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Computational thermomechanics for crystalline rock. Part II: Chemo-damage-plasticity and healing in strongly anisotropic polycrystals

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We present a new thermal-mechanical-chemical-phase field model that captures the multi-physical coupling effects of precipitation creeping, crystal plasticity, anisotropic fracture, and crack healing in polycrystalline rock at various temperature and strain-rate regimes. This model is solved via a fast Fourier transfer solver with an operator-split algorithm to update displacement, temperature and phase field, and chemical concentration incrementally. In nuclear waste disposal in salt formation, brine inside the crystal salt may migrate along the grain boundary and cracks due to the gradient of interfacial energy and pressure. This migration has a significant implication on the permeability evolution, creep deformation, and crack healing within rock salt but is difficult to incorporate implicitly via effective medium theories compared with computational homogenization. As such, we introduce a thermodynamic framework and a corresponding computational implementation that explicitly captures the brine diffusion along the grain boundary and crack at the grain scale. Meanwhile, the anisotropic fracture and healing are captured via a high-order phase field that represents the regularized crack region in which a newly derived non-monotonic driving force is used to capture the fracture and healing due to the solution precipitation. Numerical examples are presented to demonstrate the capacity of the thermodynamic framework to capture the multiphysics material behaviors of rock salt.
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Computer Methods in Applied Mechanics and Engineering manuscript No.
(will be inserted by the editor)
Computational thermomechanics for crystalline rock. Part II:1
chemo-damage-plasticity and healing in strongly anisotropic polycrystals2
Ran Ma ·WaiChing Sun3
4
Received: May 25, 2020/ Accepted: date5
Abstract
We present a new thermal-mechanical-chemical-phase field model that captures the multi-physical
6
coupling effects of precipitation creeping, crystal plasticity, anisotropic fracture, and crack healing in
7
polycrystalline rock at various temperature and strain-rate regimes. This model is solved via a fast Fourier
8
transfer solver with an operator-split algorithm to update displacement, temperature and phase field, and9
chemical concentration incrementally. In nuclear waste disposal in salt formation, brine inside the crystal
10
salt may migrate along the grain boundary and cracks due to the gradient of interfacial energy and pressure.
11
This migration has a significant implication on the permeability evolution, creep deformation, and crack
12
healing within rock salt but is difficult to incorporate implicitly via effective medium theories compared with
13
computational homogenization. As such, we introduce a thermodynamic framework and a corresponding
14
computational implementation that explicitly captures the brine diffusion along the grain boundary and
15
crack at the grain scale. Meanwhile, the anisotropic fracture and healing are captured via a high-order
16
phase field that represents the regularized crack region in which a newly derived non-monotonic driving
17
force is used to capture the fracture and healing due to the solution precipitation. Numerical examples
18
are presented to demonstrate the capacity of the thermodynamic framework to capture the multiphysics
19
material behaviors of rock salt.20
1 Introduction21
Rock salt formation has been widely considered as one of the potential repositories for nuclear waste disposal
22
for decades. The design of these salt repositories often involves re-consolidated crushed salt as buffer or
23
backfill materials to reduce excavation void space and the time required for the salt to close in around
24
the nuclear waste. Due to the high thermal conductivity, the low permeability, the self-healing properties,
25
and the ready availability of crushed salt in a repository, the re-consolidated crushed salt has attracted a
26
significant amount of interest and becomes a major focus point in many studies by the US Department of
27
Energy for heat-generating waste (e.g. Kuhlman [2013], Martin et al. [2015]). However, during the excavation
28
process, micro-cracks may form in the salt materials within the excavation damaged zone (EDZ) near the
29
repository surface. Furthermore, the reconsolidation process of the crushed rock salt as the backfill material
30
will also introduce defects and impurities such as brine, micro-cracks, pores, and a small amount of clay.
31
These imperfections will evolve under thermal-mechanical loadings via different mechanisms, such as the
32
deformation-induced perlocation [Ghanbarzadeh et al.,2015], microcrack propagation [Zhu et al.,2015],
33
and crack healing [Houben et al.,2013,Koelemeijer et al.,2012,Franssen and Spiers,1990,Heard,1972]. To
34
prevent leakage of the radioactive materials, it is necessary to understand how effective permeability of
35
rock salt evolves under different temperatures and in situ stress.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
This article is Part II of the paper series Computational thermomechanics of crystalline rock, preceded
37
by Na and Sun [2018], which focuses on the modeling of single-crystal salt. Our objective in this new
38
contribution is to propose a computational framework for polycrystalline rock salt that explicitly captures
39
the rate-dependent multi-physical coupling mechanisms that lead to a variety of anisotropic creeping,
40
fracture, healing and plasticity under different temperature and pressure ranges at the mesoscale level.
41
Due to the coupling nature of the healing, fracture, solution precipitation creep, and heat transfer across
42
length scales, it is difficult to derive macroscopic predictive models that capture the interactions of those
43
complex mechanisms via phenomenological or even microstructure-inspired path-dependent material laws.
44
As a result, our goal is to propose a model that directly simulate those multiphysics phenomena occurred
45
at the polycrystalline microstructures, rather than introducing phenomenological constitutive laws for a
46
homogenized effective medium [Pouya et al.,2016,Bryant and Sun,2019].47
1.1 Fracture and healing in rock salt: experimental evidence48
The path-dependent deformation of polycrystalline rock salt is primarily dominated by three mechanisms
49
across different length and time scales – dislocation creeping, solution precipitation, and micro-cracking
50
dilatancy [Li and Urai,2016]. The dislocation creep typically refers to the dislocation sliding along the slip
51
planes or climbing perpendicular to the slip planes. Dislocation creep is the major deformation mecha-
52
nism when the strain rate is larger than
1.0 ×107s1
[Li and Urai,2016]. When the strain rate is below
53
1.0 ×107s1
, solution precipitation becomes the major deformation mechanism due to the existence of
54
intergranular brine. Figure 1shows the typical microstructure of a polycrystalline rock salt specimen with
55
“island-channel” type grain boundaries. For a rock salt in a natural environment, the grain boundary regions
56
are usually filled with saturated brine, which assists the solution precipitation process. Meanwhile, the
57
creeping due to intergranular solution precipitation is the result of three sequential physical processes
58
(cf. Kruzhanov and St
¨
ockhert [1998]): (1) the dissolution of solid phase across solid/liquid interface at
59
high-pressure region; (2) the solute diffusion within intergranular brine due to the concentration gradient;
60
and (3) the solute precipitation from brine to the solid phase at the low-pressure region. The efficiency of the
61
solution precipitation is pre-dominated by the slowest of these three sequential processes, which is diffusion
62
rather than dissolution or precipitation for wet rock salt [Hickman and Evans,1995]. Therefore, solution
63
precipitation is largely influenced by pressure gradient [Schott et al.,2009], grain boundary structure [van
64
Noort et al.,2008], grain boundary misorientation [Van Noort et al.,2007], and concentration of aqueous
65
trace metals [Alkattan et al.,1997].66
Na+
Cl-
abRock salt
Rock salt
Fig. 1: Microstructure of rock salt with pores and intergranular brine: (a) SEM image [Desbois et al.,2012];
(b) schematic illustration.
The long-term creep behavior is also affected by the grain boundary or crack healing process [Desbois
67
et al.,2012], which not only reduces the permeability but also restores the stiffness [Houben et al.,2013].
