In the present study, three-axis dynamic tests were performed on different samples of soil, and the effect of montmorillonite based carbon nanotubes (CNT) and amorphous silica dioxide nanoparticles presence, density, hydrostatic pressure and moisture percentage on the elastic modulus and equivalent damping ratio of soil was investigated. Considering that in the present research, experimental studies and tests were carried out on three samples of sandy soil reinforced with 2 % volume fraction CNT, sandy soil reinforced with 8 % nanosilica and sandy soil reinforced with 0.5 % CNT and 4 % nanosilica. Also, the optimum moisture percentage has been determined for these types of soils. In general, with the increase of hydrostatic pressure and compaction of sand, the elastic modulus increases, and the amount of increase is different according to the type of nanoparticles. For 1 % applied strain and soil sand, with the increase of hydrostatic pressure from 100 kPa to 300 kPa, the elastic modulus of sand with 2 % CNT and 8 % nanosilica increases by 36 % and 214 %, respectively. This shows the favorable effect of silica nanoparticles on increasing the Young's modulus of sand by increasing the amount of hydrostatic pressure. The effect of CNT on improving the elastic modulus of sand in dry state is much higher than the effect of nanosilica. At 1 % strain and 4 % compaction, adding 2 % CNT to sand increases the elastic modulus of sand about 4 times compared to sand reinforced with 8 % nanosilica.

Focusing mainly on the compressive-shear (C-S) stress, the existing soil strength theories fail to well describe the soil strength under tenso-shear (T-S) stress and are not suitable for analyzing the tensile strength and T-S coupling strength. Taking the T-S strength into account, a three-dimensional nonlinear strength model of soil is established. The failure function on the pi plane is a Lade Lade-curved triangle in principal stress space, which reasonably reflects the effect of medium principal stress. The failure function on the triaxial compression meridian plane is divided into the T-S and C-S zones, in which the two failure functions are smoothly connected at the closed stress point. To easily combine it with the specific elastoplastic constitutive theory, the proposed strength model is transformed into the D-P criterion in transformed stress space using the transformation stress method, which provides a theoretical foundation for numerical calculations. Compared with the true triaxial test results and numerical slope examples, the proposed model gives a good description of the nonlinear soil strength under complex stresses and the instability failure mechanism of the slope.