Minimization and Saddle-Point Principles for the
Phase-Field Modeling of Fracture in Hydrogels
Lukas Bögera,∗, Marc-André Keipa , Christian Miehea
a Institute
of Applied Mechanics, Chair I, University of Stuttgart, 70550 Stuttgart,
Pfaffenwaldring 7, Germany
Abstract
Material modeling of hydrogels has gained importance over the last years due to
their emerging deployment in biomedical applications and as stimuli-responsive
functional materials. Their polymer network structure allows hydraulic permeability and can exhibit an extremely broad range of elastic properties. This
renders the predictive modeling of failure mechanisms a daunting task for future perspectives of this class of materials. In the present contribution, we
propose a phase-field model of hydrogel fracture embedded into a variational
framework and its implementation using a robust algorithm based on operator
splits. Saddle-point and minimization principles are outlined, their relation is
shown and representative boundary value problems are analyzed for both formulations. Phenomena specific to the slow mass transport in hydrogels are studied
by means of a diffusion-driven creep test with crack evolution and crack initiation induced by drying. The latter shows that large volume change not only
leads to buckling pattern as often studied in the literature, but also to crack
initiation and growth. The proposed model can thus be validated and its significance as a first approach to the modeling of fracture in polymeric hydrogels
is shown.
Keywords: phase-field fracture, finite elasticity, diffusion, hydrogels.
∗ Corresponding
author
Email address: [email protected] (Lukas Böger)
Preprint submitted to Elsevier
April 11, 2017
1. Introduction
Elastomers can exhibit an ability to absorb fluid while preserving their microstructural skeleton of long, possibly cross-linked polymer chains. In case of hydrophilic chains, the assembly of solvent molecules and polymeric network is
called a hydrogel. Hydrogels can be fabricated with a broad range of elastic
properties (see e.g. Naficy et al. [35] or Gong [17] for investigations on extremely tough gels) and may undergo large deformation states induced by fluid
transport phenomena, infeasible for the pure polymeric network due to its incompressible bulk response. Their innovative application perspectives exploiting
multi-physical phenomena coupled with diffusion-induced volume change raised
the attention paid to this class of materials. Use of interfacial phenomena and
self-assembly rank among the most seminal future concepts of hydrogels, see
Peppas et al. [36]. Gels can be used in biomedical applications such as drug delivery, tissue engineering or biosensors. Commercial employment in agriculture
and waste management as well as bio- and nanotechnological tasks are feasible
by means of microactuator capabilities of stimuli-responsive hydrogels, see also
Beebe et al. [2] and Ulijn et al. [38] for an overview of further applications.
A consequence of these perspectives is an increase of computational models proposed for hydrogels. Most often, the Flory-Huggins theory after Huggins
[22], Flory and Rehner [13] or Flory [12] is taken as a point of departure together
with the classical linear diffusion law attributed to Fick [11], as outlined by Hong
et al. [21], Duda et al. [10], Chester and Anand [7] or Doi [9]. Finite element
solutions are provided by e.g. Bouklas et al. [5], or Lucantonio et al. [26], see
Chester and Anand [8] for an additional thermo-mechanical coupling. Mechanical instabilities that often occur in the presence of geometrical constraints are
captured by e.g. Liu et al. [25] and Zhang et al. [41], while Krischok and Linder
[23] developed appropiate FE technologies that deal with the LBB condition.
The latter usually arises due to the classical stationary principle involved in
diffusion-deformation problems.
Despite the deployment prospects of hydrogels, no finite element study of
2
the failure mechanism in hydrogels is present in the literature to the authors’
best knowledge. To this end, we present a diffusion-deformation theory with
a diffusive crack approximation based on a fracture phase-field. Origins of the
regularization of the sharp discontinuity can be traced back to energy minimization concepts of brittle fracture mechanics as suggested by Francfort and
Marigo [14]. The regularized setting of their framework considered in Bourdin
et al. [6] is obtained by Γ-convergence inspired by the work of Mumford and
Shah [34]. Conceptually similar are recently outlined phase-field approaches to
brittle fracture, which originate from the classical Ginzburg-Landau type evolution equation, see Hakim and Karma [19] and Kuhn and Müller [24]. In a
recent sequence of works Miehe et al. [32], Miehe et al. [29] and Miehe et al. [31]
outline a general thermodynamically consistent framework for the phase-field
modeling of crack propagation. These formulations overcome certain difficulties
of the above mentioned works that are related to thermodynamical consistency
and the irreversibility of crack propagation, see also the works of Borden et al.
[4] and Verhoosel and de Borst [39]. Our phase-field formulation of fracture
is in line with gradient damage theories, however with ingredients rooted in
fracture mechanics. The phase-field approach to fracture provides a continuum
theory very well suited for the coupling to multi-field problems such as diffusiondeformation problems, see Wu and De Lorenzis [40] for a recent implementation
coupled to fluid transport in cement paste. The diffusive crack model has recently been applied in the context of Biot’s theory by Miehe et al. [30] and
Mauthe and Miehe [27]. Within this work, the coupled problem is shown to be
related to a new minimization principle complementary to classical saddle-point
formulations. This variational framework is in line with the present paper, in
which we aim at providing a first robust implementation of the phase-field modeling of fracture in hydrogels, formulated in terms of both minimization and
saddle-point principle.
The work is organized as follows. In Section 2, primary fields are introduced
along with balance equations and a motivation of the regularized crack surface.
Within Section 3, we propose minimization and saddle-point formulations, first
3
continuous in time and then in a time-discrete, incremental setting. Constitutive
functions for the elastic material response coupled to diffusion-induced volume
change and crack evolution are specified in Section 4, while Section 5 shows
numerical examples that demonstrate the features of the proposed model.
2. Finite Elasticity with Fluid Transport and Fracture
2.1. The Primary Fields and their Gradients
The boundary value problem of fracture and solvent diffusion in an elastic
solid is treated as a coupled four-field problem, governed by the deformation
field of the material body
ϕ:


 B0 × T → Bt ⊂ R∋

 (X, t) 7→ x = ϕ(X, t),
the swelling volume fraction and the chemical potential




 B0 × T → R
 B0 × T → R+
and
µ:
s:


 (X, t) 7→ µ(X, t)
 (X, t) 7→ s(X, t)
as well as the crack phase-field


 B0 × T → R+
d:

 (X, t) 7→ d(X, t).