68
Three major mechanisms account for crack healing [Houben et al.,2013,Koelemeijer et al.,2012]: (1)
69
instantaneous mechanical closure due to increased grain boundary normal pressure; (2) diffusive crack-
70
healing driven by surface energy reduction; and (3) crack healing by recrystallization. Among these healing
71
chemo-driven fracture, damage, and healing of rock salt 3
mechanisms, the diffusive crack healing driven by surface energy reduction is the most important in the
72
long term crack healing of natural rock salt. Along with the pressure-induced solution precipitation, the
73
crack tip curvature also reduces the chemical potential of the solid phase, providing an additional driving
74
force for precipitation [van Noort et al.,2008,Koelemeijer et al.,2012,Houben et al.,2013]. The crack healing
75
rate is controlled by solute diffusion instead of solution and precipitation [Houben et al.,2013], and the
76
diffusivity within the thin film is measured [Koelemeijer et al.,2012]. Moreover, a detailed observation of
77
the brine distribution in grain boundaries has revealed that the healed grain boundaries provide a threshold
78
for the solution precipitation process [Desbois et al.,2012].79
The deformation mechanism within the polycrystalline rock is so complicated that it is difficult for phe-
80
nomenological based models to quantitatively describe dislocation creep, solution precipitation, microcrack
81
dilatancy, and crack healing simultaneously. Therefore, physics-based models and explicit representation of
82
polycrystalline structure are necessary for a fundamental understanding and quantitative prediction of the
83
deformation and permeability evolution within rock salt.84
1.2 Fracture and healing in rock salt: material modeling85
Numerous researches have been conducted toward a unified field formulation to predict multi-physical
86
behaviors of rock salt and other crystalline rock. For instance, a self-consistent homogenization method is
87
used such that the crystal plasticity simulations were upscaled to predict the anisotropic plastic deformation
88
of polycrystalline rock salt [Wenk et al.,1989,Lebensohn et al.,2003]. A general kinematic framework was
89
formulated to describe the diffusion and convection of brine and air inclusions within polycrystalline rock
90
salt [Olivella et al.,1994]. Kruzhanov and St
¨
ockhert [1998] modeled the solution-precipitation creep by
91
introducing an inelastic displacement field that maximizes the creep deformation potential defined along
92
the grain boundary. Front tracking technique and adaptive mesh technique are combined at the expense
93
of numerical accuracy and efficiency to simulate grain boundary migration and diffusion in a Lagrangian
94
framework by Bower and Wininger [2004].95
Predicting crack initiation, crack propagation, and crack healing within damaged rock salt through
96
numerical simulations has received significant attention due to the ever-increasing demand for evaluating
97
the permeability of crushed salt after re-consolidation. Crack healing has traditionally been considered
98
within the framework of continuum damage mechanics [Chan et al.,1996,2000]. A phenomenological
99
model is proposed to describe the competitive effect between mechanical damage and healing in the
100
excavation damaged zone (EDZ) through continuum damage mechanics [Hou,2003]. More recently, this
101
model is extended to capture the thermo-mechanical crack healing model to simulate the competition
102
between mechanical damage and crack healing in rock salt [Zhu and Arson,2015]. Furthermore, the effect
103
of solution-precipitation on the micropore healing process within polycrystalline rock salt is simulated by a
104
diffusion-based homogenization model in Shen and Arson [2019].105
While those phenomenological models can replicate some aspects of the constitutive behaviors of rock106
salt, the number of material parameters required for curve-fitting is large and those material parameters
107
often lack significant physical underpinnings and hence over-fitting may occur when those models are
108
used for blind predictions [Liu et al.,2016,Wang et al.,2016,2019]. To circumvent this situation, multiscale
109
DEM-FEM or FEM
2
approaches are sometimes used to upscale the simplified microscopic behaviors from
110
the grain scale to the macroscopic scale [Wei and Anand,2008,Kang et al.,2015,Pouya et al.,2016,Tjioe and
111
Borja,2016,Liu et al.,2019]. However, this upscaling procedure is only meaningful if the interplay of the
112
coupled mechanisms such as microcracking dilatancy, solution precipitation creeping, crack healing, crystal
113
plasticity, and heat transfer can be sufficiently replicated at the grain scale. The objective of this paper is to
114
provide this important theoretical framework and an FFT solver that explicitly captures these multi-physical
115
coupling mechanisms, and as a result, enables us to explain, understand and upscaling these responses for
116
macroscopic predictions.117
1.3 Outlines, major contributions and notations118
In this paper, a thermodynamic framework is proposed which explicitly incorporates crystal plasticity,
119
solution-precipitation creep, strongly anisotropic cracking, and crack healing starting from the previous
120
4 Ran Ma, WaiChing Sun
work [Na and Sun,2018]. The ductile plastic deformation and dislocation creeping of each crystal grain
121
under different temperature and confining pressure are captured via crystal plasticity, whereas a high-order
122
phase-field model is introduced to predict the anisotropic crack propagation in polycrystalline rock salt
123
with a non-convex cleavage energy determined by the preferential fracture plane of rock salt. Meanwhile,
124
the solution-precipitation creep is replicated by simulating the transport of chemical species along the
125
grain boundary. The chemical concentration then induces deformation. This approach is inspired by recent
126
investigations that capture both Herring and Coble creeps at high homologous temperature [Garikipati
127
et al.,2001,Villani et al.,2014,2015]. In this work, our new contribution is that we incorporate this diffusion
128
problem into a unified framework to predict how the solution-precipitation creep affects both the fracture
129
and the healing process. Since the healing often starts at the crack tip, we introduce a measure of the
130
curvature of the phase-field into our derivation of chemical potential such that the healing mechanism
131
depends on the surface areas [Houben et al.,2013]. Finally, considering the high computational cost of the
132
coupled equations and the global
C1
continuity requirement of the strongly anisotropic phase-field model,
133
an FFT-based method is adopted to solve the coupled equations in an operator split manner.134
This paper will proceed as follows. Section 2discusses the balance law for mass, linear momentum,
135
microforce, and energy, respectively. Section 3presents the constitutive relation for small strain crystal
136
plasticity, solution precipitation, crack healing, and high-order phase-field. In Section 4, three examples are
137
presented to demonstrate the capability of the proposed framework to represent the multiphysics behavior
138
of rock salt in mesoscale. Section 5summarizes the major results and concluding remarks.139
As for notations and symbols, bold-faced letters denote tensors (including vectors which are rank-one
140
tensors); the symbol ’
·
’ denotes a single contraction of adjacent indices of two tensors (e.g.
a·b=aibi
141
or
c·d=cij djk
); the symbol ‘:’ denotes a double contraction of adjacent indices of tensor of rank two or
142
higher (e.g.
C:εe
=
Cijk l εe
kl
); the symbol ‘::’ denotes a fourth contraction of adjacent indices of tensor of
143
rank four or higher (e.g.
C:: D
=
Cijk l Dijkl
); the symbol ‘
’ denotes a juxtaposition of two vectors (e.g.
144
ab=aibj
) or two symmetric second order tensors (e.g.
(αβ)ijkl =αij βkl
); the symbol ‘
F
’ and ‘
F1
145
represent forward and backward Fourier transformation, respectively. Materials are assumed to possess
146
cubic symmetry throughout this paper unless specified.147
2 Balance laws148
In this section, the governing equations of the thermal-mechanical-chemical-phase field framework are
149
introduced. These equations constitute the boundary value problem that replicates the multiphysical
150
material behaviors of polycrystalline rock salt in the geometrically linear regime. We first introduce an
151
interfacial indicator function that defines the location of the grain boundary and regularized crack region.
152
Then the balance principles for mass, linear momentum, and microforce (material force) are derived by
153
extending the model in [Na and Sun,2018]. Finally, the energy balance equation and dissipation inequality
154
are derived for the multiphysics problem.155
2.1 Definition of interface region and indicator function156
Cryogenic experiments have revealed the island–channel structure of the rock salt grain boundary region,
157
where the saturated brine exists and solution–precipitation occurs [Desbois et al.,2012]. The solution–
158
precipitation in return will also influence the porosity of the grain boundary, and crack healing is enabled
159
by the thin brine films within the crack region [Renard et al.,2004]. However, the morphology of the grain
160
boundary region and the crack region is highly irregular and therefore difficult to capture via conformal
161
meshes or embedded discontinuities [Wang and Sun,2019,Liu and Sun,2019]. To overcome this problem, we
162
introduce regularized interfacial regions represented by phase fields to capture the multiphysical coupling
163
process occurring along the grain boundary and the crack [Garikipati et al.,2001,Sharma et al.,2018]. First,
164
the indicator for the grain boundary region is introduced as165
dGB(X) = (0, Xin the lattice region
1, Xin the grain boundary region, (1)
chemo-driven fracture, damage, and healing of rock salt 5
where
X
denotes the coordinate of the material point. A similar concept is also proposed in van Noort
166
et al. [2008] to derive a theoretical model for solution precipitation within rock salt. The indicator for the
167
interfacial region (including grain boundary region and crack region) is defined as:168
di(dc,dGB)=1h1d2
c(X)ih1d2
GB(X)i, (2)
where
dc(X)[0, 1]
is the phase field of the crack region. The square terms of the phase field
dc
and the
169
grain boundary indicator
dGB
are consistent with Sharma et al. [2018,2019]. Note that in the numerical
170
simulation, the value of
dGB
within the grain interior region is a small positive number 0
<dGB
1 instead
171
of exact zero to avoid singularity in the diffusion equation Eq. (42).172
Consider a periodic domain
with
nsd
the spatial dimensions, this domain
can be divided into the
173
bulk region bulk and the interfacial region ithrough the interfacial indicator:174
bulk ={X|di(X)<tol, X},i={X|di(X)tol, X}, (3)
where
tol (
0, 1
)
is a small enough positive number. The interfacial region contains the physical interface
175
between the solid and liquid phase, and also involves plastic deformation at the solid-solid contact region
176
[van Noort et al.,2008].177
The actual grain boundary of rock salt has an island-channel structure, as shown in Figure 1. NaCl ions
178
diffuse within the interfacial brine, together with mass transfer between the solid phase and liquid brine.
179
The grain boundary thickness in numerical simulation should be chosen as a compromise between the
180
real grain boundary, where solution-diffusion-precipitation occurs, and the computational cost. Previously,
181
either 0.1
×
grain size [Garikipati et al.,2001] or
4 µm
[Villani et al.,2015] is used as the grain boundary
182
thickness to simulate Coble creep. We choose 0.05
×
grain size as the grain boundary thickness in this paper.
183
(a) (b) (c) (d) d= 1
d= 0
Fig. 2: A schematic representation of (a) A solid body
with crack discontinuity
Γc
and grain boundary
discontinuity
ΓGB
; (b) Grain boundary region
GB
with finite thickness and grain boundary indicator
dGB
; (c) A regularized crack region
c
by fracture phase field
dc
; (d) A regularized interfacial region
i=GB cby interfacial indicator di.
2.2 Balance of mass184
Let
ρs
denotes the intrinsic density of solid halite,
ρl
denotes the intrinsic density of brine,
c
denotes the
185
intrinsic molar concentration of brine, and
φ
represents the local porosity of the interfacial domain. Let
186
D/Dt
denotes the material time derivative with respect to the solid phase, according to the mixture theory
187
[Borja,2005], the mass balance equations can be written as:188
D
Dt [(1φ)ρs]+(1φ)ρsx·v=φrcM, in i
D
Dt (φρl) + φρlx·v+x·[φρl(vlv)]=φrcM, in i
D
Dt (φc) + φcx·v+x·[φρl(vlv) + J]=φrc, in i.
(4)
Here,
v
is the velocity of the solid phase,
vl
is the velocity of the liquid phase,
J
is the diffusion flux,
rc
is the
189
molar dissolution rate, and Mis the molar mass of halite.190
6 Ran Ma, WaiChing Sun
We made several assumptions to simplify the mass balance equations. First, we assume that the bulk
191
porosity
φ
is constant in time scale and homogeneous in spatial scale. Furthermore, the material parameters
192
are measured from the effective medium which may contain voids. However, the diffusivity of the crystalline
193
material is mainly attributed to the interconnected interface domain, i.e. the grain boundaries and the cracks,
194
which can be identified from the interfacial indicator
di
[Wang and Sun,2018]. To simplify the model, we
195
follow the treatment in Garikipati et al. [2001] and Villani et al. [2015], and assume that the volumetric
196
strain of the solid constituent (and hence the porosity change) and the divergence of the Darcy’s velocity
197
(vlv)
has negligible effect on the solution-precipitation creep rate, change of the liquid density and the
198
specie concentration. While incorporating the effect of the porosity change and the fluid flux terms could be
199
important for leakage or for high porosity rock salt, such an extension is out of the scope of this study and
200
will be considered in the future. As a result, the mass balance equations (4) can be simplified as:201
˙εc=rcvnn, in i
˙
ρl=rcM, in i
˙
c+x·J=rc, in i.
