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FFT-based solver for higher-order and multi-phase-field fracture models applied to strongly anisotropic brittle materials and poly-crystals

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This paper presents the application of a fast Fourier transform (FFT) based method to solve two phase field models designed to simulate crack growth of strongly anisotropic materials in the brittle regime. By leveraging the ability of the FFT-based solver to generate solutions with higher-order and global continuities, we design two simple algorithms to capture the complex fracture patterns (e.g. sawtooth, and curved crack growth) common in materials with strongly anisotropic surface energy via the multi-phase-field and high-order phase-field frameworks. A staggered operator-split solver is used where both the balance of linear momentum and the phase field governing equations are formulated in the periodic domain. The unit phase field of the initial failure region is prescribed by the penalty method to alleviate the sharp material contrast between the initial failure region and the base material. The discrete frequency vectors are generalized to estimate the second and fourth-order gradients such that the Gibbs effect near shape interfaces or jump conditions can be suppressed. Furthermore, a preconditioner is adopted to improve the convergence rate of the iterative linear solver. Three numerical experiments are used to systematically compare the performance of the FFT-based method in the multi-phase-field and high-order phase-field frameworks.
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Computer Methods in Applied Mechanics and Engineering manuscript No.
(will be inserted by the editor)
FFT-based solver for higher-order and multi-phase-field fracture models1
applied to strongly anisotropic brittle materials2
Ran Ma ·WaiChing Sun3
4
Received: December 5, 2019/ Accepted: date5
Abstract
This paper presents the application of a fast Fourier transform (FFT) based method to solve
6
two phase field models designed to simulate crack growth of strongly anisotropic materials in the brittle
7
regime. By leveraging the ability of the FFT-based solver to generate solutions with higher-order and global
8
continuities, we design two simple algorithms to capture the complex fracture patterns (e.g. sawtooth, and
9
curved crack growth) common in materials with strongly anisotropic surface energy via the multi-phase-
10
field and high-order phase-field frameworks. A staggered operator-split solver is used where both the
11
balance of linear momentum and the phase field governing equations are formulated in the periodic domain.
12
The unit phase field of the initial failure region is prescribed by the penalty method to alleviate the sharp
13
material contrast between the initial failure region and the base material. The discrete frequency vectors
14
are generalized to estimate the second and fourth order gradients such that the Gibbs effect near shape
15
interfaces or jump conditions can be suppressed. Furthermore, a preconditioner is adopted to improve
16
the convergence rate of the iterative linear solver. Three numerical experiments are used to systematically
17
compare the performance of the FFT-based method in the multi-phase-field and high-order phase-field
18
frameworks.19
Keywords
Fast Fourier Transform; multi-phase-field fracture; higher-order phase field fracture; anisotropic
20
cracks; polycrystal; salt21
1 Introduction22
The variational approach to fracture has provided a new way to simulate crack evolution in a purely
23
elastic deforming body by finding the deformation field and evolving crack paths that minimizes an energy
24
functional [Francfort and Marigo,1998,Bourdin et al.,2008]. This approach is well suited for phase field
25
models which represents harp interfaces with via implicit functions [Miehe et al.,2010a,Lee et al.,2016,
26
Wang and Sun,2017,Na et al.,2017,Na and Sun,2018a,Geelen et al.,2018,Bryant and Sun,2018,Choo
27
and Sun,2018b,Reinoso et al.,2017,Noii et al.,2019,Heider and Sun,2019]. While thermodynamically
28
consistent formulations are proposed by Miehe et al. [2010a], a material-force-based formulation of phase
29
field model has been derived and tested in a number of literature as an alternative way to introduce phase
30
field fracture models (e.g. Borden et al. [2014b,2016], Choo and Sun [2018a]).31
Unlike the embedded discontinuity approach such as assumed strain (e.g. Regueiro and Borja [2001],
32
Wang and Sun [2018,2019a,b]), extended/generalized finite element (e.g. Mo
¨
es et al. [1999], Duarte et al.
33
[2000], Sun et al. [2017]) and strong-discontinuity surface element (e.g. Ortiz and Pandolfi [1999], Linder
34
and Raina [2013], Radovitzky et al. [2011]), the phase field fracture method does not explicitly capture
35
the jump condition with an enhanced basis for displacement. Instead, a continuous phase field is used
36
Corresponding author: WaiChing Sun
Assistant 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
to indicate the location of the damaged zone that may be interpreted as the diffusive approximation of
37
the jump conditions. While this approximated condition may not capture the evolving geometry of the
38
crack exactly, the introduction of the continuous field provides a simple way to simulate crack growth
39
without predefined crack propagation paths, allows the simulations of crack coalescence and interaction
40
and branching without the need of ad hoc branching algorithm and enables one to explicitly define crack
41
nucleation rule as a driving force.42
Recent work, such as Clayton and Knap [2015], Bryant and Sun [2018], Quintanas-Corominas et al. [2019],
43
has extended the phase field model to capture brittle fracture in anisotropic materials where the surface
44
energy depends on the orientation of the crack propagation direction [Clayton and Knap,2015], the fracture
45
mode [Bryant and Sun,2018] and the mechanism of rupture of fiber in composite [Quintanas-Corominas
46
et al.,2019]. The modification of the phase field model is relatively simple when the surface energy remains
47
convex. However, a complex microstructure may lead to strongly anisotropic surface energy that is concave
48
and cannot be described via a second-order structural tensor. Furthermore, solving this strongly anisotropic
49
phase field fracture is difficult for second-order finite element method with traditional polynomial basis.
50
Recent work has resolved this issue in a finite element framework by either introducing multiple phase
51
fields or employing finite element space spanned by isogeometric basis to preserve higher-order continuity
52
across finite elements.53
In the former case, the introduction of multiple phase field does not require significant changes in the
54
finite element models and is therefore convenient. Furthermore, the multi-phase-field formulation also
55
provides modelers a simple mechanism to incorporate different driving forces and degradation mechanisms.
56
As the total free energy can be any non-negative-valued function of deformation and the multiple phase
57
fields, this approach provides the great flexibility to evolve anisotropic damages as shown in [Nguyen
58
et al.,2017,Na and Sun,2018a,Bleyer and Alessi,2018]. Nevertheless, the multi-phase-field formulation
59
also unavoidably leads to the extra degrees of freedoms and a sizable increase of computational demand
60
[Clayton and Knap,2015,Nguyen et al.,2017,Na and Sun,2018a,Bleyer and Alessi,2018]. This increase in
61
computational demand is significant, given the high spatial resolution demanded for fracture simulations.62
Alternatively, a fourth-order isogeometric approach has been introduced in previous works, such as
63
Teichtmeister et al. [2017], Borden et al. [2014b]. In this case, the strongly anisotropic surface energy of a
64
brittle material can be captured with a single phase field since one may choose the isogeometric basis of
65
arbitrary continuity. The use of isogeometric basis is particularly convenient for a design workflow that
66
involves computer-aided design where the isogeometric analysis and the design software may employ
67
the same set of basis functions [Hughes et al.,2005,G
´
omez et al.,2008,Sun et al.,2017]. However, the
68
higher-order continuity may also lead to a denser tangential matrix that may slow down the speed of an
69
implicit iterative solver even if a pre-conditioner is used.70
1.1 Rationales of using FFT solver for resolving strongly anisotropic brittle materials71
The FFT-based method, an algorithm first introduced in Moulinec and Suquet [1994] to solve elasticity and
72
inelasticity problems via Fast Fourier Transform, is an efficient alternative to solve the anisotropic phase
73
field problems for brittle materials exhibiting strongly anisotropic surface energy in a periodic unit cell. Due
74
to the usage of the Fourier space, an FFT-based solver can easily solve problems that involve higher-order
75
derivatives. Furthermore, it also does not require the generation of a suitable mesh as it can be simply
76
solved in a Cartesian grid. This feature makes the FFT-solver a very feasible choice for solving elasticity
77
problems for heterogeneous materials with spatial properties discretized by pixels or voxels. Finally, the
78
implementation of an FFT-based solver is relatively simple. A simple implementation is often completed
79
within ten lines of code [Brisard,2015]. This combination of desirable traits makes the FFT-based solver an
80
attractive option for resolving anisotropic phase field fracture problems in unit cells where the formulation
81
demands high resolutions, sufficient continuity, and computational efficiency.82
Such an FFT solver is not only useful for simulations in periodic unit cells, but can be used for multiscale
83
simulations that are based on computational homogenization (e.g. Feyel [2003], Sun et al. [2011b,a], Liu
84
et al. [2016], Shahin et al. [2016], Wang and Sun [2018]), and global-local formation for phase field fracture
85
or gradient damage models (cf. Wu et al. [2012], Noii et al. [2019]).86
FFT strongly anisotropic phase field fracture 3
1.2 Literature review on FFT-based solvers87
The FFT-based method was first proposed to solve linear elastic inclusion problems by utilizing the
88
Lippmann–Schwinger equation defined on a periodic homogeneous reference material [Moulinec and
89
Suquet,1994,1998]. This original algorithm, which uses point collocation and truncation of Fourier series to
90
iterate between the Fourier and real space, is often referred to as the basic scheme. Based upon this basic
91
scheme, an accelerated scheme [Eyre and Milton,1999] and an augmented-Lagrangian scheme [Michel et al.,
92
1999] are independently proposed to improve the accuracy, efficiency and robustness of the FFT solvers
93
that deal with materials exhibiting jumps or sharp gradient in material properties. In the last two decades,
94
significant improvements of the FFT solver have been achieved such that they enable the applications of the
95
FFT-based solver for high-resolution simulations performed on complex microstructure and 3D images of a
96
wide spectrum of materials, such as polycrystals [Lebensohn,2001,Prakash and Lebensohn,2009,Ma and
97
Truster,2019] and composite materials [Roters et al.,2010].98
Nevertheless, since FFT solver employs trigonometric polynomials as the basis function of the solution,
99
spurious oscillations may occur when a jump condition is being approximated by the Fourier series of
100
piecewise continuous differentiable functions. This spurious oscillation is often refereed as the Gibbs
101
phenomenon or Gibbs effect Since the Gibbs phenomenon is caused by the Fourier sums overshooting at a
102
jump condition, and the decay of the Fourier coefficient at infinity is controlled by the smoothness, a sharp
103
gradient or jump is difficult to approximate even though the number of terms significantly increase, as
104
shown in Figure 1. Special numerical treatments are therefore required to dampen the Gibbs effect if the
105
solution field is expected to include sharp gradient (e.g. regularized phase field for fracture).106
0 0.2 0.4 0.6 0.8 1
-1.5
-1
-0.5
0
0.5
1
1.5
N = 10
N = 30
N = 80
Fig. 1: Fourier series approximation of a square wave. N is the number of number of terms used to
approximate the square wave. Gibbs phenomenon occurs when the Fouries series overshoot at the sharp
discontinuities and cause spurious oscillations.
