# One-Dimensional Consolidation of Multi-Layered Unsaturated Soil with Impeded Drainage Boundaries
## 3 Solution Formulations
\begin{array}{c}\textbf{X}^{(1)}\\ \boldsymbol{X}^{(2)}\\ \vdots\\ \boldsymbol{X}^{(i)}\\ \vdots\\ \boldsymbol{X}^{(n-1)}\\ \boldsymbol{X}^{(n)}\end{array} \right\}_{4t\times 1}=\left\{ \begin{array}{c}-s^{-1} \textbf{R}_{\text{t}}\widetilde{\textbf{F}}^{(1)}\\ \boldsymbol{\mathbb{V}}^{(2)}\\ \vdots\\ \boldsymbol{\mathbb{V}}^{(i)}\\ \boldsymbol{\mathbb{V}}^{(i+1)}\\ \boldsymbol{\mathbb{V}}^{(n)}\\ -s^{-1}\textbf{R}_{\text{b}}\widetilde{\textbf{F}}^{(n)}\end{array} \right\}_{4t \times 1} \tag{20}\]\[ \begin{split}\mathbf{\Xi}^{(i)}=-\mathbf{\Gamma}^{(i)}\mathbf{\Omega}^{(i)},\ \mathbf{\Gamma}^{(i)}=\left[\begin{array}{cc}\theta^{(i)}&\theta^{(i)}\\ \mathbf{\mathrm{k}}^{(i)}\theta^{(i)}\mathbf{\mathrm{p}}^{(i)}&-\mathbf{\mathrm{k}}^{(i)} \theta^{(i)}\mathbf{\mathrm{p}}^{(i)}\end{array}\right]\text{, }\mathbf{\Omega}^{(i)}=\left[ \begin{array}{cc}\mathbf{\mathrm{I}}_{1}^{(i)}&0\\ 0&\mathbf{\mathrm{J}}_{2}^{(i)}\end{array}\right]\text{, }\mathbf{\mathrm{p}}^{(i)}=h \Big{[}s/\mathbf{\Lambda}^{(i)}\Big{]}^{1/2}\\ \mathbf{\Psi}^{(i+1)}=s^{-1}\Bigg{\{}\begin{array}{cc}\widetilde{\mathbf{\mathrm{F}}} ^{(i)}-\widetilde{\mathbf{\mathrm{F}}}^{(i+1)}\\ 0\end{array}\Bigg{\}},\ \mathbf{\mathrm{X}}^{(i)}=\left\{\begin{array}{c}\mathbf{ \mathrm{A}}_{1}^{(i)}\\ \mathbf{\mathrm{A}}_{2}^{(i)}\end{array}\right\}\text{, }\mathbf{\mathrm{T}}^{(1)}=\left\{ \begin{array}{cc}\mathbf{\mathrm{R}}_{\mathbf{\mathrm{I}}}\theta^{(1)}-\theta^{(1)} \mathbf{\mathrm{\rho}}^{(1)}&\mathbf{\mathrm{R}}_{\mathbf{\mathrm{I}}}\theta^{(1)}+\theta ^{(1)}\mathbf{\mathrm{\rho}}^{(1)}\end{array}\right\}\\ \mathbf{\mathrm{B}}^{(n)}=\left\{\begin{array}{cc}\mathbf{\mathrm{R}}_{\mathbf{\mathrm{b}}} \theta^{(n)}+\theta^{(n)}\mathbf{\mathrm{p}}^{(n)}&\mathbf{\mathrm{R}}_{\mathbf{\mathrm{b} }}\theta^{(n)}-\theta^{(n)}\mathbf{\mathrm{\rho}}^{(n)}\end{array}\right\}\mathbf{ \Omega}^{(n)},\ \mathbf{\mathrm{J}}_{1}^{(i)}=\mathbf{\mathrm{E}}_{1}^{(i)}(z_{i},s)\text{.}\mathbf{ \mathrm{J}}_{2}^{(i)}=\mathbf{\mathrm{E}}_{2}^{(i)}(z_{i},s)\end{split}  \tag{21}\]  
Obviously, the coefficient vectors \(\mathbf{\mathrm{A}}_{1}^{(i)}\) and \(\mathbf{\mathrm{A}}_{2}^{(i)}\) can be directly solved from Equation (20). Then, we can obtain the excess pore pressures from the inversion of the solution (16) as  
\[\mathbf{\mathrm{u}}^{(i)}(z,t)=\frac{1}{2\pi j}\!\int_{c-j\infty}^{c+j\infty}\mathbf{ \mathrm{u}}^{(i)}(z,s)e^{st}ds \tag{22}\]  
where \(j=\sqrt{-1}\); and \(c\) is a real constant that exceeds the real part of all the singular points of \(\widetilde{\mathbf{\mathrm{u}}}^{(i)}\). For a layered unsaturated soil, Equation (22) is commonly difficult to analytically achieve. Herein, the well known Crump's method [29] is employed to approximate its result as  
\[\mathbf{\mathrm{u}}^{(i)}(z,t)=\frac{et}{T}\Bigg{\{}\frac{1}{2}\widetilde{\mathbf{ \mathrm{u}}}^{(i)}(z,c)+\sum_{k=1}^{\infty}\mathrm{Re}\bigg{[}\widetilde{\mathbf{ \mathrm{u}}}^{(i)}\bigg{(}z,c+\frac{jk\pi}{T}\bigg{)}e^{\frac{jk\pi}{T}t} \bigg{]}\Bigg{\}} \tag{23}\]  
where \(T\) is a real parameter exceeding one half of the largest time \(t_{\max}\). Once knowing the values of the excess pore-air and pore-water pressures in each soil layer, we can calculate the related volumetric strain \(\varepsilon_{\mathrm{v}}^{(i)}\) from the below expression [9; 15]:  
\[\varepsilon_{\mathrm{v}}^{(i)}(z,t)=m_{1}^{\mathrm{s}(i)}[\sigma_{z}(t)-\sigma _{z}(0)]+\mathbf{\mathrm{m}}^{\mathrm{s}(i)}\Big{[}\mathbf{\mathrm{u}}^{(i)}(z,t)- \mathbf{\mathrm{u}}^{(i)}(z,0)\Big{]} \tag{24}\]  
where \(\mathbf{\mathrm{m}}^{\mathrm{s}(i)}=\left\{ \begin{array}{cc}m_{2}^{\mathrm{s}(i )}-m_{1}^{\mathrm{s}(i)},&-m_{2}^{\mathrm{s}(i)}\end{array} \right\}\); \(m_{1}^{\mathrm{s}(i)}=m_{1}^{\mathrm{w}(i)}+m_{1}^{\mathrm{a}(i)}\) is the coefficient of the soil volume change that is caused by an evolution of the net normal stress; and \(m_{2}^{\mathrm{s}(i)}=m_{2}^{\mathrm{w}(i)}+m_{2}^{\mathrm{a}(i)}\) is the coefficient of the soil volume change that is caused by an evolution of the matrix suction.