(5)
In this equation,
˙εc
is the solution-precipitation creep rate,
v
is the molar volume of solid rock salt, and
202
n
is the normal direction of the interfacial region. The solution-precipitation creep rate
˙εc
is also used to
203
model Coble creep deformation within the grain boundary region [Garikipati et al.,2001,Villani et al.,2014,
204
2015]. The mass flux
J
is a function of the chemical potential gradient. The source term
rc
is proportional
205
to the difference between the equilibrium concentration
ceq
and the brine concentration
c
, as illustrated in
206
Section 3.2.207
2.3 Balance of linear momentum and microforce208
To complete the field theory, the balance laws of linear momentum and microforce (material force) are briefly
209
summarized. While the balance of linear momentum constrains the stress field, the microforce balance
210
provides an additional governing equation for the degradation evolution within the crystal grains and grain
211
boundaries (cf. Na and Sun [2018]). Moreover, microforce corresponding to the second-order gradient of
212
phase-field is introduced to incorporate the strongly anisotropic fracture observed in rock salt.213
First, it is assumed throughout this paper that all the governing equations and corresponding physical214
quantities are defined in a cubic shape representative volume element (RVE) with periodic boundary
215
conditions, and no body force or inertia force is considered. As a result, the balance of linear momentum
216
requires that the divergence of the Cauchy stress σvanishes everywhere:217
x·σ=0. (6)
The Cauchy stress σis, under small strain assumption, power conjugate with the strain rate ˙
ε.218
The strain energy equivalence principle is adopted here to simplify the coupling relationship between
219
the phase field and the Cauchy stress, where we assume that a fictitious undamaged body exists with
220
possibly unbalanced linear momentum corresponding to the damaged counterpart. The total stress
σ
within
221
the damaged body and the effective stress
ˆ
σ
within the undamaged body are assumed to be co-axial and
222
can be related by introducing a scalar degradation function g(dc), defined as223
σ=g(dc)ˆ
σ,g(dc) = (1k)(1dc)2+k(7)
where 0
<k
1 represents the residual portion of stiffness within the damaged region to retain the
224
well-posedness of the problem. Note that the degradation function
g(dc)
is an isotropic function of the
225
phase field
dc
, so that the fictitious effective stress
ˆ
σ
and the actual stress
σ
are coaxial. The phase field
226
anisotropy is incorporated in the phase field free energy ψdin Eq. (24).227
The existence of microforce power conjugate to the phase-field is postulated together with the balance
228
law for the microforce, such that the phase-field theory can be incorporated into the coupling equations with
229
thermodynamic consistency [Gurtin,1996]. Supposing that
π
,
ξ
, and
η
are the microforces power conjugate
230
chemo-driven fracture, damage, and healing of rock salt 7
to the phase field
˙
dc
and its first
x˙
dc
and second order gradient
xx˙
dc
respectively, then the balance
231
law of the microforces requires that232
π− ∇x·ξ+ (xx):ζ=0. (8)
It is also assumed that the body force of the microforces vanishes.233
The anisotropic microfracture can be incorporated into the crystal plasticity model to predict the brittle-
234
ductile transition in rocks [Tjioe and Borja,2015]. Here, we try to model the microcrack propagation explicitly
235
in order to provide a deep understanding of the multi-physics material process within rock salt. Note that
236
in our previous work [Na and Sun,2018], multi-phase-field is adopted for strongly anisotropic fracture.
237
While this approach is feasible to replicate strongly anisotropic fracture, the introduction of multiple phase
238
fields may significantly increase the computational resources required to solve the problems numerically.
239
To improve efficiency, we have adopted the higher-order phase field fracture model (cf. Li et al. [2015], Li
240
and Maurini [2019]) where only one phase field is required to replicate the strong anisotropy. The global
241
continuity of the interpolated phase field required to resolve the higher-order terms is fulfilled by the
242
trigonometric function basis of the FFT model [Ma and Sun,2020].243
2.4 Balance of energy and dissipation inequality244
In this section, the thermodynamic laws are presented in terms of the mechanical work, structural heating,
245
crack surface energy, and chemical potential. Our starting point is the derivation from Na and Sun [2018]
246
with the following improvements and modifications: (1) High-order phase-field is introduced instead of
247
multi-phase-field to improve computational efficiency and to reduce the number of material parameters; (2)
248
The dissipation due to fluid diffusion and those due to dissolution and precipitation are incorporated in the
249
formulation to model the solution-precipitation creeping and crack healing.250
The total strain εis decomposed into three parts:251
ε=εe+εp+εc, (9)
where
εe
is the elastic strain,
εp
is the plastic strain, and
εc
is the strain caused by the solution and precipita-
252
tion of intergranular brine.253
Define
e
as the internal energy per unit volume. The first law of thermodynamics requires that the
254
internal energy changing rate ˙
eequals to the total energy input rate:255
˙
e=σ: ˙ε− ∇x·q+rθ+π˙
dc+ξ· ∇x˙
dc+ζ:xx˙
dc+µ˙
cJ· ∇xµ+(µse µ)rc, (10)
where
q
is the thermal flux,
rθ
is the heat source,
µ
is the chemical potential of the salt solute within the
256
intergranular brine, and
µse
is the chemical potential of the rock salt in solid phase. The thermal, mechanical,
257
and chemical part of the external energy input rate is consistent with Gurtin et al. [2010] and Anand
258
[2011]. The internal energy corresponding to the phase field and its gradient is consistent with Part I of this
259
paper series [Na and Sun,2018], except that the high order term
ζ:xx˙
dc
is introduced for strong
260
anisotropy. The internal energy changing rate due to solution and precipitation across the solid-liquid
261
interface
(µse µ)rc
is also incorporated similar to the model for Li ions battery [Salvadori et al.,2018] and
262
polycrystalline rock [Lehner,1995]. Here, the chemical potential of the solid phase
µse
is a function of the
263
stress and the solid-liquid interfacial radius [Lehner,1995,van Noort et al.,2008], as shown in Eq. (39).264
Also, the Clausius-Duhem inequality takes the form [Gurtin et al.,2010]:265
ZV˙
ηrθ
θdV+ZVq
θ·ndS0, (11)
where
η
is the entropy per unit volume, and
V
is an arbitrary domain within
with
n
represents the
266
outward-pointing normal direction. The local form of the Clausius-Duhem inequality Eq. (11) reads:267
˙
η − ∇x·q
θ+rθ
θ. (12)
8 Ran Ma, WaiChing Sun
Consider that the energy balance equation Eq. (10) can be re-written as:268
− ∇x·q
θ+rθ
θ=1
θ(− ∇x·q+rθ)+1
θ2q· ∇xθ
=1
θh˙
eσ: ˙ε+1
θq· ∇xθπ˙
dcξ· ∇x˙
dcζ:xx˙
dc
µ˙
c+J· ∇xµ(µse µ)rci,
(13)
the local entropy inequality Eq. (12) can be re-written by substituting the above equation as:269
˙
eθ˙
ησ: ˙ε+1
θq· ∇xθπ˙
dcξ· ∇x˙
dcζ:xx˙
dcµ˙
c+J· ∇xµ(µse µ)rc0. (14)
The Helmholtz free energy
ψ
is introduced through the Legendre transformation of the internal energy
e270
as:271
ψ=eθη. (15)
Then, the local entropy inequality Eq. (14) can be written as the free energy inequality:272
˙
ψη˙
θσ: ˙ε+1
θq· ∇xθπ˙
dcξ· ∇x˙
dcζ:xx˙
dcµ˙
c+J· ∇xµ(µse µ)rc0. (16)
We assume that the Helmholtz free energy ψtakes the general form:273
ψ=ψ(εe,dc,xdc,xxdc,c,θ), (17)
following the treatment in Anand [2004] where the free energy
ψ
is assumed to be independent of the
274
internal variable ˜
τ. Then, the free energy inequality Eq. (16) can be written as275
D=σ∂ψ
εe: ˙εe+σ: ˙εpη+∂ψ
∂θ ˙
θ1
θq· ∇xθ
+π∂ψ
dc˙
dc+ξ∂ψ
xdc· ∇x˙
dc+ ζ∂ψ
xx˙
dc!:xx˙
dc
+µ∂ψ
c˙
cJ· ∇xµ+σ: ˙εc+(µse µ)rc0.
(18)
According to the Coleman-Noll argument, the arbitrary changing rate of the state variables ˙εe,˙
θ, and ˙
c276
requires that:277
σ=∂ψ
εe,η=∂ψ
∂θ ,µ=∂ψ
c. (19)
Furthermore, the arbitrariness of the phase field ˙
dcand its gradient x˙
dcand xx˙
dcrequires that:278
π=∂ψ
dc,ξ=∂ψ
xdc,ζ=∂ψ
xx˙
dc. (20)
With the Coleman-Noll argument Eq.
(19)
and Eq.
(20)
, the dissipation inequality Eq.
(18)
can be simplified
279
as:280
D=σ: ˙εp
|{z}
Dloc
1
θq· ∇xθ
| {z }
Dcon
J· ∇xµ
| {z }
Ddi f f
+σ: ˙εc+(µse µ)rc
| {z }
Dtran
0, (21)
where
Dloc
represents the mechanical dissipation,
Dcon
represents the thermal conduction dissipation,
Ddiff
281
represents the diffusion dissipation, and
Dtran
represents the dissipation due to dissolution and precipitation.
