An implicit gradient plasticity-damage theory for predicting size effects in hardening and softening

An implicit gradient plasticity–damage theory is constructed, which can capture size effects in hardening plasticity, as well as regularise the localisation of deformation due to softening. In hardening, boundary layers of finite thickness are formed due to constraints imposed on the plastic deformation. Upon softening, a localisation band of finite thickness emerges. Both thicknesses are governed by an intrinsic length scale which is incorporated via an implicit gradient formalism. Numerical solutions for a constrained layer in simple shear illustrate the typical responses obtained. As the thickness of the layer is varied, size effects are predicted in hardening as well as in softening. As the damage evolution progresses, the strain distribution progressively localises, culminating in a discrete crack.