Establishing a quantitative relationship between strain gradient and hetero-deformation-induced stress in gradient-structured metals

Recent experimental evidence suggests that hetero-deformation-induced (HDI) stress does not increase linearly with an increasing strain gradient, which contradicts conventional strain gradient plasticity theory. To resolve this discrepancy, a strain gradient plasticity formulation is modified in this study, taking into account the saturation of geometrically necessary dislocation (GND) accumulation due to the dynamic equilibrium between Frank–Read source formation and deactivation. It is incorporated into a crystal plasticity finite element framework, along with two analytical models accounting for the development of statistically stored and geometrically necessary dislocations at grain level, respectively, to explore the underlying relationship between deformation mechanisms and mechanical behaviors of a gradient-structured (GS) Cu material processed via single-roll angular-rolling under deformation. It is revealed that sample-level GND density contributes substantially to the total GND density in GS Cu, giving rise to significant HDI strengthening and work hardening. Furthermore, a decrease in grain size leads to an increase in strain gradient intensity and a decrease in the saturation density of GNDs at sample level, resulting in a higher initial increase rate and an earlier slowdown of sample-level GND density with increasing strain. It also leads to an increase in GND density gradient, giving rise to higher HDI stress at the sample level. Last but not least, a quantitative relationship between strain gradient and HDI stress can be established via the modified strain gradient plasticity formulation, which is capable of capturing scale-dependent mechanical responses of polycrystalline metals with heterogeneous microstructures.