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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.

a⊗b=aibj

) or two

110

symmetric second order tensors (e.g.

(α⊗β)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

Ω

in the Euclidean space deforming

124

within a time interval

I= [

0,

t]

. The deformable body

Ω

has a piece-wisely smooth boundary

Γ

such that

125

the closure of the open set is

¯

Ω=Ω∪Γ

. A displacement ﬁeld

u:Ω×I→Rd

and a twinning phase ﬁeld

126

ηt:Ω×I→R

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

Γu∪Γt=Γ

and

Γu∩Γt=∅

. 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∈Ω

, the balance law of linear momentum is written

133

as:134

∇x·σ+b=ρ¨u, in Ω

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

Ω

, 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 Ω

∇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(2−α)η3

t+(α−3)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(st⊗mt+mt⊗st), (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˙

ηt−πt˙

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−˙

ψ≥0, (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)=e−θs, (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: ˙εt−πt˙

ηt−∂ψ

∂ηt

˙

ηt+ξt−∂ψ

∂∇xηt· ∇x˙

ηt≥0. (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:(st⊗mt+mt⊗st), (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(1−ηt)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)−2Atηt1−3ηt+2η2

t−1

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)−2Atηt1−3ηt+2η2

t−¯

W+2∇x·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=[1−ϕ(ηt)]∑

α

˙

γ(α)

sP(α)+ϕ(ηt)∑

α

˙

γ(α)

s−tw Q(1)P(α)QT(1) + ˙

γtPt, (22)

where

˙

γ(α)

s

is shear strain of the slip system without twinning,

˙

γ(α)

s−tw

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×·u≡mt×u∀u∈R3. (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) = {A∈GL(3)|AAT=Iand det A=1}; (25)

233

so(3) = {A∈gl(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 180◦around the axial vector mt:236

Q(1)=2mt⊗mt−I. (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

h∆γ(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+1−εp

n−εt

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+1−1

˙

γ0∆t gn+∑

β

h∆γ(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

m−1P(α)

n+1:Cn+1:P(β)

n+1

−1

˙

γ0∆thmgm−1

n+1∆γ(α)sign ∆γ(β)−1

˙

γ0∆tgm

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

m−1D−1

αβ Cn+1:P(α)

n+1⊗P(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 (1−Y)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)−2Atηt1−3ηt+2η2

t−¯

W+2∇x·kt· ∇xηt

+βp(ηt−1)sign max

τ∈[0,t]τt−τnu +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

β∆t2un+1+1−1

2β¨un−1

β∆t˙un−1

β∆t2un≡1

β∆t2un+1+˜an+1, (40)

˙un+1=γ

β∆tun+1+1−γ

β˙un+γ∆t1−1

2β¨un−γ

β∆tun≡γ

β∆tun+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+1−ηt,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(Ω)]dim

,

ϕ(Γu) =

0

}

and the virtual phase ﬁeld

δηt∈H1(Ω)

.

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=ZΩ(δu·b−ρδu·¨u− ∇xδu:σ)dΩ+Z∂Ωδu·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=ZΩ"γ0τtδηtϕ0(ηt)−˙

ηtδηt

Mt

−2Atδηtηt1−3ηt+2η2

t−¯

W−2∇xδηt·kt· ∇xηt#dΩ=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

β∆t2

, 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

D−1

αβ mτ(β)

m−1P(β):C. (46)

∂r(α)

∂ηt

=0⇒∂γ(α)

∂ηt

=∑

β

D−1

αβ mτ(β)

m−1∂P(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)(st⊗mt+mt⊗st). (50)

∂σ

∂ηt

=ϕ0(ηt)[C(1)−C(0)] :εe−C:∂εp

∂ηt

+∂εt

∂ηt. (51)

∂τt

∂ε=Pt:Ceq. (52)

∂τt

∂ηt

=Pt:∂σ

∂ηt. (53)

The full mass matrix normalized by β∆t2for the implicit dynamic problem is written as:328

δuIMI J ∆uJ=ZΩρ

β∆t2δu·∆udΩ, (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

uu∆uJ=−DuRu,∆u=ZΩ∇xδu:∂σ

∂ε:∇x∆udΩ. (55)

δuIKIJ

ut∆ηJ

t=−DηtRu,∆ηt=ZΩ∇xδu:∂σ

∂ηt

∆ηtdΩ. (56)

δηI

tKIJ

tu∆uJ=−DuRt,∆u=ZΩϕ0(ηt)εe:[C(1)−C(0)] :"∇x∆u−∑

α ∂γ(α)

∂ε:∇x∆u!P(α)#

−γ0ϕ0(ηt)δηt∂τt

∂ε:∇x∆udΩ.