Notwithstanding the advantages the FFT-based methods have on material failure problems, the corre-
107
sponding applications on solving damage mechanics problems have not been attempted until recently. For
108
instance, Li et al. [2012] and Boeff et al. [2015] have both use FFT solvers to simulate damage of materials
109
characterized by the integral based damage model (cf. Li et al. [2012]) and the gradient-based damage
110
model (cf. Boeff et al. [2015] ). These examples indicate that the FFT solver is well suited to simulate damage
111
with regularization enabled by both nonlocal integral and gradient dependence. A similar procedure has
112
also been applied to simulate the failure of composite materials (e.g. Wang et al. [2018]) as well as the
113
interface decohesion problem in which the cohesive zone model was solved by introducing length scale
114
contrast between the interphase region and the bulk region [Sharma et al.,2018]. Finally, the FFT-based
115
damage mechanics solver has also been incorporated in a multiscale homogenization framework to model
116
brittle fracture of composite materials across length scales [Spahn et al.,2014]. The successful application of
117
the FFT-based method on continuum damage mechanics indicates its potential to solve phase-field brittle
118
fracture problems, where the crack is explicitly represented with
Γ
-convergence to the critical energy release
119
4 Ran Ma, WaiChing Sun
rate for Griffith’s theory. Although high material contrast exists between the damaged region and the intact
120
region, it is shown that the FFT-based method produces similar results as the FE method regarding the crack
121
tip stress field without suffering from the Gibbs phenomenon [Li et al.,2012,Rovinelli et al.,2019].122
FFT-based spectral method has been used to solve both the Ginzburg-Landau equation and Cahn-
123
Hilliard equation for interface migration problems in a semi-implicit fashion [Chen and Shen,1998]. In
124
these formulations, the phase field governing equations are assumed to be isotropic and homogeneous.
125
The similar numerical technique was employed recently to simulate dynamic re-crystallization [Chen et al.,
126
2015] and martensite-austinite transformation [Kochmann et al.,2016]. Recently, FFT-based method was
127
employed to solve phase-field fracture problem in both small strain elasticity case [Chen et al.,2019] and
128
finite strain elasto-plasticity case [Roters et al.,2019]129
Despite the great potential of the FFT solver to capture strongly anisotropic brittle fracture for materials
130
of complex microstructures, there is not yet any attempt to incorporate FFT solver to simulate phase field
131
fractures for anisotropic materials. This might be attributed to the facts that (1) FFT-based solver may suffer
132
from slow convergence if sharp material contract, such as bi-material interfaces, void and cavity, is presented
133
in the spatial domain [Zeman et al.,2010] and 2) one must resolve the higher order gradient required for
134
collocation method while circumventing the Gibbs phenomena that leads to spurious oscillations near crack
135
tip or other locations that exhibit sharp gradients.136
1.3 Objectives and organization of contents137
The purpose of this paper is to introduce a mathematical framework that overcomes the aforementioned
138
obstacles such that the fractures of materials with strongly anisotropic energy can be resolved via an FFT
139
solver. To close this knowledge gap, we successfully build two FFT solvers, i.e. one for the multi-phase-field
140
fracture model (which does not require higher-order continuity but introduce additional field variables),
141
and another one for the higher-order phase field fracture model (which requires only one phase field but
142
requires higher-order continuity). The strength and weakness of both modeling approaches in the FFT
143
setting are analyzed and compared in numerical experiments.144
Note that representing the initial flaw using the so-called gas phase with zero stiffness may introduce
145
sharp material contrast that deteriorates the convergence rate of the FFT-based solver. To avoid this issue,
146
we introduce a formulation such that inclusions, cavities or initial flaws can be represented as Dirichlet-type
147
boundary conditions weakly enforced by the penalty method.148
To overcome the well-known Gibbs effect exhibited in the solution obtained via FFT solver, which leads to
149
spurious high-frequency oscillations at the location where sharp material contrast presents, we approximate
150
the continuum frequency vector via a generalized version of the finite-difference based frequency operator
151
first introduced in Berbenni et al. [2014], Willot et al. [2014], Schneider et al. [2016] to calculate second-
152
and forth-order gradient while avoiding numerical instability. The discretized governing equation is then
153
solved at each grid point via a matrix-free iterative linear solver. Three numerical examples are presented to
154
compare the numerical performance of the multi-phase-field model and the high-order phase-field model.
155
Meanwhile, an interface damage model is proposed to capture the path-dependent responses of material
156
interfaces.157
This paper will proceed as follows. Section 2discusses the governing equation and the numerical
158
solution of multi-phase-field and high-order phase-field, respectively. Also, an interface failure model is
159
proposed in to represent material interface failure, for example grain boundary crack and fiber-polymer
160
de-cohesion. In Section 3, three examples are presented to compare the convergence rate and the numerical
161
performance of the multi-phase-field and the high-order phase field. Section 4summarizes the major results
162
and concluding remarks.163
As for notations and symbols, bold-faced letters denote tensors (including vectors which are rank-one
164
tensors); the symbol ’
·
’ denotes a single contraction of adjacent indices of two tensors (e.g.
a·b=aibi
or
165
c·d=cij djk
); the symbol ‘:’ denotes a double contraction of adjacent indices of tensor of rank two or higher
166
(e.g.
C:εe
=
Cijk l εe
kl
); the symbol ‘::’ denotes a fourth contraction of adjacent indices of tensor of rank four or
167
higher (e.g.
C:: D
=
Cijk l Dijkl
); the symbol ‘
’ denotes a juxtaposition of two vectors (e.g.
ab=aibj
) or
168
two symmetric second order tensors (e.g.
(αβ)ijkl =αij βkl
); the symbol ‘
F
’ and ‘
F1
’ represent forward
169
FFT strongly anisotropic phase field fracture 5
and backward Fourier transformation, respectively. Materials are assumed to be simple cubic symmetry
170
throughout this paper unless specified.171
2 Strongly anisotropic phase field theories172
In this section, we review the two most common phase field models, i.e., the multi-phase-field model and
173
the higher-order phase field model, used to replicate crack growth of materials with strongly anisotropic
174
surface energy and introduce a modified version of phase field model designed to capture fractures and
175
damage across interfaces and/or grain boundaries.176
The following assumptions are made in both models:177
1. The inertial effects are negligible.178
2. The material is under the isothermal condition.179
3. Rate dependence is neglected.180
4. The deformation is infinitesimal such that the effect of geometrical nonlinearity can be neglected.181
5. The boundary conditions are periodic.182
6. The material is either elastic or in the brittle fracture regimes.183
In addition, we consider a general case in which the surface energy can be non-convex. In the case of the
184
multiphase field model, we follow the treatment in Nguyen et al. [2017] and Na and Sun [2018a,b] such
185
that the strongly ansiotropic birttle fracture is manifested by using multiple phase fields, each phase field
186
is associated with damage accumulated on the corresponding plane orthogonal to a pre-defined normal
187
vector. In the case of the high-order phase field, the governing equation of the phase field is modified to
188
incorporate the higher-order terms such that the nonconvex surface energy can be characterized. Both
189
phase field models are solved via a fast Fourier transform solver (cf. Zeman et al. [2010], Ma and Truster
190
[2019], Eghtesad et al. [2018]). As a result, the spatial domain of the boundary value problem is idealized
191
as a unit cell subjected to periodic boundary conditions. This unit cell, denoted as
herein, represents a
192
microstructure discretized by cubic voxels whose centroids are FFT points in a regular Cartesian grid, as
193
shown in Figure 2. Meanwhile, the periodic boundary conditions provide a mean for us to employ solution
194
space spanned by the trigonometric basis functions, while also generates homogenized responses that are
195
bounded by both the traction- and displacement- based boundary conditions [Zohdi and Wriggers,2001,
196
Sun et al.,2011a,b,Fish,2013,Wang and Sun,2015,Sun et al.,2013,Liu et al.,2016].197
The crack surface is denoted as
Γ
. As such, the total energy potential within the cracked body can be
198
decomposed into the elastic stored energy and the free surface energy:199
Ψ=Zψed+ZΓGcdΓ, (1)
where
ψe
is the elastic strain energy density, and
Gc
is the cleavage energy per unit area. According to the
200
Griffith’s brittle fracture theory, brittle crack propagates when the elastic strain energy release rate
ψed
is
201
larger than the interface energy of the newly generated crack GcdΓ.202
In the phase field brittle fracture theory, the sharp crack surface
Γ
is represented by a diffusive regularized
203
implicit indicator functio called phase field (denoted as
d
herein), where
d=
0 represents the intact region
204
and
d=
1 represents the completely damaged region [Bourdin et al.,2008,Miehe et al.,2010b,Wang and
205
Sun,2017,Na et al.,2017,Bryant and Sun,2018,Qinami et al.,2019]. This damage phase field
d
is regularized
206
such that the infinitesimal unit surface area
dΓ
is smeared to an infinitesimal domain
γd
, as shown in
207
Figure 2(b), and the total cleavage energy can be approximated by domain integral:208
lim
l00ZGcγ(d,d)d=ZΓGcdΓ, (2)
where
l0
is the regularization length scale. The strongly anisotropic phase field theory can be introduced by
209
using an anisotropic non-convex crack surface density
γ
in two approaches. The first approach uses multiple
210
phase fields with each phase field representing one preferential crack propagation direction [Clayton and
211
Knap,2015]. The second approach uses one phase field with high order gradient terms [Teichtmeister et al.,
212
2017].213
6 Ran Ma, WaiChing Sun
Ω
(a) (b)
Γ
𝛾(𝑑, ∇𝑑 )
Ω
x
y
zx
y
z
d= 1
d= 0
𝜕Ω 𝜕Ω
l0
Fig. 2: A schematic demonstration of phase field fracture in the deformed configuration of RVE subjected to
periodic boundary condition. (a) A solid body with sharp crack; (b) A solid body with regularized crack.
