# Multi-Field Models of Fiber Reinforced Concrete for Structural Applications
## 6 Conclusions  
This research has produced semi-discrete models of fiber-assisted transport in cement-based materials. Voronoi cell lattice models (VCLM) serve as a framework for the proposed developments. The models are used, in combination with analogous models of the mechanical behavior of short-fiber reinforcement, to simulate multi-field phenomena. The proposed models are verified through comparisons with theory and results based on discrete (volumetric) discretizations of the fibers. Capabilities of the semi-discrete approach are demonstrated through electro-mechanical analyses, with applications to self-sensing, and hygro-mechanical analyses of plastic settlement. Computational efficiency of the semi-discrete approach allows for larger amounts of fibers, or larger spatial dimensions, to be considered, which is important for structural applications. The following conclusions can be made:  
1. At low fiber dosages, both the discrete and semi-discrete approaches to modeling conductive fibers offer precise estimates of conductivity of the composite material, in comparison with the equivalent inclusion method (EIM) of Hatta and Taya. 2. The semi-discrete approach is computationally efficient in its representation of the conductivity contributions of individual fibers. Fiber additions do not increase the number of computational degrees of freedom of the model. For \(V_{f}=0.01\), which is one of the fiber dosages considered herein, the number of computational nodes was nearly one order of magnitude smaller, relative to the number of nodes used for modeling the composite with discrete (volumetric) fibers. The computational savings would be even more dramatic when considering higher fiber dosages or aspect ratios. 3. When comparing with the EIM results, the semi-discrete approach requires sufficient resolution of the background lattice such that the fibers are electrically isolated from one another by matrix material. Increasing \(V_{f}\), while keeping nodal density constant, eventually causes composite conductivity estimates of the semi-discrete model to be higher than the corresponding EIM values. 4. For a given fiber content, conductivity of the composite varies with directionality of the fiber distribution. The discrete and semi-discrete models capture the expected result: conductivity of the composite material increases with preferential orientation of fibers with respect to the electrical field. This capability should enable the study of fiber distribution effects in actual members, where boundary effects and other sources of fiber distribution non-uniformity are present. 5. The coupled electro-mechanical model captures the changes in composite resistance due to axial loading. An updating of nodal point positions leads to a composite gauge factor of unity. Other composite gauge factors can be simulated by introducing a parameter associated with the sensitivity of fiber conductivity to applied strain. 6. With respect to the simulations of plastic settlement of an unreinforced matrix, the results are in agreement with previously published results based on finite element analysis. Moreover, the results are shown to be equivalent, regardless of the dimensionality of the VCLM. 7. The proposed model enables study of the relative benefits of introducing short fibers to reduce plastic settlement of concrete. Both the mechanical and transport modifications of the material are considered. The simulation results support some experimental observations that higher fiber content leads to reduced settlement and increased amounts of bleed water due to the stiffness and conductivity enhancements of the fibers, respectively. 8. Model extensions and validation efforts, through comparisons with physical test data, are needed to address a variety of practical simulation needs. With respect to the validation efforts, it is difficult to measure some of the model input parameters (e.g., the mechanical and hydraulic conductivity properties of individual fibers within fresh concrete).