# Modelling the Structure and Anisotropy of London Clay Using the SA_BRICK Model
## 3 Developments of the SA_BRICK Model
### Modelling of the Anisotropy  
A comprehensive overview of the anisotropy of London clay and its modelling was given by [PERSON] et al. [42]. As other soils are deposited over a large lateral extent, London clay is best described as cross-anisotropic; it responds differently when compressed in the vertical and horizontal directions but responds in the same way if compressed in either of the two horizontal directions. In terms of elastic behaviour, only five constants are required to define cross-anisotropy [43]. [PERSON] et al. [44] modified the basic BRICK model by adding the plastic strain reduction through the volumetric strain hardening by brick repositioning and formulating the model in the general strain space. Additionally, the stiffness anisotropy was introduced by transforming the coordinate system in which the model was based and adding an anisotropic stiffness matrix, as explained bellow. The same approach to account for the anisotropy was chosen here for upgrading the S_BRICK model to the SA_BRICK model. The important difference between the work of [PERSON] et al. [44] and this research was that the bricks were not repositioned in the SA_BRICK \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **BRICK** & **S\_BRICK** & **SA\_BRICK** \\  & [8] & [15] & **This Study** \\ \hline String lengths & \(L_{b}=L_{b}^{0\ 1+\beta^{\prime}\ \phi_{c}}\) & \(L_{b}^{mod}=L_{b}^{0\ 1+\beta^{\prime}\ \phi_{c}}\) & \(L_{b}^{mod}=L_{b}^{0\ 1+\beta^{\prime}\ \phi_{c}}\) & \(L_{b}^{mod}=L_{b}^{0\ 1+\beta^{\prime}\ \phi_{c}}\) & \(\alpha\) \\ State parameter & \(\psi_{c}=\varepsilon_{v}-\varepsilon_{v0}-\lambda\ \ln(p^{\prime}/p^{\prime} _{0})\) & \(\psi_{B}=\varepsilon_{v}-\varepsilon_{v0}-\lambda\ln(p^{\prime}/p^{\prime} _{0})+\omega\) & \(\psi_{c}=\varepsilon_{v}-\varepsilon_{v0}-\lambda\ \ln(p^{\prime}/p^{\prime} _{0})\) \\ Stiffness parameter & \(t_{curr}=\frac{t}{1+\beta^{\prime}\ \phi_{c}}\) & \(t_{curr}=\frac{t}{1+\beta^{\prime}\ \phi_{c}}\) & \(t_{curr}^{mod}=\frac{t}{1+\beta^{\prime}\ \phi_{c}}\ \omega\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of the formulations of structure effects in the BRICK models in the form of modification of string lengths, state and stiffness parameters. Figure 4: Concept of the additional structure parameter \(\omega\) in the S_BRICK model [16]. model but remained in the same original shape and form as in the original BRICK model (i.e., as defined by the shape of the S curve, that is, by the string lengths and corresponding normalised shear stiffnesses \(G/G_{max}\), as given in Table 1). The introduction of anisotropy was carried out following [PERSON] et al. [17], by using the transformation matrix \([M]\) defined by Equation (10):  
\[[M]=\left[D_{xyz}^{iso}\right]^{-1}\left[D_{xyz}^{aniso}\right] \tag{10}\]  
where \(\left[D_{xyz}^{iso}\right]^{-1}\) is the isotropic compliance matrix and \(\left[D_{xyz}^{aniso}\right]\) the anisotropic stiffness matrix. \(\left[D_{xyz}^{iso}\right]^{-1}\) and \(\left[D_{xyz}^{aniso}\right]^{-1}\) take the following form in Cartesian coordinates by using the bulk modulus \(K\) and Poisson's ratio at state on the normal consolidation line \(\
u^{NC}\):  
\[\left[D_{xyz}^{iso}\right]^{-1}=\frac{1}{3K\left(1-2\
u^{NC}\right)} \begin{bmatrix} 1&-\
u^{NC}&-\
u^{NC}&0&0&0\\ -\
u^{NC}&1&-\
u^{NC}&0&0&0\\ -\
u^{NC}&-\
u^{NC}&1&0&0&0\\ 0&0&0&2\left(1+\
u^{NC}\right)&0&0\\ 0&0&0&0&2\left(1+\
u^{NC}\right)&0\\ 0&0&0&0&0&2\left(1+\
u^{NC}\right)\end{bmatrix}  \tag{11}\]  
\[\left[ \begin{bmatrix}&\frac{1}{E_{h}}&-\frac{\
u_{ch}}{E_{v}}&-\frac{\
u_{hh }}{E_{h}}&0&0&0\\ &-\frac{\
u_{ch}}{E_{v}}&\frac{1}{E_{v}}&-\frac{\
u_{ch}}{E_{v}}&0&0&0\\ \left[D_{xyz}^{aniso}\right]^{-1}=-\frac{\
u_{hh}}{E_{h}}&-\frac{\
u_{ch}}{E_{h }}&\frac{1}{E_{h}}&0&0&0\\ 0&0&0&\frac{1}{\zeta_{ch}}&0&0\\ 0&0&0&0&\frac{1}{\zeta_{ch}}&0\\ 0&0&0&0&\frac{2\left(1+\
u_{hh}\right)}{E_{h}}\end{bmatrix}