# Feedback-Forward Neural Networks for Failure Mechanics Problems
## 4 Neural Network (NN) Based Elasticity for Fracture Problems
### Phase-Field Modeling of Brittle Fracture  
In this section, we summarize the variational formulations for phase-field modeling of brittle fracture in elastic solids at small strains. The constitutive work density function consists of the following sum:  
\[\Psi(\varepsilon,d,\
abla d)=g(d)\ \psi(\varepsilon)+[1-g(d)]\psi_{c}+2\frac{ \psi_{c}}{\zeta}\ l\ \gamma(d,\
abla d)\, \tag{19}\]  
of a degrading elastic bulk energy \(\psi\) depicted in (13) and a contribution due to fracture which represents the accumulated dissipative energy. Hereby, the crack phase-field \(d(\mathbf{x},t)=0\) represents the unbroken state of the solid and \(d(\mathbf{x},t)=1\) refers to the fully fractured state. The function \(g(d)=(1-d)^{2}\) models the degradation of the stored elastic energy of the solid due to fracture. The crack surface density function is defined as \(\gamma(d,\
abla d)=\frac{1}{2l}d^{2}+\frac{l}{2}|\
abla d|^{2}\) in terms of the fracture length scale \(l\) that governs the regularization. The formulation (19) depends on two additional fracture parameters, namely, the critical fracture energy \(\psi_{c}\) and \(\zeta\), which controls the post-critical range after crack initialization, as well documented in [61]. Based on the above-introduced work density function, we derive two governing equations for the coupled problem. The first equation is the stress equilibrium or the quasi-static form of the balance of linear momentum defined as follows:  
\[\text{Div}[\mathbf{\sigma}]=\mathbf{0}\quad\text{with}\quad\mathbf{\sigma}=\partial_{ \varepsilon}\Psi=g(d)\ \widetilde{\mathbf{\sigma}}\quad\text{and}\quad\widetilde{\mathbf{\sigma}}=\lambda\ \text{tr}[ \varepsilon]\ \mathbf{1}+2\ \mu\ \varepsilon \tag{20}\]  
in terms of the effective stress tensor \(\widetilde{\mathbf{\sigma}}\) and by neglecting volume forces. Following [79], the evolution of the crack phase-field in the domain \(\Omega\) represents the second governing equation as follows:  
\[[d-l^{2}\Delta d]+\eta d+(d-1)\mathcal{H}=0\quad\text{with}\quad\mathcal{H}= \max_{s\in[0,l]}D(\mathbf{x},s)\geq 0\quad\text{and}\quad D=\zeta\Big{\langle} \frac{\psi^{+}(\varepsilon^{+})}{\psi_{c}}-1\Big{\rangle}_{+} \tag{21}\]  
along with its homogeneous Neumann boundary condition \(\
abla d\cdot\mathbf{n}=0\) on \(\partial\Omega\). Here, \(\mathbf{n}\) is the outward normal on \(\partial\Omega\) and \(\eta\geq 0\) is a material parameter that characterizes the artificial/numerical viscosity of the crack propagation. The crack driving force \(\mathcal{H}\) is introduced as a local history variable that accounts for the irreversibility of the phase-field evolution by filtering out a maximum value of what is known as the crack driving state function \(D\). This is achieved by introducing the Macaulay bracket \(\langle x\rangle_{+}=(x+|x|)/2\). Note that only the tensile/positive part of the _elastic_ energy in (13) is considered for computing the crack driving force. It is defined in terms of the positive strain tensor \(\varepsilon^{+}=\sum_{a=1}^{\delta}\langle\varepsilon_{a}\rangle_{+}\ N_{a}\otimes N_{a}\) with \(\delta=2,3\). Here, \(\{\varepsilon_{a}\}_{a=1..\delta}\) are the principal elastic strains and \(\{\mathbf{N}_{a}\}_{a=1..\delta}\) are the principal strain directions; for further details on energy decomposition, see [56].