282
A sufficient condition for the total dissipation
D
to be non-negative is that all the dissipation components
283
are non-negative individually. The diffusion dissipation and solution-precipitation dissipation are given by
284
(Ddiff =J· ∇xµ0
Dtran =σ: ˙εc+rc(µse µ)=rc[µse µvσ:(nn)] 0. (22)
chemo-driven fracture, damage, and healing of rock salt 9
These two inequality conditions pose restrictions to the admissible mass diffusion and solution precipitation
285
constitutive relations as shown in Section 3.2. The thermal conduction dissipation
Dcon
is guaranteed
286
positive by the Fourier’s law and a positive scalar thermal conductivity
κ
(or a positive definite thermal
287
conductivity tensor). The Fourier’s law is used to correlate the heat flux qand the temperature θ:288
q=κxθ(23)
where the scalar variable κis the isotropic thermal conductivity.289
2.5 A specific form of free energy290
The following explicit expression is adopted for the total free energy
ψ
, which is split into the elastic free
291
energy
ψe
, the crack surface energy
ψd
, the chemical free energy
ψc
, and the thermal contribution of the
292
stored energy ψθ:293
ψ=ψe(εe,θ,dc) + ψd(dc,xdc,xx˙
dc) + ψc(c,θ) + ψθ(θ). (24)
A specific form of free energy is proposed based on our previous work [Na and Sun,2018] with the
294
following modifications: (1) Chemical free energy
ψc
is included for solute diffusion within interfacial brine;
295
(2) High-order phase-field cleavage energy is used instead of multi-phase-field cleavage energy; (3) An
296
initial phase-field penalizing term is included to avoid the sharp material contrast between the crack region
297
and the intact region. The five parts of the Helmholtz free energy ψtake the following form:298
ψe=g(dc)we
+(εe,θ) + we
(εe,θ)
ψd=Gc1
2l0d2
c+l0
4xdc· ∇xdc+l3
0
32 xx˙
dc:A:xx˙
dc+1
2βip f (dc1)2
ψc=µ0c+Rθcln c
c01
ψθ=cv[(θθ0)θln(θ/θ0)]
(25)
Here,
Gc
is the cleavage energy per unit mass,
l0
is the character length,
βip f
is the penalty coefficient for the
299
initial phase field,
θ0
is the initial temperature, and
cv
is the specific heat coefficient per unit mass. The 4th
300
order tensor
A
makes the phase field free energy
ψd
an anisotropic function of the phase field
dc
as shown in
301
Figure 3, where the spatial distribution of the phase field
dc
is a non-convex function with two preferential
302
cleavage direction. The isotropic phase field free energy
ψd
can be recovered by replacing
A
with the 4th
303
order identity tensor
I
. Note that in the mesoscale Coble creep models, either statistical thermodynamics
304
based chemical free energy density
ψc
[Garikipati et al.,2001] or classical lattice-void free energy density
ψc
305
[Villani et al.,2014,2015] is utilized. For the interfacial brine, the chemical free energy density
ψc
for ideal
306
fluid is used, although other choices remain applicable.307
The positive part
we
+
and negative part
we
+
of the elastic strain energy in equation Eq.
(25)
are also
308
defined to avoid crack propagation under volumetric compression:309
(we
+=1
2Khεe
vi2
++µεe
d:εe
d3αK(θθ0)hεe
vi+
we
=1
2Khεe
vi2
3αK(θθ0)hεe
vi,(26)
where
εe
v
is the volumetric elastic strain,
εe
d
is the deviatoric elastic strain,
α
is the thermal expansion
310
coefficient, and Kand µare Lam´
e constants.311
To further simplify the balance of energy equation, we assume that the entropy change due to chemical
312
diffusion and crack propagation is negligible compared with plastic dissipation, such that:313
θ2ψ
∂θc˙
c=Rθln c
c0
˙
c Dloc,θ2ψ
∂θdc
˙
dc=3αθKg0(dc)hεe
vi+˙
dc Dloc. (27)
Substitute the Legendre transformation Eq.
(15)
and the specific form of free energy
ψ
Eq.
(24)
into the
314
energy balance equation (Eq.
(10)
), and consider the Coleman–Noll arguments (Eq.
(19)
) and (Eq.
(20)
) and
315
the above equation, the energy balance equation can be written as:316
cv˙
θ=θ2ψ
∂θεe: ˙εe+σ: ˙εp− ∇x·q+rθ=3αKθI: ˙εe+σ: ˙εp− ∇x·q+rθ. (28)
10 Ran Ma, WaiChing Sun
(a)
0
0.5
1
(b)
0
0.5
1
(c)
0
0.5
1
Fig. 3: Phase field distribution when the phase field value at the center is enforced as unity. The phase field
distributions minimize the total phase field free energy
ψd
. (a) Phase field distribution corresponding to
isotropic phase field free energy (
A=I
); (b) Phase field distribution corresponding to low-order phase field,
where phase field is a convex function with one preferential cleavage direction [Ma and Sun,2020]; (c) Phase
field distribution corresponding to high-order phase field in Eq.
(25)
, where phase field is a non-convex
function with two perpendicular preferential cleavage directions determined by the fourth order anisotropic
tensor A.
Considering that the phase field
dc
is non-conserved and brittle fracture (instead of creep damage) is
317
rate-independent, Ginzburg-Landau type phase-field equation is derived based on the specific form of free
318
energy
ψ
(24). Substitute the Coleman–Noll relation (20) into the microscopic force balance equation (8), and
319
assuming that the fourth order anisotropic tensor Ais piecewise constant:320
Gc
l0
dcGcl0
2x· ∇xdc+Gcl3
0
16 A:: 4dc+βi p f (dc1)=2(1dc)H,dcH4
#, (29)
where
H4
#
denotes the Sobolev space of
-periodic functions [Vond
ˇ
rejc et al.,2014]. Note that the 4th order
321
gradient of the phase field
dc
exists in the strong form Eq.
(29)
, the 4th order derivative of the solution field
322
should be quadratically integrable. Here, the phase field driving force
H
is a function of the fictitious stored
323
strain energy without degradation:324
H=max
τ[0,t]we
+, (30)
where the positive part of the elastic stored energy
we
+
and the accumulated plastic work
wp
will be
325
introduced in the following section. The Macaulay brackets
h·i
represents the ramp function. The plastic
326
deformation threshold
wp
0
is introduced to control the contribution from accumulated plastic work
wp
to
327
ductile fracture. Note that the phase-field driving force
H
in equation (30) is monotonically increasing,
328
which will be modified in Section 3.3 to enable diffusion controlled crack healing.329
3 Constitutive relations330
This section introduces the constitutive relations used in this paper. First, a small strain crystal plasticity
331
model is revisited with a Voce type hardening relation. Then, a diffusion model is introduced which allows
332
the solution, diffusion, and precipitation of NaCl solute along the grain boundary region and the crack
333
region. Chemical potential depending on pressure and solid/liquid interface curvature is considered which
334
enables pressure gradient driven and crack tip driven solution and precipitation. Finally, a fourth-order
335
anisotropic tensor representing the preferential cleavage direction is presented for the high-order phase-field
336
model, together with a modified phase-field driving force to enable diffusion-controlled crack healing.337
3.1 Small strain crystal plasticity338
The current small strain crystal plasticity model is re-formulated based on the finite strain counterpart
339
[Messner et al.,2015,Ma and Truster,2019]. The elastic constitutive relation is derived based on the effective
340
chemo-driven fracture, damage, and healing of rock salt 11
stress
ˆ
σ
defined in Equation Eq.
(7)
, the additive decomposition Eq.
(9)
, the Coleman-Noll argument Eq.
341
(19), and the elastic free energy Eq. (25) as:342
ˆ
σ=ˆ
C:εe,εe=εεpεc(31)
where
σ
is the Cauchy stress,
ε
is the total strain rate,
εθ
is the thermal expansion,
εc
is the chemical
343
deformation, and
C
is the 4th order elastic stiffness tensor. The chemical deformation is an explicit function
344
of the solution-precipitation rate rc, as defined in Eq. (5).345
In polycrystalline material, the plastic strain is achieved by dislocation slide on each slip system. Let
n(s)
346
and
b(s)
represent the normal and slip direction of the
(s)
th slip system, Then the total plastic strain is the
347
tensorial summation of the shear strain on each slip system:348
˙εp=
nsli p
s=1
˙
γ(s)m(s),m(s)=sym b(s)n(s). (32)
For small strain problem, the anti-symmetric part of the slip system does not contribute to the plastic strain.
349
In this investigation, the Voce model is adopted as the constitutive relation considering that rock salt is
350
strain rate sensitive. Note that the Voce model is temperature independent, and temperature dependence
351
can be introduced by replacing the Voce model by other crystal plasticity models such as the mechanical
352
threshold (MTS) model [Kok et al.,2002]. The relationship between the shear strain rate of the
(s)
th slip
353
system ˙
γ(s)and the resolved shear stress τ(s)is assumed to follow the power law:354
˙
γ(s)=˙
γ0
˜
τ
τ(s)
˜
τ
n1
τ(s). (33)
Here,
˙
γ0
is the reference slip rate, and
˜
τ
is the isotropic hardening variable. In this model, Taylor hardening
355
is assumed where all slip systems contribute equally to the hardening variable
˜
τ
. The slip system resistance
356
˜
τis decomposed into the intrinsic (yield) resistance τyand the extrinsic (hardening) resistance τw
357
˜
τ=τy+τw. (34)
The extrinsic resistance τwevolves as a function of the slip system activity ˙
γ(s)
358
˙
τw=h01τw
τvmnslip
s=1˙
γ(s)(35)
where θ0is the initial hardening rate, and the work hardening saturation strength τvsets the upper bound359
of
τw
. The exponents
m
and
n
are separate parameters. Note that the hardening variable
˜
τ
is monotonically
360
increasing, so the current crystal plasticity model is not suitable for creep loading. Dislocation creep
361
deformation can be modeled by introducing a dislocation annihilation model [Barton et al.,2013].362
3.2 Constitutive relation for interfacial diffusion363
This section provides the derivation of the constitutive relation for the diffusion problem. To capture the
364
fully coupled chemical-mechanical effect, the constitutive relation for the chemical potential
µ
is derived
365
from a corresponding Gibbs free energy [Garikipati et al.,2001]. We assume that the chemical potential of
366
the solute in brine does not explicitly depend on stress. Hence, the chemical potential
µ
[unit:
J mol1
] for
367
ideal solution is derived based on the Coleman-Noll argument Eq.
(19)
and the chemical free energy Eq.
(25)368
as:369
µ=µ0+Rθln c
c0
,xµ=Rθ
cxc+Rln c
c0
xθ, (36)
where µ0and c0are the reference chemical potential and reference concentration, respectively.370
The diffusion of brine along grain boundaries and cracks contributes to the solution-precipitation creep
371
and crack healing. It is assumed that the diffusion coefficient
D0
is independent of the grain boundary
372
12 Ran Ma, WaiChing Sun
normal pressure, so that the diffusion equation remains linear. Assuming that the temperature-gradient
373
driven diffusion is negligible compared with the concentration-gradient driven diffusion, the flux of the salt
374
solution Jalong the grain boundary is proportional to the gradient of the chemical potential µ, i.e.,375
J=di(dc,dGB)D0c
Rθxµ=di(dc,dGB)D0xc(37)
where
D0
[unit:
mm2s1
] is the diffusion coefficient of saturated salt solution. This diffusion coefficient
D0
is
376
a positive scalar such that the diffusion dissipation
Ddiff
is non-negative. The interface indicator
di(dc,dGB)
377
ensures that the diffusion flux outside the interfacial region
i
approximately vanishes [Sharma et al.,2018].