(57)

δηI

tKIJ

tt ∆ηJ

t=−DηtRt,∆ηt=ZΩ−γ0∂τt

∂ηt

ϕ0(ηt) + τtϕ00(ηt)δηt∆ηt

+1

Mt∆tδηt∆ηt+2At1−6ηt+6η2

tδηt∆ηt

+∂¯

W

∂ηt

δηt∆ηt+∂¯

W

∂εe:∂εe

∂ηt

δηt∆ηt

+2∇xδηt·kt· ∇x∆ηtdΩ.

(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)−2Atηt1−3ηt+2η2

t−¯

W, (60a)

373

˙

ηt

Mt

=2∇x·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 δηt∈H1(Ω)is introduced to derive the weak form of equation (60b):385

˜

Rloc =γ0τt(εn+1,˜

ηt)ϕ0(˜

ηt)−˜

ηt−ηt,n

Mt∆t−2At˜

ηt1−3˜

ηt+2˜

η2

t−¯

W=0, (62a)

386

˜

Rt=ZΩ−(ηt,n+1−˜

ηt)δηt

Mt∆t−2∇xδηt·kt· ∇xηt,n+1dΩ=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

Mt∆t≤0, ˜

Rloc(1) = −1−ηt,n

Mt∆t≥0, (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)−2At1−6˜

ηt+6˜

η2

t−∂¯

W

∂ηt−∂¯

W

∂εe:∂εe

∂ηt−1

Mt∆t

, (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 ˜

ηt−1˜

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 ×10−12 s−1

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 ×10−1J m−2

ltLength scale 5.0 ×10−9m

MtMobility parameter 1000.0 J−1s−1

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 m−2

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

x−y

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

References555

1.

Gr

¨

assel O, Kr

¨

uger L, Frommeyer G, Meyer L. High strength Fe–Mn–(Al, Si) TRIP/TWIP steels

556

development—properties—application. International Journal of plasticity 2000; 16(10-11): 1391–1409.557

2.

Barnett M. Twinning and the ductility of magnesium alloys: Part I:“Tension” twins. Materials Science and

558

Engineering: A 2007; 464(1-2): 1–7.559

3. Humphreys FJ, Hatherly M. Recrystallization and related annealing phenomena. Elsevier . 2012.560

4. Kelly A, Knowles KM. Crystallography and crystal defects. John Wiley & Sons . 2020.561

5.

Guo Y, Abdolvand H, Britton T, Wilkinson A. Growth of

{

112 2

}

twins in titanium: A combined

562

experimental and modelling investigation of the local state of deformation. Acta Materialia 2017; 126:

563

221–235.564

6.

Tom

´

e C, Lebensohn RA, Kocks U. A model for texture development dominated by deformation twinning:

565

application to zirconium alloys. Acta metallurgica et materialia 1991; 39(11): 2667–2680.566

7.

Kalidindi SR. Incorporation of deformation twinning in crystal plasticity models. Journal of the Mechanics

567

and Physics of Solids 1998; 46(2): 267–290.568

8.

Cheng J, Ghosh S. A crystal plasticity FE model for deformation with twin nucleation in magnesium

569

alloys. International Journal of Plasticity 2015; 67: 148–170.570

9.

Cheng J, Ghosh S. Crystal plasticity ﬁnite element modeling of discrete twin evolution in polycrystalline

571

magnesium. Journal of the Mechanics and Physics of Solids 2017; 99: 512–538.572

10.

Qinami A, Bryant EC, Sun W, Kaliske M. Circumventing mesh bias by r-and h-adaptive techniques for

573

variational eigenfracture. International Journal of Fracture 2019; 220(2): 129–142.574

11.

Na S, Bryant EC, Sun W. A conﬁgurational force for adaptive re-meshing of gradient-enhanced porome-

575

chanics problems with history-dependent variables. Computer Methods in Applied Mechanics and Engineer-

576

ing 2019; 357: 112572.577

12.

Ardeljan M, McCabe RJ, Beyerlein IJ, Knezevic M. Explicit incorporation of deformation twins into

578

crystal plasticity ﬁnite element models. Computer Methods in Applied Mechanics and Engineering 2015; 295:

579

396–413.580

13.

Jin T, Mourad HM, Bronkhorst CA, Beyerlein IJ. A single crystal plasticity ﬁnite element formulation

581

with embedded deformation twins. Journal of the Mechanics and Physics of Solids 2019; 133: 103723.582

14.

