# Settlement Foundations by Exploring the Collapse of Unsaturated Soils
## 2 Adopted Numerical Framework Using EBBM Theoretical to Predict Collapse
### Plastic Deformation  
A plastic strain matrix can be derived using \(\lambda\mathrm{p}\), the plastic multiplier; g, the plastic potential; and \(\mathrm{q}\), the deviator stress:  
\[\dot{\epsilon_{\mathrm{p}}}=\ \lambda_{\mathrm{p}}\biggl{(}-\frac{1}{3}\frac{ \partial\mathrm{g}}{\partial\mathrm{p}}\mathrm{I}+\frac{\partial\mathrm{g}}{ \partial\mathrm{q}}\,\frac{3}{2\mathrm{q}}\mathrm{dev}(\sigma)\biggr{)} \tag{6}\]  
Based on the associated flow rule, g, the plastic potential is considered identical to the yield function F.  
The yield function (F) of EBBM (see Figure 1) is as follows:  
\[\mathrm{F}=\mathrm{q}^{2}-\mathrm{M}^{2}\bigl{(}\mathrm{p}+\mathrm{p_{s}} \bigr{)}\bigl{(}\mathrm{p_{c}}-\mathrm{p}\bigr{)}+\ \mathrm{p_{ref}}^{2}\biggl{(}\mathrm{e}^{\frac{\mathrm{b}(s-\mathrm{s})}{ \mathrm{p_{ref}}}}-\ \mathrm{e}^{\frac{-\mathrm{b}}{\mathrm{p_{ref}}}}\biggr{)} \tag{7}\]Here, M represents the slope of the critical state line, \((\rm{p_{s}}=k.s)\), \(\rm{p_{s}}\) relates to the impact of suction on the soil's cohesiveness, \(k\) is a parameter characterizing the increase in cohesion due to suction, \(\rm{P_{ref}}\) indicates the reference pressure, \(\rm{b}\) is described as a smoothing parameter, and \(\rm{s_{y}}\) is the suction yield value (Figure 1).  
\(p_{c}\) represents the preconsolidation pressure corresponding to the current suction value:  
\[\rm{\frac{P_{cs}}{P_{ref}}=\left(\frac{P_{c}}{P_{ref}}\right)^{[\frac{\lambda( 0)-K}{\lambda(S)-K}]};\;\lambda(s)=\;\lambda(0)\left[(1-w)\exp\left(\frac{-s}{m }\right)+w\;\right]} \tag{8}\]  
\(\rm{P_{cs}}\) is the consolidation pressure in a saturated sate (zero suction), \(\rm{K}\) is the swelling index, \(\lambda(s)\) is the slope of the normal compression line corresponding to the current suction (s), and \(\lambda(0)\) represents the slope at zero suction. \(\rm{m}\) and \(\rm{w}\) refer to soil stiffness and weighting parameters, respectively.  
Equation (8) considers the effect of suction, which is linked to the degree of saturation using SWRC. When the suction decreases in response to infiltration, the yield surface reduces, crossing the LC curve and leading to irreversible volume change, and consequently, the soil undergoes collapse.  
The volumetric plastic strains \(\varepsilon_{p,v}\) directly influence the change in the consolidation pressure, which governs the hardening characteristics in the stress-strain relationship:  
\[\rm{\dot{P_{c}}}=-\frac{1+e_{0}}{\lambda(0)-K}P_{c}\dot{\varepsilon_{p,v}} \tag{9}\]  
However, by increasing the suction, when the suction reaches the yield suction \(\rm{s=s_{y}}\), the plastic volumetric strain resulting from yielding on the SI yield surface can be expressed as follows:  
\[\rm{\dot{P_{c}}}=-\frac{1+e_{0}}{\lambda_{S}-K_{S}}(s_{y}+P_{atm})\epsilon_{p,v} \tag{10}\]  
Appendix A provides a detailed description of the Richards equation. The relative water permeability function has been characterized using the [PERSON] (1980) [50] and [PERSON] (1976) models [51] (see Appendix A). SWRC demonstrates a hysteresis effect, reflecting that the relationship between the suction and water content is not unique. During the drying path, increasing suction induces pore water to be displaced by air from the voids. In contrast, during the wetting process, water infiltrates the voids, leading to a reduction in suction [39; 52; 53]. Since the majority of soil compression occurs as the soil becomes saturated, it is usually more suitable to apply the wetting path of SWRC rather than the drying path when analyzing the soil foundation in the context of rainfall infiltration [54]. To establish the wetting path curve of the investigated soil, the drying path curve was initially fitted using the [PERSON] [PERSON] (2018) method [55]. Subsequently,  
Figure 1: Unique yield surface of the EBBM [48].  
the wetting path curve was derived using their approach and by applying the parameters recommended in the research of [PERSON] et al. (2005) [56].