# Modelling the Structure and Anisotropy of London Clay Using the SA_BRICK Model
## 3 Developments of the SA_BRICK Model  
### Modelling of the Structure  
The state of overconsolidation affects the behaviour of the BRICK model by the modification of string lengths and elastic shear modulus \(G^{\varepsilon}\) (kPa) according to the following formulae (Equations (1) and (2)):  
\[L_{b}=L_{b}^{0}\frac{\chi^{\phi}}{\chi^{\mathbb{G}}}, \tag{1}\]  
\[G^{\varepsilon}=\chi^{G}\;K^{\varepsilon}\;\frac{3\left(1-2\;v^{nc}\right)} {2\left(1+v^{nc}\right)} \tag{2}\]  
where \(L_{b}^{0}\) is the initial string length of brick \(b\) for the normally consolidated material, \(\chi^{\phi}\) are \(\chi^{G}\) are the variables that take into account the increase in stiffness and strength due to the overconsolidation, respectively, while \(K^{\varepsilon}\) (kPa) is the current isotropic elastic bulk modulus and \(v^{nc}\) is [PERSON]'s ratio on the normal consolidation line as a material parameter. The extent of overconsolidation is measured by the difference in the volumetric strain between a current state and the state on the normal compression line (NCL) in the form of a state parameter \(\psi_{\varepsilon}\):  
\[\psi_{\varepsilon}=\varepsilon_{v}-\varepsilon_{v0}-\lambda\;\ln\left(p^{ \prime}/p^{\prime}_{\;0}\right), \tag{3}\]  
where \(\varepsilon_{v}\) and \(p^{\prime}\) are the current volumetric strain and mean effective stress, respectively, while the subscript \(0\) refers to the initial reference state at the NCL. The state parameter \(\psi_{\varepsilon}\) affects the evaluation of variables \(\chi^{\phi}\) and \(\chi^{G}\) by the following formulae:  
\[\chi^{\phi}=1+\beta^{\phi}\;\psi_{\varepsilon}, \tag{4}\]  
\[\chi^{G}=1+\beta^{G}\;\psi_{\varepsilon} \tag{5}\]  
where \(\beta^{\phi}\) and \(\beta^{G}\) are material parameters. Additionally, the state in the \(\varepsilon_{v}-\ln p^{\prime}\) plane influences the current value of the volumetric stiffness variable \(\iota_{curr}\) as  
\[\iota_{curr}=\frac{\iota}{\chi^{G}} \tag{6}\]  
where \(\iota\) is the volumetric stiffness material parameter. [PERSON] [15] and [PERSON] [16] introduced the influence of structure in the original BRICK model and named the improved model S_BRICK. The structure was incorporated into the model by adding two variables, namely \(\alpha\) and \(\omega\). The variable \(\alpha\) was used to proportionally increase or decrease the original string lengths \(L_{b}\), having theeffect of changing the area beneath the S-shaped curve \(G_{t}/G_{max}-\varepsilon_{s}\). Parameter \(\alpha\) was introduced into Equation (1) as follows:  
\[L_{b}^{mod}=L_{b}^{0}\frac{\chi^{\phi}}{\chi^{\mathrm{C}}}\;\alpha \tag{7}\]  
Given that the area under the S-shaped curve is proportional to \(\sin\phi^{\prime}\), this modification directly influences the value of the maximum angle of the shearing resistance, i.e., the strength response of the model. The applicable range of \(\alpha\) for natural materials was chosen so that the angle of shearing resistance \(\varphi\) lies within the range from \(18^{0}\) (\(\alpha=0.6\)) to \(36^{0}\) (\(\alpha=1.2\)) for the normally consolidated states (Figure 3). The second introduced variable to consider the effect of the structure on stiffness in the BRICK model was \(\omega\). It is introduced to modify the state parameter \(\psi_{\epsilon}\) (see Equation (3)) as follows:  
\[\psi_{B}=\psi_{\epsilon}+\omega=\varepsilon_{\varphi}-\varepsilon_{\varphi 0}- \lambda\ln\left(\frac{p^{\prime}}{p^{\prime}{}_{0}}\right)+\omega \tag{8}\]  
The concept of \(\omega\) is graphically represented in Figure 4 in the \(\varepsilon_{\varphi}-\ln p^{\prime}\) plane, in which NCL and the critical state line (CSL) are shown for a structured and a reconstituted sample (marked with *). Its effect can easily be understood as an increase in the distance between the normal compression line and the critical state line in terms of volumetric strain as a consequence of structure, in comparison with that distance on the reconstituted material.