378
Assuming that the brine concentration is close to the halite equilibrium concentration
ceq
, the solution-
379
precipitation rate
rc
is proportional to the difference between the equilibrium concentration
ceq
and current
380
concentration
c
based on the experimental observations on mineral solution-precipitation [Alkattan et al.,
381
1997,Schott et al.,2009]:382
rc=di(dc,dGB)αsksceq c, (38)
where
αs
[unit:
mm1
] is a material coefficient which is inversely proportional to the grain boundary
383
thickness, and
ks
[unit:
mm s1
] is the solution-precipitation coefficient which denotes the speed of salt
384
migrating across the solid/liquid interface. Note that the solution coefficient
ks
is positive such that the
385
second part of
Dtran
0. In Eq.
(38)
, the equilibrium concentration
ceq
depends on the pressure, temperature,
386
and solid/liquid interface curvature [Driesner and Heinrich,2007]. If the current concentration
c
is lower
387
than the equilibrium concentration
ceq
, the solution rate is larger than the precipitation rate and the source
388
term
rc
is positive; otherwise, the source term
rc
is negative. Note that the new crack region is assumed to
389
be filled promptly with saturated salt solution.390
Remark 1.The material coefficient αsis introduced to fix the difference of the source term between Eq. (38)391
and the expression in Alkattan et al. [1997]. In Eq.
(38)
, the source term
rc
has the unit
mol m3s1
, while in
392
Alkattan et al. [1997] the source term
rc
has the unit
mol m2s1
representing the amount of salt migrating
393
across the solid/liquid interface. In this paper, the material coefficient
αs
[unit:
mm1
], which is inversely
394
proportional to the grain boundary thickness, converts the surface source term to the volumetric source
395
term by assuming that the mass migration across the interface region is evenly distributed across the grain
396
boundary region. This coefficient also includes the ratio between the actual grain boundary area and the
397
homogenized grain boundary area.398
Furthermore, the crack tip curvature may also influence the chemical potential of the solid phase and
399
provide the driving force for diffusion-controlled crack healing. In the meantime, the equilibrium chemical
400
concentration
Ceq
of the interfacial brine can be derived through the equality between the solid-phase
401
chemical potential
µse
and the brine chemical potential
µ
at equilibrium status. At the room temperature,
402
the chemical potential of the solid phase
µse
is a function of the pressure
p
and principle curvature 1
/r
(only
403
one component for 2D):404
µse =f+pv+vγsl
r=µ0+2Rθln Ceq
C0
, (39)
where
f
is the free energy of the solid phase under atmospheric pressure,
v
is the molecular volume
405
of the solid,
γsl
is the interfacial energy between solid and liquid phase, and
µ0
and
C0
are the reference
406
chemical potential and concentration. The crack tip radius is positive for convex a solid interface. With
407
the expression in equation (36) for chemical potential, the equilibrium concentration
ceq
is defined such
408
that the chemical potential of the solute in brine equals to that of the adjacent solid-phase
µse
. Therefore,
409
the equilibrium concentration (
ceq
) at crack tip can be written as an explicit function of the pressure
p
and
410
principle curvature 1/r(2D):411
ceq =c0
eq exp pv
2RT +γsl v
2rRT c0
eq 1+pv
2RT +γsl v
2rRT , (40)
where
c0
eq
is the equilibrium concentration at atmosphere pressure and room temperature with a straight
412
solid-liquid interface.413
When using the phase-field model to predict the fracture behavior, it is usually assumed that the crack
414
tip is sharp and the crack tip radius is zero when the length-scale parameter
l0
approaches zero. Therefore, it
415
chemo-driven fracture, damage, and healing of rock salt 13
is impossible to compute the crack tip curvature numerically. To overcome this problem, a user-input crack
416
tip radius is assigned to the crack tip region with the help of the Heaviside function
H(x)
for 2-dimensional
417
case:418
r=(r0,l3
0
xxdc− ∇2dcI· ∇xdc
tol.
, otherwise (41)
where tol is the tolerance to differentiate the crack tip region from the crack region. Note that the second-
419
order gradient of the phase-field is difficult to compute in finite element method, since the gradient of
420
the polynomial shape function is not continuous across the element edge. The effect of equation (41) in
421
capturing the crack tip region is shown in Figure 4, where the phase field
dc
is computed by the penalty
422
term with
H=
0 in Eq.
(29)
. Figure 4(a) and (b) show the phase field and crack tip indicator of a ‘C’ shape
423
crack, while Figure 4(c) and (d) show the phase field and crack tip indicator of two crossed cracks. The
424
proposed criterion is effective in capturing the crack tip region even for a curved crack. Note that Eq.
(41)425
can be only applied for high-order phase-field model. For second-order phase-field model, the second-order
426
gradient has singular values, which will cause numerical issues when determining the crack tip region.427
1.0
0.8
0.6
0.4
0.2
0.0
(a) (b) (c) (d)
Fig. 4: Effectiveness of using equation (41) to detect the crack tip region. Both the phase field and the crack
tip indicator are computed numerically with spectral basis functions. The crack region detection criteria
takes the value
l3
0
xxdc− ∇2dcI· ∇xdc
. (a) Initial phase field distribution of curved crack using
high-order phase-field model; (b) Crack tip determination criteria of curved crack; (c) Initial phase field
distribution of intersecting crack using high-order phase-field model; (d) Crack tip determination criteria of
intersecting crack.
Combining equations Eq.
(5)
), Eq.
(37)
, and Eq.
(38)
, the final form of the diffusion equation can be
428
written as:429
˙
cD0x·(dixc)=di(dc,dGB)αsksceq c, (42)
where the equilibrium concentration ceq depends on local pressure and solid-liquid interface curvature.430
3.3 Constitutive relations for anisotropic phase field and crack healing431
In this section, the anisotropic tensor
A
for the high-order phase field problem (29) is introduced. Further-
432
more, a crack propagation driving force Htaking account of both cracking and healing is proposed.433
In the high-order phase-field problem (29), the fourth-order anisotropic tensor
A
forms non-convex
434
cleavage energy in the polar plot. For material with cubic symmetry, the anisotropic tensor
A
adopts the
435
general form [Teichtmeister et al.,2017]:436
A=I+αap f (A1A1+A2A2) + βa p f sym(A1A2), (43)
where
I
is the fourth order identity tensor,
αap f
and
βap f
are material parameters penalizing the anisotropy,
437
and
A1
and
A2
are second order anisotropic tensors determined by the preferential cleavage plane of rock
438
salt. The phase-field problem reduces to an isotropic case if the coefficients
αap f
and
βap f
vanish. In order
439
for the anisotropic tensor Ato be positive definite, the following criteria shall be met:440
αap f >1, |βap f |<2|1+αap f |(44)
14 Ran Ma, WaiChing Sun
Let
a1
and
a2
represent the normal directions of two perpendicular cleavage planes of single crystal rock
441
salt which are determined by the initial orientation, then the second order anisotropic tensors
A1
and
A2
442
can be defined as:443
A1=a1a1,A2=a2a2. (45)
In the typical phase-field based brittle fracture model, the driving force
H
is forced to be monotonically
444
increasing by keeping its maximum historical value, since the crack healing process is not prevented by
445
the thermodynamic laws. Herein, it is assumed that the healing process is activated when the total strain
446
is volumetric compression by allowing the phase-field driving force to decrease. The phenomenological
447
model for halite cleavage plane healing based on solution-precipitation kinetics is proposed in this paper,
448
considering that the stiffness recovery rate becomes slower along with the healing process [Shen and Arson,
449
2019]:450
Hn+1=max [HnαhHnh−rciVmt,we
+]. (46)
Here,
wp
0
is the reference plastic work controlling the contribution of accumulated plastic work
wp
on phase
451
field evolution, and
αh
is a non-dimensional coefficient indicating the percentage of contribution from
452
precipitation to stiffness recovery.453
Note that the crack healing is also possible when the crack is opening [Koelemeijer et al.,2012]. In this
454
case, the healing process is accomplished through solution precipitation in a thin water film coated on the
455
crack wall. This process is much slower than the crack closure case when the crack region is filled with brine,
456
and therefore is not considered in this model.457
4 Numerical aspect on FFT-based method458
The boundary value problem consists the following governing equations: balance of linear momentum
459
equation Eq.
(6)
, the energy balance equation Eq.
(28)
, the phase field problem Eq.
(29)
, and the diffusion
460
equation Eq.
(42)
. These coupled equations are solved in an operator-split manner by a collocation FFT-based
461
solver to take advantage of its globally
C
continuous basis functions and more efficient computational
462
cost.463
The mechanical equation can be efficiently solved by the matrix-free conjugate gradient method [Zeman
464
et al.,2010,Brisard and Dormieux,2010], even though the equivalent stiffness matrix is non-symmetric.
465
Balance of linear momentum (6) can be re-formulated in a periodic domain with the help of the Green’s
466
operator Gindependent of the reference material [Zeman et al.,2017]:467
(Gσ=0in spatial domain
F1ˆ
G:F(σ)=0in frequency domain. (47)
Here, the operator
denotes convolution, which can be computed conveniently in the frequency domain.
468
The Green’s operator
G
used in this paper is independent of the reference material, which projects an
469
arbitrary strain field to its compatible part. The major idea of the Newton-Krylov method is to use an
470
iterative linear solver to solve the linearized form of equation (47), but instead of assembling the stiffness
471
matrix, the convolution operation is performed utilizing FFT. Compared with the traditional fixed-point
472
scheme, accelerated scheme, and the augmented Lagrangian scheme, the Newton-Krylov method generally
473
exhibits better numerical efficiency. However, a major trade-off is that the Newton-Krylov method fails to
474
converge when the spatial domain contains jump conditions such as a sharp contract of material properties.
475
This issue can be alleviated numerically by replacing the sharp material contract with regularized interfacial
476
representation via implicit function and introducing a residual stiffness for a completely damaged zone.
