Abstract
The necessary condition for multiple deformation rate was given by Hill. The condition is ΔsijΔvji=O, where Sij denotes the nominal stress rate, and Δ shows the difference between any two multiple solutions. We discovered that the necessary condition makes the difference ΔΛ of proportional coefficient in the constitutive equations under the associated flow theory indefinite at the critical state. The sufficient condition for the multiplicity is deduced from the requirement which makes the coefficient Λ indefinite. That is.sijεij=O, which requires that the hardening modulus h vanishes, and gives the relationship which influences upon the gradient CL of the critical curve. Since the relationship involves the third invariant J3 of deviatoric stress, the critical state cannot be uniquely determined by the pressure p and equivalent stress q alone. In the plane strain compression, the mode of shear plane can be predicted similar to the one observed in such experiments, based on the gradient CL of the critical state curve.