(1)
(2)
(3)
The deformation field ϕ maps points X ∈ B0 of the undeformed reference
configuration B0 ⊂ R∋ onto points x ∈ Bt of the current, deformed configuration
Bt ⊂ R∋ at time t ∈ T . The material deformation gradient is defined as
F := ∇ϕ(X, t).
(4)
Its cofactor and Jacobian describe the deformation of infinitesimal line, area
and volume elements dx = F dX, da = det[F ]F −T dA and dv = det[F ]dV ,
respectively. The related condition J := det[F ] > 0 constrains the deformation
map ϕ. Additionally, with the metrics of current and reference configurations
4
H ·n0 = h̄0
P ·n0 = t̄0
X ∈ B0
n0
X ∈ B0
ϕ
ϕ = ϕ̄
deformation field
X ∈ B0
n0
∇d·n0 = 0
X ∈ B0
µ
s
n0
d
Γ
µ = µ̄
swelling field chemical potential field fracture phase-field
Figure 1: Boundary value problem of diffusion-mechanics with a fracture phase-field. The
boundary ∂B0 is decomposed into Dirichlet and Neumann parts ∂B0ϕ ∪∂B0t for the deformation
and ∂B0µ ∪ ∂B0h for the chemical potential, while the fracture phase-field is constrained by a
possible Dirichlet boundary condition d = 1 on Γ and the Neumann condition ∇d · n0 = 0 on
the full surface ∂B0 .
g, G ∈ Sym+ (3), C := F T gF and c := F −T GF −1 are denoted as the right
and left Cauchy-Green tensor.
The swelling volume fraction s is the volume v per solvent molecule times
the number c of solvent molecules per reference unit volume,
s := vc,
(5)
describing the volume resulting from solvent concentration and intuitively captures volume effects as one of the central phenomenon of interest in hydrogels.
Fluid transport in the solid is governed by the material chemical potential gradient, which we define as
M := −∇µ(X, t).
(6)
Furthermore, the crack phase-field characterizes the unbroken state of the
material with d = 0, and with d = 1 the fully broken state at X ∈ B0 . It can in
general be obtained by minimization of a crack surface functional. This functional, according to Miehe et al. [29], regularizes a sharp crack surface topology
Γ → Γl ,
Γl (d) :=
Z
γ(d, ∇d) dV,
(7)
B0
and is governed by the crack surface density function γ. A linear evolution of
the fracture phase-field can be prescribed by
γ(d, ∇d) :=
1 2 l
d + ∇d · ∇d,
2l
2
5
(8)
B0
b)
a)
c)
Figure 2: Exemplary solutions of diffusive crack topology for a given sharp crack Γ, prescribed
by the Dirichlet condition d = 1. The sequence a) - c) visualizes the limit Γl → Γ of the
regularized crack surface functional (7) towards the sharp crack with different length scales
la > lb > lc .
which also coincides with terms obtained in Γ-convergent regularizations of free
discontinuity problems outlined in Ambrosio and Tortorelli [1]. The parameter
l is a length scale controlling the regularization.
2.2. Stress Tensors and Volume Flux Vectors
Consider now a part P0 ⊂ B0 of the reference configuration B0 with the
boundary ∂P0 and the deformed part Pt ⊂ Bt with boundary ∂Pt . The stress
theorem attributed to Cauchy links the traction t acting on da ⊂ ∂Pt to the
outward surface normal as t(x, t; n) := σ(x, t)n by the Cauchy stress tensor σ.
Considering the identity T dA = tda induces the nominal stress tensor
P := (Jσ)F −T .
(9)
Similarly, an outflux of species molecules through da of ∂Pt is associated with
a macroscopic volume change. This diffusion-induced volume flux is denoted
by h, and a linear dependency on the outward normal vector is required by
h
h(x, t; n) := (x, t) · n through the spatial volume flux vector
h. A referential
volume flux H is then given by the condition HdA = hda, and the material
volume flux can be identified as
H := F −1(J h).
6
(10)
2.3. Equations in Diffusion-Deformation Problems with Fracture
The equations of the coupled problem are given in a Lagrangian description
as
H
1. Balance of volume
ṡ = −Div[ ]
2. Balance of linear momentum
Div[P ] + γ̄ = 0
3. Phase-field evolution
gc
l (d
4. Constitutive stresses
b , s, d)
P = ∂F ψ(F
− l2 ∆d) − 2(1 − d)H + K ∋ 0
b , s, d)
µ = ∂s ψ(F
5. Constitutive chemical potential
6. Constitutive volume flux
7. Constitutive crack driving quantity
H = ∂Mφb∗ (M, ḋ; F , s)
H = ψbel (F )
M, d;˙ F , s)
K = ∂ḋ φb∗ (
8. Constitutive crack resistance
(11)
defined on the reference configuration B0 for a quasi-static problem. Here, the
free energy function ψb introduced in 2.4 is assumed to be additively composed of
multiple parts, from which one exclusively describes the elastic energy storage
of the undamaged solid, denoted as ψel (F ).
2.4. Constitutive Free Energy Function and Dissipation Potential
Let ψ be the free energy density per unit volume of the reference configuration. It is presumed to depend on the primary fields (1) – (3) and their
gradients. By claiming its invariance with respect to rigid deformation of the
current configuration and consistency with a local dissipation postulate, the free
energy density has the form
b , s, d, ∇d) = ψ̄(C, s, d, ∇d).
ψ = ψ(F
(12)
Nominal stresses, chemical potential and the crack driving quantity can be constituted according to (11)4,5,7 from (12).
