On a mixed energetic-dissipative constitutive law for non-proportional loading, with focus on small-scale plasticity

We analyse the Mixed energetic-dissipative Potential (MP) recently proposed by our group to predict, within higher-order strain gradient plasticity (SGP), reliable size-dependent response under general loading histories. Such MP follows former proposals by Chaboche, Ohno, and co-workers for nonlinear
kinematic hardening in the context size-independent metal plasticity. The MP is given by M quadratic addends that each transitions, at a different threshold value, into a linear dissipative contribution. Hence, the MP involves 2M positive material parameters, given by the M threshold values and M moduli weighing each quadratic recoverable term. We analytically demonstrate that, under proportional loading, the MP limit for M->infinity converges to a less-than-quadratic
potential with well defined properties. This result is of crucial importance to identify the material parameters of any model adopting the MP.
Moreover, our analysis provides a formula for the characterisation of the energetic and dissipative parts of any possible MP limit, showing that,
regarding the capability to describe the effect of diminishing size within SGP, the MP can be selected such as its contribution to the strengthening (i.e., an increase in yield point) is mostly dissipative, whereas its contribution to the increase in strain hardening is mostly recoverable.