In this section, a multi-phase-field based small strain elasticity model is first introduced, which provides
214
the driving force for both multi-phase-field and high-order phase-field. Then the multi-phase-field theory
215
and high-order phase-field theory are derived in a periodic unit cell, with the governing equations solved
216
by the FFT-based method. Also, an interface model is presented to accommodate both phase field theories.
217
2.1 Small strain isotropic elasticity218
The strain free energy ψein equation (1) is a function of the mechanical constitutive relation.219
The classical isotropic linear elasticity model is used to replicate the elastic responses. This approximation
220
is acceptable for crystalline materials such as rock salt which does not exhibit strong anisotropy in the elastic
221
regime [Clayton,2010,Liu et al.,2010]. Meanwhile, we assume that the surface energy for the fracture is
222
of cubic symmetry for both multi-phase-field and high-order phase-field, which is suitable for crystalline
223
materials which exhibits four three-fold rotational axes oriented at 109.5
with respect to each other. The
224
elastic strain energy reads:225
ψe(ε,d) = 1
2κε2
v+µε:ε1
3ε2
v. (3)
Here, εvis the volumetric strain εv=tr(ε), and κand µare the Lam´
e constants.226
The total material degradation Dat point xis defined as:227
D=k+ (1k)
n
i=1
(1di)2, (4)
where
k
1 is a small portion of the residual stiffness to retain well-posedness,
n
is the total number of
228
phase fields, and diis the ith damage phase field.229
Assume, without loss of generality, that
n>
1 for multi-phase-field and
n=
1 for high-order phase-field
230
throughout this paper. Then for the multi-phase-field method
(n>
1
)
, each phase field
di
is corresponding
231
to the damage along a preferential propagation direction, following the treatment in Nguyen et al. [2017],
232
Na and Sun [2018a]. The Lam´
e constant κand µare functions of the local damage phase field:233
µ(d) = Dµ0,κ(d) = κ0,εv0
Dκ0,εv>0(5)
where
κ0
and
µ0
are the undamaged Lam
´
e constants. Note that the elastic constitutive relation is anisotropic
234
in that the volumetric modulus
κ
depends on the current volumetric strain
εv
. The strain energy functional
235
for the ith phase field within totally nnumber of phase fields is defined as:236
Hn
i= (1k)max
τ[0,t]n
j=1,j6=i1dj21
2κ0hεvi2+µ0ε:ε1
3ε2
v, (6)
FFT strongly anisotropic phase field fracture 7
where
h·i := (·+| · |)/
2. For multi-phase-field, nis a positive integer, while for high-order phase field n
237
equals to one.238
In this equation, the monotonically increasing of damage is enforced by using the maximum strain energy
239
functional in time
τ[
0,
t]
to form the energy functional for the variational problem. The corresponding
240
thermodynamic driving force associated to direads (cf. Nguyen et al. [2017]),241
Fi= (1di)Hn
iGcδγi(di,xdi) = 0 (7)
where the second term in Eq. (7) is the thermodynamic force related to the growth of the phase field di.242
2.2 Multi-phase-field model243
In multi-phase-field theory, the total cleavage energy is split to multiple anisotropic cleavage energies, each
244
represented by a seperate phase field di:245
ZGcγd=ZGc
n
i=1
γi(di,di)d. (8)
In this way, the total cleavage energy is non-convex and can be applied to simulate strongly anisotropic
246
brittle fracture.247
2.2.1 Phase field theory in periodic unit cell248
Crack density per unit volume γiis defined as a function of the damage phase field and its gradient:249
γi(di,di)=
n
i=11
2l0
d2
i+l0
2di·Ai· ∇di+1
2β1(di1)2,diH1
#(), (9)
where
l0
is the regularization length scale,
A
is the second order anisotropic tensor, and
β1
is the penalty
250
parameter to enforce the phase field within the initial damage region. Function space
Hp
#()
is the space of
251
all periodic functions with pth order square integrable weak gradient. The anisotropic tensor
A
is usually
252
formulated as:253
Ai=[I+β2(Inini)] , (10)
where
ni
is the normal direction of the preferential crack propagation direction corresponding to the ith
254
phase field,
I
is the second order identity tensor. The penalty parameter
β2
serves to penalize the anisotropic
255
propagation, where β2=0 recovers the isotropic phase field theory.256
Total crack surface area
Γ
in the unit cell
is defined as the domain integration of the crack surface
257
density γ:258
Γi(d) = Zγi(di,di)d, (11)
which is a functional that depends on the damage phase field
d
and its gradient. The initial balanced phase
259
field distribution di(t=0)minimizes the total crack surface energy:260
di(x) = Arg "inf
diH1
#()
Γl(di,di)#. (12)
The current equilibrium mechanical field and damage field shall minimize the total potential
Ψ
defined
261
in equation (1), where the elastic strain energy
ψe
is defined in (3) and the crack surface density
γ
is defined
262
in (9). Therefore, the first order variation of the total potential functional should vanish, which is a necessary
263
condition of total potential stationary condition. The mechanical part of the variational equations yields the
264
equilibrium equation265
∇ · σ=0, (13)
8 Ran Ma, WaiChing Sun
where the body force is neglected. Then, the ith phase field distribution can be determined such that the
266
first order variation with respect to divanishes:267
Z2(di1)Hn
iδdi+Gcdi
l0
δdi+l0di·Ai· ∇δdi+β1(di1)δdid=0, δdiH1
#, (14)
which holds for arbitrary phase field perturbation
δdi
in the
H1
#
space. Note that the FFT approach is
268
a collocation method which directly obtain the numerical solution from the strong form. Therefore, we
269
perform the integration by parts on (14) to obtain the Euler-Largange equation, i.e.,270
Z2(di1)Hn
iδdi+Gc
l0
diδdi−∇·(Gcl0Ai· ∇di)δdi+β1(di1)δdid
+ZGcl0di·Ai·nδdidS=0,
(15)
where
is the boundary of the unit cell and
n
is the unit normal. Note that the integration-by-part operation
271
increases the continuity requirement of the cleavage energy
Gc
, the length scale
l0
, the anisotropic tensor
Ai
,
272
and the phase field
di
. Assuming that the cleavage energy
Gc
, the length scale
l0
, and the anisotropic tensor
273
Ai
are periodic within the unit cell, then the surface integration term in equation (15) vanishes considering
274
that the phase field gradient
di
is periodic while the surface normal
n
is anti-periodic. The Euler-Lagrange
275
equation of the multi-phase-field functional yields:276
2(di1)Hn
i+Gc
l0
di−∇·(Gcl0Ai)· ∇diGcl0Ai:(∇⊗∇di)+β1(di1)=0, diH2
#. (16)
In practice, piece-wise constant cleavage energy
Gc
, length scale
l0
, and anisotropic tensor
Ai
are mostly
277
used, for example, in polycrystalline material or composite material. However, setting material parameters
278
to be piecewise constant will introduce jump condition in equation (16), and therefore not feasible. This
279
problem is circumvented in equation (17) by adopting the following approximations: (1) use
δ
function at
280
the grid points as the test function
δdi
; (2) assume the gradient of
Gc
,
l0
, and
Ai
are zero at all grid points.
281
Then, the simplified Euler-Lagrange equation yields:282
2(di1)Hn
i+Gc
l0
diGcl0Ai:(∇⊗∇di)+β1(di1)=0, diH2
#, (17)
where the phase field
di
lives in a high order smooth function space. The solution of the simplified strong
283
form (17) is an approximation to the real solution which minimizes the total energy potential Ψ.284
Remark 1
Remark on the length scale parameters in multi-phase-field models. In general, the length scale
285
parameter of the phase field models equipped with a specific degradation function can be identified from
286
experiments where a validation exercise that relates peak stress to material parameters, length scales (cf.