7
For the prescription of the volume flux, the irreversibility constraint and a
viscosity of the fracture phase-field, an objective dissipation potential can be
introduced as
M, ḋ; F , s) = φ̄∗(M, d;˙ C, s)
φ∗ = φb∗ (
(13)
at given deformation and swelling state. Volume flux and crack resistance are
then given by (11)6,8 , and the local dissipation inequality is satisfied if φb∗ is a
convex function with respect to the arguments
M and d.˙
3. Continuous and Incremental Variational Framework
In the following, we outline continuous and time-discrete minimization and
saddle-point principles on the basis of Miehe et al. [30], that are shown to imply
the set of equations (11)1−3 as Euler equations.
3.1. Continuous Minimization and Saddle-Point Principles
3.1.1. Rate of Energy, Dissipation Potential and Load Functionals
The canonical rate-of-energy functional depends on the rates of the primary
fields at a given state {ϕ, s, d} and reads
Z
d
˙
{ ∂F ψb : ∇ϕ̇ − ∂s ψb Div[ ] + ∂d ψb d˙ + ∂∇d ψb · ∇d˙ } dV. (14)
E(ϕ̇, ṡ, d) =
dt
B0
H
Here, we used the evolution (11)1 to inject a dependency on the volume flux
H.
This balance equation is thus taken as a local constraint condition yielding the
evolution ṡ.
Next, a dissipation potential function is formulated in terms of
H and d.˙ It
can be related to its dual φb∗ in (13) by the Legendre transformation
H
b , d;
˙ ∇ϕ, s) = sup {
φ(
M
M · H − φb∗(M, d;˙ ∇ϕ, s)}.
(15)
Thus, at given {ϕ, s, d}, the definition of the canonical dissipation potential
functional reads
H
D( ) :=
Z
B0
H
b , ḋ; ∇ϕ, s) dV.
φ(
8
(16)
The functional for external loads splits into a mechanical contribution and
one associated with volume fluxes over the boundary,
Z
Z
Z
Pext (ϕ̇, ) =
t̄0 · ϕ̇ dA −
γ̄ · ϕ̇ dV +
H
∂B0µ
∂B0t
B0
µ̄
H · n0 dA
(17)
with body forces γ̄ and surface tractions t̄0 , both independent of the deformation. The prescribed chemical potential µ̄ is similarly assumed to not depend
on the primary fields.
3.1.2. The Canonical Three-Field Minimization Principle
Equipped with the rate-of-energy functional and the dissipation as well as
load functionals (14), (16) and (17), the rate potential
Π(ϕ̇,
H, ḋ) := dtd E(ϕ̇, H, d)˙ + D(H, d)˙ − Pext(ϕ̇, H)
(18)
can be introduced at a given state {ϕ, s, d}. It may be split up into an internal
and an external term,
Π(ϕ̇,
H
˙ =
, d)
Z
B0
π(∇ϕ̇,
H, Div[H], d,˙ ∇d)˙ dV − Pext (ϕ̇, H),
(19)
where the internal rate potential density appears as a quantity per unit volume,
defined as
π(∇ϕ̇,
H, Div[H], d,˙ ∇d)˙ = ∂F ψb : ∇ϕ̇ − ∂sψb Div[H] + ∂d ψb ḋ
˙ ∇ϕ, s).
b H, d;
+ ∂∇d ψb · ∇d˙ + φ(
(20)
Then, the deformation rate, volume flux and fracture phase-field rate at given
variables {ϕ, d, s} are the solution of the three-field minimization principle
{ϕ̇,
H, ḋ} = Arg{ϕ̇∈W
inf
ϕ̇
inf inf
˙
H∈W
H d∈W
Π(ϕ̇,
˙
H, d)}
(21)
ḋ
with admissible spaces Wϕ̇ , WH and Wd˙ for the rates of deformation, volume
flux and phase-field rate, respectively. Evaluating the variation of (19) for virtual rates of deformation δ ϕ̇ and fracture phase-field δ d˙ and virtual volume flow
9
δ
H results in the Euler equations
b + γ̄
Div[∂F ψ]
1. Balance of linear momentum
2. Inverse Fick’s law
=0
in B0
b + ∂H φb = 0
∇[∂s ψ]
in B0
b + ∂ ˙φb ∋ 0
∂d ψb − Div[∂∇d ψ]
d
3. Phase-field evolution
4. Prescribed tractions
b · n0 − t̄0
(∂F ψ)
−∂s ψb + µ̄
5. Prescribed chemical potential
∂∇d ψb · n0
6. Phase-field outflux
in B0
=0
on ∂B0t
=0
on ∂B0µ
=0
on ∂B0
(22)
These equations include the local equation of motion (11)2,4 and an inverse
form of the linear diffusion law attributed to Fick (11)5,6 as well as the phasefield evolution (11)3 and Neumann boundary conditions. The balance of volume
(11)1 is part of the variational statement as a constraint condition, i.e., ṡ is given
by (11)1 for volume fluxes
H that stem from the solution of the minimization
principle.
3.1.3. The Extended Four-Field Saddle-Point Principle
Consider the dissipation potential function φb resulting from the generalized
Legendre transformation (15) in terms of the dual function φb∗ . Deploying (15)
in (19) leads to an extended four-field potential
Z
˙ )=
˙ ∇d,
˙
Π+ (ϕ̇, , d,
π + (∇ϕ̇, , Div[ ], d,
H M
B0
that introduces a dependency on
H
H
M) dV − Pext(ϕ̇, H)
(23)
M as a mixed variable dual to H. The mixed
rate potential density then reads
π + (∇ϕ̇,
H, Div[H], d,˙ ∇d,˙ M) = ∂F ψb : ∇ϕ̇ − ∂sψb Div[H] + ∂d ψb d˙
+ ∂∇d ψb · ∇d˙ + M · H − φb∗ (M, ḋ; ∇ϕ, s).
In this potential,
(24)
M is replaced by its definition (6) and H by its relationship
to the rate ṡ via (11)1 . After insertion of both terms into the potential Π+ in
(23), integration by parts and use of the boundary condition µ = µ̄ on ∂B0µ , we
10
obtain the reduced mixed potential
Z
˙ ∇d)
˙ dV − P ∗ (ϕ̇, µ),
˙ =
π ∗ (∇ϕ̇, ṡ, µ, ∇µ, d,
Π∗ (ϕ̇, ṡ, µ, d)
ext
(25)
B0
where the integrand of the first term is the potential density
˙ ∇d)
˙ = ∂F ψb : ∇ϕ̇ + ∂s ψbṡ + ∂d ψb d˙ + ∂∇d ψb · ∇d˙
π ∗ (∇ϕ̇, ṡ, µ, ∇µ, d,
˙ ∇ϕ, s).
