# Discrete Lattice Element Model for Fracture Propagation with Improved Elastic Response
## 2 Novel Discrete Lattice Element Model
### Kinematics of Mechanical Lattice Elements  
The formulation of our generalized beam lattice elements is based on the kinematics of rigid-body spring models, where rigid material particles, usually obtained by Voronoi tessellations, are connected by links, as shown in Figure 2. In the rigid-body spring model, each link may be viewed as comprising discrete springs positioned at the midpoint of the edge shared by two adjacent Voronoi cells (point P in Figure 2), representing interacting rigid particles, and two rigid links joined by these springs (segments 1-P and 2-P in Figure 2). The generalized beam lattice element is defined by nodes 1 and 2 and its local coordinate frame x-y. The stiffness of the rigid-body spring in Figure 2 is expressed using elastic constants rather than spring constants. This choice is made to ensure the equivalence of the models. Given the displacement and rotation of the particle \(I=1,2\), \(\mathbf{u}_{I}= \begin{bmatrix}u_{I}&v_{I}&\theta_{I}\end{bmatrix} ^{T}\), the displacement at an arbitrary point (\(x\),\(y\)) of each particle \(I\) is given as  
\[ \begin{array}{l}u=u_{I}-(y-y_{I})\theta_{I},\\ v=v_{I}+(x-x_{I})\theta_{I},\,I=1,\,2\end{array}  \tag{1}\]  
The nodal displacement vectors associated with particles \(\mathbf{u}_{I}\) are defined with respect to the local coordinate axes of the lattice element, shown in Figure 2, whereby nodes are usually placed at the nuclei of the Voronoi diagram [10,11]. For each generalized beam element, generalized strains \(\mathbf{\epsilon}= \begin{bmatrix}\epsilon&\gamma&\kappa\end{bmatrix} ^{T}\) may be defined in the local coordinate system  
Figure 1: (**a**) Structure of discrete lattice model with Voronoi cells as particles of material and lattice elements. (**b**) The proposed enhanced beam element connecting neighboring Voronoi cells. Parameter \(h\) is extracted from the Voronoi diagram and represents the height of the beam cross section. Figure 2: Discrete lattice element model composed of two rigid particles 1 and 2 linked with generalized beam element (red) or rigid-body spring (blue). (Figure 2), with \(\epsilon\), \(\gamma\), and \(\kappa\) corresponding to the normal strain, transversal shear strain, and beam curvature, respectively. They can be computed from the vector of element nodal displacements \(\mathbf{u}= \begin{bmatrix}u_{1}&v_{1}&\theta_{1}&u_{2}&v_{2}&\theta_{2}\end{bmatrix} ^ {T}\) as  
\[\mathbf{\epsilon}=\mathbf{B}\mathbf{u}, \tag{2}\]  
where matrix \(\mathbf{B}\) represents the strain-displacement matrix in the local coordinate form  
\[\mathbf{B}=\frac{1}{l_{e}} \begin{bmatrix}-1&0&y_{p}&1&0&-y_{p}\\ 0&-1&-l_{e}/2&0&1&-l_{e}/2\\ 0&0&-1&0&0&1\end{bmatrix} . \tag{3}\]  
The generalized beam lattice element represents an extension of the linear Timoshenko beam element used in discrete lattice models [32; 33; 34; 35; 36]. A closer inspection and comparison with the corresponding expressions obtained for a linear Timoshenko beam element shows that the curvatures of these two elements are identical, while the shear strain is equal to the value computed at the sample point of a Timoshenko element with selective reduced integration (the Gaussian point positioned at the middle of the element). The only significant difference is encountered in the normal strains (see non-zero terms \(y_{p}\)), arising because here the normal strain is associated with the normal spring force acting at point P, which is in a general case shifted from the straight line connecting nodes by \(y_{p}\) (see Figure 2). Consequently, these two elements possess different stiffness matrices, which become identical only when point P aligns with the local coordinate axis \(x\), as shown in [10]. Motivated by these similarities, in the following, we depart from the kinematics of the rigid-body spring model and consider the links between two adjacent nodes (particles) as a generalized beam model, whose kinematics is defined by Equations (2) and (3).