A note on some features of the theory of localization of deformation

This note examines two aspects of the theory which treats localization of deformation as a bifurcation from homogeneous deformation. The results are obtained for solids modelled as elastic-plastic and having smooth yield and plastic potential surfaces, but it is not required that inelastic strain increments be normal to the yield surface. First, it is demonstrated that discontinuous bifurcations, for which elastic unloading occurs outside the zone of incipient localization, first become possible at the point of continuous bifurcation, for which further plastic deformation is assumed to occur both inside and outside of the zone of localization. Second, we investigate an apparent paradox which arises in the rigid plastic limit of an elastic-plastic localization calculation if normality does not apply. This is resolved by consideration of the relative amounts of the bifurcation mode corresponding to elastic and to plastic deformation and it is demonstrated that even for very small amounts of elasticity, bifurcation modes which are inadmissible in the rigid plastic case become possible.