# Modelling the Structure and Anisotropy of London Clay Using the SA_BRICK Model
## 3 Developments of the SA_BRICK Model
### Modelling of the Anisotropy
\right] \tag{12}\]  
where the meaning of the above terms is as follows:  
\[E_{h}=K\left(2+\left(\frac{E_{h}}{E_{v}}\right)\right)\left(1-2\
u^{NC}\right) \tag{13}\]  
\[E_{v}=E_{h}/\left(\frac{E_{h}}{E_{v}}\right) \tag{14}\]  
\[G_{vh}=\frac{E_{h}}{2(1+\
u_{hh})(G_{hh}/G_{vh})} \tag{15}\]  
\[\
u_{vh}=\frac{1}{4}\left[1+2(1-\
u_{hh})\left(\frac{E_{v}}{E_{h}}\right)- \frac{E_{v}}{K}\right] \tag{16}\]  
As can be seen from Equations (12)-(16), only three additional parameters are needed to fully define anisotropy \(\left[D_{xyz}^{aniso,ref}\right]\), namely [PERSON]'s modulus ratio \(E_{h}/E_{v}\), the shear modulus ratio \(G_{hh}/G_{vh}\) and [PERSON]'s ratio in the horizontal plane \(\
u_{hh}\). By utilising the matrix \([M]\), the strain increments are transformed into the modified strain increments \(\{\dot{\epsilon}^{mod}\}\) by the following expressions [17]:  
\[\left\{\dot{\sigma}\right\}=\left[D^{aniso}\right]\left\{\dot{\epsilon}\right\} =\left[D^{iso}\right]\left\{\dot{\epsilon}^{mod}\right\}=\left[D^{iso}\right] \left[D^{iso}\right]^{-1}\left[D^{aniso}\right]\left\{\dot{\epsilon}\right\}= \left[M\right]\left\{\dot{\epsilon}\right\} \tag{17}\]  
\[\left\{\dot{\epsilon}^{mod}\right\}=\left[M\right]\left\{\dot{\epsilon}\right\} \tag{18}\]  
The BRICK model operates in the BRICK coordinate system with the stress and strain components \(\{p,t_{xx},t_{y},\tau_{xy},\tau_{yz},\tau_{zx}\}\) and \(\{\epsilon_{vol},g_{zxx},g_{y},\tau_{xy},\tau_{yz},\tau_{zx}\}\), and transformationsbetween Cartesian and BRICK coordinate systems are needed by utilising the transformation matrices \([T^{\sigma}]\) and \([T^{\varepsilon}]\) as follows:  
\[\left[D^{iso}\right]=[T^{\sigma}]\left[D^{iso}_{xyz}\right][T^{\varepsilon}]^{-1} \tag{19}\]  
\[\left[D^{aniso}\right]=[T^{\sigma}]\left[D^{aniso}_{xyz}\right][T^{\varepsilon }]^{-1} \tag{20}\]  
The BRICK formulation takes into account that when a soil material undergoes shearing, the bricks start to slide past one another until the condition of constant volume (i.e., critical state) is reached. Therefore, the modelling of coupling between shear and volumetric components must evolve to reach the state of no coupling. To achieve this, the evolutional variable \(0\leq\Omega\leq 1\) was introduced into the matrix \([M]\) from Equation (11), which is designated as \([M^{*}]\):  
\[[M^{*}]= \begin{bmatrix}\frac{\partial e^{mod}_{\text{sol}}}{\partial e_{vol}}& \Omega\frac{\partial e^{mod}_{\text{sol}}}{\partial\{T\}}\\ \Omega\frac{\partial\left\{T^{\eta}\right\}}{\partial e_{vol}}&\frac{\left\{T ^{\eta}\right\}}{\partial\{T\}}\end{bmatrix}  \tag{21}\]  
in which \(\Omega\) represents the degree of string mobilisation in the direction of strain increment:  
\[\Omega= \begin{cases}\frac{1-\eta}{\zeta}\ if\ \eta>1-\zeta\\ 1\ if\ \eta\leq 1-\zeta\end{cases}  \tag{22}\]  
where \(\zeta\) is the additional parameter and \(\eta\) is a measure of the alignment of strings in the direction of strain increment \(\left\{\hat{\varepsilon}\right\}\):  
\[\eta=\sum_{b=1}^{B}R_{b}\frac{L^{mob}_{b}}{\frac{L_{b}}{\zeta}} \tag{23}\]  
The number \(B\) is the number of bricks in each stress point, \(R_{b}\) is the proportion of material represented by brick \(b\), \(L_{b}\) is the string length of brick \(b\), \(L^{mob}_{b}\) is the mobilised string length of brick \(b\) in the direction of the strain increment and \(\zeta\) is a parameter which considers the Lode angle effects on string lengths. More details of this transformation are given in [45]. By using the parameter \(\zeta\), the threshold string alignment \(\eta\) is defined with the value \(1-\zeta\). If \(\eta\) is reduced below \(1-\zeta\), then parameter \(\Omega=1\), meaning that strings are becoming less aligned in the direction of strain increment. In this case, full anisotropy is taking place. On the other hand, by increasing \(\eta\) over \(1-\zeta\), the parameter \(\Omega\) starts to reduce according to the transformation \(\frac{1-\eta}{\zeta}\) from 1 towards 0.