# Multi-Field Models of Fiber Reinforced Concrete for Structural Applications
## 4 Electro-Mechanical Analyses with Applications to Self-Sensing
### Verification of Conductivity Simulations  
As primary conditions for simulating electrical behavior of fiber-reinforced composites, the model should account for the amount, aspect ratio, and orientation of the fibers, as well as the differing conductivities of the fiber and matrix materials. Effective medium theories have been useful in estimating fiber composite properties (including conductivity) when basic requirements are met regarding the orientation distribution of the fibers. In this study, the equivalent inclusion method (EIM) of [PERSON] and [PERSON] [14] provides benchmark values for verifying the lattice models. Whereas the EIM was developed for heat conduc  
Figure 5: Profile view of fiber distribution within the 3D unit cell. Figure 6: Flow through a cluster of conductive fibers embedded within a lowly conductive matrix. The fiber aspect ratio \(l_{f}/d_{f}=30\). tion problems, it is analogous to [PERSON]'s equivalent inclusion method in elasticity [69]. The EIM applies to potential flow problems, in general, including the electric conductivity simulations presented herein. The expression for determining composite conductivity is given in Appendix A. For the assumed model parameters, Figure 7 compares estimates of composite conductivity according to the EIM of Hatta and Taya with bounds of the Hashin-Shtrikman solution [70]. To verify the lattice models, a potential difference \(\Delta\phi=1\) is applied across the unit cell shown in Figure 5. As previously mentioned, the potential is assigned to nodal arrays along each surface normal to the \(x\)-direction, separated by distance \(L_{e}\), according to \(\phi(x)=\Delta\phi x/L\). Fibers were not placed close to the boundaries, which facilitated domain discretization and the assignment of nodal potentials. The influence of this condition on the conductivity simulations, which appears to be secondary, was modeled using parallel and series constructions of the regions with and without fibers [71]. Conductivity of the composite material, \(\sigma\), was determined using  
\[J=-\sigma\frac{\phi(L)-\phi(0)}{L} \tag{14}\]  
where \(J\) is the average current density and \(\phi(L)-\phi(0)\) is the potential difference between the faces of the unit cell. Average current density was found by summing nodal flux values over either of the \(x\)-faces, and dividing by the face area. The procedure for calculating nodal flux values is outlined in [20]. Results of the discrete (volume element) approach to modeling fibers are compared with the EIM values in Figure 8, for small volume fractions of fibers up to \(V_{f}=0.01\). The limitation on fiber dosage is due, in part, to the computational expense of models with larger volume fractions. For higher fiber contents, contact between fibers is difficult to avoid, along with the eventual formation of percolation clusters. Seven numerical specimens, differing only in their random distributions of fibers, were produced for each fiber volume fraction considered. Several observations can be made regarding the conductivity values plotted in Figure 8. The lattice model results exhibit the expected trend: conductivity of the composite material increases as the dosage of conductive fibers increases. The mean values given by the lattice model agree well with those of the EIM of Hatta and Taya. In a separate study, experimental measurements of AC composite conductivity of steel-fiber reinforced cementitious materials agreed well with the EIM, when some degree of preferential alignment of the fibers was considered [13]. Figure 7: Relationship between fiber content and conductivity of the composite material according to two effective medium theories (for conductivity ratio \(\sigma_{f}/\sigma_{m}=1000\) and aspect ratio \(l_{f}/d_{f}=50\)). The simulated conductivity values exhibit scatter about the respective mean values. The error bars in Figure 8a indicate one standard deviation from the mean value.