The bifurcation analysis was derived in three-dimensional stress states by using a critical state constitutive model by Yao et al. (2004)

which based on the nonassociated flow rule. Theoretical analysis shows that

bifurcation is dependent closely on the stress path. Bifurcation occurs in the hardening regime under the Lode angle changing between minus twenty-five degree and fifteen degree

and the stress ratio

principal strain at bifurcation and inclination angle of shear bands first increase and then decrease with increasing of the Lode angle. But there is no bifurcation occurring in other stress path under constant mean stress. Finally

the return mapping algorithm is adopted in order to implement the model into a nonlinear finite element analysis software ABAQUS through the user material subroutine (UMAT) interface. Numerical simulation results show that the bifurcation also exists in the numerical analysis. The comparison between the numerical results and theoretical solutions indicates that both are in good agreement.