## 2 LEFM Problem Statement  
To formulate the linear elastic fracture mechanics (LEFM) problem, we consider the two dimensional cracked domain \(\Omega\) shown in Figure 1. The boundary \(\Gamma\) consists of the parts \(\Gamma_{0}\), where free surface boundary conditions apply, \(\Gamma_{u}\), where displacements \(\bar{u}\) are prescribed and \(\Gamma_{t}\) where the surface tractions \(\bar{t}\) are applied as Neumann conditions. The domain includes a crack under the assumption of free surface conditions \(\Gamma_{c}\). As depicted in Figure 1, the domain boundary is decomposed as \(\Gamma=\Gamma_{0}\cup\Gamma_{u}  
where \(\mathbf{\sigma}\) is the Cauchy stress tensor, \(\mathbf{n}\) is the unit outward normal to the boundary, \(\mathbf{b}\) is the applied body force per unit volume, \(\mathbf{u}\) is the displacement field and \(\mathbf{\
abla}\) is the gradient operator.  
If small deformations are assumed, then the strain field \(\mathbf{\varepsilon}\) can be described as as the symmetric gradient of the displacement field:  
\[\mathbf{\varepsilon}=\mathbf{\
abla}_{s}\mathbf{u} \tag{2}\]  
Furthermore, if linear elastic material behavior is assumed, stresses can be obtained from strains through Hooke's law:  
\[\mathbf{\sigma}=\mathbf{D}:\mathbf{\varepsilon} \tag{3}\]  
where \(\mathbf{D}\) is the elasticity tensor, which in case of two dimensional problems assumes the following form:  
\[\mathbf{D}=\frac{E}{1-\
u^{2}}\left[\begin{array}{ccc}1&\
u&0\\ \
u&1&0\\ 0&0&\frac{1-\
u}{2}\end{array}\right],\quad\text{for plane stress} \tag{4a}\] \[\mathbf{D}=\frac{E}{\left(1+\
u\right)\left(1-2\
u\right)}\left[\begin{array}{ cccc}1-\
u&\
u&0\\ \
u&1-\
u&0\\ 0&0&\frac{1-2\
u}{2}\end{array}\right],\quad\text{for plane strain} \tag{4b}\]  
with \(E\) and \(\
u\) denoting Young's modulus and Poisson's ratio respectively.  
A decisive quantity in classic fracture mechanics [86] is the energy release rate, defined as:  
\[\mathcal{G}=-\frac{\partial\Pi}{\partial a} \tag{5}\]  
where \(\Pi\) is the total potential energy and \(a\) is the crack length (or area for three dimensional problems). Then, based on the energy release rate criterion, crack propagation will occur when:  
\[\mathcal{G}\geq\mathcal{G}_{c} \tag{6}\]  
where \(\mathcal{G}_{c}\) is the critical energy release rate or fracture toughness, which can be considered as a material parameter.  
For a pure mode I problem, the mode I stress intensity factor (SIF) is related to the energy release rate as follows:  
\[\mathcal{G}=\frac{K_{I}^{2}}{E^{\prime}} \tag{7}\]  
where \(K_{I}\) is the stress intensity factor and \(E^{\prime}\) is the effective elastic modulus:  
\[E^{\prime}=\left\{\begin{array}{cc}E&\text{for plane stress}\\ \frac{E}{1-\
u^{2}}&\text{for plane strain}\end{array}\right. \tag{8}\]  
Figure 1: Cracked Body and boundary conditions.  
Based on this, the critical stress intensity factor is defined as:  
\[K_{c}=\sqrt{E^{\prime}\mathcal{G}_{c}} \tag{9}\]  
The corresponding relation for mixed mode planar problems is:  
\[\mathcal{G}=\frac{1}{E^{\prime}}\left(K_{I}^{2}+K_{II}^{2}\right) \tag{10}\]  
where \(K_{II}\) is the mode II SIF. The square root of the quantity \(\left(K_{I}^{2}+K_{II}^{2}\right)\) can be considered as an equivalent SIF:  
\[K_{eq}=\sqrt{K_{I}^{2}+K_{II}^{2}} \tag{11}\]  
Then, the energy release rate criterion of Equation (6) can be written in terms of the SIFs as:  
\[K_{eq}\geq K_{c} \tag{12}\]  
Several criteria have been proposed to determine the direction of crack propagation, such as the maximum circumferential stress or maximum hoop-stress criterion [87], the maximum energy release rate criterion [88] and the minimum strain energy density criterion [89]. To what concerns the XFEM/SBFEM analyses carried out in this work, we adopt the latter criterion, which results in the following expression for the angle of crack propagation:  
\[\theta_{c}=2\tan^{-1}\left[\frac{-2K_{I}/K_{II}}{1+\sqrt{1+8(K_{I}/K_{II})^{2} }}\right] \tag{13}\]