A note on nonassociated plastic flow rules

The analytical properties of the constitutive equations in plasticity with a nonassociated flow rule are investigated. Under the assumption of small deformations the directional stiffness (and compliance) rule is considered and the relevant spectral properties of the tangent stiffness tensor are assessed. It is shown that the directional stiffness may be larger than the elastic. It may also be negative in the case of a formally perfectly plastic material and so the nonassociative flow rule represents (spurious) softening in terms of an associated flow rule. The issue of uniqueness at finite strains is briefly addressed, whereby use is made of the tangent stiffness tensor relating the velocity gradient to the first Piola-Kirchhoff stress rate. The relevant spectral properties, which generalise those from the small deformation case, are found explicit. A sufficient condition for uniqueness is given in terms of a critical (upper bound) value of the hardening modulus.