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 183 184<!-- Local IspellDict: english --> 185