− µṡ − φb∗ (−∇µ, d;
(26)
The load functional in (25) is again composed of a mechanical and a chemical
term,
∗
(ϕ̇, µ)
Pext
=
Z
γ̄ · ϕ̇ dV +
Z
t̄0 · ϕ̇ dA +
∂B0t
B0
Z
∂B0h
µh̄0 dA
(27)
where the first two summands can identically be found in (17) and h̄0 denotes
a prescribed volume flow on the referential surface part ∂B0h .
Hence, {ϕ, d, s} given, the mixed four-field saddle-point principle determines
the deformation rate, the evolution of the swelling volume fraction, the chemical
potential as well as the fracture phase-field by
˙ = Arg{ inf
{ϕ̇, ṡ, µ, d}
inf sup
˙
inf Π∗ (ϕ̇, ṡ, µ, d)}.
˙
ϕ̇∈Wϕ̇ ṡ∈L2 µ∈Wµ d∈W
ḋ
(28)
Thus, the Euler equations are an outcome of the first variation of (25) for
admissible virtual rates δ ϕ̇, virtual chemical potential δµ and swelling volume
fraction rate δ ṡ as well as the virtual fracture phase-field rate δ ḋ, and they read
b + γ̄
Div[∂F ψ]
1. Balance of linear momentum
3. Balance of volume
4. Prescribed tractions
in B0
=0
in B0
ṡ + Div[∂M φb∗ ] = 0
in B0
∂s ψb − µ
2. Constitutive chemical potential
3. Phase-field evolution
=0
b + ∂ ˙φb ∋ 0
∂d ψb − Div[∂∇d ψ]
d
b · n0 − t̄0
(∂F ψ)
=0
on ∂B0t
=0
on ∂B0h
∂∇d ψb · n0
=0
on ∂B0
(∂M φb∗ ) · n0 − h̄0
5. Prescribed volume flow
6. Phase-field outflux
11
in B0
(29)
This again covers the local equation of motion (11)2,4 , a constitutive relation
for the chemical potential (11)5 , the volume balance (11)1,6 and the phase-field
evolution (11)3 together with Neumann boundary conditions.
3.2. Incremental Variational Principles for the two Formulations
Now consider a time interval with discrete step size τ := t − tn and require
the fields to be known at time tn . The objective is to obtain these quantities at
time t (associated variables have no subscript) by means of variational principles
formulated for the discrete time increment. In what follows, we restrict ourselves
to pure Dirichlet problems, i.e., we assume for simplicity that Pext = 0 and
∗
Pext
= 0 in (18) and (25).
3.2.1. Incremental Three-Field Minimization Principle
The incremental potential in [tn , t] is specified as the equivalence of (18),
which is discrete in time and reads
Z
Πτ (ϕ, , d) =
π τ (∇ϕ,
H
B0
H, Div[H], d, ∇d) dV
(30)
without Neumann boundary conditions. π τ is an incremental potential density
per unit volume linked to the rate potential density π (20) by
Z t
˙ }.
π(∇ϕ̇, , Div[ ], ḋ, ∇d)dt
π τ (∇ϕ, , Div[ ], d, ∇d) = Algo{
H
H
tn
H
H
(31)
Algo represents a time integration algorithm of the argument, that also includes
H
an integration of (11)1 as for example s = sn − τ Div[ ] and d˙ = (d − dn )/τ .
Combined with an implicit integration of (31) the incremental potential density
reads
π τ (∇ϕ,
b
b H, d; ∇ϕ , sn , dn ).
H, Div[H], d, ∇d) = ψ(∇ϕ,
H, sn−τ Div[H], d, ∇d)+τ φ(
n
(32)
Then, the finite-step-sized incremental three-field minimization principle reads
{ϕ,
H, d} = Arg{ϕ∈W
inf
ϕ
inf inf
H∈W
H d∈W
Πτ (ϕ,
d
H, d)},
(33)
with Wϕ , WH and Wd being the admissible spaces of the fields to be determined.
12
3.2.2. Incremental Four-Field Saddle-Point Principle
The time-discrete mixed incremental potential in [tn , t] is an analog counterpart to (25) and, for pure Dirichlet problems, reads
Z
π ∗τ (∇ϕ, s, µ, ∇µ, d, ∇d) dV.
Π∗τ (ϕ, s, µ, d) =
(34)
B0
Here, the mixed incremental potential density π τ per unit volume is linked to
the rate potential density π ∗ (26) by
π ∗τ (∇ϕ, s, µ, ∇µ, d, ∇d) = Algo∗ {
Z
t
π ∗ (∇ϕ̇, ṡ, µ, ∇µ, d, ∇d)dt }.
(35)
tn
Using an implicit integration of (35) yields
b
π ∗τ (∇ϕ, s, µ, ∇µ, d, ∇d) = ψ(∇ϕ,
s, d, ∇d)−µ(s−sn )−τ φb∗ (−∇µ, d; ∇ϕn , sn , dn ).
(36)
Then, the finite-step-sized incremental four-field saddle-point principle can be
formulated as
inf sup
{ϕ, s, µ, d} = Arg{ inf
inf Π∗τ (ϕ, s, µ, d)}
ϕ∈Wϕ s∈L2 µ∈Wµ d∈Wd
(37)
and is solvable in two steps. First, s is determined locally for given state {ϕ, µ, d}
by the local minimization problem
(L) :
{s} = Arg{ inf2 π ∗τ (∇ϕ, s, µ, ∇µ, d, ∇d)}
(38)
s∈L
together with a local Newton solution scheme. This local minimization induces
a reduced incremental potential density
∗τ
πred
(∇ϕ, µ, ∇µ, d, ∇d) = inf2 π ∗τ (∇ϕ, s, µ, ∇µ, d, ∇d),
(39)
s∈L
and the global incremental potential Π∗τ (34) boils down to the reduced threefield potential
Π∗τ
red (ϕ, µ, d) =
Z
B0
∗τ
(∇ϕ, µ, ∇µ, d, ∇d) dV.
