## 3 Theoretical Model  
### Governing Equations  
Based on the experimental results (Series A), a theoretical model was proposed to describe the behaviour of partially saturated non-cohesive soil in an initially contractive state subjected to axisymmetric monotonic loading under undrained conditions [38]. The core element of the model is a differential equation the integration of which allows the reproduction of any stress path within the stress space, including the generation of pore pressure in relation to the value of [PERSON]'s parameter \(B\) characterizing the saturation level. A significant role in the proposed equation is played by two compressibility coefficients, the first corresponding to the compressibility of fluid treated as a mixture of incompressible water and compressible gas, the second to the compressibility of the soil skeleton. In general, these coefficients, which also depend on the type of loading (isotropic, deviatoric), are functions of soil state (void ratio and stress state expressed by mean effective stress and the stress deviator). The functions should be determined experimentally for each type of non-cohesive soil on the basis of isotropic consolidation and pure shearing tests under triaxial conditions. The presented solutions are based on an idea of a semi-empirical incremental model for non-cohesive soils proposed originally by [57], which makes it possible to predict the liquefaction of fully saturated sandy soils, whereas the idea of developing a model for partially saturated soils was presented originally by [PERSON] in an internal report [58] issued in 2015 and then published in the form of a research paper [38]. In general, the model can predict the full spectrum of soil behaviors shown in Figure 1 (from contractive to dilative ones); however, in this paper we focused on analyzing and theoretically reproducing the liquefaction phenomenon, therefore only the contractive state is analyzed. The basic assumptions of the model proposed are detailed below. Volumetric strains of soil skeleton, \(\varepsilon_{v}\), under triaxial compression conditions are dependent on mean effective stress and the stress deviator, \(q\). \[\varepsilon_{v}=f\big{(}p^{\prime},q\big{)}, \tag{4}\]  
or alternatively  
\[\varepsilon_{v}=f\big{(}p^{\prime},\eta\big{)}, \tag{5}\]  
where \(\eta\) is the stress ratio expressed by the following equation:  
\[\eta=\frac{q}{p^{\prime}}. \tag{6}\]  
The stress ratio, \(\eta\), is a useful and convenient variable that makes it possible to define precisely the loading and unloading process in the stress space. Functions describing strain-stress dependence (Equations (4) and (5)) take different forms for loading and unloading. Consequently, with a change in load direction, respective parameters have to be adopted. Stress invariants \(p\prime\) or \(\eta\) and the strain invariant \(\varepsilon_{v}\) are linked by functions of soil skeleton compressibility [38], where: \(x_{s}^{p^{\prime}}\) corresponds to isotropic compression (\(\eta=\text{const. }\)), denoted as \(x_{s}^{p^{\prime}J}\) under loading conditions and \(x_{s}^{p^{\prime}Jan}\) for unloading, and \(x_{s}^{\eta}\) corresponds to pure shearing (\(pt=\text{const.