# Multi-Field Models of Fiber Reinforced Concrete for Structural Applications
## 5 Hygro-Mechanical Analyses of Plastic Settlement
### Formulation of the Matrix Phase Model  
The modeling of the matrix phase, implemented within the VCLM, relies on several simplifying assumptions that are listed below. In future work, it may be possible to include the effects of early-age solidification associated with cementitious materials hydration [93; 94; 95]. * The load-deformation behavior of the mixture follows small-strain elastic theory. * No hydration takes place during the time period covered by the analyses, such that heat generation, chemical shrinkage, or hardening do not occur. * The material is isothermal and isolated from ambient conditions. * The surface of the formwork is considered to be frictionless and no wall effect is considered. As a continuum, every point in the solid body of plastic concrete satisfies the equilibrium condition  
\[\
abla\mathbf{\sigma}+\rho\mathbf{g}=\mathbf{0} \tag{15}\]  
where \(\rho\) is the density of the two phase porous medium, taken as \(\rho=nS_{w}\rho_{w}+(1-n)\rho_{s}\), in which \(\rho_{w}\) is the density of water and \(\rho_{s}\) is the density of solid skeleton; \(n\) is the porosity; \(S_{w}\) (=1, herein) is degree of saturation and \(\mathbf{g}\) is the acceleration of gravity. Stress tensor \(\mathbf{\sigma}\) can expressed in terms of effective stress \(\mathbf{\sigma}^{\prime}\) and pore water pressure \(p\) under isothermal conditions  
\[\mathbf{\sigma}=\mathbf{\sigma}^{\prime}+\alpha_{b}S_{w}\mathbf{m}p \tag{16}\]  
where Biot's coefficient \(\alpha_{b}=1-K_{d}/K_{s}\), in which \(K_{d}\) and \(K_{s}\) are bulk moduli of the drained porous material and solid portion of the matrix, respectively. Because the solid itself can be treated as almost incompressible when compared to the drained mixture, we can assume \(\alpha_{b}=1\) here. For the VCLM, \(\mathbf{m}=[1\ 0\ 0\ 0\ 0\ 0]^{T}\). Neglecting other strain sources, such as temperature variation, effective stress \(\mathbf{\sigma}^{\prime}=\mathbf{D}\mathbf{\epsilon}\), where \(\mathbf{D}\) is the stiffness matrix of drained solid skeleton. Transport of water in the porous media is driven by gravity and can be described using Darcy's Law in rate format. For the applications considered herein, [PERSON]'s law is written in terms of hydraulic conductivity  
\[\mathbf{q}=-\frac{K_{h}}{\gamma_{w}}(\
abla p-\rho_{w}\mathbf{g}) \tag{17}\]  
where \(\mathbf{q}\) is the volumetric flux density, hydraulic conductivity \(K_{h}=\kappa\gamma_{w}/\mu\), in which \(\kappa\) is the permeability of the porous media, \(\mu\) is the viscosity of water, and \(\gamma_{w}\) is the specific weight of water. The total mass of water of any phase stays constant during the transport process. For the case of settlement, only the liquid phase of water is considered, and the mass conservation equation of liquid water and solid skeleton together considering the mass diversion of hydrates [96] can be deduced:  
\[ \begin{split} S_{w}\
abla\cdot\dot{\mathbf{u}}+n\frac{\partial S _{w}}{\partial t}+\
abla\cdot&[\frac{K_{h}}{\gamma_{w}}(\
abla p -\rho_{w}\mathbf{g})]\\ &=-\frac{S_{w}}{\rho_{s}}[(1-n)\dot{\rho}_{s}-\dot{n}_{hyd}]- \frac{\dot{n}_{hyd}}{\rho_{w}}\end{split}  \tag{18}\]  
where \(\dot{\mathbf{u}}\) is the velocity of the solid skeleton and \(\dot{n}_{hyd}\) is the production rate of new hydrates, which is assumed to be zero herein. Equations (16) and (18) can be transformed into the following matrix form by applying a [PERSON] residual method:  
\[ \begin{bmatrix}\mathbf{K_{uu}}&-\mathbf{K_{up}}\\ \mathbf{0}&\mathbf{K_{pp}}\end{bmatrix}\begin{Bmatrix}\mathbf{u}\\ \mathbf{p}\end{Bmatrix} + \begin{bmatrix}\mathbf{0}&\mathbf{0}\\ \mathbf{C_{pu}}&\mathbf{C_{pp}}\end{bmatrix} \frac{d}{dt} \begin{Bmatrix} \mathbf{u}\\ \mathbf{p}\end{Bmatrix} = \begin{Bmatrix}\mathbf{f_{u}}\\ \mathbf{f_{p}}\end{Bmatrix}