Dynamic size effects across the scales: Localization in round notched bars

Abstract

The strengthening and hardening of metals increase with diminishing size due to plastic strain gradient accompanied by Geometrically Necessary Dislocations (GNDs). In contrast, the inertia forces under severe dynamic loading conditions decrease. To shed the light on the interaction between the two mechanisms, the present work focuses on the localization in notched bars subject to high strain-rate tensile loading and first lays out a new computational model where the Fleck and Willis strain gradient plasticity theory is implemented into a dynamic finite element framework. The new framework allows accounting for strain gradient strengthening through the non-local plasticity material model and models the “smaller is stronger” mechanism (controlled by a built-in length parameter), while material inertia governs the “larger is more dynamic” effect. The work by Needleman (2018) [Effect of size on necking of dynamically loaded notched bars, Mech. Materials, 116:180-188] is taken as a starting point, and the present study first conveys the message that the scale-dependent dynamic effect observed in visco-plastic tensile bars also exists for rate-independent materials across scales (questioned by Needleman, 2018). Subsequently, the notched bar’s size and corresponding loading are scaled to the micrometer size range where plastic strain gradients influence the material response. Focusing attention on the shift between plastic localization in the notch and away from the notch (in the region subject to uni-axial tension) allows cementing this shift at the micron scale. The study shows that gradient strengthening owing the GNDs stabilizes the early localization in the notch and delays the shift between the localization mechanisms when the size of the notched bar diminishes while keeping the material length parameter constant. Moreover, severe gradient strengthening gives rise to a slow-growing neck-like (micron scale) localization mechanism where a neck-type deformation evolves near the notch — but not in the notch as oppose to larger bar sizes.