287
Pham et al. [2017], Choo and Sun [2018a]) after quantifying the amplification factor of the fracture energy due
288
to spatial discretization (cf. [Bourdin et al.,2008]. In principle, this validation exercise could be extended to
289
multi-phase-field and higher-order-phase field models. In the case of the higher-order-phase field models, it
290
might be possible to identify the fracture energy as a function of orientation by having multiple experiments
291
for the same specimen of different orientations. In the case of the multi-phase-field models, the multiple
292
phase field governing equations lead to great flexibility where a wide spectrum of fracture behaviors can be
293
replicated by different combinations of degradation mechanism, driving force, length scale and anisotropy
294
of fracture energy, as demonstrated in Quintanas-Corominas et al. [2019], Bleyer and Alessi [2018]. In this
295
work, we assume that all the damage mechanisms for the multi-phase-field model employs the same form
296
of degradation function, regularization term and length scale parameter. This treatment reduces the number
297
of material parameters required for the models and therefore simplifies the inverse problems required to
298
identify the material parameters. However, it is possible that there are situations in which enhancing the
299
multi-phase-field models by enabling different length scale parameters, regularized profiles or forms of
300
degradation functions can be beneficial for replicating more complex crack nucleation and propagation
301
behaviors. While the calibration and validation of phase field fracture is an ongoing research topic that
302
rightfully attracts considerable interest, the design of inverse problems, experimental procedure and the
303
experiments themselves for the strongly anisotropic phase field models are out of the scope of these studies
304
and will be considered in the future.305
FFT strongly anisotropic phase field fracture 9
2.2.2 Spatial discretization and numerical solution306
The coupled mechanical and phase field equations can be solved in a staggered fashion. In this work,
307
the mechanical step that solves the balance of linear momentum (13) is incrementally updated via the
308
Fourier-Galerkin method [Zeman et al.,2017,Ma and Truster,2019]. The FFT-based solution method of the
309
anisotropic phase field evolution equation (17) is presented in detail in this paper.310
The phase field evolution equation (17) is a linear equation with respect to the ith phase field. These
311
equations are solved in a semi-implicit way, where the phase field (
dj
,
j6=i
) of the last iteration is used
312
to calculate the partial damaged strain energy
Hn
i
. Then, the strong form (17) can be re-formulated by
313
calculating the second order gradient in the Fourier space, which leads to the discretized form:314
2Hn
i+Gc
l0
+Gcl0Ai:F1ξξF+β1di=2Hn
i+β1. (18)
Since the stiffness matrix is generally non-symmetric, this equation can be solved relatively efficiently by a
315
matrix-free iterative linear solver, for example biconjugate gradient method (BiCG) or generalized minimal
316
residual method (GMRES).317
Note that in the discretized form (17), second order gradient is involved. Therefore, the solution of the
318
discrete equation (17) suffers from the Gibbs effect when the continuum frequency vector is used, especially
319
at the boundary of the fully damaged region and the partially damaged region where the first order gradient
320
is discrete. The discrete frequency vector from Willot et al. [2014] is generalized to higher order gradient case
321
in order to overcome this problem. The backward finite difference scheme is used to calculate the discrete
322
frequency vector ki
323
k=exp(iξ)1, (19)
where iis the complex number unit and ξis the continuum frequency vector defined as324
ξi=2πni
N,ni=N/2, . . . , N/2 +1 , mod (N) = 0
(N1)/2, · · · ,(N1)/2 , mod (N) = 1(20)
Here,
N
is the total number of grid points. The discrete frequency vector (19) is generalized to enable second
325
order gradient based on the central difference rule, which in the 2D case reads:326
kxx =2(cos ξx1)
kyy =2cos ξy1
kxy =sin ξxsin ξy
(21)
The second order gradient of a scalar field
φ
can be calculated as
φxx =F1[kxx ˆ
φ]
,
φyy =F1[kyy ˆ
φ]
, and
327
φxy =F1[kxy ˆ
φ],328
where ˆ
φdenotes the Fourier coefficient of φ.329
The second order discrete frequency vector (21) can be easily generalized to 3D case.330
2.3 High-order phase-field model331
In high-order phase-field theory, the total potential
Ψ
is split into the strain potential and the cleavage
332
energy potential in the same way as (1). Instead of using multiple phase field for each preferential crack
333
propagation direction, the crack density contains a high-order phase field gradient term to construct a
334
non-convex functional. In this section, the fourth order phase-field theory is first re-derived in a periodic
335
unit cell together with corresponding Euler-Lagrange equation. Then details are presented to solve the
336
coupled mechanical and high-order phase-field in a staggered way using FFT-based method.337
10 Ran Ma, WaiChing Sun
2.3.1 Phase field theory in periodic unit cell338
In the high-order phase-field fracture theory, crack density function depends on the phase field as well as
339
the first and second gradients of the phase field itself, i.e.,340
γ(d,d,∇∇d)=1
2l0
d2+l0
4d· ∇d+l3
0
32 (∇∇d):A:(∇∇d)+1
2β1(d1)2,dH2
#(). (22)
Again,
H2
#()
denotes the space of periodic functions with square integrable second order gradient,
A
is a
341
fourth order anisotropic tensor to accommodate more complex anisotropies, and the numerical parameter342
β1
1 serves to enforce the initial phase field. The structural tensor
A
is invariant with respect to rotations
343
from the cubic group. For materials belongs to cubic symmetry group, for example rock ralt, the fourth
344
order anisotropic tensor can be defined as345
A=I+α(A1A1+A2A2) + β2sym(A1A2), (23)
where
I
is the fourth order symmetric identity tensor,
α
and
β2
are two penalty parameters, and
A1
and
A2
346
are two second order structural tensors which are also invariant with respect to rotations from the cubic
347
symmetry group. These second order tensors A1and A2are defined as348
A1=a1a1,A2=a2a2, (24)
where the unit vectors
{ai}i=1,2,3
represents an orthonormal basis, which are related to the material point
349
orientation and satisfy
ai·aj=δij
,
ai×aj=eijk ak
. The isotropic high-order phase-field theory can be
350
recovered by setting α=β2=0.351
Then, the total crack surface area is the integral of the crack surface density over the periodic unit cell
:
352
Γl(d) = Zγ(d,d,∇∇d)d. (25)
Substitute this expression into the total potential (1) and take first order variation to minimize the total
353
potential Ψdefined in equation (1).354
In order to find the phase field
d
which minimizes the total potential
Ψ
defined in equation (1), the
355
stationary condition is reached by taking first order variation over d:356
Z"2(d1)H1
1δd+Gc 1
l0
dδd+l0
2d· ∇δd+l3
0
16 ∇∇d:A:∇∇δd+β1(d1)δd!#d
=0, δdH2
#,
(26)
where
δd
is aribtrary phase field perturbation from
H2
#
space. Note that the partial strain energy
H1
1
is
357
defined in equation (6) with total number of phase fields
n
equals to 1. Then integration by part is performed
358
to get the Euler equation of this problem.359
Z"2(d1)H1
1+Gc
l0
d−∇·Gcl0
2d+∇ · ∇ · Gcl3
0
16 A:∇∇d!+β1(d1)#δdd
+Z"Gcl0
2d·n+Gcl3
0
16 ∇∇d:A:(δdn)Gcl3
0
16 ∇ · (A:∇∇d)·nδd#dS=0.
(27)
Note that, similar to Equation
(15)
, the surface integral term only vanishes when the anisotropic tensor
360
A
and the cleavage energy density
Gc
are periodic, and the length scale
l0
is constant. Then we get the
361
following Euler equation which minimizes the total potential Ψ:362
2(d1)H1
1+Gc
l0
d−∇·Gcl0
2d+∇ · ∇ · Gcl3
0
16 A:∇∇d!+β1(d1)=0. (28)
FFT strongly anisotropic phase field fracture 11
This strong form can be further simplified by assuming that Gcand Aare piecewise constant:363
2(d1)H1
1+Gc
l0
dGcl0
2∇·∇d+Gcl3
0
16 A:: ∇∇∇d+β1(d1)=0, dH4
#. (29)
The reason for such simplification can be found in Section 2.2.1. This assumption is true for many material
364
models, for example crystal plasticity, where the preferential cleavage direction is constant within the grain,
365
while the transition between two adjecent grains is sharp. This assumption doesn’t hold for functionally
366
graded material where a smooth variation of the preferential crack propagation direction exists. This
367
equation, when solved by finite element method in its weak form, requires
C1
continuous shape functions.
368
However, when this equation is solved by collocation method in its strong form, fourth order differentiable
369
interpolation functions are required. The FFT-based method offers a way to solve this equation in its strong
370
form, where the trigonometric polynomial shape functions are arbitrary order differentiable.371
2.3.2 Spatial discretization and numerical solution372
The strong form (29) is a linear equation and can be re-formulated when the fourth order gradient is
373
calculated in Fourier space:374
2H1
1+Gc
l0
+Gcl0
2F1ξ·ξF+Gcl3
0
16 A:: F1ξξξξF+β1!d=2H1
1+β1. (30)
Note that different from multi-phase-field where the updated phase field
di
can only be solved semi-
375
implicitly, the high-order phase-field equation can be solved fully implicitly since the partial strain energy
376
H1
1
is independent of phase field. Since the stiffness matrix is generally non-symmetric, this equation can be
377
solved by matrix-free iterative linear solver, for example biconjugate gradient method (BiCG) or generalized
378
minimal residual method (GMRES).379
Similarly to the multi-phase-field, continuous frequency vector and discrete frequency vector are defined
380
in (20) and (19), respectively. In order to calculate fourth order gradient without suffering from Gibbs effect,
381
the following discrete frequency vector is suggested382
kxxxx =4(cos ξx1)2
kxxx y =2 sin ξxsin ξy(cos ξx1)
kxxyy =4(cos ξx1)cos ξy1
kxxyy =2 sin ξxsin ξycos ξy1
kyyyy =4cos ξy12,
(31)
where
ξ
is the continuous frequency vector defined in (20). The fourth order gradient of a scalar field
φ383
can be calculated as
φxxxx =F1[kxxxx ˆ
φ]
,
φxxx y =F1[kxxxy ˆ
φ]
,
φxxyy =F1[kxxyy ˆ
φ]
,
φxyyy =F1[kxyyy ˆ
φ]
,
384
and φyyyy =F1[kyyyy ˆ
φ],385
where
ˆ
φ
denotes the Fourier coefficient of
φ
. The fourth order discrete frequency vector (31) can be easily
386
generalized to 3D case.387
Remark 1.The iterative linear solver can be accelerated by a suitable preconditioner
M
. In this paper, a finite
388
difference matrix is constructed based on the isotropic part of the linear equations (17) and (29). The Laplace
389
operator is approximated by the five-point stencil with periodic boundaries. The incomplete Cholesky
390
decomposition of this finite difference matrix is utilized as the preconditioner
M
. Also, the phase field of the
391
previously converged step is used as the initial guess to further accelerate the linear solver.392
2.4 Regularized interface fracture model for grain boundaries393
The macroscopic failure of materials with microstructures formed by assemblies of grains, layers, laminates,
394
or other forms of building blocks can be triggered by a combination of intergranular fracture and intra-
395
granular fracture. Examples of these materials include polycrystal, sedimentary rock, biological tissues,
396
12 Ran Ma, WaiChing Sun
and composite materials [Aifantis and Willis,2005]. As a result, a good combination of bulk and interfacial
397
fracture models is essential to capture the essence of failures.398
In the phase field fracture literature, there are different approaches to simulate the competition between
399
intergranular fracture and intragranular fracture. One approach is to explicitly define an additional interface
400
region in a discretized mesh. An interface crack propagation model with interface-normal based anisotropic
401
tensor
A=nn
and reduced cleavage energy
Gc
is assigned to the interface region [Clayton and Knap,
402
2015]. This treatment is relatively simple, as both the fracture inside the grain and along the grain boundaries
403
can be captured by the same phase field. Another approach is to approximate the cohesive fracture displace-
404
ment jump
[[u]]
with a regularized auxiliary field
v
by utilizing the initial equilibrium grain boundary phase
405
field, such that explicit mesh discretization for an additional interface region is avoided [Nguyen et al.,2017,
406
Verhoosel and de Borst,2013,Radovitzky et al.,2011,Paggi et al.,2018]. In this approach, the phase field
407
model is used to model the crack growth inside the grain, while a cohesive zone model is used to replicate
408
the complex crack behaviors of grain bonduaries with richer descriptions enabled by the cohesive zone
409
models. [Paggi et al.,2018].410
For simplicity, we adopt the first approach is to represent the sharp interface via a regularized diffusive
411
region associated with a characteristic length scale. As such, an interface region is explicitly identified via an
412
implicit function and the spatial domain is partitioned after a realization of polycrystal structure is generated
413
via a mesh generator. The major assumption is that the interface phase field fracture model is transverse
414
isotropic with respect to the normal direction of an interface or grain boundary. For multi-phase-field, a
415
simple approach is used by assigning one of the principal directions of the symmetric anisotropic tensor
A416
parallel to the interface normal for the governing equations of all the phase field variables. For high-order
417
phase-field, a fourth order anisotropic tensor
A
invariant to the rotation from transverse isotropy group is
418
utilized (cf. Li and Maurini [2019]):419
C11 C11 2C66 C13 0 0 0
C11 2C66 C11 C13 0 0 0
C13 C13 C33 0 0 0
0 0 0 C44 0 0
0 0 0 0 C44 0
0 0 0 0 0 C66
. (32)
In order for the 4th order stiffness tensor (32) to be positive definite, the following conditions have to be
420
met [Mouhat and Coudert,2014]:421
2C2
13 <C33 (C11 +C12),C44 >0, C66 >0. (33)
Triple junctions and quadruple points are assumed to be isotropic with 4th order identity tensor.422
2.5 Discrete algorithm423
The coupled mechanical and phase field equations are solved in a sequential manner, and the algorithm is
424
shown in Algorithm 1. Within each step, the mechanical problem is first solved with the help of the Green’s
425
operator
G
. The essential boundary conditions and natural boundary conditions are satisfied in two coupled
426
Newton-Raphson iterations. Detailed theory and implementation of the mechanical part can be found in
427
Zeman et al. [2017], Ma and Truster [2019]. After the mechanical problem converges within the required
428
tolerance, the linear phase field equations are then solved by an iterative linear solver, for example conjugate
429
gradient or generalized minimal residual method (GMRES). Although the two governing equations are
430
solved sequentially, it has been shown that the final solution could converge to the exact solution when the
431
time step is smaller than the critical time step, a important feature shared by phase field equations solved
432
via finite element methods e.g. Wheeler et al. [2014], Miehe et al. [2016], Ambati et al. [2015], Wang and Sun
433
[2017], Bryant and Sun [2018].434
FFT strongly anisotropic phase field fracture 13
Algorithm 1: FFT-based algorithm for strongly anisotropic phase field problem.
