## 4 Basic Statements of the Regularized XFEM Formulation  
We consider a regularized XFEM developed for strain localization and cohesive interfaces [13, 17, 27]. The displacement field is approximated by enriching the space of the shape functions with additional functions that fulfill the partition of unity property of the standard finite element method. The enrichment functions consist of the product of the Heaviside function by the standard shape functions \(\mathbf{N}\)  
\[\mathbf{u}(\mathbf{x})=\mathbf{N}(\mathbf{x})\mathbf{V}+\mathcal{H}(d( \mathbf{x}))\mathbf{N}(\mathbf{x})\mathbf{J} \tag{25}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\varphi(\xi)\) & \(\hat{\varphi}(\kappa)\) & \(\hat{\varphi}(0)\) & \(\dfrac{\partial\hat{\varphi}}{\partial\kappa}\bigg{|}_{\kappa=0}\) & \(\dfrac{\partial^{2}\hat{\varphi}}{\partial\kappa^{2}}\bigg{|}_{\kappa=0}\) \\ \hline \(\tilde{\varphi}_{L}=\frac{1}{2}\,e^{-|\xi|}\) & \(\dfrac{1}{1+4\,\pi^{2}\,\kappa^{2}}\) & 1 & 0 & \(\
eq 0\) \\ \(\tilde{\varphi}_{G}=2\,\sqrt{\pi}e^{-4\pi^{2}\xi^{2}}\) & \(e^{\frac{-\kappa^{2}}{4}}\) & 1 & 0 & \(\
eq 0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Fourier transforms of two weight functions. In Equation (25), \(\mathbf{V}\) is the vector containing the degrees of freedom of the standard part of the displacement field \(\mathbf{v}\), while \(\mathbf{J}\) collects the enriched degrees of freedom related to the displacement jump across the interface. The strain field is computed via compatibility from Equation (25) as  
\[\boldsymbol{\varepsilon}(\mathbf{x})=\mathbf{B}(\mathbf{x})\mathbf{V}+\mathcal{ H}(d(\mathbf{x}))\mathbf{B}(\mathbf{x})\mathbf{J}+\delta(d(\mathbf{x}))\,\bar{ \mathbf{N}}(\mathbf{x})\mathbf{J} \tag{26}\]  
where \(\mathbf{B}\) is the compatibility matrix, \(\bar{\mathbf{N}}\) is related to the term in Equation (3) containing the (generalized) gradient \(\
abla\mathcal{H}=\delta\mathbf{n}\). Following the methodology proposed in [13, 17] regularization is carried out. The regularized displacement and strain fields become  
\[\mathbf{u}_{\rho}(\mathbf{x}) =\mathbf{N}(\mathbf{x})\mathbf{V}+\mathcal{H}_{\rho}(d(\mathbf{x} ))\mathbf{N}(\mathbf{x})\mathbf{J} \tag{27a}\] \[\boldsymbol{\varepsilon}_{\rho}(\mathbf{x}) =\boldsymbol{\varepsilon}_{a}(\mathbf{x})+\boldsymbol{ \varepsilon}_{b}(\mathbf{x}) \tag{27b}\]  
where  
\[\boldsymbol{\varepsilon}_{a}(\mathbf{x}) =\mathbf{B}(\mathbf{x})\mathbf{V}+\mathcal{H}_{\rho}(d(\mathbf{x }))\mathbf{B}(\mathbf{x})\mathbf{J} \tag{28a}\] \[\boldsymbol{\varepsilon}_{b}(\mathbf{x}) =\delta_{\rho}(d(\mathbf{x}))\,\bar{\mathbf{N}}(\mathbf{x}) \mathbf{J} \tag{28b}\]  
Let the material occupying volume \(V\) be elastic with Young's modulus \(\mathbf{E}\). The regularized interface is characterized by the Young's modulus by unit length \(\bar{\mathbf{E}}/\delta_{\rho}=\mathbf{E}/(\beta\delta_{\rho})\), where \(\beta<1\) and \(\beta>1\) mean that the regularized interface is weaker or stronger, respectively, than the bulk, while for \(\beta=1\) the Young's moduli of the interface and the bulk coincide. The dimensional consistency and the motivations for assuming \(\bar{\mathbf{E}}\) in this way were thoroughly discussed in [17]. Let us omit the spatial dependence.