## 2 Thick Level Set Damage Model
### Energetic form and Regularization  
The system of Equations (2) can be alternatively expressed using the energetic formulation.  
If \(P\) denotes the potential energy, the variational form is obtained by minimizing \(P\) with respect to \(\delta u\) and \(\delta\Phi\):  
\[P(u,\Phi)=\int_{\Omega}W(\varepsilon(u),\dot{d}(\Phi))d\Omega-\int_{\partial \Omega_{u}}f.ud\partial\Omega \tag{6}\]  
Using the relation \(\delta W=\frac{\partial W}{\partial\varepsilon}\delta\varepsilon+\frac{ \partial W}{\partial d}\delta d\), the potential energy \(P\) can be differentiated as follows:  
\[DP(u,\Phi)[(\delta u,\delta\Phi)]=\int_{\Omega}\sigma:\varepsilon(\delta u)d \Omega-\int_{\partial\Omega_{u}}f.\delta ud\partial\Omega-\int_{\Omega}Yd^{ \prime}(\Phi)\delta\Phi d\Omega \tag{7}\]  
The equilibrium equation is obtained by zeroing (7) with respect to \((\delta u,0)\):  
\[DP(u,\Phi)[(\delta u,0)]=\int_{\Omega}\sigma:\varepsilon(\delta u)d\Omega- \int_{\partial\Omega_{u}}f.\delta ud\partial\Omega=0 \tag{8}\]  
The amount of dissipated energy as the front moves is obtained by analyzing (7) for \((0,\delta\Phi)\):  
\[DP(u,\Phi)[(0,\delta\Phi)]=-\int_{\Omega}Yd^{\prime}(\Phi)\delta\Phi d\Omega \tag{9}\]  
The variational problem defined above is known to face issues such as spurious mesh dependencies. The latter may provoke damage growth with negligible energy. Regularization methods are used to prevent such problems.  
The TLS model, by definition (1), is a regularization method using the characteristic length \(l_{c}\) to bound the damage evolution to the band limited by \(\Phi=l_{c}\). In addition and since the damage derivative \(d^{\prime}(\Phi)\) is non-zero only in the damaged zone (i.e., \(0<\Phi\leq l_{c}\)), the integration is limited to the damaged band unlike other non-local classical damage models. Thus, the regularization and the \"non-local\" treatment of the TLS are activated only on this damaged band.  
In addition, a change of variables can be introduced. The dissipated energy in the new basis \((\Phi,s)\) is hence expressed as:  
\[DP(u,\Phi)[(0,\delta\Phi)]=-\int_{\Gamma_{0}}\int_{0}^{0\leq l\leq l_{c}}Y( \Phi,s)d^{\prime}(\Phi)\left(1-\Phi/\rho(s)\right)d\Phi\delta\Phi ds \tag{10}\]  
where \(s\) and \(\rho\) are respectively the curvilinear coordinate and the curvature radius along \(\Gamma_{0}\). The latter is the boundary corresponding to the zero-isovalue of \(\Phi\).  
Damage growth is then activated when:  
\[\int_{\Gamma_{0}}\int_{0}^{0\leq l\leq l_{c}}Y(\Phi,s)d^{\prime}(\Phi)\left(1- \Phi/\rho(s)\right)d\Phi\delta\Phi d\mathrm{s}=\int_{\Gamma_{0}}\int_{0}^{0\leq l \leq l_{c}}Y_{c}(\Phi,s)d^{\prime}(\Phi)\left(1-\Phi/\rho(s)\right)d\Phi\delta \Phi d\mathrm{s} \tag{11}\]  
Regarding the regularization, the authors of [22] propose averaging \(Y\) and \(Y_{c}\) by introducing the homogenized quantities \(\bar{Y}\) and \(\bar{Y}_{c}\):  
\[\bar{Y}=\frac{\int_{0}^{0\leq l\leq l_{c}}Y(\Phi,s)d^{\prime}(\Phi)(1-\Phi/ \rho(s))d\Phi}{\int_{0}^{0\leq l\leq l_{c}}d^{\prime}(\Phi)(1-\Phi/\rho(s))d \Phi}\quad\bar{Y}_{c}=\frac{\int_{0}^{0\leq l\leq l_{c}}Y_{c}(\Phi,s)d^{ \prime}(\Phi)(1-\Phi/\rho(s))d\Phi}{\int_{0}^{0\leq l\leq l_{c}}d^{\prime}( \Phi)(1-\Phi/\rho(s))d\Phi} \tag{12}\]  
Note that if \(Y_{c}\) is uniform all over the domain and independent of \(d\); \(\bar{Y}_{c}=Y_{c}\). The new homogenized quantities fulfill the propagation condition; the regularized front evolution reads:  
\[\bar{Y}-\bar{Y}_{c}\leq 0;\quad\dot{a}\geq 0;\quad(\bar{Y}-\bar{Y}_{c})\dot{a}=0 \tag{13}\]  
where \(a\) denotes the front advance to be computed as described in Section 2.4, Figure 2.  
Adopting this strategy means that non-locality appears only when \(d>0\) and that the non-local model tends to the local model when \(l_{c}\) tends to zero:  
\[\lim_{l_{c}\to 0}\bar{Y}=Y\quad\text{and}\quad\lim_{l_{c}\to 0}\bar{Y}_{c}=Y_{c} \tag{14}\]