A potential for higher-order phenomenological strain gradient plasticity to predict reliable response under non-proportional loading

We propose a plastic potential for higher-order (HO) phenomenological strain gradient plasticity (SGP), predicting reliable size-dependent response for general loading histories. By constructing the free energy density as a sum of quadratic plastic strain gradient contributions that each transitions into linear terms at different threshold values, we show that we can predict the expected micron-scale behaviour, including increase of strain hardening and strengthening-like behaviour with diminishing size. Furthermore, the anomalous behaviour predicted by most HO theories under non-proportional loading is avoided. Though we demonstrate our findings on the basis of Gurtin (Gurtin 2004 J. Mech. Phys. Solids 52, 2545-2568, doi:10.1016/j.jmps.2003.11.002) distortion gradient plasticity, adopting Nye's dislocation density tensor as primal HO variable, we expect our results to hold qualitatively for any HO SGP theory, including crystal plasticity.