πred
(40)
Then, the second, global solution step is based on the saddle-point problem
(G) :
{ϕ, µ} = Arg{ inf
sup
inf Π∗τ
red (ϕ, µ, d)}.
ϕ∈Wϕ µ∈Wµ d∈Wd
13
(41)
4. Isotropic Constitutive Model and Algorithmic Treatment
4.1. Constitutive Energy Storage and Dissipation Potential Functions
An isotropic model for the material response is considered, contained in the
two constitutive functions, free energy and dissipation potential. The latter may
have two forms, one suited for the saddle-point and one for the minimization
principle.
4.1.1. Free Energy Storage
The energy storage function (12) is additively composed of an elastic bulk response, solvent molecule mixing, a penalty summand incorporating a volumetric
constraint and a fracture contribution that contains an accumulated dissipative
work density, i.e., a regularized fracture surface energy density. In this sense, ψb
is the total pseudo-energy density as usually considered in fracture mechanics,
b , s, d, ∇d) = ψbbulk (F , d) + ψbche (s) + ψbcon (det F , s) + ψbf rac (d, ∇d). (42)
ψ(F
The neo-Hookean material response is assumed to model the elastic contribution, and the corresponding energy function is degraded by a prefactor that
represents the fracture state of the solid and still conserves a stabilizing elastic
rest energy density,
γ
ψbbulk (F , d) = [(1−d)2 +r]ψbel (F ) = [(1−d)2 +r] [ F : F −3−2 ln(det F ) ], (43)
2
where γ := N kT is the shear modulus with N being the monomer number
in one polymer chain and the product of absolute temperatur and Boltzmann
constant kT . The stabilizing parameter r ≈ 0 is chosen as small as possible.
The chemical free energy term goes back to Flory and Rehner [13] and reads
χs
s
)+
],
ψbche (s) = α[ s ln(
1+s
1+s
(44)
where the modulus of mixing has been denoted by α := kT /v and where the
additional parameter of interaction χ is named Flory-Huggins parameter. Recall
that the swelling volume fraction s is positive (see (2)1 ), but has no upper bound.
14
The volumetric penalty term is quadratic and reads
ǫ
ψbcon (det F , s) = [ f (det F , s) ]2
2
with
f (det F , s) = det F − 1 − s.
(45)
It provides an approximate constraint for the incompressibility of polymer chain
network and the interstitial solvent. Here, ǫ acts as a penalty parameter.
Finally, the regularized fracture surface energy density reads
gc 2
d + l2 ∇d · ∇d ,
ψbf rac (d, ∇d) = gc γl (d, ∇d) =
2l
(46)
where the material parameter gc is Griffith’s critical energy release rate of the
solid.
4.1.2. Dissipation Potential for Minimization and Saddle-Point Formulation
The dissipation potential is additively composed of a bulk part incorporating
the constitutive relation between volume flux and chemical potential and a fracture part that ensures the crack phase-field irreversibility and has an additional
viscous term,
H
H
b , d; ∇ϕ , sn , dn ) = φbbulk ( ; ∇ϕ , sn ) + φbf rac (d˙ = (d − dn )/τ )
φ(
n
n
and
M, d; ∇ϕn, sn, dn) = φb∗bulk (M; ∇ϕn, sn) + φbf rac (d˙ = (d − dn )/τ )
φb∗ (
(47)
(48)
For the minimization principle at a given state {∇ϕn , sn , dn } and at time tn ,
the dissipation potential reads
H
b , d; C n , sn , dn ) =
φ(
H ⊗ H) + I((d − dn )/τ ) + η2 ((d − dn )/τ )2 ,
1
Cn : (
2M sn
(49)
with the volume mobility M := vD/kT , and where the non-smooth indicator
function reads
I(d˙ = (d − dn )/τ ) =


0 for d˙ ≥ 0

∞ otherwise.
(50)
Within the saddle-point principle, the dissipation potential depends on
M
˙ At time t, it is defined as
and the time-discrete rate d.
η
2
M, d;˙ C n , sn) = M2sn C −1
n : (M ⊗ M)+I((d−dn )/τ )+ ((d − dn )/τ ) , (51)
2
φb∗ (
15
again at given chemo-mechanical state. Two observations valid for both formulations can be made at this point. Firstly, these dissipation potentials model
isotropic diffusion. Secondly, the fracture phase-field enters the potential functions only via the irreversibility condition and a quadratic viscous term – the
bulk diffusivity is thus independent of the crack evolution. This assumption
is valid for the boundary value problems of this contribution, as the slow fluid
transport would not cause remarkable changes of the macroscopic material response. A more fine-grained mechanism that couples the fracture state to the
diffusivity is nevertheless intended for future studies.
4.2. Alternative Representations of the Phase-Field Equation
4.2.1. Generalized Ginzburg-Landau Equation
It is possible to recast equation (22)3 or (29)3 to
1 b
1
b
,
∂d ψ − Div[∂∇d ψ]
d˙ = − δd ψb := −
η
η
(52)
where we identify hxi := (x + |x|)/2 as the ramp function of R+ . This form of
the phase-field evolution equation is based on Gurtin [18] and has been understood as a Ginzburg-Landau evolution equation for d with a local irreversibility
constraint, see Miehe et al. [33]. The rate d˙ is proportional to the functional
derivative of the “total” pseudo-energy density ψb (42). For η → 0, equation
(52) collapses to
d˙ ≥ 0,
−δd ψb ≤ 0,
d˙ −δd ψb = 0.
