# Multi-Field Models of Fiber Reinforced Concrete for Structural Applications
## 5 Hygro-Mechanical Analyses of Plastic Settlement
### Verification of the Matrix Phase Model  
The settlement component of the proposed model is verified with the numerical results of [PERSON] and [PERSON] (1979), who simulated settlement of a mortar specimen using planar finite elements. The virtual prismatic specimen has dimensions \(b\times l\times h\) equal to \(800\times 800\times 400\,\mathrm{mm}\). The upper surface of the specimen is exposed to the environment and is free to deform, while the other surfaces are constrained by a rigid mold with frictionless walls. As previously noted, moisture exchange with the environment and hydration of the cementitious materials are not considered in this simulation. The parameter values used for the simulations are indicated in Table 1. Hydraulic conductivity \(K_{h}\) is considered to be a function of the initial hydraulic conductivity and porosity:  
\[K_{h}=\frac{K_{h0}}{(1-n)} \tag{25}\]in which porosity is assumed to evolve with volume change due to settlement [97]  
\[n=1+(\frac{n_{0}-1}{1+\varepsilon_{v}}) \tag{26}\]  
where \(n_{0}\) is initial porosity, which can be calculated from the mixture design; \(\varepsilon_{v}\) is the volumetric strain that is approximated as the sum of the principle strains, i.e., \(\varepsilon_{v}\approx\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}\).  
Since the same constitutive model is applied in both modeling frameworks, the settlement computed using the VCLM should be comparable to that of the finite element method when applying identical material parameters (as given by [PERSON] and [PERSON] [92]). The proposed model is examined with one-, two-, and three-dimensional Voronoi cell discretizations. The latter two cases are shown in Figure 12. The settlement is measured as the top-center node displacement for all cases.  
As seen in Figure 13, the VCLM results are practically identical to those of the two-dimensional FE model, regardless of the dimensionality of the VCLM. Since the interface between the specimen and the mold is frictionless, settlement and pore pressure should be a function only of the vertical coordinate of a material point, provided the mesh density is adequate. Figure 14 displays the vertical displacement and pore pressure distributions over the specimen height at different times. The proposed settlement model produces consistent results regardless of the dimensionality of the VCLM. The smooth, curve-like nature of the distributions of nodal point values suggests the random discretization of the specimen does not introduce significant bias within the simulation results. Considering the close agreement between the 2D and 3D simulation results, 2D versions of the VCLM are used for the parametric analyses in the next section.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Material Parameter** & **Value** \\ \hline Elastic Modulus \(E\) (MPa) & 0.2778 \\ Density of Solid Skeleton \(\rho_{s}\) (kg/m\({}^{3}\)) & 2691 \\ Density of Pore Water \(\rho_{w}\) (kg/m\({}^{3}\)) & 1000 \\ Initial Porosity \(n_{ini}\) & 0.25 \\ Initial Hydraulic Conductivity \(K_{\text{A0}}\) (mm/s) & \(5.1\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameter values for the plastic settlement analyses.  
Figure 12: Discretization for VCLM analyses: (**a**) two-dimensional; and (**b**) three-dimensional representations of the virtual specimen.