Surface effects, boundary conditions and evolution laws within second strain gradient plasticity

•Inner PVP for microforces and evaluation of the energy residual for second strain gradient plasticity.•Surface effects in second strain gradient plasticity and formation of a microstructural boundary layer.•Insulation condition as energy feature of the body/environment system.•“Rest” boundary conditions for the elastic/plastic interface besides the usual continuity ones.•Mixed modeling scheme for the evolution laws to avoid rigid-plastic behaviors.

The principle of the virtual power (PVP) is used in conjunction with the concepts of “energy residual” and “insulation condition” to address second strain gradient plasticity. The energy residual with its typical divergence format is an extra stress power playing the role of basic state variable to describe the gradient effects, whereas the insulation condition constitutes a global energy characterization of the body as part of the body/environment system. The microstructure of a second strain gradient material (but not of a first strain gradient one) is shown to exhibit surface effects with the formation of a thin boundary layer. This boundary layer is in local (and global) equilibrium according to the principles of the material surface mechanics and supports the boundary microtractions, except a part (Cauchy-like traction) transmitted to the bulk microstructure; it works as a structured two-dimensional manifold replacing the conventional purely geometrical concept of boundary surface. By the insulation condition the higher order boundary conditions are determined (for first and second strain gradient plasticity), including those for the moving elastic/plastic interface. Besides the usual continuity boundary conditions, some extra (at most three) “rest” boundary conditions are required to fix the current location of the interface. The restrictions on the constitutive equations and the evolution laws are also addressed, whereby a mixed modeling scheme is used to model the dissipative stresses. An application to an elastic-softening bar in extension shows the ability of the proposed model to capture and describe the formation of a multi-waved deformation pattern within the localization band.