(53)
Equations (52) and (53) are clearly characterized by their non-smoothness. This
trait can suitably be formulated in terms of a non-smooth history variable H
that renders the crack propagation driving force. In line with Miehe et al. [29],
the Ginzburg-Landau evolution equation (52) is turned into a dimensionless
form
η̃ d˙ = (1 − d)H − [ d − l2 ∆d ]
(54)
which has a geometric character as it links the diffusive crack driving force to
a geometric resistance stemming from the regularized crack surface. This crack
16
driving field can be written as
H(X, t) = max D(X, w)
w∈[0,t]
(55)
and enforces the irreversibility of d˙ by permanently collecting the maximum of
a so-called crack driving state function
D=
2ψbel (F )
.
gc /l
(56)
The function D relates the elastic energy ψbel to the critical energy release rate
gc /l smeared out over the length scale l of the regularized crack. Note that
the dimensionless phase-field equation (54) allows for high versatility, once D
is understood as a constitute ingredient for regularized crack propagation in
solids.
4.2.2. Tensile Crack Driving
The isotropic crack driving state function (56) provides no distinction between compression and tensile stresses. This is a shortcoming in boundary value
problems in which compression occurs, and can be overcome by an additive split
of the energy density in (56) as
+
−
ψbbulk = [(1 − d)2 + r]ψbel
(F ) + ψbel
(F ),
(57)
using a tensile-compressive split of the undamaged bulk energy storage ψbel =
+
−
ψbel
+ ψbel
. Miehe et al. [33] formulated a crack driving state function in terms
of Lagrangian logarithmic strains that can additively be decomposed into compression and tensile parts, resulting in
D=
+
2ψbel
(F )
.
gc /l
(58)
4.2.3. Crack Driving with Threshold
This last constitutive crack state function has no threshold for the crack evolution. Due to the irreversibility constraint in (50) and (54) however, degrading
can occur even before a localization of d due to crack growth takes place. A
17
modified crack driving function can be formulated that accounts for an elastic
range and prevents this unphysical behavior,
l2
ψbf rac (d, ∇d) = 2ψc [d + ∇d · ∇d].
2
(59)
Here, ψc denotes a critical fracture energy per unit volume, and contrary to
(46), d appears within a linear term. Frémond and Nedjar [16], Frémond [15],
Pham et al. [37] or Miehe [28] have used such a constitutive assumption for
gradient damage formulations. With (59) at hand, the crack driving function
D takes the form
D=
*
+
+
ψbel
(F )
−1 .
ψc
(60)
As a consequence, crack growth onsets are allowed if the tensile strain energy
+
ψbel
exceeds ψc . As the latter parameter is defined per unit volume, this criterion
is unaffected by the fracture length scale l.
4.2.4. Crack Driving by Effective Stress
The formulation (60) with a threshold induces a further crack driving state
as a direct function of the tensile stresses. In line with Miehe et al. [31], the
undamaged Cauchy stress σ = P F /J is split into compression and tensile parts
σ = σ+ + σ−
with
σ + :=
3
X
hσ a i na ⊗ na .
(61)
a=1
+
A Legendre transformation of the elastic energy density ψbel
(F ) used so far
∗+
leads to an enthalpy function ψbel
(σ + ) dependent on the tensile stresses, which,
+
∗+
applied to one-dimensional linear elasticity with images ψbel
= ψbel
and
1
E
∗+
+
=
hσi2
ψbel
= hεi2 = ψbel
2
2E
and ψc =
1 2
σ ,
2E c
(62)
reveals the identification of the brittle driving force contribution in (60) with
2
+
ψbel
hσi+
=
,
ψc
σc
(63)
in terms of hσi+ and a critical tensile fracture stress σc > 0. When transferred
to two or three dimensions, this function suggests the use of an isotropic stress
18
ϕ
Fd
F0
Bd
X
F
Bt
B
Figure 3: Dry, undeformed configuration Bd with s = 0, stress-free pre-swollen configuration
B with s = s0 acting as a reference state and deformed configuration Bt . Pre-swelling is
1/3
described by F 0 = J0 1 .
criterion in terms of the principal tensile stresses hσ a i (61). An associated crack
driving state function
D=
*
2
3 X
hσ a i
a=1
σc
−1
+
(64)
yields an isotropic failure surface in the space of principal stresses and, though
incompatible with a variational formulation outlined in the present work, is
rooted in a basic quantity responsible for crack growth. In addition, the threshold prevents a crack onset in regions with d = 0, which has a positive influence
on convergence rates for material failure with steep phase-field gradient zones
of sharp cracks.
4.3. The Pre-Swollen and Stabilizing Reference Configuration
A zero swelling volume fraction describes a dry body and represents a physically meaningful state, in particular suitable for problems that involve swelling
effects. This state however suffers from a challenge related to the chemical
potential: while ψche itself approaches zero for s → 0, the chemical potential
goes towards negative infinity. To prevent these numerical difficulties, the reference configuration is characterized as pre-swollen and free from stresses, an
assumption first used by Hong et al. [20]. In what follows, the dry reference
configuration (Bd in Fig. 3) shall be identified with the subscript d. The pre1/3
swollen configuration is subject to a homogeneous volume increase F 0 = J0 1
together with the condition ∂F ψd = 0 . From this latter condition, the incipient
volume fraction s0 can be deduced as
γ
1
−1/3
s0 =
J0
−
+ J0 − 1.
ǫ
J0
19
(65)
A multiplicative decomposition F d = F F 0 can now be used to compare dry
and pre-swollen configurations and postulate identical energetic states, which
leads to
Z
ψ̂d (F d , s, d, ∇d) dVd =
Bd
Z
ψ̂(F , s, d, ∇d) dV.
(66)
B
Here, ψ̂(F , s) is a quantity of the pre-swollen configuration and ψ̂d (F d , s) of the
dry one. The two free energy densities can be related to each other via
b , s, d, ∇d) = J −1 ψbd (F , s, d, ∇d)
ψ(F
0
F =F
d =F F 0
(67)
as a result from (66).