1for n1to nstep do
2if Strain boundary condition then ¯ε=¯εBC
3else ¯ε=¯
C1
n¯
σBC
4while true do
5εn+1=εn+1+¯ε
6solve for ˜ε:G(Cn:˜ε)=G(Cn:¯ε)
7update εn+1:εn+1=εn+1+˜ε
8while R>tol. do
9update σn+1:σn+1=f(ε,σn, history)
10 solve for ˜ε:G(Cn+1:˜ε)=Gσn+1
11 update εn+1:εn+1=εn+1+˜ε
12 update residual: R=k˜εk
13 end
14 ¯
σ=hσn+1i
15 if k¯
σσBC k
k¯
σk<tol. then break
16 update ¯
Cn+1
17 update ¯ε:¯ε=¯
C1
n+1(¯
σσBC )
18 end
19 Update phase field
20 if Multi-phase-field then
21 2Hn
i+Gc
l0+Gcl0Ai:F1ξξF+β1di=2Hn
i+β1.
22 else if High-order phase-field then
23 2H1
1+Gc
l0+Gcl0
2F1ξ·ξF+Gcl3
0
16 A:: F1ξξξξF+β1d=2H1
1+β1
24 end
3 Analysis and numerical examples435
In this section, we compare the accuracy and efficiency of the FFT-based solvers for multi-phase-field
436
and high-order phase-field First, the one-dimensional equilibrium phase field with central concentrated
437
failure is solved using the FFT-based method for both the low-order and high-order phase field. Special
438
attention is paid to compare the Gibbs effect manifested in both models, as well as the convergence rate and
439
performance. Then, a two-dimensional benchmark case is used to demonstrate and benchmark the ability of
440
the FFT-based method in solving strongly anisotropic fracture problems, more specifically the sawtooth
441
crack pattern observed in strongly anisotropic fracture. Again, the convergence progress of the multi-phase
442
field and high-order phase-field is compared. Finally, a three-dimensional polycrystalline rock salt example
443
is presented to illustrate the competing between intergranular and intragranular fracture in polycrystals.444
3.1 One-dimension verification and convergence analysis445
A one-dimensional phase-field distribution problem from Borden et al. [2014b] is used as an example to
446
assess the accuracy of the FFT solver as well as examine the numerical solutions obtained via the FFT-based
447
method in solving phase field problems in periodic domains. Consider a rod with a concentrated flaw in the
448
middle. This concentrated flaw can be represented by prescribing the phase field value at this point to be 1.
449
The governing equation of the phase field model then yields the initial phase field distribution. Without
450
applying any external loading, there exists an exact solution that governs the phase field distribution in
451
the one-dimensional setting. The multi-phase-field governing equation and the corresponding analytical
452
14 Ran Ma, WaiChing Sun
solution reads,453
d4l2
0d00 +β1d=β1,d(x) = exp −|x|
2l0. (34)
Meanwhile, the high-order phase-field governing equation together with its analytical solution reads,454
d2l2
0d00 +l4
0d(4)+β1d=β1,d(x) = 1+|x|
l0exp −|x|
l0. (35)
In our numerical example, the length of the rod is 200 to be consistent with the results in Borden et al.
455
[2014b]. The initial phase field penalty
β1
is 1000 at
x=
0 and 0 elsewhere. The length scale parameter
l0
is
456
0.08.457
These phase-field partial differential equations are solved numerically in FFT-based method (17) and
458
(29). The numerical solution is then integrated over the rod to get the numerical potential
Ψ
based on (1).
459
The surface energy potential that leads to the exact solution of the phase field distribution in (34) and (35)
460
reads,461
Ψc=2GcZ100
0d2
4l0
+l0(d0)2dxGc, (36)
for multi-phase-field, and the surface energy potential for the exaction solution for the high order phase
462
field problem reads,463
Ψc=2GcZ100
0"d2
4l0
+l0
2(d0)2+l3
0
4(d00)2#dxGc, (37)
which again corresponds with the high-order phase-field solution in Borden et al. [2014b].464
Fig. 3: Verification of 1D phase field distribution against analytical solution. The second order derivative
is required by multi-phase-field, while the fourth order derivative is required by high-order phase field.
(a) Phase field (
d(x)
); (b) First derivative (
l0d0(x)
); (c) Second oderivative (
l2
0d00(x)
); (d) Fourth derivative
(l4
0d(4)(x)).
Figure 3compares the numerical and analytical solutions, the corresponding first order derivatives (
l0d0
),
465
the second order derivatives (
l2
0d00
), as well as the fourth order derivatives (
l4
0d(4)
). Both the phase field and
466
FFT strongly anisotropic phase field fracture 15
its high-order derivative fits with the analytical solutions, except at the position
x=
0 where the fourth
467
order derivative of the analytical solution is supposed to be undefined whereas the numerical solution
468
remains finite due to the differentiability of the sine and cosine functions.469
The Gibbs phenomenon is not observed even for the fourth order derivative near the initial failure point
470
where large phase field gradient exists. These results suggest that the backward discrete frequency scheme
471
[Willot et al.,2014] combined with the generalized discrete frequency vector for high-order gradient (21)
472
and (31) is effective in eliminating the Gibbs phenomenon that might otherwise cause spurious oscillations.
473
It needs to be mentioned that the solution based on continuous frequency vector suffers from the Gibbs
474
phenomenon, while the solution based on other discrete frequency vector schemes, for example the forward
475
scheme and the central scheme from Willot et al. [2014], does not converge to the analytical solution upon
476
refinement. Corresponding results are not presented for brevity.477
The error in the surface energy potential,
Gnum Gc
, is presented in Figure 4(a). Under the same
478
mesh refinement, the high-order phase-field has lower surface energy error than multi-phase-field. The
479
h-convergence rate is about 0.72 for multi-phase-field, while 1.01 for high-order phase-field. The faster
480
convergence rate of the high-order phase-field upon mesh refinement is at the expense of the lower linear
481
solver convergence rate. The linear solver convergence rate of the strong form (17) and (29) is presented
482
in Figure 4(b). The convergence efficiency of both models becomes lower upon each mesh refinement,
483
indicating that the condition number of the stiffness matrix becomes larger each upon mesh refinement.
484
Furthermore, our numerical experiment indicates that the high-order phase-field requires significantly
485
more iterations (more than one order) to converge than the low-order phase-field counterpart. Note that,
486
in the multi-dimensional cases, the number of independent phase field required to replicate anisotropy of
487
the surface energy likely increases according to the complexity of the anisotropic responses. Hence, the
488
2nd-order phase field governing equation must be solved multiple time, while there is only one 4th-order
489
phase field governing equation required for the higher-order case. This trade-off is further analyzed in the
490
next two numerical examples.491
0.0067 0.0200 0.0666 0.1998
10-4
10-2
100
Surface Energy Error
(a)
2nd order
4th order
0.0067 0.0200 0.0666 0.1998
102
104
Iterations
(b) 2nd order
4th order
58.4
34.5
86.9
127.8
Fig. 4: Convergence behavior of phase field under no external loading: (a) Convergence plot upon mesh
refinement; (b) Convergence progress of iterative linear solver (without preconditioning).