477
Note that due to the periodic nature of the trigonometric basis functions and the numerical efficiency
478
of the FFT-based method, the FFT-based method is frequently used for homogenization and concurrent
479
multiscale modeling [Kochmann et al.,2018]. The numerical efficiency of the microscale simulation within
480
the multiscale modeling can be improved by reducing the number of basis functions in FFT-based method
481
[Kochmann et al.,2019], or using discrete harmonics based homogenization [Barton et al.,2015].482
The temperature field is updated in a semi-implicit fashion using the FFT-based method [Zhu et al.,
483
1999]. The temperature field can be directly updated semi-implicitly when the thermal conductivity
κ
and
484
chemo-driven fracture, damage, and healing of rock salt 15
specific heat
Cv
are homogeneous and temperature independent, where the temperature increasing rate
485
˙
θ
is approximated by first-order backward difference and the heat source is approximated by first-order
486
Adams-Bashforth approximation. By taking the strain energy dissipation from the last converged step, the
487
temperature field update at step (n+1)can be performed in the frequency domain as:488
ˆ
θn+1=Cvˆ
θn+ˆ
rθ+Fhσn: ˙εp
n3αKθnI: ˙εe
ni
Cv+κk·kt(48)
where
k
is the frequency vector,
t
is the time step increment, and
ˆ·
denotes quantities in the frequency
489
domain.490
The continuum form of the high-order phase-field (29) defined in the periodic RVE can also be solved
491
by the FFT-based spectral method. The gradient operator and Laplacian operator can be conveniently
492
computed in the frequency domain, and the Gibbs effect can be alleviated utilizing the finite-difference
493
based frequency vector. Assuming that the anisotropic tensor
A
, the cleavage energy
Gc
, and length scale
l0
494
are piecewise constant and periodic, the continuous linear equation (29) can be discretized as:495
2H+Gc
l0
+Gcl0
2F1k·kF+Gcl3
0
16 A:: F1kkkkF+βi p f !dc=2H+βi p f . (49)
The stiffness matrix is not Hermitian unless the anisotropic tensor
A
is homogeneous, therefore the gen-
496
eralized minimal residual (GMRES) method is used to solve this equation. A modified driving force
H497
based on equation (46) instead of a typical monotonically increasing driving force (30) is used to allow
498
diffusion-controlled crack healing. Also, the initial defect region is enforced by the penalty method to avoid
499
sharp material contrast, which leads to deteriorated convergence behavior of the mechanical equation (47).
500
The diffusion equation (42) can be discretized by estimating the gradient operator in the frequency
501
domain:502
(1+αskstdi¯
λF1(ik)·(ik)F − F1(ik)·F˜
λ(x)F1(ik)Fcn+1=cn+αskstdiceq
λ(x) = ¯
λ+˜
λ(x) = diD0t.(50)
The stiffness matrix is Hermitian, so the conjugate gradient (CG) method is used to solve this equation. The
503
equilibrium concentration
ceq
depends on the pressure field and phase-field of last converged time step. The
504
diffusion equation is more involved to be solved by the FFT-based spectral method in two aspects. First,
505
the material parameters are continuously varying making it impossible to update the concentration field
506
semi-implicitly. Although the varying diffusivity can be split into a volume average part and a perturbation
507
part [Zhu et al.,1999], a fully implicit scheme is adopted considering that the solution-diffusion-precipitation
508
process requires a long time step [Sharma et al.,2018]. Second, the large diffusivity contrast between the
509
interfacial region and the bulk region is highly heterogeneous and will introduce numerical issues. This
510
problem can be alleviated by selecting a proper diffusivity residual for the bulk region.511
The coupled thermo-chemo-mechanical-phase field equations are solved in a staggered iterative scheme.
512
The detailed algorithm is presented in Algorithm 1. For the mechanical problem, either average stress
513
components or strain components are provided at the start of each time step, and the stress/strain compo-514
nents are mutually exclusive. For example, the following average stress/strain could serve as the boundary
515
condition for displacement-controlled uniaxial tension:516
¯
σ=
0 0
0 0 0
0 0 0
, ¯ε=
¯
ε11 ∗ ∗
∗ ∗
∗ ∗
. (51)
Two Newton iterations are utilized to solve this problem. The outer iteration updates the average strain
517
components based on the strain boundary condition and the deviation between average stress and stress
518
boundary condition. The inner iteration solves the Lippmann-Schwinger equation with the most updated
519
strain components. Then, the thermal equation, the diffusion equation, and the phase-field equation are
520
solved successively. The internal variables and the equilibrium concentration
ceq
are updated at the end of
521
each time step.522
16 Ran Ma, WaiChing Sun
Algorithm 1: FFT-based thermal mechanical 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
9ε=εn+1εnα(θnθn1)Iεc
10 update σn+1:σn+1=f(ε,σn,dc, history);
11 solve for ˜ε:G(Cn+1:˜ε)=Gσn+1;
12 update εn+1:εn+1=εn+1+˜ε;
13 update residual: R=k˜εk;
14 end
15 ¯
σ=hσn+1i;
16 if k¯
σσBC k
k¯
σk<tol. then break;
17 update ¯
Cn+1using equation;
18 update ¯ε:¯ε=¯
C1
n+1(¯
σσBC );
19 end
20 Update temperature θby solving Eq. (48) ;
21 Update phase field dcby solving equation Eq. (49) ;
22 Update chemical concentration cand chemical source rcby solving equation Eq. (50) ;
23 Update internal variables ˜
τfrom step n+1 to step n;
24 compute ceq from Eq. (40), Hfrom Eq. (46), εcfrom Eq.(5) ;
25 end
5 Examples523
In this section, numerical examples are presented to demonstrate the capability of the proposed numerical
524
framework in capturing the coupled physical process in rock salt. In particular, the interplay among crystal
525
plasticity, strongly anisotropic cracking, solution precipitation, and crack healing is, for the first time,
526
replicated explicitly in numerical simulations. We first calibrate the material model for crystal plasticity with
527
data available from the literature. Then we introduce physics-based material parameters for the phase-field
528
fracture, thermal diffusion and chemical transport. Then, a polycrystalline RVE creep simulation with
529
constant stress boundary condition is performed to illustrate the solution precipitation creep within the
530
interfacial region. A polycrystalline RVE simulation with monotonically increasing loading is performed
531
to capture the competition between intergranular and intragranular fracture in polycrystalline rock salt.
532
Finally, a cyclic loading numerical example is used to demonstrate the capability of the proposed crack
533
healing model in capturing the diffusion-controlled crack healing.534
5.1 Simulation setup and material model calibrations535
Rock salt single crystal exhibits a face-centered-cubic (FCC) structure. Along with the typical
{
111
}h
1
¯
1
0
i
slip
536
systems, two other slip systems are also observed including
{
110
}h
1
¯
1
0
i
and
{
100
}h
1
¯
1
0
i
.The
{
110
}h
1
¯
1
0
i537
slip system has the lowest critical resolved stress at room temperature [Carter and Heard,1970], while the
538
other two slip systems have 6 times larger critical resolved shear stress than
{
110
}h
1
¯
1
0
i
[Wenk et al.,1989,
539
Lebensohn et al.,2003]. Note that the
{
110
}h
1
¯
1
0
i
slip system provides only two independent variants, while
540
five independent variants are required to accomodate an arbitrary plastic strain. In this paper, only the
541
{110}h1¯
10islip system is considered to be consistent with our previous work [Na and Sun,2018].542
chemo-driven fracture, damage, and healing of rock salt 17
Table 1: Material properties for crystal plasticity
Parameters Description Value Unit Reference
EElastic modulus 38.0 GPa Na and Sun [2018]
νPoisson’s ratio 0.25 - Na and Sun [2018]
nRate sensitivity exponent 15.0 - -
¯
γRate normalization factor 1.0 ×1010 s1-
τyInitial yield stress 0.5 MPa -
τvSaturation stress 10.0 MPa -
mHardening exponent 1.0 - -
h0Initial hardening rate 30.0 MPa -
The single crystal uniaxial compression experiments from Carter and Heard [1970] are used to calibrate
543
the crystal plasticity parameters. The elastic constants of single crystal rock salt are from Carter and Norton
544
[2007]. Although rock salt has a cubic symmetry crystal structure, its elastic anisotropy factor is almost one,
545
so isotropic elastic constants are used. Since the Voce model Eq.
(33)
– Eq.
(35)
is independent of temperature,
546
only room temperature stress-strain curves are used with three different strain rates. A single crystal RVE
547
is loaded in the
[
001
]
direction with uniaxial compression boundary conditions. The calibrated material
548
parameters for crystal plasticity are shown in Table 1. The comparison between the experiment and the
549
simulation is shown in Figure 5.550
0 0.02 0.04 0.06 0.08 0.1 0.12
Vertical strain
0
10
20
30
40
Differential stress (MPa)
Fig. 5: Confined compression stress-strain response with different strain rates at room temperature [Carter
and Heard,1970].
The material parameters for the phase-field fracture model are shown in Table 2. The cleavage energy
551
Gc
is taken from our previous paper [Na and Sun,2018], and other parameters are adjusted such that the
552
strongly anisotropic cleavage behaviors observed experimentally are satisfied. Typical crack tip radius and
553
crack opening angle are taken from Koelemeijer et al. [2012]. The preferencial cleavage plane is {100}.554
The thermal parameters [Na and Sun,2018] and the diffusional parameters [Alkattan et al.,1997] are
555
shown in Table 3and 4, respectively. The crack healing coefficient
αh
is manually adjusted to qualitatively
556
meet the experimental crack healing rate [Houben et al.,2013], although the experiment is designed for
557
opening crack adsorbed with thin brine film which makes quantitative comparison impossible. Simplifica-
558
tions are adopted such that the interfacial region has the same thermal conductivity and thermal expansion
559
as the grain bulk region. Also, it is assumed that the thickness of the brine film is much larger than several
560
hundred nanometers such that the diffusion coefficient
D0
and solubility
ceq
of the macroscale brine are
561
representative.562
18 Ran Ma, WaiChing Sun
Table 2: Material properties for strongly anisotropic phase field
Parameters Description Value Unit Reference
GcCleavage energy 1.15 J m2Na and Sun [2018]
l0Length scale 1.0 ×105mNa and Sun [2018]
βip f Initial phase field penalty 1000.0 - -
αap f Anisotropy factor 1.2 - -
r0Crack tip radius 0.5-5 µm Koelemeijer et al. [2012]
βap f Anisotropy factor 1000.0 - -
Table 3: Material properties for thermal problem
Parameters Description Value Unit Reference
αThermal expansion coefficient 11.0 ×106K1Na and Sun [2018]
cvSpecific heat 2.0 ×106J m3K1Na and Sun [2018]
κThermal conductivity 2.0 W m1K1Na and Sun [2018]
Table 4: Material properties for diffusion
Parameters Description Value Unit Reference
D0Diffusion coefficient 2.2 ×103m2s1Alkattan et al. [1997]
ksSolution-precipitation coefficient 5.0 ×101m s1Alkattan et al. [1997]
γsl Solid-liquid interfacial energy 0.129 J m2Houben et al. [2013]
vMolecular volume of solid NaCl 2.7 ×105m3mol1Koelemeijer et al. [2012]
RIdeal gas constant 8.314 J mol1K1Koelemeijer et al. [2012]
αsGrain boundary thickness coefficient 1000.0 m1-
αhCrack healing coefficient 10.0 - -
ceq
0Equilibrium concentration 5416.0 mol m3Alkattan et al. [1997]
5.2 Long-term creeping due to solution precipitation563
In this numerical example, a series of creep simulations are performed to illustrate the solution precipitation
564
creep observed in polycrystalline rock salt [Schott et al.,2009]. The numerical set-up is shown in Figure 6(a).
