isotropic elasticity.
isotropic. The implication is that both the elastic and plastic constitutive responses may be influenced by the
504
spin due to the deformation twinning.505
The elastic stiffness tensor
C(
0
)
in Equation
(14)
is a push-forward of the elastic stiffness tensor
C0
506
defined in the crystal configuration, with
C0
11 =
80.8
GPa
,
C0
12 =
34.6
MPa
, and
C0
44 =
50.0
GPa
, where the
507
shear stiffness is increased compared with the previous section.508
Figure 10 shows the relationship between the homogenized shear stress and shear strain. Again, the
509
operator splitting scheme yields earlier twinning propagation than the monolithic scheme and the alternating
510
minimization scheme, indicating that the phase field propagation progress is faster in the operator splitting
511
20 Ran Ma, WaiChing Sun
scheme. Also, the monolithic scheme fails to converge after the last step shown in Figure 10, while the
512
alternating minimization scheme and operator splitting scheme could converge with better numerical
513
robustness.514
0 0.005 0.01 0.015 0.02
shear strain
0
200
400
600
800
1000
shear stress (MPa)
monolithic
alternating minimization
operator splitting
Fig. 10: Stress-strain relation of a polycrystal shear test with anisotropic elasticity.
Figure 11 shows the phase field distribution of the three numerical schemes. It is observed that all three
515
numerical schemes yield similar phase field distribution pattern. The elastic anisotropy also influences
516
the phase field distribution by comparing Figure 7and Figure 11. Figure 12 shows the final orientation
517
distribution of the three numerical schemes. Again, due to the Lie algebra interpolation of the crystal
518
orientation, the orientation transition from the parent region to the twinning region is smooth. It is also
519
observed that the phase field propagation progress of the operator splitting scheme is faster than the other
520
two schemes, and the twinning region propagates to the neighbor grains earlier.521
(a) monolithic scheme (b) alternating minimization (c) operator splitting
Fig. 11: Twinning phase field distribution of different numerical schemes with anisotropic elasticity.
Figure 13 shows the equivalent plastic strain within the polycrystalline domain from all the numerical
522
schemes. Similar as the single crystal test, the monolithic scheme and the alternating minimization scheme
523
produce the same equivalent plastic strain, while the operator splitting scheme produces higher equivalent
524
plastic strain.525
By comparing the numerical results of the polycrystalline shear test with either isotropic elasticity
526
(Section 5.2) or anisotropic elasticity (Section 5.3), we have the following observations. First, the operator
527
splitting scheme is less accurate than the monolithic scheme and the alternating minimization scheme.
528
However, our observation suggests that the operator splitting scheme is more robust than the monolithic
529
scheme in the sense that the monlithic scheme has failed to converge, perhaps due to the higher condition
530
Solvers for crystal plasticity/twinning phase field models 21
(a) monolithic scheme (b) alternating minimization (c) operator splitting
Fig. 12: Crystal reorientation of different numerical schemes with anisotropic elasticity. The inverse pole
figures are with respect to the xaxis as shown in Figure 6.
(a) monolithic scheme (b) alternating minimization (c) operator splitting
Fig. 13: Equivalent plastic strain distribution of different numerical schemes with anisotropic elasticity.
number whereas the operator splitting scheme is able to generate converged updates for the sub-problems.
531
In the meantime, the operator splitting scheme also exhibits higher computational efficiency than the other
532
two schemes in the numerical experiments.533
6 Conclusion534
A coupled crystal plasticity and twinning phase field model is proposed to simulate deformation twinning
535
observed in polycrystalline materials. The proposed model is thermodynamically consistent. Lie algebra is
536
utilized to interpolate the crystal reorientation due to deformation twinning within the phase field interfacial
537
region, such that the orientation transition from the parent region to the twinning region is smooth and
538
there is no need to introduce an additional set of slip systems to characterize the dislocation slip within
539
the twinning region. Three numerical schemes are proposed to solve the coupled problem, including a
540
monolithic scheme, an alternating minimization scheme, and an operator splitting scheme. The results show
541
that all three numerical schemes produce similar twinning pattern for both the single-crystal simulation and
542
the polycrystal simulation. Future work will target the following challenges. First, the accuracy, stability, and
543
computational cost of the three schemes will be discussed. The time-step and mesh-resolution dependent
544
accuracy and stability of the operator splitting scheme will be discussed in detail. Second, the coupling
545
among crystal plasticity, twinning phase field, and fracture phase field will be investigated. It is expected
546
that this model can be applied to predict the fracture behavior of polycrystalline material with deformation
547
twinning, e.g. HMX (octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine).548
22 Ran Ma, WaiChing Sun
7 Acknowledgments549
The authors thank the two anonymous reviewers for the timely feedback and insightful suggestions. The
550
authors are supported by the Dynamic Materials and Interactions Program from the Air Force Office of
551
Scientific Research under grant contracts FA9550-17-1-0169 and FA9550-19-1-0318, and the NSF CAREER
552
grant from Mechanics of Materials and Structures program at National Science Foundation under grant
553
contracts CMMI-1846875. These supports are gratefully acknowledged.554
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... Also, spectral method is mostly used for solving linear problems [54]. In [52,55,56], the proposed phase-field simulations for deformation twinning and dislocation induced plasticity in hexagonal closed-pack materials were formulated on small strain theory; still, the twin evolution is usually accompanied by large interface orientation and large shear deformations [57] even under small strains [58]. Thus, coupling between twin evolution and fracture is of importance to achieve high accuracy in the numerical solution. ...