3.2 Two-dimensional plane strain single-crystal rock salt492
A plane strain single crystal rock salt is used to demonstrate the numerical behavior of the FFT-based
493
method in solving 2D phase field crack propagation problems. The room temperature elastic constants and
494
phase field parameters are listed in Table 1[Na and Sun,2018a]. Although single crystal rock salt belongs to
495
cubic symmetry group, the elastic anisotropic factor is almost 1, therefore isotropic elasticity model (3) is
496
used here. The cleavage plane of single crystal rock salt is
{
100
}
plane. In order for the two-dimensional
497
simulation to be able to represent the crack propagation, the
[
001
]
axis is set to coincide with the
z
axis while
498
the single crystal deforms in the xyplane.499
16 Ran Ma, WaiChing Sun
Table 1: Material properties of the specimens for the numerical simulations
Parameters Description Value Unit
EElastic modulus 38.0 GPa
νPoisson’s ratio 0.25 -
β1Initial phase field penalty 1000.0 -
GcCleavage energy 1.15 J m2
l0Length scale 1.0 ×105m
β2Anisotropy factor for multi-phase-field 40.0 -
αAnisotropy factor for high-order phase-field 1.2 -
β2Anisotropy factor for high-order phase-field 1000.0 -
In this section, a single crystal rock salt with a central concentrated flaw is first utilized to demonstrate
500
the initial equilibrium phase field pattern and the convergence behavior of multi-phase-field and high-order
501
phase-field. Then plane-strain tension test is presented to show the ability of both methods to represent the
502
sawtooth crack pattern.503
3.2.1 Influence of frequency vector on numerical accuracy and stability504
A two-dimensional plate with a central concentrated flaw is utilized to compare the numerical behavior of
505
multi-phase-field and high-order phase-field. The initial equilibrium phase field of both methods is solved
506
using BiCG solver, with the relative tolerance 1.0 ×108.507
Figure 5compares the continuum frequency vector (left), rotated discrete frequency vector (middle),
508
and forward-backward discrete frequency vector (right), regarding the accuracy and stability of the solution
509
field. In all three cases, the initial equilibrium phase field is solved with the anisotropic factor
β2=
0. When
510
the continuum frequency vector (
k=iξ
is used in equation 18, the phase field
d
and its Laplacian appears
511
to be smooth without spurious oscillation. Actually, when a smaller length scale parameter
l0
is used or
512
high-order gradient is involved in the governing equation, the usage of continuum frequency vector often513
suffers from spurious oscillation. When the rotated scheme is used Chen et al. [2019], Roters et al. [2019],
514
numerical instability is observed in the solution, which is also reported elsewhere Chen et al. [2019]. The
515
instability is caused by the construction of the discrete frequency vector instead of the Gibbs phenomenon.
516
When the rotated scheme is adopted, the real solution is metastable, while the solution shown in Figure
517
5appears to be more stable. This instability is easy to understand when centered scheme
k=isin ξ
is
518
used, where the instability is also observed, but less obvious for the rotated scheme. This also explains
519
why the instability is less obvious as phase field propagates as reported Chen et al. [2019]. Note that such
520
instability is specific for this type of governing equation (17) and equation (29) where the unknown field
521
and its Laplacian co-exist in one equation. This instability can be overcome by using the backward scheme
522
(19) such that the unknown field and its gradient are smooth even when lower length scale
l0
and higher
523
order gradient is used, as already presented in Figure 3.524
Figure 6(a) and (b) shows the initial phase field distribution of multi-phase-field and high-order phase-
525
field, respectively. For the multi-phase-field result, only one of the two phase field variables is shown.
526
The high-order phase-field can represent the non-convex phase-field distribution observed in strongly
527
anisotropic crack propagation. More importantly, the phase field distribution is smooth over the domain
528
without local perturbation or Gibbs phenomenon observed.529
Figure 7(a) shows the convergence rate of the iterative linear solver for multi-phase-field (17) and
530
high-order phase-field (29) without preconditioning. For lower mesh resolution, the computational costs of
531
both methods are comparable, while for higher mesh resolution, high-order phase-field is more expensive.
532
Note that only one linear equation is solved for high-order phase-field, while for multi-phase-field the
533
number of linear equations depends on the total number of independent phase fields. It is suggested that
534
for lower mesh resolution, high-order phase-field is more efficient, while for higher mesh resolution, multi-
535
phase-field is more efficient. Figure 7(b) shows the corresponding convergence rate with preconditioning.
536
The convergence rate of both multi-phase-field and high-order phase-field can be accelerated by using the537
aforementioned preconditioner in Remark 1. But the convergence rates still deteriorate with an increasing
538
FFT strongly anisotropic phase field fracture 17
  

          
1.0
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
Fig. 5: Influence of discrete frequency vector on the accuracy and stability of the initial phase field distri-
bution. Top: 2D phase field distribution; Middle: phase field distribution along the central line; Bottom:
Laplacian of phase field. The anisotropic factor β2equals to zero for all three cases.
(a)
0
0.2
0.4
0.6
0.8
1
(b)
0
0.2
0.4
0.6
0.8
1
Fig. 6: Initial equilibrium phase field distribution with central point flaw: (a) Multi-phase-field method; (b)
High-order phase-field method.
18 Ran Ma, WaiChing Sun
number of grid points, indicating that the condition number increases with more refined discretization.
539
Without proper remedies, such as a suitable pre-conditioner, this attribute may limit the applications of
540
the current FFT-based scheme to solve phase field equations in fine grids that lead to sizable large linear
541
systems, for example image-based simulations. A comprehensive study on the remedy for this issue will be
542
explored in a future study but is out of the scope of the research covered in this article.543
55 89 149 245 403
47
125
365
992
2919
Iterations
(a)
2nd order
4th order
55 89 149 245 403
20
39
80
169
369
Iterations
(b)
2nd order
4th order
1.05
2.01
1.50
0.73
Fig. 7: Convergence rate of strongly anisotropic phase field upon mesh refinement: (a) Without precondi-
tioning; (b) With preconditioning. Only one phase field variable is considered in multi-phase-field. Both
multi-phase-field and high-order phase-field are solved by conjugate gradient. Note that the anisotropic
factor (tensor) also influence the convergence behavior.
3.2.2 Sawtooth crack pattern in single crystal rock salt544
A two-dimensional single crystal rock salt simulation to illustrate the capability of both methods to reproduce
545
sawtooth crack pattern which is typically observed in strongly anisotropic phase field models [Teichtmeister
546
et al.,2017]. Figure 8shows the geometrical setup. A
3 mm
by
1 mm
single crystal rock salt is discretized into
547
303
×
101 equally distributed grid points, so that the grid spacing is about
0.01 mm
which is comparable to
548
the length scale parameter
l0
. A
0.09 mm
diameter central hole is assigned unit initial phase field and serves
549
as the crack nucleation site. The initial orientation of the single crystal rock salt is
(
44
, 0
, 0
)
in Bunge
550
Euler angle convention, such that one cleavage plane is
(
44
to the loading direction while the other is
(
46
551
to the loading direction. Two phase fields with perpendicular crack propagation direction are utilized in
552
multi-phase-field, and a cubic symmetry fourth order structural tensor is used in high-order phase-field.
553
The cleavage energy of the clamping region (top and bottom part in Figure 8) is slightly increased such
554
that the crack will not penetrate through the boundary. The coupled mechanical and phase field equations
555
are solved in a staggered manner, therefore a small strain increment is required to get convergent solution.
556
Average strain boundary condition is prescribed:557
¯ε=3.75 ×1040.0
0.0 1.5 ×103. (38)
The total strain is equally divided into 1000 steps.558
The sawtooth crack pattern, observed experimentally in Yuse and Sano [1993],
?
], Takei et al. [2013], is
559
replicated in the current simulations for both the multi-phase-field case (Figure 9(a)) and rgw high-order
560
phase-field case (Figure 9(b)).561
Recall that the preferential cleavage plane of single crystal rock salt is
{
001
}
, and the initial orientation
562
of the single crystal sample is shown in Figure 8. Therefore, the two preferential cleavage planes in the
563
FFT strongly anisotropic phase field fracture 19
44o
Na
Cl
3 mm
1 mm
d= 0.09 mm Gc= 1.15 J m2
Gc= 1.50 J m2
Gc= 1.50 J m2
x
y
Fig. 8: Simulation setup of a two dimensional single crystal tension test with increased cleavage energy
Gc
in the top and bottom region. The diameter of the initial flaw is
0.09 mm
. The NaCl crystal structure denotes
the initial orientation of the single crystal sample.
plane strain tension simulation are oriented at 44
counter-clockwise and 46
clockwise from the horizontal
564
direction. Upon loading, the crack initiates at the central hole where stress concentration exists, and
565
propagates along the 46
energetically preferred direction. After the crack reaching the clamping region, the
566
higher cleavage energy
Gc
of the material in the clamping region make propagating through the material a
567
less energetic favorable configuration thank kinking with a different direction. Hence a kink is formed to
568
allow the crack propagating along the second preferred direction (44
counter-clockwise from the horizontal
569
direction). However, since the second preferred direction is not the most energetically preferred one once
570
the crack tip is not crossing the tougher layer, the crack therefore kinks back to the first preferred direction
571
before it reaches the clamping region. This process keeps repeating until the crack propagates through
572
the specimen. In total, 6 kinks are observed in the multi-phase-field case, and 8 kinks are observed in the
573
high-order phase-field case. In both cases, the crack kinks as it reaches the clamping boundary the first time
574
while propagating along the first preferential direction. It then kinks before reaching the clamping boundary
575
when propagating along the second preferential direction.576
Fig. 9: Sawpath crack patterns simulated via the two strongly anisotropic phase field models (a) Multi-
phase-field case; (b) High-order phase-field case.
20 Ran Ma, WaiChing Sun
However, one key difference in the crack pattern is that the crack propagates less along the second
577
preferential direction in high-order phase-field case than in the multi-phase-field case. Such a difference may
578
be attributed to the different phase field anisotropy parameters as well as the periodic boundaries which
579
may lead to different crack patterns than the non-periodic counterparts as reported in [Na and Sun,2018a,580
Li and Maurini,2019,Li et al.,2015]. More details about the relations among preferred crack propagation
581
direction, anisotropy of surface energy and the resultant kinking patterns on phase field simulations have
582
been discussed in Li and Maurini [2019], Li et al. [2015] based on the quenching experiments reported in
583
Yuse and Sano [1993], Takei et al. [2013].584
In Figure 9, the smearing damage region in the multi-phase-field model is wider than the high-order
585
counterpart, although both models employ the same length scale parameter
l0
to regularize the damage
586
field. This difference can be attributed by the following factors. First, the multi-phase-field and high-order587
phase-field models are formulated via different energy functionals. The high-order phase-field has a more
588
regularized spatial profile as the additional higher-order term of the governing equation controls the
589
forth-derivatives, while the lower-order governing equations of the multiple-phase-field model does not
590
regulate the third- and forth-derivative. The difference in profile shapes of the two models lead to different
591
amounts of energy dissipated even if the driving force is identical [Borden et al.,2014a]. Furthermore, since
592
the multi-phase field model contains multiple phase field evolving around multiple driving forces, the
593
phase field corresponding for different mechanisms may nucleate at different time and grow differently.