In order to construct the initial chemical potential of the pre-swollen configuration, the energy density (67) is employed,
µ0 = −
α
1
χ
s0
ǫ
[J0 − 1 − s0 ] +
)+
+
.
ln(
J0
J0
1 + s0
1 + s0
(1 + s0 )2
(68)
With the pre-swollen state at hand, fields like chemical potential, volume flux,
deformation, swelling volume fraction and fracture phase-field depend on preswollen positions X. For dissipative potential terms associated with the volume
flux, no such distinction seems to be established at this point. The flux
H
however is an intrinsic quantity of the pre-swollen configuration, because of its
introduction as a pull-back of the associated flux in the deformed configuration,
see (10). Thus, only the energetic contribution of (67) is altered within the timediscrete potentials. With this technique, good convergence rates are achieved
both for small values of J0 , intended to represent approximately dry states,
and for a higher pre-swelling parameter, which arise in the context of boundary
value problems with saturated initial specimen. Local variables s and sn must be
initially set in the finite element context, and for the saddle-point formulation,
the chemical potential at the nodes has to be initialized to µ0 .
5. Analysis of Representative Boundary Value Problems
In the following, we aim to analyze the capabilities of both minimization and
saddle-point principle in terms of three representative boundary value problems.
20
ϕ̄y (t)
µ̄(t) > µ0
µ̄(t) > µ0
J0 = 1.5
J0 = 4
d¯ = 1
ϕ̄x = 0
ϕ̄x = 0
J0 = 7
ϕ̄x = 0
ϕ̄y = 0
a)
ϕ̄y = 0
b)
c)
Figure 4: The three boundary value problems, where a) is mechanically driven with a prescribed notch and b), c) are drying processes without any defects.
The solution scheme is based on an operator split and the energetic history
field (55) as described in Miehe et al. [29]. This procedure has proven to be
a robust and effective method for solving the coupled problem and eliminates
the subdifferential in the Euler equation (29)3 and (22)3 as a side effect. It has
been implemented in conjunction with a sequential direct solver, which turned
out to be the most reliable option for the coefficient matrices suffering from
a mediocre conditioning due to the highly nonlinear Flory-Rehner term (44)
and different time scales of diffusion and fracture. The latter stems from the
slow nature of non-equilibrium diffusion processes, that competes with small
time step sizes necessary for the solution of phase-field fracture simulations.
Fortunately, the boundary value problems 5.1, 5.2 and 5.3 exhibit very slow
crack evolution, which permits an accurate solution even for large time step
sizes. For the following examples, the criterion based on effective stresses (64)
is chosen as the constitutive input for (55).
5.1. Delayed Crack Propagation in 2-Dimensional Block of Hydrogel
An implementation of the minimization formulation is employed for the following boundary value problem, in which a failure mechanism characteristic to
hydrogels as described in Bonn et al. [3] is captured. A two-dimensional square
block of hydrogel is given as B0 = {X ∈ R∈ | X ∈ [′, ⌊] × [′, h]} with b = 75 mm
and h = 150 mm. It is subject to no external load and in equilibrium with its
external environment, i.e., zero tractions for the mechanical part, and µ0 be21
No.
1
2
3
4
5
6
7
8
9
Parameter
γ
α
χ
M
J0
ǫ
σc
η
lf
Name
shear modulus
mixing modulus
mixing control parameter
volume diffusivity parameter
pre-swelling parameter
penalty parameter
critical stress
crack viscosity
length scale parameter
Value
1.0
24.2
0.1
10−4
1.5
10
3.0
0
2he
Unit
N/mm2
N/mm2
–
mm4 /N s
–
N/mm2
N/mm2
N/mm2 s
mm
Table 1: Material parameters for boundary value problems, where he is the side length of a
typical finite element triangulation
ing prescribed as a Neumann boundary condition on the whole boundary. The
specimen is horizontally fixed at the right edge by zero Dirichlet boundary conditions. At the top and the bottom facets, a non-zero displacement is applied,
linearly increasing from zero to a finite value ±ϕ̄∗2 at time t∗ and then being
held constant. The loading curve is depicted in Fig. 6. The horizontal notch
¯
is realized by Dirichlet boundary conditions for the phase-field, d(X,
t) = 1 for
X2 = h/2 and 0 ≤ X1 ≤ b/5. A non-uniform mesh has been chosen to ensure a sufficient resolution of the regularized crack surface where the crack is
expected to grow while keeping the computational cost of the simulation affordable by non-parallel processing. In addition, the symmetry of the specimen is
exploited and only one half of the problem is discretized, together with proper
symmetric Dirichlet boundary conditions. In terms of finite element spaces,
non-conforming Q1 Q1 Q1 elements have been chosen to provide a first efficient
approach, leaving the implementation of Raviart-Thomas elements for future
investigations. Material parameters for the simulations are listed in Table 1.
The maximum displacement ϕ̄∗2 and the loading speed controlled by t∗ (the
time from which on the applied displacement is held constant) influence the
onset of crack growth. Here, a time step size of τ = 10−1 s has been chosen
together with t∗ = 10 s and ϕ̄∗2 = 0.11h. The crack does not propagate during
loading – instead, in the phase with constant displacement, a diffusion-driven
viscous effect slowly increases the tension at the crack tip, until the specimen
22
t0 = 0 s
t1 = 5 s
t2 = 80 s
t3 = 100 s
t3 = 115 s
Figure 5: Contor of the nominal stress P22 for 5 different time steps. Parts of the specimen
with d > 0.98 are blanked out for better visibility of the crack pattern. As symmetry has been
exploited, the lower half of the contour plots are mirrored.
cracks relatively quick. Snapshots taken during the simulation are depicted
in Fig. 5, while loading curve and the normalized sum of mechanical reaction
forces in x2 -direction at the top of the specimen are shown in Fig. 6. Clearly,
the relaxation of nodal forces keep record of the diffusion impact and strongly
change during crack evolution. As can be seen in Fig. 6, the crack suddenly
starts propagating at t ≈ 90 s, which is a large delay compared to the loading
of the hydrogel specimen.
5.2. Drying 2-Dimensional Rectangle with Geometrical Constraints
While the first example has been mechanically driven and realized through an
implementation of the minimization formulation, we now proceed to the saddlepoint principle and consider boundary value problems with constant mechanical
constraints, where the crack evolution is exclusively caused by stresses that arise
from a drying process.