Jin T, Mourad HM, Bronkhorst CA, et al. Three-dimensional explicit ﬁnite element formulation for

583

shear localization with global tracking of embedded weak discontinuities. Computer Methods in Applied

584

Mechanics and Engineering 2019; 353: 416–447.585

15.

Borst dR, Verhoosel CV. Gradient damage vs phase-ﬁeld approaches for fracture: Similarities and

586

differences. Computer Methods in Applied Mechanics and Engineering 2016; 312: 78–94.587

16.

Zhang X, Krischok A, Linder C. A variational framework to model diffusion induced large plastic

588

deformation and phase ﬁeld fracture during initial two-phase lithiation of silicon electrodes. Computer

589

methods in applied mechanics and engineering 2016; 312: 51–77.590

17.

Hu S, Henager Jr CH, Chen L. Simulations of stress-induced twinning and de-twinning: a phase ﬁeld

591

model. Acta Materialia 2010; 58(19): 6554–6564.592

18.

Clayton JD, Knap J. A phase ﬁeld model of deformation twinning: nonlinear theory and numerical

593

simulations. Physica D: Nonlinear Phenomena 2011; 240(9-10): 841–858.594

19.

Clayton JD, Knap J. Phase ﬁeld modeling and simulation of coupled fracture and twinning in single

595

crystals and polycrystals. Computer Methods in Applied Mechanics and Engineering 2016; 312: 447–467.596

20.

Kondo R, Tadano Y, Shizawa K. A phase-ﬁeld model of twinning and detwinning coupled with

597

dislocation-based crystal plasticity for HCP metals. Computational materials science 2014; 95: 672–683.598

Solvers for crystal plasticity/twinning phase ﬁeld models 23

21.

Liu C, Shanthraj P, Diehl M, et al. An integrated crystal plasticity–phase ﬁeld model for spatially resolved

599

twin nucleation, propagation, and growth in hexagonal materials. International Journal of Plasticity 2018;

600

106: 203–227.601

22.

Choo J, Sun W. Coupled phase-ﬁeld and plasticity modeling of geological materials: From brittle fracture

602

to ductile ﬂow. Computer Methods in Applied Mechanics and Engineering 2018; 330: 1–32.603

23.

Ma R, Sun W. FFT-based solver for higher-order and multi-phase-ﬁeld fracture models applied to

604

strongly anisotropic brittle materials. Computer Methods in Applied Mechanics and Engineering 2020; 362:

605

112781.606

24.

Aldakheel F, Hudobivnik B, Hussein A, Wriggers P. Phase-ﬁeld modeling of brittle fracture using

607

an efﬁcient virtual element scheme. Computer Methods in Applied Mechanics and Engineering 2018; 341:

608

443–466.609

25.

Bryant EC, Sun W. A mixed-mode phase ﬁeld fracture model in anisotropic rocks with consistent

610

kinematics. Computer Methods in Applied Mechanics and Engineering 2018; 342: 561–584.611

26.

Na S, Sun W. Computational thermomechanics of crystalline rock, Part I: A combined multi-phase-

612

ﬁeld/crystal plasticity approach for single crystal simulations. Computer Methods in Applied Mechanics

613

and Engineering 2018; 338: 657–691.614

27.

Suh HS, Sun W, O’Connor D. A phase ﬁeld model for cohesive fracture in micropolar continua. Computer

615

Methods in Applied Mechanics and Engineering 2020; 369.616

28.

Liu G, Li Q, Msekh MA, Zuo Z. Abaqus implementation of monolithic and staggered schemes for

617

quasi-static and dynamic fracture phase-ﬁeld model. Computational Materials Science 2016; 121: 35–47.618

29.

Strang G. On the construction and comparison of different splitting schemes. SIAM J. Numer. Anal. 1968:

619

506–517.620

30. MacNamara S, Strang G. Operator splitting. In: Springer. 2016 (pp. 95–114).621

31.

Cheng Y, Kurganov A, Qu Z, Tang T. Fast and stable explicit operator splitting methods for phase-ﬁeld

622

models. Journal of Computational Physics 2015; 303: 45–65.623

32.

Tegze G, Bansel G, T

´

oth GI, Pusztai T, Fan Z, Gr

´

an

´

asy L. Advanced operator splitting-based semi-

624

implicit spectral method to solve the binary phase-ﬁeld crystal equations with variable coefﬁcients.