565
A four-grain RVE is constructed with
1 mm
edge length. The initial orientations of the grains are denoted in
566
Figure 6(a), with
[
001
]
axis perpendicular to the
xy
plane. Constant average stress rate is enforced during
567
the loading period until the destinate stress is reached, and then constant stress boundary condition is
568
enforced during the creep deformation period.569
Grain boundary layer with
0.05 mm
thickness is introduced between each pair of grains, and initial grain
570
boundary indicator
dGB
is assigned to the grain boundary region. Crystal plasticity constitutive relation
571
is assigned to the grain boundary region, and the initial orientation inherits from the neighboring grains.
572
In the current model, grain boundary thickness is also a key material parameter. The grain boundary
573
thickness in numerical simulation should be chosen as a compromise between the real grain boundary,
574
where solution-diffusion-precipitation occurs, and the computational cost. The solution-precipitation creep
575
rate increases as the grain boundary thickness increases. A constant grain boundary thickness should be
576
used for different RVEs to reveal size effect instead of using an arbitrary fraction of the RVE edge length.577
The triple junctions of the grain boundary region also require proper treatment regarding the grain
578
boundary normal and the solution precipitation strain mode. Here, the solution precipitation creep defor-
579
mation
˙ε
equals to zero at the triple junctions, but the initial phase field is enforced as 1 for compensation.
580
Otherwise, the triple junctions become rigid inclusions as creep deformation increases.581
In this paper, only solution precipitation creep is considered instead of the competition between dislo-
582
cation creep and solution precipitation creep, since the hardening variable
˜
τ
is monotonically increasing
583
in the Voce type crystal plasticity model Eq.
(33)
– Eq.
(35)
. The dislocation creep can be incorporated by
584
introducing a dislocation annihilation mechanism to allow decreasing hardening variables.585
chemo-driven fracture, damage, and healing of rock salt 19
Figure 6(b) shows the creep rate evolution during the creep loading under different stress level. Three
586
constant stress levels are tested:
0.1 MPa
,
0.5 MPa
,
1.0 MPa
. During the constant stress-rate loading period, a
587
monotonically increasing strain rate is observed which represents the transition from the elastic region to
588
the plastic region. In the constant stress period, the strain rate gradually decreases due to the monotonically
589
increasing hardening variable. Then, the creep strain rate reaches a constant value which depends on the
590
stress level. It is also observed that the strain rate constant
n=
1, which is consistent with the solution
591
precipitation creep experiment [Schott et al.,2009].592
2040
6080
Grain Boundary
(Thickness = 0.05L) x
y
(a) The set-up for creep test
10-2 100102104
Time (s)
10-8
10-6
10-4
Creep rate (1/s)
(b) Creep rate evolution
Fig. 6: Analysis of creep loading effect on solution-precipitation creep rate: (a) The numerical set-up for the
creep simulation; (b) Creep rate evolution for each creep loading condition.
Figure 7shows the evolution of grain boundary brine concentration (a-c), longitudinal strain (d-f), and
593
transverse strain (g-i) for the
σ=1.0 MPa
case. It is observed that as time increases, the grain boundary
594
brine concentration almost remains constant. A concentration gradient exists along the grain boundary,
595
where the high concentration region corresponds to the grain boundary region with higher grain boundary
596
normal compression. Concentration flux normal to the grain boundary is almost negligible, although
597
slight concentration increasing is observed at the grain bulk adjacent to the grain boundary. On the other
598
hand, creep deformation is observed in the grain boundary region as time increases. A compressive grain-
599
boundary-normal strain
εxx
is observed in the vertical grain boundary since the grain boundary pressure
p
is
600
much higher than the horizontal grain boundary where tensile grain-boundary-normal strain
εyy
is observed.
601
It is also observed that the compressive strain
εxx
in the vertical grain boundary region approximately equals
602
the tensile strain
εyy
in the horizontal grain boundary region, indicating that the solution precipitation model
603
is mass conservative assuming that density is constant. Note that the strain values at the triple junctions
604
have large perturbations, since four grains with different orientations interact with each other. The stiffness
605
of the triple junction is reduced by imposing unit initial phase field dcin this region.606
5.3 Anisotropic cracking607
The third example is designed to illustrate the anisotropic crack initiation and propagation in polycrystalline
608
rock salt and the competition among intergranular and intragranular fracture, and plastic deformation.609
A 2D polycrystalline RVE with 40 grains is generated by Neper [Quey et al.,2011], and the RVE is equally
610
divided into 399
×
399 grid points, as shown in Figure 8(a). The RVE edge length is
1 mm
, and the average
611
grain size is
0.2 mm
. Random initial orientations are assigned to each grain with the
[
001
]
axis perpendicular
612
to the
xy
plane, such that the cleavage planes are also perpendicular to the
xy
plane. The fracture energy
Gc
613
of the material point within the bulk region is
1.15 J m2
, and the preferential fracture plane is
{001}
. Reduced
614
fracture energy (
1.0 J m2
) is assigned to the grain boundary region, and the grain boundary is assumed to
615
20 Ran Ma, WaiChing Sun
(a) Concentration t=75 s (b) Concentration t=475 s (c) Concentration t=975 s
(d) Longitudinal strain t=75 s (e) Longitudinal strain t=475 s (f) Longitudinal strain t=975 s
(g) Transverse strain t=75 s (h) Transverse strain t=475 s (i) Transverse strain t=975 s
Fig. 7: Brine concentration and strain distribution at different creep stages for the
σ=1 MPa
simulation in
Figure 6: (a-c) Evolution of grain boundary brine concentration [unit:
mol m3
]; (d-f) Evolution of strain in
the loading direction εxx ; (g-i) Evolution of strain in the transverse direction εyy.
be isotropic in crack propagation. Small strain crystal plasticity model is applied to both the grain interior
616
region and the grain boundary region. The grain boundary layer thickness approximately equals to 0.014
L
,
617
where
L
represents the RVE edge length. The deformation process is assumed to be isothermal at room
618
temperature, which is reasonable considering that the specimen size is relatively small.619
Pure shear average strain is enforced as the boundary condition, with a constant average strain rate
˙
γ=620
1.0 ×104s1:621
ε=γ0.0
0.0 γ. (52)
Figure 8(b) shows the homogenized stress-strain response in the axial direction. It is observed that the
622
fracture process of polycrystal rock salt is more ductile compared with corresponding single crystal results
623
[Na and Sun,2018]. One major reason is that the grain boundary region and intergranular anisotropy
624
prevent the crack from propagating through the specimen, as shown in Figure 9(a-c). It is observed that the
625
as external loading increases, both intragranular crack and intergranular crack initiate and gradually form a
626
network within the specimen.627
chemo-driven fracture, damage, and healing of rock salt 21
(a)
Initial configuration of polycrystal
RVE
(b) Homogenized stress-strain curve
Fig. 8: Analysis of crack initiation and propagation in polycrystalline rock salt: (a) Initial configuration of
polycrystal RVE and grain boundary. The RVE edge length
L
is
1 mm
, and the grain boundary thickness is
l=
0.014
L
, the initial orientation is random with
[
001
]
axis perpendicular to the
xy
plane. (b) Homogenized
stress-strain curve.
The Von-Mises stress distributions at different loading stages are shown in Figure 9(d-f). As expected,
628
stress concentration is observed at the crack tip and the grain boundary regions. The Gibbs effect is also
629
observed mainly for two reasons: (1) regular frequency vector is used instead for solving the mechanical
630
problem of the finite difference based frequency vector; (2) large material stiffness contrast exists within the
631
interfacial region.632
5.4 Chemical-diffusion-controlled crack healing633
In the last example, diffusion-controlled crack healing is simulated through a prescribed loading-unloading-
634
reloading strain path. A two-dimensional single-crystal RVE is divided into 399
2
grid points, and the edge
635
length of the specimen is
1 mm
by
1 mm
. A circular flaw with
0.1 mm
radius is introduced in the center of
636
the specimen for crack initiation. The initial Euler angle is
(
0
, 0
, 0
)
in Bunge notation, such that the
[
100
]637
axis is parallel to the loading direction. The loading-unloading-reloading strain path is shown in Figure 10
638
(a). A uniaxial tension boundary condition is conducted with a constant strain rate.639
In this section, the linear elastic constitutive relation is used instead of crystal plasticity for several
640
reasons. First, creeping deformation during the healing process due to the residual stress can be avoided,
641
such that the crack healing simulation could be more comparable to the experiment [Houben et al.,2013].
