# Discrete Lattice Element Model for Fracture Propagation with Improved Elastic Response
## 2 Novel Discrete Lattice Element Model
### Failure Mechanics  
The failure mechanism of a lattice element is controlled through the incorporation of an embedded strong discontinuity [17; 18]. The discontinuity is placed in the middle of the generalized lattice beam finite element, corresponding to the position of the Gauss integration point and the location of the crack between the Voronoi cells. This involves introducing a jump in the displacement field along both the longitudinal and transversal directions of the element, achieved by applying a Heaviside function [32; 33; 34; 35; 36]. The formulation of the lattice element is expressed in the fashion of the finite element method, utilizing linear interpolation functions. This finite element formulation is implemented in the FEAP finite element code. The model is capable of representing both Mode I and Mode II failure mechanisms by introducing embedded discontinuities along both the longitudinal and transversal directions of the element. The Heaviside function is defined as \(H_{x_{p}}=0\) for \(x\leq x_{p}\) and \(H_{x_{p}}=1\) for \(x>x_{c}\). The vector \(\mathbf{\alpha}= \begin{bmatrix}\alpha_{u}&\alpha_{v}&0\end{bmatrix} ^{T}\) represents the jump in the displacement field at the position of discontinuity (in the middle of the element), as shown in Figure 3. The enhanced displacement field can be written as the sum of the regular and discontinuous (singular) part:  
\[\mathbf{u}(x)=\overline{\mathbf{u}}(x)+\mathbf{\alpha}H_{x_{p}}= \begin{bmatrix} \overline{u}(x)\\ \overline{v}(x)\\ \overline{\theta}(x)\end{bmatrix} + \begin{bmatrix}\alpha_{u}\\ \alpha_{u}\\ 0\end{bmatrix} H_{x_{c}}, \tag{11}\]  
The displacement field can be written in the fashion of finite element interpolations:  
\[\mathbf{u}(x)=N_{1}(x)\mathbf{u}_{1}+N_{2}(x)\mathbf{u}_{2}+\mathbf{\alpha} \underbrace{(H_{x_{c}}-N_{2}(x))}_{M(x)}, \tag{12}\]  
where the regular part is interpolated with the linear shape functions \(N_{1}(x)=1-\frac{x}{L}\), \(N_{2}(x)=\frac{x}{L}\) and \((H_{x_{c}}-N_{2}(x))\) represents the interpolation function \(M(x)\) in the fashion of the embedded strong discontinuity:  
\[M(x)= \begin{cases}-\frac{x}{L_{c}};x\;\in\;[0,\;x_{c})\\ 1-\frac{x}{L};\;x\;\in\;\langle x_{c},\;l_{e}\rangle\end{cases} . \tag{13}\]  
The discontinuity parameters \(\mathbf{\alpha}\) are computed locally. The enhanced displacement field can be written in matrix form as:  
\[\mathbf{u}=\mathbf{N}\mathbf{u}+\mathbf{M}\mathbf{\alpha}, \tag{14}\]  
where \(\mathbf{N}\) represents the matrix of interpolation functions \(N_{1}(x)\) and \(N_{2}(x)\) with the dimension \(3\times 6\), while the \(3\times 3\) matrix \(\mathbf{M}=\text{diag}[\mathbf{M}(\mathbf{x}),\mathbf{M}(\mathbf{x}),0]\) contains \(M(x)\) discontinuity functions. The enhanced deformation field can be written as the sum of regular \(\overline{\mathbf{\varepsilon}}(x)\) and singular part \(\mathbf{\alpha}\delta_{x_{c}}\)  
\[\mathbf{\epsilon}(x)=\overline{\mathbf{\varepsilon}}(x)+\mathbf{\alpha}\delta_{x_{c}}= \begin{bmatrix}\overline{\mathbf{\varepsilon}}(x)\\ \overline{\mathbf{\gamma}}(x)\\ \overline{\mathbf{\kappa}}(x)\end{bmatrix} + \begin{bmatrix}\alpha_{u}\\ \alpha_{v}\\ 0\end{bmatrix} \delta_{x_{c}}. \tag{15}\]The Dirac function \(\delta_{x_{c}}\) tends to infinity for \(x=x_{c}\); otherwise, it is zero.