625

Journal of Computational Physics 2009; 228(5): 1612–1623.626

33.

Lee HG, Shin J, Lee JY. First and second order operator splitting methods for the phase ﬁeld crystal

627

equation. Journal of Computational Physics 2015; 299: 82–91.628

34.

Shin J, Lee HG, Lee JY. First and second order numerical methods based on a new convex splitting for

629

phase-ﬁeld crystal equation. Journal of Computational Physics 2016; 327: 519–542.630

35.

Wu JY, Nguyen VP, Nguyen CT, Sutula D, Bordas S, Sinaie S. Phase ﬁeld modeling of fracture. Advances

631

in applied mechancis: multi-scale theory and computation 2018; 52.632

36.

Proust G, Tom

´

e CN, Jain A, Agnew SR. Modeling the effect of twinning and detwinning during

633

strain-path changes of magnesium alloy AZ31. International Journal of Plasticity 2009; 25(5): 861–880.634

37.

Anand L. Single-crystal elasto-viscoplasticity: application to texture evolution in polycrystalline metals

635

at large strains. Computer methods in applied mechanics and engineering 2004; 193(48-51): 5359–5383.636

38.

Mota A, Sun W, Ostien JT, Foulk JW, Long KN. Lie-group interpolation and variational recovery for

637

internal variables. Computational Mechanics 2013; 52(6): 1281–1299.638

39.

Heider Y, Wang K, Sun W. SO (3)-invariance of informed-graph-based deep neural network for

639

anisotropic elastoplastic materials. Computer Methods in Applied Mechanics and Engineering 2020; 363:

640

112875.641

40.

Miehe C, Schr

¨

oder J. A comparative study of stress update algorithms for rate-independent and

642

rate-dependent crystal plasticity. International Journal for Numerical Methods in Engineering 2001; 50(2):

643

273–298.644

41.

Bangerth W, Hartmann R, Kanschat G. deal. II—a general-purpose object-oriented ﬁnite element library.

645

ACM Transactions on Mathematical Software (TOMS) 2007; 33(4): 24.646

42. Bangerth W, Heister T, Heltai L, et al. The deal. II library. Version 2013; 8: 1–5.647

43.

Sun W, Ostien JT, Salinger AG. A stabilized assumed deformation gradient ﬁnite element formulation

648

for strongly coupled poromechanical simulations at ﬁnite strain. International Journal for Numerical and

649

Analytical Methods in Geomechanics 2013; 37(16): 2755–2788.650

44.

Sun W, Chen Q, Ostien JT. Modeling the hydro-mechanical responses of strip and circular punch

651

loadings on water-saturated collapsible geomaterials. Acta Geotechnica 2014; 9(5): 903–934.652

24 Ran Ma, WaiChing Sun

45.

Sun W. A stabilized ﬁnite element formulation for monolithic thermo-hydro-mechanical simulations at

653

ﬁnite strain. International Journal for Numerical Methods in Engineering 2015; 103(11): 798–839.654

46.

Na S, Sun W. Computational thermo-hydro-mechanics for multiphase freezing and thawing porous

655

media in the ﬁnite deformation range. Computer Methods in Applied Mechanics and Engineering 2017; 318:

656

667–700.657

47.

Salinger AG, Bartlett RA, Bradley AM, et al. Albany: using component-based design to develop a

658

ﬂexible, generic multiphysics analysis code. International Journal for Multiscale Computational Engineering

659

2016; 14(4).660

48.

Ambati M, Gerasimov T, De Lorenzis L. A review on phase-ﬁeld models of brittle fracture and a new

661

fast hybrid formulation. Computational Mechanics 2015; 55(2): 383–405.662

49.

Ambati M, Gerasimov T, De Lorenzis L. Phase-ﬁeld modeling of ductile fracture. Computational Mechanics

663

2015; 55(5): 1017–1040.664

50.

Armero F, Simo J. A new unconditionally stable fractional step method for non-linear coupled thermo-

665

mechanical problems. International Journal for numerical methods in Engineering 1992; 35(4): 737–766.666

51.

Wick T. Modiﬁed Newton methods for solving fully monolithic phase-ﬁeld quasi-static brittle fracture

667

propagation. Computer Methods in Applied Mechanics and Engineering 2017; 325: 577–611.668

52.

Quey R, Dawson PR, Barbe F. Large-scale 3D random polycrystals for the ﬁnite element method:

669

Generation, meshing and remeshing. Computer Methods in Applied Mechanics and Engineering 2011;

670

200(17–20): 1729-1745.671