To this end, we consider a rectangular block B0 = {X ∈ R∈ | X ∈
[′, ⌊] × [′, h]} with b = 300 cm and h = 60 cm, in equilibrium with its external environment at time t = 0. Fluid will be forced to diffuse out of the gel at
the top edge by application of a linearly increasing draining condition on the
top surface µ̄(t) > µ0 . Additionally, a fluid-saturated initial configuration is
23
P
Ry,i
Ry,max
[-]
1.0
1.0
0.8
0.8
0.6
0.6
b)
c)
d)
0.4
ϕ̄y
ϕ̄max
[-]
0.4
prop. loading
reac. forces
0.2
0.0
e)
0
20
40
0.2
60
80
100
0.0
120
t [s]
Figure 6: Plots of loading scheme and normalized nodal reaction forces of the upper boundary
in y−direction over time. The rapid decrease of reaction forces clearly indicates the delayed
and viscous-like failure of the specimen. Points in time at which snapshots in Fig. 5 are taken
are marked by the vertical lines, i.e. b) 5 s, c) 80 s, d) 100 s and e) 115 s.
a)
b)
c)
d)
Figure 7: Drying-induced crack evolution in a hydrogel rectangle horizontally fixed at both
vertical edges. The chemical potential contour lies between µ0 (J0 ) (red) and µ̄ (blue). Snapshots are taken at a) 0 s, b) 1.85 · 105 s, c) 1.30 · 106 s and d) 1.97 · 106 s, regions with d > 0.91
are blanked out for the sake of better crack pattern visibility.
24
required, where the preswelling comes in handy: choosing a high parameter J0
renders a homogeneous swollen state of the hydrogel. Vertical displacement is
blocked at the bottom, horizontal displacement at left and right facets. Material
parameters are as before (see Table 1), however with J0 = 4 and a critical stress
σc = 1.0 N/mm2 . A uniform mesh is used to discretize the specimen, and for
simplicity, linear Q1 Q1 Q1 elements have been chosen, bilinearly interpolating
all nodal primary values. While the deployment of Taylor-Hood elements for
displacement and chemical potential clearly is a better choice, again the usage
of more advanced finite element techniques is beyond the scope of this paper.
We observe that due to draining-induced tractions, multiple crack paths
develop from top to bottom, which is a similar observation that can be made for
water-saturated soils during drying. As no defects are taken into consideration,
the crack initiation has instability-like characteristics – how many cracks develop
and where they are placed over the width of the specimen cannot be controlled.
Between the first visible crack evolution and the arrival of cracks at the bottom
of the specimen 1.97·106 s pass by. This again underlines the slow nature of fluid
transport and diffusion-driven fracture in the solid (that would crack in one or
at most in a few time steps without the consideration of species diffusion). For
this reason, the simulation could be run with a large time step size of 7 · 102 s.
5.3. 2-Dimensional Circle of Hydrogel with a Prescribed Draining Boundary
The third example is again implemented as the saddle-point formulation
and similar to the previous one, however without the harsh geometrical constraints that force crack-driving tension. Instead, we consider a disk B = {X ∈
R∈ | ||X|| ≤ ∇} with r = 2 mm, that is subject to mechanical Dirichlet boundary conditions allowing for unconstrained homogeneous shrinking: nodes on the
vertical and horizontal axes through the circle center are fixed horizontally and
vertically, respectively. Quadrilateral elements with bilinear shape functions for
all degrees have be chosen for simplicity. The chemical potential µ̄(t) > µ0 is
applied everywhere on ∂B and again linearly increasing starting from µ0 . Here,
a heavily swollen initial configuration is assumed by setting J0 = 7. While σc is
25
Figure 8: Geometrically unconstrained diffusion-driven crack pattern for two different discretizations at different instances of time. The instability-like crack initiation is clearly meshdependent. Contour plots for the chemical potential are shown with a range between µ0 (J0 )
(red) and µ̄ (blue), illustrating the homogeneous initial and end states of the disk. Again,
elements with d > 0.95 are blanked out for the sake of crack pattern visibility.
0.00
−0.50
−1.00
µ [MPa]
−1.50
−2.00
−2.50
−1.0
−0.5
0.0
0.5
1.0
X1 /r [-]
Figure 9: Profile of the chemical potential for nodes with X2 ≈ 0 and at 0 s, 25 s, 50 s, 75 s,
90 s, 95 s, 100 s, 105 s, 115 s and 150 s (top to bottom). It starts at µ0 and increases at the
outer edges as prescribed by boundary conditions until a homogeneous state is reached. The
gradient of µ causes the circle to shrink.
26
0.1 N/mm2 and M = 10−2 mm4 /N s, all other material parameters can be found
in Table 1.
The volume decrease induces pressure in the inner and tension in the outer
regions of the disk. As soon as the tension reaches a critical value, radial crack
initiation occurs at various locations at the surface. Again, the onset of crack
growth is an instability-like mechanism, and thus the simulation turned out to
be mesh-dependent, which is illustrated in Fig. 8 for two different discretization. In this plot, the chemical potential is depicted as a contour plot, showing
the gradient from outer to inner regions as the non-equilibrium state variable
responsible for shrinking. At the end (t ≈1500 s), the hydrogel disk reaches a homogeneous state. Crack evolution however stopped earlier, because the critical
stress was not reached anymore. This boundary value problem illustrates that
the possibly large volume change characteristic to hydrogels can not only lead
to the regularly studied instabilities in the presence of geometrical constraints,
but also to crack initiation and growth in unconstrained specimen.
6. Conclusion
We outlined a thermodynamically consistent variational formulation of phasefield fracture in elastic solids undergoing diffusion-induced volume change and
implemented a specific material law for hydrogels. A minimization as well as a
saddle-point formulation has been presented and studied with three boundary
value problems. While the solution scheme based upon an operator split turned
out to be robust and reliable, the finite element shape functions still were chosen with ease of implementation and computational efficiency in mind, leaving
conforming FE spaces and those known to be more adequate with respect to the
LBB condition for future work. Additionally, the extension to three dimensions
is desirable, among other things for a more precise validation and adjustment
of the model by closer comparisons to experimental results. For the time being,
we could present the successful computational modeling of crack evolution in
hydrogels, which may serve as a starting point for predictive failure analysis of
27
this seminal class of materials, that will be employed increasingly often as the
promising functional material in biomedical applications.
28
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