594
Meanwhile, the damage region of the higher-order phase field model is driven by only one phase field with
595
one driving force. As a result, the difference of the results is expected.596
3.3 Three-dimensional polycrystaline microstructure597
In the third example, a multi-axial tension test performed on a three-dimensional polycrystalline rock salt
598
specimen is simulated to examine the ability of the FFT-based method in solving strongly anisotropic crack
599
propagation in polycrystals. The initial microstructure of the polycrystalline RVE is shown in Figure 10,
600
which is generated using the open source software Neper [Quey et al.,2011]. The 1.0
×
1.0
×
1.0 mm domain
601
is equally discretized into 99
3
grid points, containing 40 equiaxed grains with random initial orientation.
602
Therefore, the grid point spacing is about
0.01 mm
which is comparable to the length scale parameter
603
l0
. The material parameters are the same as the two-dimensional example and are listed in Table 1. For
604
multi-phase-field method, three independent phase field variables are required to represent three
{
100
}605
cleavage plane, and the preferential propagation direction is determined by the initial orientation of the
606
grain. The grain boundary region with
0.03 mm
thickness is explicitly meshed. All three phase field variables
607
share the same crack propagation direction determined by the grain boundary normal for multi-phase-field,
608
while transverse-isotropic structural tensor
A
is assigned to the grain boundary region for high-order
609
phase-field. Again, triple junctions and quadruple points are assumed to be isotropic for simplicity and are
610
assigned with identity structural tensor. A spherical void of diameter
0.06 mm
is assigned with a prescribed
611
phase field equal to unity and serves as the nucleation site. This initial flaw can be considered as the porous
612
flaw typically exists at rock salt grain boundary. The coupled mechanical and phase field equations are
613
solved in a staggered manner. Therefore, the strain increment has to be small for the solution to converge.
614
An homogenized strain boundary condition is imposed onto the polycrystalline RVE.615
¯ε=
3.75 ×1040.0 0.0
0.0 1.5 ×1030.0
0.0 0.0 0.0
. (39)
This total strain of the RVE is equally divided into 1000 incremental steps for numerical simulations.616
The simulated three-dimensional strongly anisotropic crack propagation obtained from both models are
617
shown in Figure 11. While the multi-phase-field and high-order phase-field models do not yield the same
618
crack pattern, both of them exhibit the influence of strongly anisotropic tensor
A
and
A
on the crack pattern.
619
The difference in crack patterns may be attributed to different types of regularization (2nd-order vs.
620
4th-order) used in both models, as well as the different ways damage are accumulated in the single- and
621
multi-phase-field models. Nevertheless, since the grain boundary fracture remains the dominated damage
622
mechanism in the RVE, the results of this simulation is likely to be even more sensitive to the choice of
623
FFT strongly anisotropic phase field fracture 21
x
y
z
φ0.06 mm
initial flaw
(a) (b)
Fig. 10: Initial polycrystalline microstructure and simulation setup: (a) Polycrystalline RVE with finite
thickness grain boundary region; (b) The spatial dimension of the RVE with a spherical flaw of 0.06mm-
diameter at the centroid of the RVE.
model used to replicate the grain boundary fracture [Wei and Anand,2008] than the choice of the phase
624
field fracture models for the bulk volume.625
Furthermore, the anisotropic tensor penalizing the preferential crack direction, the reduced grain
626
boundary cleavage energy may also influence the crack propagation pattern, as shown in Figure 3.3. The
627
grain boundary cleavage energy is reduced from
1.15 J m2
to
1.0 J m2
. As observed in Figure 3.3 (c)
628
and (f), the crack has a stronger tendency to propagate along the grain boundary region with a reduced
629
grain boundary cleavage energy. Whether it is possible to enforce the equivalence of the higher-order and
630
multi-phase field models with the corresponding material parameters and degradation function is currently
631
unknown, but will be examine in future study.632
(a) (b) (c)
(d) (e) (f)
Fig. 11: Three-dimensional crack propagation in polycrystalline rock salt. (a-c) multi-phase-field case; (d-f)
high-order phase-field case. The gray region corresponds to the grain boundary region, while the red region
corresponds to the crack region.
22 Ran Ma, WaiChing Sun
(a) (b) (c)
(d) (e) (f)
Fig. 12: Three-dimensional crack propagation in polycrystalline rock salt with reduced grain boundary
cleavage energy. (a-c) multi-phase-field; (d-f) high-order phase-field. The gray region corresponds to the
grain boundary region, while the red region corresponds to the crack region.
Figure 13 shows the Von Mises stress evolution within the unit cell. In this figure, ‘MPF’ stands for
633
multi-phase-field, ‘HPF’ stands for high-order phase-field, and ‘low GB’ stands for reduced grain boundary
634
cleavage energy. The stress-strain curves represent the evolution of homogenized Cauchy stress versus
635
the axial strain in the tension direction. It is observed that the multi-phase-field generally has lower
636
homogenized stress than the high-order phase-field in the elastic region. Two reasons may account for such
637
observation. First, it has been argued in Figure 4that the second order phase-field has larger discretization
638
error than high-order phase-field under the same mesh resolution. Also, such discretization error is further
639
amplified when the elastic stiffness is determined by the combining effect of three phase fields. Therefore,
640
the equivalent phase field is more distributed and the phase-field of the undamaged region is also larger
641
than high-order phase-field, which results in more reduced elastic stiffness in multi-phase-field. A similar
642
effect is observed in the Von Mises stress contour of multi-phase-field with intact grain boundary (Figure 13
643
(a-c)) and with reduced grain boundary cleavage energy Figure 13 (g-i). The stress distribution is relatively
644
homogenized compared with typical crack tip stress field.645
The accuracy of multi-phase-field might be increased with smaller regularization length scale
l0
and
646
more refined grid points. On the other hand, high-order phase-field offers higher accuracy under the same
647
mesh resolution at the expense of higher computational cost. The phase field is less distributed within the
648
unit cell, and stress relaxation region is observed besides the crack. Also, stress concentration is observed
649
near the crack tip.650
Another interesting aspect is that, when the grain boundary cleavage energy is reduced, the stress-strain
651
response of the elastic region is not affected for both multi-phase-field and high-order phase-field. But the
652
sudden-drop of the load carrying ability occurs earlier, and crack has a larger possibility to propagate along
653
the grain boundary (Figure 3.3). Note that the stress within the cracked region can be further reduced by
654
using lower residual stiffness
k
. Residual stiffness
k
as low as 1.0
×
10
6
has been used in FFT-based method
655
Chen et al. [2019] without introducing convergence issue typically observed in FFT-based method with
656
sharp material contrast.657
Finally, it should be noted that FFT solver is not the only feasible method that leverages the Green’s
658
function to accelerate simulations of polycrystals. For instance, Zhang and Oskay [2017] has established a
659
reduced-order multiscale homogenization procedure based on eigenstrain that is both sparse and scalable.
660
The comparison among different approaches to accelerate simulations of fracture is an important topic, and
661
will be considered in future studies.662
FFT strongly anisotropic phase field fracture 23
A B C
D E F
G H I
J K L
18
15
10
5
2
29
25
20
15
10
5
1
19
15
10
5
1
27
25
20
15
10
5
1
MPF, high GB
HPF, high GB
MPF, low GB
HPF, low GB
Fig. 13: Von Mises stress (in MPa) of the mid-plane of the RVE evolving during the tension test, where
‘mpf’ stands for multi-phase-field, ‘hpf’ stands for high-order phase-field, and ‘gb’ stands for reduced grain
boundary cleavage energy. (a-c) Multi-phase-field; (d-f) High-order phase-field; (g-i) Multi-phase-field with
reduced grain boundary cleavage energy; (j-l) High-order phase-field with reduced grain boundary cleavage
energy.
24 Ran Ma, WaiChing Sun
4 Conclusion663
We introduce an FFT-based solver to solve the two most commonly used strongly anisotropic phase
664
field fracture models, i.e., the multi-phase-field and high-order phase-field models in a periodic cell. The
665
governing equations of both methods are derived in a periodic unit cell originated from the same total
666
potential
Ψ
. Since Dirichlet boundary condition cannot be explicitly imposed in the FFT domain, a penalty
667
term is added to the total potential
Ψ
to enforce the phase field in the initial failure region. The discrete
668
frequency vector is generalized to enable calculating second order and fourth order gradient, which is
669
required to solve the phase field model using a collocation method. Three numerical examples are utilized to
670
demonstrate the accuracy and numerical behavior of the FFT-based method in solving strongly anisotropic
671
phase field. Major conclusions are summarized below:672
1.
The FFT-based method is capable of consistently solve both multi-phase-field and high-order phase-field
673
governing equations. Upon each mesh refinement, the surface energy error of both methods decreases
674
toward zero.675
2.
With the same mesh resolution, the high-order phase-field model has lower surface energy error than
676
the multi-phase-field counterpart.677
3.
No local perturbation or Gibbs phenomenon is observed in both cases even in the presence of the
678
fourth-order gradient.679
4.
When mesh resolution is sufficiently low, the FFT solvers for high-order phase-field and multi-phase-
680
field models have similar computational costs. As mesh resolution increases, the high-order phase-field
681
FFT solver requires much more linear solver iterations to converge to the same relative error.682
5.
When solved by FFT-based method, both multi-phase-field and high-order phase-field can represent
683
sawtooth crack pattern typically observed in strongly anisotropic crack problem. Moreover, crack
684
coalescence and branching are observed in three-dimensional polycrystalline microstructure.685
Future work will target the following challenges. First, a more sophisticated preconditioner will be
686
derived to improve the convergence rate of the linear solver. Currently, the condition number of the stiffness
687
matrix increases with mesh resolution for both the multi-phase-field and high-order phase-field models. It
688
is expected that a preconditioner or a Green’s operator can be proposed such that the condition number can
689
become independent of discretization.690
Second, the strongly anisotropic phase field method will be incorporated in a multiphysics framework691
for simulating the material behavior of polycrystalline rock salt at various temperature and strain rate. More
692
specifically, the thermodynamic coupling of crystal plasticity [Chen et al.,2015] , twinning [Clayton and
693
Knap,2011], ductile fracture [
?
