Summary: The class of generalised standard materials is not relevant to model the non-associative constitutive equations. The possible generalisation of Fenchel’s inequality allows the recovery of flow normality rule for non-associative behaviour. Here the normality rule is written in weak form as an implicit relation. This leads to the introduction of the class of implicit standard materials. This formulation is applied to constitutive equations involving nonlinear kinematic hardening, indispensable to describe accurately and realistically the cyclic plasticity of metallic materials. For these plastic flow rules we extend the shakedown bound theorems; an analytical example of the shakedown of a thin-walled tube under constant traction and alternate cyclic torsion is considered, and the obtained solution is proved to be exact.