1% Phase field approach of brittle fracture with the `Cast3M` finite element solver
2% T. Helfer
3% 13/01/2017
4
5\newcommand{\Frac}[2]{\displaystyle\frac{\displaystyle #1}{\displaystyle #2}}
6\newcommand{\deriv}[2]{\Frac{\partial #1}{\partial #2}}
7\newcommand{\Psiel}{\Psi^{\mathrm{el}}}
8\newcommand{\pPsiel}{\Psi^{\mathrm{el}\,+}}
9\newcommand{\nPsiel}{\Psi^{\mathrm{el}\,-}}
10\newcommand{\tenseur}[1]{\underline{#1}}
11\newcommand{\tsigma}{\underline{\sigma}}
12\newcommand{\tepsilonel}{\underline{\epsilon}^{\mathrm{el}}}
13\newcommand{\trace}[1]{{\mathrm{tr}\paren{#1}}}
14\newcommand{\paren}[1]{\left(#1\right)}
15\newcommand{\ppart}[1]{\left<#1\right>_{+}}
16\newcommand{\npart}[1]{\left<#1\right>_{-}}
17\newcommand{\dtot}[1]{\mathrm{d}}
18
19This article describes how to implement the phase field approach using
20`MFront` and the `Cast3M` finite element solver.
21
22# Description of the phase field approach
23
24The physical idea underlying the phase field approach of brittle
25fracture dates back to the pioneering work of Francfort and Marigo
26(see @francfort_revisiting_1998). Bourdin et al. were the first to
27propose a numerical implementation of the approach by regularizing the
28cracks (see @bourdin_numerical_2000).
29
30The description of the phase field approach used here relies on the
31work of Miehe et al. (see @miehe_phase_2010)
32
33## Defining a volumic energy associated to fracture
34
35The crack network \(\Gamma\) contained in a body \(B\) can be
36regularized by the following operator defining a damage field \(d\):
37
38\[
39  \left\{
40    \begin{aligned}
41      d(x)-l^2 \triangle d(x) &= 0 \ \text{on} \ B \\
42      d(x) &= 1 \ \text{on} \ \Gamma \\
43      \nabla d(x).n &= 0 \ \text{on} \ \partial B
44    \end{aligned}
45  \right.
46\]
47
48where (\partial B\) the boundary of \(B\). This operator introduces a
49characteristic length \(l\).
50
51The crack surface \(\Gamma\) can be computed by the following
52integral:
53
54\[
55\Gamma\paren{d}=\int_{B}\gamma\paren{d}\,\dtot\,V
56\]
57
58where \(\gamma (d)=\frac{d^{2}}{2\,l}+\frac{l}{2}\,\vec{\nabla} d\,\cdot\,\vec{\nabla}{d}\)
59
60Let \(g_{c}\) the energy associated with the creation of a unit
61surface of crack. The amount of energy \(E_{c}\) that was required to
62create the crack network is:
63
64\[
65E_{c}=g_{c}\,\int_{B}\gamma\paren{d}\,\dtot\,V=\int_{B}g_{c}\,\gamma\paren{d}\,\dtot\,V
66\]
67
68\(g_{c}\,\gamma\paren{d}\) is an energy *per unit of volume*
69associated with the creation of crack network. This quantity is the
70fundamental brick when building a thermodynamically consistent
71approach to the phase field modelling of brittle failure.
72
73In pratice, the crack network is an unknown of the mechanical problem,
74so the idea of the phase field approach is to introduce the damage
75field \(d\) and add \(g_{c}\,\gamma\paren{d}\) in the free energy of
76the material. Application of the Claussis-Duhem inegality then leads
77to an explicit equation satisfied by the damage field, as described
78below.
79
80## Choice of the free energy
81
82To motivate the choice of the free energy for the phase-field
83behaviour, we begin to recall some classical results about the
84decomposition of the free energy of an isotropic elastic material into
85a positive and negative parts.
86
87### Decomposition in positive and negative parts of the free energy of an isotropic elastic material
88
89The free energy \(\Psiel\) of an isotropic elastic material is
90given by:
91
92\[
93\Psiel=\lambda\,\trace{\tepsilonel}^{2}+2\,\mu\,\tepsilonel\,\colon\,\tepsilonel
94\]
95
96where \(\lambda\) and \(\mu\) are the Lamé coefficients of the
97material.
98
99This free energy can be decomposed in a positive part associated with
100tension and a negative part associated with compression:
101
102\[
103\Psiel=\pPsiel+\nPsiel
104\]
105
106where \(\pPsiel\) and \(\nPsiel\) are defined by:
107\[
108\left\{
109\begin{aligned}
110\pPsiel &= 2\,\mu\,\ppart{\tepsilonel}\,\colon\,\ppart{\tepsilonel}+\lambda\,\ppart{\trace{\tepsilonel}}^{2}\\
111\nPsiel &= 2\,\mu\,\npart{\tepsilonel}\,\colon\,\npart{\tepsilonel}+\lambda\,\npart{\trace{\tepsilonel}}^{2}
112\end{aligned}
113\right.
114\]
115
116In this expression, the positive part of a scalar is defined by:
117\[
118\ppart{x}=
119\left\{
120\begin{aligned}
121x & \text{ if } x>0 \\
1220 & \text{ if } x\leq0
123\end{aligned}
124\right.
125\]
126
127The positive part of a tensor is defined as an isotropic function as
128follows:
129\[
130\ppart{\tenseur{x}}=\sum_{i=1}^{3}\ppart{\lambda_{i}}\,\tenseur{n}_{i}
131\]
132
133## Definition of the free energy for the phase-field behaviour
134
135To take into choi The free energy \(\Psi\) is choosen to have the
136following form:
137
138\[
139\Psi=\paren{m(d) + k}\,\pPsiel{} + \nPsiel{}+g_{c}\,\gamma\paren{d}
140\]
141
142where:
143
144- \(m(d) = \paren{1-d}^2\) describe the effect of damage on the
145  positive part of the energy.
146- \(k\) is a small constant to ensure a minimal stiffness to the fully
147  damaged material.
148
149## Definition of the stress
150
151The stress is defined as the thermodynamic force associated with the elastic strain:
152
153\[
154\tsigma
155=\deriv{\Psi}{\tepsilonel}
156=\paren{m(d) + k}\,\paren{2\,\mu\,\ppart{\tepsilonel}+\lambda\,\ppart{\trace{\tepsilonel}}\,\tenseur{I}}+
1572\,\mu\,\npart{\tepsilonel}+\lambda\,\npart{\trace{\tepsilonel}}\,\tenseur{I}
158\]
159
160## Thermodynamic force associated with the damage
161
162Inspired by thermodynamic arguments, namely the respect of the
163Clausis-Duhem inegality, Miehe introduces the following equation:
164
165\[
166	\frac{g_{c}}{l}\left[d - l^2 \triangle d\right] = 2\,(1 - d)\,H
167\]
168
169where
170\[
171  H(x,t) = \max_{\tau \in [0,t]}\left[\pPsiel\paren{\tau}\right]
172\]
173
174This definition of \(H\), which appears as the driving force of the
175damage evolution, has been introduced to described the irreversibility
176of the crack propagation, altough the demonstration that this choice
177leads to an increasing value of the damage is not straightforward.
178
179# Implementation
180
181
182# References
183
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