... In terms of validating the phase-field results of transmission mechanisms of deformation twins, atomistic simulations (e.g., molecular dynamics simulations [50,55] and density functional theory [52]) and experimental results [51,59] are the most widely used. Some drawbacks to these validations exist such as • discrepancies of the peak stress value from the simulation and experimental data [51], • qualitative comparison of distribution of order parameter using the isotropic gradient energy parameter [52,55,56], • adopting empirically determined large non-physical values for the phase-field parameters (e.g., twin-twin interfacial energy, initial twin nucleus, and energy barrier heights between the matrix and the twinning [50,51]), and • validating at the different length-scales [50,60]. ...
... We verify the proposed implementation of the timeresolved continuum-based model for magnesium by the static phase-field model [68] and molecular dynamics (MD) simulations [69] (Fig. 1). By choosing the same length-scale for the phase-field model and MD simulations, we assure the compatibility of MD results with our implementation, which is often left aside in the literature [51,56,60]. It is also worth stating that all MD simulations use extremely high deformation rates, making it difficult to understand whether a phenomenon results from the rate sensitivity of the material or is a numerical artifact [70,71]. ...
Article
Full-text available
Crack initiation and propagation as well as abrupt occurrence of twinning are challenging fracture problems where the transient phase-field approach is proven to be useful. Early-stage twinning growth and interactions are in focus herein for a magnesium single crystal at the nanometer length-scale. We demonstrate a basic methodology in order to determine the mobility parameter that steers the kinetics of phase-field propagation. The concept is to use already existing molecular dynamics simulations and analytical solutions in order to set the mobility parameter correctly. In this way, we exercise the model for gaining new insights into growth of twin morphologies, temporally-evolving spatial distribution of the shear stress field in the vicinity of the nanotwin, multi-twin, and twin-defect interactions. Overall, this research addresses gaps in our fundamental understanding of twin growth, while providing motivation for future discoveries in twin evolution and their effect on next-generation material performance and design.
... More specifically, the deformation twinning is identified as a volume-preserving stretch followed by a rigid-body rotation (where the rotation is contained in the elastic part of the deformation gradient). This is in contrast to the conventional approach adopted in the literature, in which twinning is characterized by a simple shear deformation (Clayton and Knap, 2011;Kondo et al., 2014;Liu et al., 2018b;Grilli et al., 2020;Ma and Sun, 2021;Hu et al., 2021). Although the two approaches are equivalent in the sharp-interface description, they are not necessarily equivalent in the diffuse-interface description. ...
... (13) and (14) for = 0 and = 1, respectively. Formulae for the inelastic velocity gradient L in resembling that in Eq. (20) (or, analogously, for the total plastic strain rate in the small-strain theory) can be found in several phase-field models of twinning and plasticity (Kondo et al., 2014;Liu et al., 2018b;Grilli et al., 2020;Hu et al., 2021;Ma and Sun, 2021), as well as in numerous meso-scale crystal plasticity formulations in which twinning is treated as a pseudo-slip system (e.g., Kalidindi, 1998;Staroselsky and Anand, 2003;Kowalczyk-Gajewska, 2010;Izadbakhsh et al., 2011;Zhang and Joshi, 2012;Chang and Kochmann, 2015). The difference with respect to all these formulations, and the distinctive feature of the present model, is the form of the twinning contribution in Eq. (20), which is a consequence of treating the twinning as a displacive transformation characterized by the stretch U tw , as discussed in Sections 2.2 and 2.3. ...
