# Shear Strength of Sand: Integrated Analysis of Initial Porosity and Stress Effects
## 1 Introduction  
Shear strength of soil is a fundamental material property used in calculations for foundation bearing capacity, earth pressure, and slope stability analysis. One of the criteria for soil failure is the Coulomb-Mohr criterion, which is defined by two geotechnical parameters: the internal friction angle and soil cohesion. The most important parameter determining the strength of non-cohesive soils in this model is the internal friction angle, which is widely used in geotechnical structure analyses. There are many methods for determining soil shear strength, which are detailed in [1]. They stated that different methods of testing soil shear strength yield results with significant discrepancies, regardless of the shearing mechanism. The greatest differences concern the value of cohesion, which can vary up to sixfold depending on the measurement equipment used. Common tests for soil strength include direct shear tests and triaxial compression tests. [PERSON] et al. [2] compared results from triaxial compression and direct shear tests and concluded that the direct shear apparatus provides lower values of the internal friction angle. Issues related to comparing the internal friction angle in triaxial compression and extension tests and direct shear tests are further discussed in [3, 4]. Laboratory triaxial compression and direct shear tests clearly show that shear stresses can reach a maximum value during shearing. After reaching this value, the stresses either remain constant or decrease to a critical value with continued shearing. The same type of soil in direct shear testing can sometimes exhibit significant strength surpassing and then dropping to a residual value, while under different testing conditions, it may behave differently. This means that the peak value achieved is not solely dependent on the type of soil. This phenomenon necessitates a precise definition of the soil's internal friction angle. Two concepts are commonly accepted: \(\phi_{p}\), the peak internal friction angle corresponding to the maximum shear stresses during testing, and \(\phi_{cv}\), the critical internal friction angle corresponding to the strength obtained at constant sample volume. Ref. [5] indicated that the peak internal friction angle was influenced by soil compaction, average effective stresses, stress path, and basic soil shear strength defined by the critical state internal friction angle. Many studies [6; 7; 8] indicated that soil changed its volume during shearing. Initially highly compacted soils exhibited an increase in porosity during shearing, while initially loose soils showed a decrease in porosity. Shearing soil in the critical state occurred at a porosity index known as the critical porosity index. Studies indicated that the higher the average effective stresses, the lower the critical porosity index. Analyses of this phenomenon allowed for the determination of the critical state line (CSL). Soils above the CSL exhibited contractive behaviour, while soils below it exhibited dilative behaviour. Ref. [6] showed how the soil porosity index changes depending on the logarithm of average soil stresses. It was indicated that the change in grain shape significantly affects the value of the critical internal friction angle and the slope of the CSL. The CSL also allowed for a deeper analysis of phenomena related to soil liquefaction, as described in [9; 10]. The CSL pertains to the critical state and is directly related to the critical internal friction angle. The issue of the peak internal friction angle in relation to the critical internal friction angle is described in the literature by the state parameter defined as the difference between the initial porosity index and the critical porosity index [7]. Research results indicated that the difference between the peak and critical friction angles, reaching up to 20 degrees, was greatest at negative state parameter values and decreased to a range of 0 to 4 degrees at state parameter values greater than or equal to 0 [7; 11; 12].