642
Also, the focus of this section is to demonstrate the effectiveness of the precipitation, diffusion, and crack
643
healing model, and to determine whether crack healing in rock salt is diffusion-controlled or precipitation644
controlled. Note that the usage of elastic constitutive relation is solely for illustration convenience without
645
losing generality.646
The homogenized stress-strain response of the loading-unloading-reloading process is shown in Figure
647
10 (b). Reduced stress due to crack propagation is observed after the external loading reaches a critical point,
648
and unloading boundary condition is initiated before the crack propagates through the specimen. This is
649
possible for the staggered coupling scheme, where crack initiation and propagation are delayed compared
650
with the corresponding monolithic coupling scheme. After the unloading process, the specimen is held
651
at stress-free status for 300 seconds which is long enough for diffusion induced crack healing. Stiffness
652
recovery is observed during the reloading process.653
The phase-field evolution during the loading-unloading-reloading process is shown in Figure 11. The
654
initial circular flaw with
0.1 mm
radius is prescribed by the initial phase field, as shown in Figure 11 (a), to
655
provide crack initiation spot. The fracture phase-field distributions after the loading process, the stress-free
656
holding process, and the reloading process are shown in Figure 11 (b), (c), and (d), respectively. The crack
657
22 Ran Ma, WaiChing Sun
(a) Phase field ε=0.00205 (b) Phase field ε=0.02455 (c) Phase field ε=0.04955
(d) Von Mises stress ε=0.00205 (e) Von Mises stress ε=0.02455 (f) Von Mises stress ε=0.04955
Fig. 9: Phase field and Von Mises stress distribution within the polycrystal specimen at different load stages:
(a-c) Phase field; (d-f) Von Mises stress ([unit: MPa]).
εxx
t (s)
0.0014 30 30.0014
1.103×10-3
(a) Strain path
0 0.2 0.4 0.6 0.8 1 1.2
Vertical strain 10-3
0
1
2
3
4
Deviatoric stress (MPa)
107
Loading
Unloading
Reloading
(b) Stress strain curve
Fig. 10: Strain path and stress strain curve of the crack healing simulation. (a) Strain path of the loading-
unloading-reloading process. (b) Uniaxial stress-strain curve during the loading-unloading-reloading
process. The RVE holds for a while with no external loading between the unloading process and the
reloading process to provide enough time for crack healing.
does not penetrate through the specimen after the unloading process, as shown in Figure 11 (b). Crack
658
healing is observed after the stress-free holding process as shown in Figure 11 (c), which explains the
659
stiffness recovery observed in the homogenized stress-strain response in Figure 10 (b). During the reloading
660
process, the crack continues to propagate along the original path when the external loading reaches a critical
661
point.662
chemo-driven fracture, damage, and healing of rock salt 23
1.0
0.8
0.6
0.4
0.2
0.0
(a) (b) (c) (d)
Fig. 11: Phase field evolution during the loading-unloading-reloading process. (a) Initial phase-field repre-
senting the initial circular flaw with
0.1 mm
radius; (b) Fracture phase-field distribution after the loading and
unloading process; (c) Crack healing after the stress-free holding process; (d) Crack propagation continues
to penerate the specimen after the reloading process.
The chemical concentration distributions corresponding to the time step described in Figure 11 (a-d) are
663
shown in Figure 12 (a-d), respectively. The initial chemical concentration (Figure 11 (a)) is homogeneous
664
and equals to the equilibrium chemical concentration at stress-free states. After the loading and unloading
665
process, the chemical concentration remains unchanged, since the time is very short and both solution-
666
precipitation and diffusion are time dependent. After holding the sample at stress-free states for 300 seconds
667
which is long enough for solution-precipitation and diffusion to occur, the precipitation is observed within
668
the crack tip region and the initial flaw region, as shown in Figure 12 (c). This precipitation causes the crack
669
healing observed in Figure 11 (c) and the stiffness recovery observed in Figure 10 (b).670
5416.0
5415.8
5415.6
5415.4
5415.2
5415.0
(a) (b) (c) (d) [mol·m-3]
Fig. 12: Chemical concentration distribution during the loading-unloading-reloading process. (a) Homo-
geneous initial chemical concentration equals to the equilibrium concentration
ceq
at stress free status; (b)
Chemical concentration after the loading and unloading process corresponding to Figure 11 (b); (c) Chemical
concentration distribution after the stress-free holding process corresponding to Figure 11 (c); (d) Chemical
concentration after the reloading process corresponding to Figure 11 (d).
6 Conclusions671
A mathematical framework is proposed to simulate the long-term creep, fracture and healing coupling
672
process in rock salt under a variety of thermal, mechanical and chemical conditions. An FFT-based method
673
is employed to solve the coupled equations in a staggered scheme. By leveraging the numerical efficiency
674
and globally
C
continuous basis function, the strongly anisotropic crack growth and healing are simulated.
675
Our numerical examples demonstrate that the proposed model is capable of capturing the physical behavior
676
observed in rock salt, including solution-precipitation creep, strongly anisotropic cracking, and diffusion
677
controlled crack healing with stiffness restoration.678
24 Ran Ma, WaiChing Sun
7 Acknowledgments679
This research is supported by the Nuclear Energy University program from the Department of Energy under
680
grant contract DE-NE0008534, the Earth Materials and Processes program from the US Army Research
681
Office under grant contract W911NF-18-2-0306, the Dynamic Materials and Interactions Program from the
682
Air Force Office of Scientific Research under grant contracts FA9550-17-1-0169 and FA9550-19-1-0318, as well
683
as the Mechanics of Materials and Structures program at National Science Foundation under grant contract
684
CMMI-1462760 and the NSF CAREER grant CMMI-1846875, CSE-1940203, These supports are gratefully
685
acknowledged. The views and conclusions contained in this document are those of the authors, and should
686
not be interpreted as representing the official policies, either expressed or implied, of the sponsors, including
687
the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce
688
and distribute reprints for government purposes notwithstanding any copyright notation herein.689
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... For example, Ma and Sun [137] assumed that the healing process is activated when the material experiences volumetric compression, while the stiffness recovery rate becomes slower along the healing process. This extension is out of scope of this study, and hence, we assume that cracking is irreversible. ...
... Heat exchange is an important mechanism for selecting the candidate materials for the nuclear waste geological disposal such as clay and salt. Long-term disposal such as the Yucca Mountain Project in New Mexico, for instance, relies on the combination of low permeability, high thermal conductivity, and self-healing mechanisms to ensure the isolation of the radioactive wastes [137,169,170]. ...
Thesis
Full-text available
Global challenges associated with extreme climate events and increasing energy demand require significant advances in our understanding and predictive capability of coupled multi- physical processes across spatial and temporal scales. While classical approaches based on the mixture theory may shed light on the macroscopic poromechanics simulations, accurate forward predictions of the complex behavior of phase-changing geomaterials cannot be made without understanding the underlying coupling mechanisms among constituents at the microstructural scale. To precisely predict the multi-physical behaviors originated by smaller scales, fundamental understandings of the micromechanical interactions among phase constituents are crucial. Hence, this dissertation discusses mathematical and computational frameworks designed to capture coupled thermo-hydro-mechanical-fracture processes in phase-changing porous media that incorporate necessary microscopic details. To achieve this goal, this dissertation aims to introduce a practical way to investigate how phase transition and evolving microstructural attributes at small scales affect the applicability of meso- or macroscopic finite element simulations, by leveraging the phase field method to represent the regularized interfaces of phase constituents. Firstly, a multi-phase-field microporomechanics model is presented to model the growth and thaw of ice lenses. In specific, we extend the field theory for ice lens that is not restricted to one-dimensional space. The key idea is to represent the state of the pore fluid and the evolution of freezing-induced fracture via two distinct phase field variables coupled with balance laws, which leads to an immersed approach where both the homogeneous freezing and ice lensing are distinctively captured. Secondly, a thermo-hydro-mechanical theory for geological media with thermally non-equilibrated constituents is presented, where we develop an operator-split framework that updates the temperature of each constituent in an asynchronous manner. Here, the existence of an effective medium is hypothesized, in which the constituents exhibit different temperatures while heat exchange among the phases is captured via Newton’s law of cooling. Thirdly, an immersed phase field model is introduced to predict fluid flow in fracturing vuggy porous media, where crack growth may connect previously isolated voids and form flow conduits. In this approach, we present a framework where the phase field is not only used as a damage parameter for the solid skeleton but also as an indicator of the pore space, which enables us to analyze how crack growth in vuggy porous matrix affects the flow mechanism differently compared to the homogenized effective medium while bypassing the needs of partitioning the domain and tracking the moving interface. Finally, we present a new phase field fracture theory for higher-order continuum that can capture physically justified size effects for both the path-independent elastic responses and the path-dependent fracture. Specifically, we adopt quasi-quadratic degradation function and linear local dissipation function such that the physical size dependence are insensitive to the fictitious length scale for the regularized interface, which addresses the numerical needs to employ sufficiently large phase field length scale parameter without comprising the correct physical size effect.
... Note that the material could have melted if (1) the specific heat is low and/or (2) the energy dissipation is large such that the local temperature may rise without significant diffusion (Goldsby and Tullis, 2011; Ma and Sun, 2020). Furthermore, a more profound temperature increase may also trigger the brittleductile transition that affects the mechanical responses and fracture patterns (Choo and Sun, 2018). ...
Article
Full-text available
We propose a material point method (MPM) to model the evolving multi-body contacts due to crack growth and fragmentation of thermo-elastic bodies. By representing particle interface with an implicit function, we adopt the gradient partition techniques introduced by Homel and Herbold 2017 to identify the separation between a pair of distinct material surfaces. This treatment allows us to replicate the frictional heating of the evolving interfaces and predict the energy dissipation more precisely in the fragmentation process. By storing the temperature at material points, the resultant MPM model captures the thermal advection-diffusion in a Lagrangian frame during the fragmentation, which in return affects the structural heating and dissipation across the frictional interfaces. The resultant model is capable of replicating the crack growth and fragmentation without requiring dynamic adaptation of data structures or insertion of interface elements. A staggered algorithm is adopted to integrate the displacement and temperature sequentially. Numerical experiments are employed to validate the diffusion between the thermal contact, the multi-body contact interactions and demonstrate how these thermo-mechanical processes affect the path-dependent behaviors of the multi-body systems.
... For mechanics problems, the major issues include the time and labor cost for physical experiments, the lack of facilities or equipment to complete the required tests and the difficulties to obtain specimens [54,63,95]. As a result, an alternative approach, which is adopted in this study, is to use sub-scale simulations as the digital representation that generates auxiliary data sets to build material laws or forecast engine for the macroscopic material responses [15,47,49,90]. Classic methods for causal discovery are based on probabilistic graphical modeling [61], the structure of which is a directed acyclic graph (DAG) with nodes representing random variables and edges representing conditional dependencies between variables. ...