Miehe et al.,2016,Choo and Sun,2018a], solution-precipitation creep [Urai
694
et al.,1986], and diffusion-controlled crack healing [Urai et al.,2008] will be addressed.695
5 Acknowledgments696
The authors would like to thank Dr. Timothy Truster from the University of Tennessee for the fruitful
697
discussion. The two anonymous reviewers have provided helpful suggestions and feedback that improve
698
the manuscript. Their efforts are gratefully acknowledged. This research is supported by the Nuclear Energy
699
University program from the Department of Energy under grant contract DE-NE0008534, the Earth Materials
700
and Processes program from the US Army Research Office under grant contract W911NF-18-2-0306, the
701
Dynamic Materials and Interactions Program from the Air Force Office of Scientific Research under grant
702
contract FA9550-17-1-0169, as well as the Mechanics of Materials and Structures program at National
703
Science Foundation under grant contract CMMI-1462760 and the NSF CAREER grant CMMI-1846875. These
704
supports are gratefully acknowledged. The views and conclusions contained in this document are those of
705
the authors, and should not be interpreted as representing the official policies, either expressed or implied,
706
of the sponsors, including the Army Research Laboratory or the U.S. Government. The U.S. Government is
707
authorized to reproduce and distribute reprints for government purposes notwithstanding any copyright
708
notation herein.709
FFT strongly anisotropic phase field fracture 25
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H
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... Within the first class of models, the so-called second order models account for an anisotropic non-local fracture energy involving the gradient of damage variable(s) [25][26][27][28][29][30]. On the other hand, the so-called fourth order models involve the Hessian of the damage variable [31][32][33][34][35][36][37]. It was shown that second order models predict a reciprocal surface energy that is convex with respect to the orientation. ...
... Li et al. [31], Nguyen et al. [10], Li and Maurini [34], Ma and Sun [36] and Petrini et al. [68] investigated zig-zag crack propagation [4,7] via phase-field damage models. Nguyen et al. [10], Ma and Sun [36] and Petrini et al. [68] used a second-order multiphase-field model with an anisotropic fracture energy similar to the model presented in Section 2.2, while Li et al. [31], Li and Maurini [34] and Ma and Sun [36] employed a fourth-order phase-field damage model. ...
... Li et al. [31], Nguyen et al. [10], Li and Maurini [34], Ma and Sun [36] and Petrini et al. [68] investigated zig-zag crack propagation [4,7] via phase-field damage models. Nguyen et al. [10], Ma and Sun [36] and Petrini et al. [68] used a second-order multiphase-field model with an anisotropic fracture energy similar to the model presented in Section 2.2, while Li et al. [31], Li and Maurini [34] and Ma and Sun [36] employed a fourth-order phase-field damage model. These studies have shown that models accounting for an anisotropic fracture energy are able to reproduce the phenomenological behaviour of zig-zag cracking. ...
Article
Full-text available
In several classes of ductile and brittle materials consisting of different cleavage planes, an orientation dependency of the fracture process is observed. It leads for instance to complex failure behaviours and crack paths in polycrystalline or architected materials. This paper focuses on modelling anisotropy of brittle fracture by means of a variational phase-field approach. More precisely, we study different models including several phase (or damage) variables corresponding to different damage mechanisms. First, we recall a multi-mechanism gradient damage model based on an anisotropic non-local fracture energy. We then consider a model accounting for an anisotropic degradation of the elasticity stiffness tensor. Both types of anisotropies are compared in terms of their influence on analytical homogeneous solutions under uniaxial and biaxial tensile loadings. Weak and strong anisotropies are captured via the chosen multi-mechanism damage framework. The models are implemented numerically by using a finite element discretization. In order to improve numerical performance, we implement an algorithm based on a hybrid direct–iterative resolution of the displacement sub-problem. Accuracy of model prediction is assessed by comparing numerical results to theoretical solutions under uniaxial loading. Benchmark numerical tests on notched and perforated plates highlight the role of material parameters on the fracture anisotropy. Furthermore, both models are able to retrieve zig-zag crack patterns observed in prior numerical and experimental studies. Finally, we discuss the predictions of a model combining both types of anisotropies.
... A phase-field model for brittle fracture in materials with strongly anisotropic surface energy was proposed by Li et al. (2015) [22] and then further simplified and improved [23]. The work has prompted further developments extending the original higher-order phase-field model to finite deformation setting [25], dynamic brittle fracture with material point method [26], and a fully three-dimensional (3D) fast Fourier transform based implementation accounting for polycrystalline microstructure [27]. However, these contributions all assumed isotropic elasticity, which is questionable for many real materials [20,21]. ...
... In the isotropic setting, fracture toughness G c is independent of θ , thus the GMERR reduces to the classical MERR criterion. Recent experiments [19] on tearing brittle thin films with strongly anisotropic surface energy have interrogated the GMERR criterion, finding results inconsistent with the criterion Eq. (27). Furthermore, these experiments report crack propagation along metastable directions, suggesting a principle based on local maximization rather than global maximization of G(θ )/G c (θ ). ...
Article
Full-text available
Anisotropy is inherent in many materials, either because of the manufacturing process, or due to their microstructure, and can markedly influence the failure behavior. Anisotropic materials obviously possess both anisotropic elasticity and anisotropic fracture surface energy. Phase-field methods are elegant and mathematically well-grounded, and have become popular for simulating isotropic and anisotropic brittle fracture. Here, we developed a variational phase-field model for strongly anisotropic fracture, which accounts for the anisotropy both in elastic strain energy and in fracture surface energy, and the asymmetric behavior of cracks in traction and in compression. We implement numerically our higher-order phase-field model with mixed finite element, inspired by formulations for plate/shell elements, where similar continuity requirements exist. For strongly anisotropic materials, as reported in the recent experiments, one could obtain several crack propagation directions for a given loading configuration, depending on imperfections of the initial crack. From an energy point of view, the selection of crack propagation direction is dictated by local principle of the generalized maximum energy release rate. Herein, for the first time we examine numerically this local principle, reproduce the crack behaviors observed in recent experiments. Numerical simulations exhibit all the features of strongly anisotropic fracture.
... The first cause is the Gibbs phenomenon or Gibbs effect, initially proposed by Henry Wilbraham in 1848 and rediscovered by J. Willard Gibbs in 1898 [139] and well known in signal processing. It was reported in the mechanical field in [140][141][142][143]. Fourier series approximate periodic functions by summing numbers of normal trigonometric functions with different frequencies and and up to a given truncation frequency. ...
... Illustration of Fourier series approximation using N frequencies for a square wave[141]. ...
Thesis
This PhD thesis addresses numerical modeling of the fracture of heterogeneous materials using a Fast Fournier Transform (FFT) based method and a phase-field model. FFT-based numerical methods relying on the voxel type mesh, show higher computational efficiency and similar accuracy as the finite element method for the same mesh type. These methods, however, are well-known to lead to numerical artifacts (spurious oscillations). The first part focuses on these artifacts and their causes. A neighbor voxel average and an improved composite voxel method relying on a signed distance function are proposed to reduce those oscillations. The second part of the thesis focuses on damage modeling using a length-insensitive phase-field model. The correct implementation of this model for heterogeneous materiasl within the FFT solver is presented, and it is shown that this model suppresses the influence of the characteristic length on local and global responses as compared to classical models.
... Coupling the EVP-FFT-based method with a phasefield model [2,22] enables the simulation of different additional physical processes. Examples are simulations to capture the recrystallization of polycrystals [33, 236], Fig. 8 Strain distribution in a periodic glass fiber reinforced plastic microstructure [92] phase transformations [100] and fracture within polycrystalline microstructures [135], respectively. Further extensions concern the modelling of fatigue crack growth [175,176], modelling interface decohesion in terms of a nonlocal interphase [190] or a gradient damage approach [191]. ...
Article
The overall, macroscopic constitutive behavior of most materials of technological importance such as fiber-reinforced composites or polycrystals is very much influenced by the underlying microstructure. The latter is usually complex and heterogeneous in nature, where each phase constituent is governed by non-linear constitutive relations. In order to capture such micro-structural characteristics, numerical two-scale methods are often used. The purpose of the current work is to provide an overview of state-of-the-art finite element (FE) and FFT-based two-scale computational modeling of microstructure evolution and macroscopic material behavior. Spahn et al. (Comput Methods Appl Mech Eng 268:871–883, 2014) were the first to introduce this kind of FE-FFT-based methodology, which has emerged as an efficient and accurate tool to model complex materials across the scales in the recent years.
... A matrix-free geometric multigrid preconditioner was developed in [307], and its parallelized variant in [312]. Furthermore, we mention the development of a FFT (fast Fourier transform) solver for higher-order phase-field fracture problems [313]. ...
Article
Full-text available
Computational modeling of the initiation and propagation of complex fracture is central to the discipline of engineering fracture mechanics. This review focuses on two promising approaches: phase-field (PF) and peridynamic (PD) models applied to this class of problems. The basic concepts consisting of constitutive models, failure criteria, discretization schemes, and numerical analysis are briefly summarized for both models. Validation against experimental data is essential for all computational methods to demonstrate predictive accuracy. To that end, the Sandia Fracture Challenge and similar experimental data sets where both models could be benchmarked against are showcased. Emphasis is made to converge on common metrics for the evaluation of these two fracture modeling approaches. Both PD and PF models are assessed in terms of their computational effort and predictive capabilities, with their relative advantages and challenges are summarized.
... The proposed method inherits the advantages of classical FFT methods in terms of simplicity of mesh generation and parallel implementation. In addition, Ma and Sun [62] proposed an FFT-based solver for higher-order and multi-phase-field fracture, while the model was applied to strongly anisotropic brittle materials. ...
Article
This paper presents an overview of the theories and computer implementation aspects of phase field models (PFM) of fracture. The advantage of PFM over discontinuous approaches to fracture is that PFM can elegantly simulate complicated fracture processes including fracture initiation, propagation, coalescence, and branching by using only a scalar field, the phase field. In addition, fracture is a natural outcome of the simulation and obtained through the solution of an additional differential equation related to the phase field. No extra fracture criteria are needed and an explicit representation of a crack surface as well as complex track crack procedures are avoided in PFM for fracture, which in turn dramatically facilitates the implementation. The PFM is thermodynamically consistent and can be easily extended to multi-physics problem by ‘changing’ the energy functional accordingly. Besides an overview of different PFMs, we also present comparative numerical benchmark examples to show the capability of PFMs.
... In addition, FFT approaches are very efficient, with a computational cost which grows as n log(n), improving the FEM computational efficiency in the homogenization of bulk heterogeneous materials by orders of magnitude [9,10]. There are also other tangential benefits coming from the use of spectral approaches for studying lattice materials like the possibility of studying brittle fracture or ductile damage using phase-field fracture or gradient damage approaches, methods that show a very efficient performance in FFT [11][12][13]. ...