A three-dimensional fractional elastoplastic constitutive model for rocks within ductile domain

Under the influence of high three-dimensional geostress, rocks transition into the ductile domain, undergoing continuous plastic hardening and volumetric contraction. Accurately describing the three-dimensional anisotropic deformation of rocks within ductile domain is of great significance for deep underground engineering. Therefore, a three-dimensional fractional elastoplastic constitutive within ductile domain is proposed in this study, including yield function and fractional flow rule. The ductile yield function is based on the modified Mohr-Coulomb criterion and generalized Matsuoka-Nakai deviatoric function. The deviatoric stress of yield surface is negatively correlated to hydrostatic pressure, but positively correlated to Lode angle. The yield surfaces in both meridian and deviatoric planes evolve with the plastic internal variable, accurately capturing the stress state during hardening. Two different fractional orders are used to control the plastic flow direction within meridian and deviatoric planes, represented by dilation angle and plastic deflection angle, respectively. These fractional orders are determined based on the relationship between plastic shear strain and volumetric strain, and they vary with the plastic internal variable, effectively capturing the plastic flow direction throughout hardening. The proposed model is validated using green sandstone data from hydrostatic compression and true-triaxial tests. The effect of fractional orders on the dilation angle and plastic deflection angle is discussed. Under the influence of fractional orders, both dilation angle and plastic deflection angle range from $0^{\circ }$ to $-{90}^{\circ }$. Besides, a comparison between the non-orthogonality and orthogonality flow rules is made. These results indicate that the fractional flow rule significantly improves the applicability and accuracy of constitutive model.