Shakedown theorems for elastic–plastic solids in the framework of gradient plasticity

Static and kinematic shakedown theorems are given for a class of generalized standard materials endowed with a hardening saturation surface in the framework of strain gradient plasticity. The so-called residual-based gradient plasticity theory is employed. The hardening law admits a hardening potential, which is a C1-continuous function of a set of kinematic internal variables and of their spatial gradients, and is required to satisfy a global sign restriction (but not to be necessarily convex). The totally produced, the accumulated and the freely moving dislocations per unit volume, distinguished as statistically stored and geometrically necessary ones, are in this way accounted for. Like for a generalized standard material, the shakedown safety factor is found to depend on the (generally size dependent) yield and saturation limits, but not on the particular differential-type hardening law of the material.