Localization analysis via a geometrical method

geometrical technique is proposed in order to solve explicitly the critical conditions at localization far a quite general constitutive behaviour with isotropic elastic properties. These critical conditions are shown to be closely related to the spectral properties (eigenvalues and eigenvectors) of the sum and difference of two tensors describing the inelastic effects. When these two tensors are coaxial, it is shown that the normal to a potential localization plane always lies in one of their principal planes. It is also demonstrated that, depending on the constitutive behaviour and the loading conditions, several expressions for the critical hardening modulus at localization are available and their respective domain of validity well defined. The roles and interactions of both deviatoric and hydrostatic non-associativities in the critical conditions of localization are emphasized