Cyclic spherical stresses are prevalent in dynamic stress fields and significantly influence the dynamic behavior of loess, a material characterized by high compressibility and anisotropy. Previous research has primarily focused on shear stresses, often overlooking the impact of spherical stresses. This study investigated the deformation induced by cyclic spherical stress under different initial states. Irreversible and reversible components were identified from both volumetric and shear strains, and their variation patterns were analyzed. Shear strain is found to be generated by the material's anisotropy. The results indicate that the volume of the sample shrinks significantly under cyclic spherical stress, with irreversible volumetric strain increasing nonlinearly as the number of cycles increases. Irreversible shear strains can be categorized into two types based on their formation mechanisms. The first is when significant initial anisotropy leads to radial deformation greater than axial deformation under spherical stress, resulting in shear strain increasing in the negative direction. As consolidation stress increases, the initial anisotropy gradually diminishes. The second is when stress-induced anisotropy results in positive shear strain because consolidation deviatoric stress contributes to an increase in shear strain in the positive direction. As the stress ratio rises, the induced anisotropy is further enhanced. The axial reversible strain of the sample is minor, and the reversible components of volumetric and shear strains primarily arise from radial contraction and expansion. As the spherical stress increases, the sample volume shrinks (positive volumetric strain), whereas the initial anisotropy leads to negative shear strain, resulting in opposite signs. Finally, a method for predicting irreversible strain under cyclic spherical stress is established based on a memoryless geometric distribution.

Under various stress paths, the deformation characteristics represented great differences. In this paper, a series of cyclic triaxial tests have been conducted with Fujian standard sand. By comparing the constant deviatoric (CDS) and constant axial stress paths (CAS), the influence mechanism of the cyclic amplitude of the deviatoric stress was discussed. The test results showed that the stress path significantly influenced the volumetric and shear strains. The increasing and decreasing trend in the volumetric strain (epsilon v) was consistent with the spherical stress (lnp). Compared with the two stress paths, the slope of the epsilon v-lnp curve during the loading and unloading stages was larger under the CAS path. In the CDS path, qc almost did not affect the cumulative volumetric strain, and in the CAS path, the effect was obvious. The shear strain curve was in accordance with the direction of the stress path. As the cyclic number increased, the shear strain gradually accumulated. The shear strain accumulation under the CAS path was larger. The shear strain largely depended on the relative position between the critical state line (CSL) and the stress state of the soil during cyclic loading and unloading. In practical engineering, the soil will experience various stress paths. For example, in slope or earth-rock dam engineering, where the water level rises and falls repeatedly, the soil often goes through the stress path of constant deviational stress with the cyclic increase and decrease in the spherical stress. In foundation pit engineering, the soil often experiences the stress path of the constant axial stress (CAS) with cyclic loading and unloading of the lateral stress. The stress path greatly influences the deformation and strength of soil. Therefore, the previous two stress paths are compared in this paper to discuss the influence of the cyclic amplitude of deviatoric stress. Under three different consolidation states, the cyclic amplitude of the deviatoric stress significantly influenced the volumetric and shear strains. The shear strain largely depended on the relative position between the critical state line (CSL) and the stress state of the soil during cyclic loading and unloading. Therefore, in practical engineering, if the stress path in the experiment differs from the actual value, the influence of the stress path should be properly considered. The results should be modified according to the degree of influence of each stress condition.