## 1 Introduction  
It is well known that monotonically loaded non-cohesive saturated soils under undrained conditions may be prone to liquefaction due to a static [1] or cyclic [2, 3] load. It was originally thought that, in order for soil to liquefy, voids should be completely filled with water which corresponds to full saturation of the medium. This was due to the fact that any presence of air would significantly limit or prevent the process of pore pressure build-up. Such an approach was physically justified because the presence of air results in full or partial dissipation of the pore pressure excess and consequently scales it down by the Skempton's parameter \(B\) decreasing liquefaction potential [4]. However, observations in areas particularly vulnerable to earthquakes have shown that liquefaction can also occur in soils that are not fully saturated. The first studies proving that partially saturated soils may also undergo liquefaction concerned the response of the soil to cyclic loads [5, 6]. There followed many others related to the same issue, e.g., [7, 8, 9, 10, 11, 12, 13]. Meanwhile, the large impact of saturation level on the liquefaction potential of monotonically loaded soils was proved by a series of laboratory tests [14, 15, 16, 17, 18, 19]. In order to theoretically predict the behaviour of partially saturated non-cohesive media, the MODSOL model proposed by [PERSON] and [PERSON] [20] has been elaborated, though it is dedicated mainly to cyclic loads as well. An interesting approach to modelling the response of soil to monotonic loading is the elasto-viscoplastic model proposed by [PERSON] et al. [21], which was derived for media characterized by relatively high suction values and low saturation levels. Another theoretical study on the behaviour of unsaturated soils can be found in [22; 23; 24], in which the authors also generally focus on suction pressure at low saturation levels. The implementation of constitutive laws governing the behaviour of soil in a partially saturated state in a SaniSand-Z model built by [PERSON] et al. can be found in [25]. There are some other theoretical approaches to constitutive modeling of partially saturated soils, e.g., a deviatoric hardening model which can predict static liquefaction proposed by [PERSON] et al. [26], solutions based on thermodynamic equations by [PERSON] et al. [27; 28] or a hypo-plastic model of sand developed by [PERSON] et al. [29]. According to [4; 30; 31], in partially saturated media, the location of the instability line tends to increase with a decrease in the saturation level, which causes a change in the moment of triggering of the liquefaction process. The problem of liquefaction of partially saturated soils seems to be important in many areas, also, from an engineering point of view, e.g., wave-inducted seabed liquefaction [32; 33], blast-induced densification in liquefiable sediments [34], submarine large-scale landslides triggered by gas dissociation [4], etc. Another interesting case relates to wet mine tailings storage facilities, where the zone of partial saturation could reach a dozen meters [35]. This is a consequence of the successive deposition of sediments in the form of soil-water mixtures by spigotting, which causes cyclic wetting and drying of stored waste mass [36]. As a result, relatively thick zones of partially saturated sediments appear above the full saturation line. In such a situation, the potential generation of pore pressure within this zone could, in an extreme case, lead to the liquefaction of tailing deposits, significantly reducing the stability of the surrounding dams. This is the reason why post-flotation cooper tailings have been selected as one of the investigated soils in this analysis. In order to examine the influence of partial saturation on the undrained shear strength of soil, an extensive research programme has been carried out in recent years at the Institute of Hydro-Engineering of the Polish Academy of Sciences.