A cross-scale stress gradient plasticity theory for length-scale effects on hardening behaviors of microbeam bending

Abstract

Understanding and defining intrinsic length-scales is the key to developing continuum plasticity theories that accurately capture size-dependent behaviors. This study presents a cross-scale stress gradient plasticity (C-σGP) theory that couples the dynamics of soft dislocation pile-up in stress gradients with continuum mechanics without resorting to phenomenological evolution (hardening) laws. The theory explicitly incorporates four material length-scales: slip-band spacing, dislocation source length, dislocation pile-up length, and boundary layer thickness. We implemented the C-σGP model using an implicit algorithm to simulate the pure bending behavior of single-crystalline microbeams. Results show that only two intrinsic length-scales are required to capture the size-dependent bending strength at different strain stages. One is the source length that controls the yield strength, and the other is the slip-band spacing which governs the post‑yield hardening. Moreover, this study reveals for the first time how the evolution of slip-band spacing with plastic strain significantly affect the strain‑hardening rate and flow intermittency observed at micro‑ and sub‑micron scales. By identifying and quantifying these intrinsic lengths, the C-σGP framework provides a physically grounded foundation for future gradient‑enhanced plasticity models of small‑scale structures.