# Dilatant Nature of Sand Shear Strength
## 1 Introduction  
The strength of sand (the secant friction angle) depends on its initial density, stress level, anisotropy, and deformation mode. The oldest and simplest Mohr-Coulomb failure criterion is most frequently used in engineering practices [1]. In the Mohr-Coulomb criterion, the influence of the intermediate principal stress on the sand strength is ignored, and its application gives a conservative solution. Among the many other criteria, the best known are those described by [PERSON] and [PERSON] [2], and [PERSON] and [PERSON] [3], which take into account the influence of the intermediate principal stresses on the strength of the sand. [PERSON] and [PERSON] [4], [PERSON] and [PERSON] [5], and [PERSON] et al. [6] investigated the effect of the initial anisotropy of sand on its strength. In geotechnical laboratories, conventional drained triaxial compression tests are most commonly performed to determine the strength of sand. The strength of sand for drained triaxial compression can be expressed as a function of the critical state friction angle (\(\phi^{\prime}_{cv}\)), the initial density index (\(I_{D}\)), and the mean normal stress (\(p^{\prime}\)) at failure [7]. In engineering practice, the most common are plane strain conditions, which are simulated in biaxial compression (BXC), direct and simple shear tests. In general, for dilative behavior, the shear strength of sand under plane strain conditions is greater than in triaxial compression at the same stress level [1; 8; 9; 10; 11]. The drained triaxial extension tests are sensitive to small errors, owing to the necking phenomenon, which makes the extension tests unstable [12; 13]. The angle of friction is slightly greater in most triaxial extension tests, compared with the results of triaxial compression tests under one set of initial conditions [14; 15; 16; 17]. The influence of the intermediate principal stress on the sand strength was investigated using various true triaxial shear apparatuses. The results of these true triaxial tests, which were collected by [PERSON] et al. [18] and [PERSON] [12], show very different indirect principal stress effects on a sand's shear strength. The main influences on the sand strength includethe shear banding, the boundary conditions, the initial sample anisotropy, the sample slenderness, and the experimental technique stability [12]. The influence of the intermediate principal stress on the sand strength is better expressed by the ratio of the friction angle to the friction angle in triaxial compression, as opposed to the friction angle alone [19]. Shear deformations cause volumetric changes and directly affect the strength of sand. The relationship between the strength and dilatancy for direct shear was given by [PERSON] [20]. Simple relationships between the sand strength and the dilatancy for triaxial compression (TXC), plane strain (biaxial compression (BXC)), and triaxial extension (TXE) were given by [PERSON] [21]. [PERSON] [7] analyzed the TXC and BXC test results of 17 different sands and found a relationship between the sand strength, the dilatancy, and the relative dilatancy index. The relative dilatancy index combines the effects of the initial densities and the stress levels on the strength and dilatancy of sand. The general stress-dilatancy relationship was derived from the friction state concept (FSC) of [PERSON] [22]. For failure states, a purely frictional behavior can be observed for sand shearing without the breakage effect [23; 24; 25; 26]. The effect of breakage on the relationship between the strength and the dilatancy during the shearing of railway ballast, limestone gravel, and rockfill materials was observed by [PERSON] [27] and [PERSON] [28; 29]. This study shows that, under general conditions, the sand shear strength can be expressed as a function of the critical state angle and the dilatancy for TXC using the FSC. First, it is shown that the shear strength of sand can be correctly expressed by the critical state angle and the dilatancy for TXC, BXC, and TXE.