A Physically Grounded Model for Size Effects in the Initial Yielding of Metallic Materials with Deformation Heterogeneity

The size effect in the initial yielding of metallic materials with deformation heterogeneity has garnered significant attention. However, the underlying physics of this effect remains unclear, and physically grounded models that quantify the relationship between microstructure and mechanical properties are still lacking. Here, we revisit both stress and strain gradient plasticity models, focusing particularly on the stress gradient model due to its physical material length scale and straightforward numerical implementation. By deriving yield stress models based on single-ended dislocation pileup, we identify a critical issue in stress gradient models: the assumption of dislocation pile-up configurations significantly affects yield stress predictions. To elucidate the dislocation mechanisms driving the size-dependent yielding behavior, we investigate two benchmark cases in gradient theories: homogeneous materials undergoing nonuniform deformation and heterostructured materials undergoing uniform deformation, utilizing nonlocal crystal plasticity and discrete dislocation dynamics simulations, respectively. The results not only clarify the issue raised in stress gradient theory, but also suggest the mechanism that pileup-induced stress plays a dominant role in governing the size effect during initial yielding for both homogeneous materials and heterostructured materials. These insights lead to the development of a new physically grounded model based on pileup-induced internal stress, i.e., back stress, which quantitatively predicts the size effect in the initial yielding of heterostructured material under tension and homogeneous material under torsion. This work clarifies the dislocation mechanisms governing extra strengthening in metallic materials with deformation heterogeneity and introduces a physically-based model quantitatively correlating the microstructures with the mechanical properties of heterostructured materials.