# Multi-Field Models of Fiber Reinforced Concrete for Structural Applications
## 2 Voronoi-Cell Lattice Model
### Mechanical Behavior  
The mechanical behavior of the concrete is represented by a Rigid-Body-Spring Network (RBSN) [22; 23; 24], which is akin to other lattice models of concrete [25; 26; 27; 28; 29; 30; 31; 32]. This section briefly describes its definition, with additional details given elsewhere [20]. Homogeneous regions of the material body are typically represented by randomly placed nodes, subject only to a minimum distance constraint. To construct material features within the domain, subsets of nodes are sequentially placed to define the vertices, edges, and faces of the desired features, prior to random filling of the feature volumes [33]. The RBSN is an assembly of rigid-body-spring elements. Each element is composed of a zero-size spring set, located at the centroid \(C\) of the corresponding Voronoi facet, connected to the nodes via rigid-arm constraints (Figure 1b). As mentioned above, element connectivity is determined by the Delaunay tessellation of the nodes; the dual Voronoi tessellation provides geometric measures that are used to determine element properties. With respect to the \(n-s-t\) coordinate axes, defined local to Voronoi facet \(ij\) (Figure 1b), the stiffnesses of the axial and rotational springs composing the spring set are  
\[k_{n}=\beta k_{s}=\beta k_{t}=E\frac{A_{ij}}{h_{ij}} \tag{1}\]  
\[k_{\phi n}=E\frac{I_{p}}{h_{ij}};\;\;k_{\phi s}=E\frac{I_{s}}{h_{ij}};\;\;k_{ \phi t}=E\frac{I_{t}}{h_{ij}} \tag{2}\]  
where \(E\) is the elastic modulus of the material; \(A_{ij}\) is the area of Voronoi facet; \(h_{ij}\) is the distance between \(i\) and \(j\); \(I_{s}\) and \(I_{t}\) are the second moments of the facet area about axes \(s\) and \(t\), respectively; \(J_{p}\) is the polar second moment of area of the facet; and \(\beta\) is a factor related to macroscopic Poisson ratio \(\
u\). Herein, \(\beta=1\) renders the lattice elastically uniform, albeit with \(\
u=0\)[34]. Complete representation of the Poisson effect, both macroscopically and in an element-local sense, has recently been accomplished using an iterative procedure [35]. The spring coefficients of Equations (1) and (2) form the diagonal of the material matrix  
\[\mathbf{D}=(1-\omega)\text{diag}\big{[}k_{n},k_{s},k_{t},k_{\phi n},k_{\phi s },k_{\phi t}\big{]} \tag{3}\]  
where \(0\leq\omega\leq 1\) is a scalar damage parameter used to model material fracture. Prior to fracture initiation, \(\omega=0\). The element stiffness matrix (with respect to element local coordinates) is  
\[\mathbf{K_{e}}=\mathbf{B^{T}}\mathbf{D}\mathbf{B} \tag{4}\]where **B** relates the spring generalized displacements to the nodal degrees of freedom of the element [22]  
\[\textbf{B}= \begin{bmatrix}-\textbf{I}&\textbf{B}_{\textbf{12}}&\textbf{I}&\textbf {B}_{\textbf{14}}\\ \textbf{0}&-\textbf{I}&\textbf{0}&-\textbf{I}\end{bmatrix}  \tag{5}\]  
in which submatrices \(\textbf{B}_{\textbf{12}}\) and \(\textbf{B}_{\textbf{14}}\) are defined by  
\[\textbf{B}_{\textbf{12}}= \begin{bmatrix}0&-z_{c}&y_{c}\\ z_{c}&0&-h/2\\ -y_{c}&h/2&0\end{bmatrix} ;\ \ \textbf{B}_{\textbf{14}}= \begin{bmatrix}0&z_{c}&-y_{c}\\ -z_{c}&0&-h/2\\ y_{c}&-h/2&0\end{bmatrix}  \tag{6}\]  
Recently, this elemental formation has been extended to accommodate large displacement analyses [36,37]. After transforming \(\textbf{K}_{\textbf{e}}\) to global coordinates, the direct stiffness approach is used to assemble element stiffness matrices and internal force contributions into the structural equation set  
\[\textbf{K}\Delta\textbf{u}=\textbf{R}+\textbf{F} \tag{7}\]  
in which \(\Delta\textbf{u}\) is the increment in generalized nodal displacements; **K** is the system stiffness matrix assembled from the elemental components; **R** and **F** are the external and internal nodal force vectors, respectively. One advantage of this form of discrete model is in its ability to simulate tensile fracture with reduced mesh bias. In particular, implementations of the crack band model of [PERSON] et al.