## 2 Generation of Pore Pressure in Partially Saturated Soils  
The phenomenon of soil liquefaction is closely related to the volume changes which dry soil is subjected to during shearing (see the critical state theory [39; 40; 41; 42; 43; 44]). In general, the behaviour of non-cohesive soil during shearing depends on its initial state, which determines its contractive or dilative response and/or contractive-dilative intermediate states [39]. Contractive and dilative states are separated by the critical state line (CSL) [39], which is also identified as a steady-state line (SSL) [45] at which sheared soil reaches its critical state [46] (Figure 1a). As a measure of the intensity of soil response (contractive or dilative behaviour), a state parameter \(\Psi\)[40] is usually used, which is defined as a change in void ratio that must occur for the medium to reach a critical state (steady state) at a constant mean effective stress. Differences between undrained shear stress paths corresponding to each initial state (schematically shown in Figure 1b) are caused by changes in pore pressure build-up and subsequent redistribution between mean effective and total stresses. For initially contractive soils, this results in a decrease in the mean effective stress and subsequently a reduction of their shear strength. This reduction occurs after crossing an instability line [4; 47], where the stress deviator reaches its maximum value. In the extreme case, effective stress vanishes, the soil is unable to transfer the shear stress and liquefies [48], which can be formally written as:  
\[p^{\prime}=p-u=0. \tag{1}\]  
The degree to which voids are filled with water may change between full saturation, represented by degree of saturation \(S_{r}=1\), and a completely dry medium (no water), \(S_{r}=0\). In between there occur intermediate states identified with unsaturated soils, which also include partial saturation. Partial saturation should be distinguished from an unsaturated state in which the gaseous phase of different levels is present, partial saturation being characterised by the lack of a continuous gas phase and the presence of only isolated air bubbles surrounded by water [18]. The model presented in this paper refers to partially saturated soils in which the saturation degree is usually above approximately 80% (Figure 2). Figure 1: (**a**) Critical state line with definition of state parameter. (**b**) Typical responses of saturated soil during shearing under undrained conditions. Furthermore, in non-cohesive media with a degree of saturation corresponding to a partial saturation state, suction pressure is negligibly small or does not occur at all [20; 49; 50]. In Figure 3, water retention curves for some selected non-cohesive soils, i.e., Hostun sand [20] and Singapore fine sand [51], as well as suction pressures for tested soils (determined in Sand Apparatus for the range of high saturation degree) together with their grain size distribution curves, are presented. It can be seen that for a degree of saturation above 0.8 we are dealing with very small suction pressure values of the order of 1 or 2 kPa and therefore the application of [PERSON]'s law given by Equation (1) can be justified [52]. Figure 3: Suction pressures for tested soils and water retention curve for: Hostun sand [20] and Singapore fine sand [51]. Figure 2: Saturation distribution in a soil medium. Soils in a state of partial saturation are characterized by values of [PERSON]'s parameter \(B\)[53] in a range from \(B=1\), which is equivalent to full saturation, to \(B\approx 0^{+}\), corresponding to the lower boundary of a partially saturated state. Isotropically loaded saturated soil has a tendency to decrease in volume, but compaction under undrained conditions is prevented by bonds imposed by hardly compressible pore fluid, which leads to the generation of excess pore pressure, \(\Delta u\). [PERSON]'s parameter \(B\) is defined as a portion of generated pore pressure, \(u\), caused by corresponding isotropic stress.