## 1 Introduction
### Background of an \"Estimation Procedure\"
#### 1.1.2 Unsaturated Shear Strength  
Considering the Mohr-Coulomb failure criterion in saturated soil mechanics, the problem is analyzed in two orthogonal planes. The effective normal stress \((\sigma-\mathrm{u}_{\mathrm{w}})\) is represented on the abscissa axis, whereas the shear stress \(\tau\) on the failure plane is represented on the ordinate axis [23], as expressed in Equation (2):  
\[\tau=c^{\prime}+(\sigma-\mathrm{u}_{\mathrm{w}})\times\tan\phi^{\prime} \tag{2}\]  
where:  
\(c^{\prime}=\) effective cohesion in saturated conditions  
\(\phi^{\prime}=\) effective angle of internal friction in saturated conditions  
Regarding unsaturated soil mechanics, [PERSON] et al. [24] extended the Mohr-Coulomb criterion by employing a similar approach, considering the shear stress \(\tau\) and net normal stress \((\sigma-\mathrm{u}_{\mathrm{a}})\) on the failure plane for two of three orthogonal planes. A third orthogonal plane representing the matric suction \((\mathrm{u}_{\mathrm{a}}-\mathrm{u}_{\mathrm{w}})\) is incorporated, which is defined by Equation (3) and illustrated in Figure 3a:  
\[\tau=c^{\prime}+(\sigma-\mathrm{u}_{\mathrm{a}})\mathrm{tan}\,\phi^{\prime}+( \mathrm{u}_{\mathrm{a}}-\mathrm{u}_{\mathrm{w}})\,\mathrm{tan}\phi^{\mathrm{b}} \tag{3}\]  
In Figure 3, \(\phi^{\prime}\) relates net normal stress to the shear strength, and \(\phi^{\mathrm{b}}\) represents the increase in shear strength with respect to the matric suction. However, studies have shown that the relationship between shear strength and matric suction deviates from linearity [26], leading to the conclusion that the development of the internal friction angle \(\phi^{\prime}\) follows an approximately linear trend, whereas \(\phi^{\mathrm{b}}\) exhibits nonlinear behavior [27], as illustrated in Figure 3b. [PERSON] et al. [28] developed an analytical model to estimate nonlinear shear strength using two equations: one with a fitting parameter \(\mathrm{k}\), based on the normalized volumetric water content, and one without this parameter, but based on the SWCC for input values such as gravimetric water content at residual suction and in a saturated state. [PERSON] and [PERSON] [29] employed an effective stress variable to estimate the unsaturated shear strength by comparing the estimated shear strength with the measured values, thereby achieving high accuracy with the proposed equation. This variable is derived from the parameters of saturated shear strength, suction, and air-entry value. [PERSON] [30] developed an empirical hyperbolic fitting equation that requires the shear strength of a saturated sample and an air-dried sample tested without suction control, or  
Figure 3: (**a**) Extended Mohr-Coulomb graph (modified from [PERSON] et al. [25]); (**b**) hyperbolic function based on the extended Mohr-Coulomb equation. samples tested under controlled suction greater than the maximum expected suction. Both alternatives were compared with the results obtained for various previously published soils. [PERSON] and [PERSON] [1] considered the stress equilibrium conditions and multiphase interactions of soil in an unsaturated state in order to develop a micromechanical model. This model predicts shear strength as a nonlinear function of the degree of saturation, volumetric water content, matric suction, and SWCC input parameters. [PERSON] et al. [31] propose a hyperbolic empirical equation for the apparent cohesion, represented in the plane formed by suction and shear strength, using an artificial neural network. This equation is based on cohesion and friction in the saturated state of different soils and equipment. Subsequently, a comparison was made between the estimated shear strength and the measured values, achieving high precision. [PERSON] et al. [12] investigated the effect of soil type on the shear strength over a wide suction range. They concluded that in sand, the shear strength increases until it reaches a maximum value and then decreases significantly.