... which has been employed in phase-field models for twinning only (Clayton and Knap, 2011) and for twinning with plasticity (Kondo et al., 2014;Liu et al., 2018b;Ma and Sun, 2021;Hu et al., 2021). By construction, the rank-one mixing rule (21) ensures a compatible planar diffuse matrix-twin interface with the normal m (1) for all 0 ≤ ≤ 1. ...
Article
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A finite-strain phase-field model of coupled deformation twinning and crystal plasticity is developed in the paper. Twinning is treated as a displacive transformation characterized by a volume-preserving stretch rather than a simple shear, the latter considered in the conventional approach. It is shown that the two approaches are equivalent in the sharp-interface description, but not in the diffuse-interface description. In the proposed stretch-based kinematics, each pair of conjugate twinning systems is represented by a single twin deformation variant, and thus a single order parameter suffices to consistently describe the two conjugate twinning systems, thereby treating them equally. The model is formulated in the framework of incremental energy minimization, which, upon time discretization, leads to a quasi-optimization problem due to the specific form of the incremental potential within the diffuse interfaces. To facilitate finite-element implementation, a micromorphic formulation of the model is employed. As an application, tensile twinning in HCP magnesium alloys is examined, and a set of comprehensive 2D plane-strain problems is studied to illustrate the features of the proposed approach.
... Although these models present twin bands qualitatively, to the best of the authors' knowledge, they have never been compared directly with experimental twin maps, and their limitations have not been investigated. The only possible alternative is to couple the phase field simulation with the CPFE simulation (Kondo et al. [75], Liu et al. [76], Liu et al. [77], Ma and Sun [78]). While the CPFE handles the stress by solving the equilibrium equations, the phase-field simulation governs the twin nucleation and growth. ...
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Crystal plasticity simulation is an important tool for advanced Integrated Computational Materials Engineering for metals and alloys. The current work presents a calibration and validation framework for crystal plasticity finite element (CPFE) simulation of extension twinning in the Mg alloy WE43 using the scanning electron microscopy with digital image correlation (SEM-DIC) technique. Rolled Mg alloy WE43 was subjected to in situ uniaxial compression along its rolling direction. Full-field displacement maps were captured using SEM-DIC during load pauses, and twin variant maps were obtained from the strain maps using post-processing analysis. CPFE was used to investigate the experimental results via a multi-scale twinning model developed for HCP polycrystals. In addition to macroscopic stress-strain curves, crystal plasticity parameters were calibrated using the variation of twin fraction area versus the applied strain obtained from the SEM-DIC results to accurately capture the twinning parameters. A new SEM-DIC pipeline was created for the open-source PRISMS-Plasticity CPFE software that can read in the precise deformation map generated by SEM-DIC as an input boundary condition for the finite element simulation and conduct the CPFE simulation. The performance of CPFE was evaluated versus the SEM-DIC obtained strain and twin maps. The results show that the CPFE can successfully model the macroscopic stress-strain response and the twin area fraction and that it can additionally capture microscale strain and twinning.
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We present a hybrid model/model-free data-driven approach to solve poroelasticity problems. Extending the data-driven modeling framework originated from \citet{kirchdoerfer2016data}, we introduce one model-free and two hybrid model-based/data-driven formulations capable of simulating the coupled diffusion-deformation of fluid-infiltrating porous media with different amounts of available data. To improve the efficiency of the model-free data search, we introduce a distance-minimized algorithm accelerated by a k-dimensional tree search. To handle the different fidelities of the solid elasticity and fluid hydraulic constitutive responses, we introduce a hybridized model in which either the solid and the fluid solver can switch from a model-based to a model-free approach depending on the availability and the properties of the data. Numerical experiments are designed to verify the implementation and compare the performance of the proposed model to other alternatives.
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We present a hybrid model/model-free data-driven approach to solve poroelasticity problems. Extending the data-driven modeling framework originated from Kirchdoerfer and Ortiz (2016), we introduce one model-free and two hybrid model-based/data-driven formulations capable of simulating the coupled diffusion-deformation of fluid-infiltrating porous media with different amounts of available data. To improve the efficiency of the model-free data search, we introduce a distance-minimized algorithm accelerated by a k-dimensional tree search. To handle the different fidelities of the solid elasticity and fluid hydraulic constitutive responses, we introduce a hybridized model in which either the solid and the fluid solver can switch from a model-based to a model-free approach depending on the availability and the properties of the data. Numerical experiments are designed to verify the implementation and compare the performance of the proposed model to other alternatives.
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