# Multi-Field Models of Fiber Reinforced Concrete for Structural Applications
## 4 Electro-Mechanical Analyses with Applications to Self-Sensing
### Verification of Conductivity Simulations
In Figure 8b, the simulated conductivity values have been normalized by the corresponding EIM values, \(\sigma_{EIM}\), such that the entire set of results can be plotted together. These results indicate that there is correlation between the degree of fiber alignment in the \(x\)-direction and the conductivity values. The degree of alignment is expressed by the average of \(\cos\alpha\), where \(\alpha\) is the angle between each fiber axis and the \(x\)-direction. As expected, preferential alignment produces higher conductivity values. The differing random realizations of the fiber-matrix pathways for conductive flow may also contribute to the scatter in the conductivity values. The model capabilities are potentially useful in understanding the effects of alignment bias due to the production/casting processes and container boundaries. Non-uniform or poorly aligned fiber distributions may strongly influence the fracture properties of fiber-reinforced concrete [57, 60, 61, 72]. The severity of alignment bias on composite conductivity has not been examined herein, since all fibers distributions are, in a nominal sense, uniformly at random. The same set of parameters is used for simulating composite conductivity using the proposed semi-discrete modeling approach. Regardless of the fiber dosage, the number of computational nodes representing the composite material, \(n_{c}\), does not change and is equal to \(n_{b}\) (\(\approx\)16,000), which is the number of nodes defining the background lattice representing the matrix material. This is in contrast to the discrete approach for which computational cost grows with fiber dosage, as shown in Figure 9. For perspective, nodal counts for a case of a higher fiber aspect ratio are included in the figure. It is apparent that the full volumetric discretization of the fibers becomes cost prohibitive as the volume fraction or aspect ratio of the fibers increases. In other words, the proposed semi-discrete approach has a clear advantage with respect to its lower computational complexity. For the parameter set and maximum fiber dosages considered herein, \(n_{c}/n_{b}\) = 7.8, which is nearly an order of magnitude difference in the number of nodes. As seen in Figure 10, similarly good agreement with the EIM curve is observed for lower \(V_{f}\) values, but the model results tend to drift upward as \(V_{f}\) grows. At lower \(V_{f}\) values, the background mesh is sufficiently fine such that the fibers are electrically isolated from one another. In other words, fiber elemental contributions to diffusivity (as per Equation (12)) are introduced to different matrix nodes, such that any two fibers are effectively separated by matrix material. The discrete approach satisfies this condition by design of the fiber cross-section, as shown in Figure 2b. The semi-discrete fibers could remain electrically isolated for much larger \(V_{f}\) provided the density of nodes representing  
Figure 8: (**a**) Composite conductivity values based on discrete (volumetric) fiber models. Seven nominally identical, randomly generated models were tested for each \(V_{f}\) value; and (**b**) dependence of simulated conductivity measurements on the orientation of fibers with respect to the direction of the electric field. the matrix was increased accordingly. However, the computational efficiency advantages of the semi-discrete approach would then be lessened. Nonetheless, these simulations verify accuracy of the semi-discrete approach for modeling the transport behavior of fibers. Notably, the semi-discrete approach also captures the influence of directional bias of the fibers on composite conductivity, as shown in Figure 10b. In a separate study, the mechanical analogue of this fiber model was used to study the potential influences of fiber distribution non-uniformity within steel-fiber reinforced concrete slabs [61]. The modeling approach captured the salient influences of fiber amount and orientation on the fracture behavior of beam samples extracted from the slabs.