# Multi-Field Models of Fiber Reinforced Concrete for Structural Applications
## 2 Voronoi-Cell Lattice Model
### Transport Behavior  
Flow behavior is simulated using the same lattice topology, based on the Delaunay tessellation of the nodal points. The lattice elements serve as conduits of one-dimensional flow between the element nodes [20]. The cross-section area of a given conduit element is equal to the area of the Voronoi facet associated with the element nodes (Figure 1c). From a nodal point perspective, the lattice accurately simulates uniform flow through homogeneous materials subjected to a potential difference (i.e., the random lattice design does not affect the condition of uniform flow). The lattice model accommodates both steady-state and transient solutions. In cases where crack-assisted flow is of interest, a dual  
Figure 1: Lattice model: (**a**) domain discretization based on Delaunay and Voronoi tessellations; (**b**) mechanical lattice element _ij_ based on the rigid-body-spring concept; and (**c**) conductive lattice element _ij_.  
lattice approach is effective [41; 42; 43; 44]. The field of displacements (for mechanical behavior) is represented at the Voronoi generator points, whereas the potential field is represented at the Voronoi vertices. Cracking is not considered in this study, so a conventional approach (where the mechanical and flow network nodes are coincident with the Voronoi generator points [20]) is employed.  
The case of moisture diffusion, which is used to describe the modeling approach, is analogous to the cases of electrical and hydraulic conductivity considered later. Established approaches are used for modeling moisture diffusion in cement composites, the convective boundary conditions at exposed surfaces, and the coupling of the diffusion and strain analyses [45]. For isothermal conditions, with no sinks or sources present, the governing equation for transient diffusion problems is  
\[\frac{\partial h}{\partial t}=\text{div}[D(h)\text{grad}h] \tag{8}\]  
where \(h\) is the pore relative humidity and \(D\) is a humidity dependent diffusion coefficient, which is a scalar quantity for isotropic conditions. The problems considered hereafter do not involve the dependence of \(D\) on the field quantity. The semi-discrete form of Equation (8) is  
\[\mathbf{M}\dot{\mathbf{H}}+\mathbf{K}\mathbf{H}=\mathbf{f} \tag{9}\]  
where \(\mathbf{H}\) is the vector of nodal relative humidities and the dot over \(\mathbf{H}\) indicates time derivative. The system matrices, \(\mathbf{M}\) and \(\mathbf{K}\), are assembled from the elemental capacity and diffusivity matrices, respectively:  
\[\mathbf{M_{e}}=\frac{h_{ij}A_{ij}}{6d}\left[\begin{array}{cc}2&1\\ 1&2\end{array}\right], \tag{10}\]  
\[\mathbf{K_{e}}=D\frac{A_{ij}}{h_{ij}}\left[\begin{array}{cc}1&-1\\ -1&1\end{array}\right] \tag{11}\]  
as would be done for the finite element analysis of potential flow problems using two-node elements [46]. In the elemental capacity matrix, \(d=1.0\), \(2.0\), and \(3.0\) for \(1\)D, \(2\)D, and \(3\)D networks, respectively. The ensuing discussions pertain to electrical or hydraulic conductivity, rather than diffusivity, but the equations representing potential flow take the same general form.