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, 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 (2004) distortion gradient plasticity, adopting Nye’s dislocation density tensor as primal HO variable, we expect our results to hold qualitatively for any HO strain gradient plasticity theory, including crystal plasticity.