To investigate the deformation characteristics of loess under different stress paths

a bi-directional triaxial apparatus was used to perform constant compression tests ( is deviatoric stress)

constant shear tests ( is spherical stress)

and constant stress ratio tests. The cross-influence mechanism of spherical stress and deviatoric stress on volumetric strain and shear strain was systematically analyzed. The experimental results indicate that the spherical stress not only generates volumetric strain

but also results in radial strain being greater than axial strain (about 5.79 times) due to anisotropy

leading to negative shear strain; while the deviatoric stress generates shear strain

it is also accompanied by approximately 1%―3% shear-induced volumetric shrinkage. This indicates that the deformation of loess is jointly dominated by spherical stress and deviatoric stress

forming 4 types of cross stress-strain relationships:

and (where and are the volumetric strains caused by spherical stress and deviatoric stress

respectively; sp and sq are the shear strains caused by spherical stress and deviatoric stress

respectively). Given that traditional models are difficult to describe such cross relationships

this study extends the model framework (where K is the bulk modulus and G is the shear modulus) by adding cross moduli and on the basis of the original bulk modulus and shear modulus

quantifying the volumetric strain caused by deviatoric stress and the shear strain caused by spherical stress

respectively. By introducing the stress ratio hardening coefficient

a four-modulus incremental nonlinear constitutive model capable of reflecting the anisotropy and shear shrinkage characteristics of loess is developed. By solving the model parameters and comparing them with experimental data

the accuracy of the model in predicting the mechanical response of loess under different loading paths was validated. The research provides new methods and theoretical foundations for a deeper understanding of the mechanical behavior of loess and its engineering applications.