Statistical mechanics of plasticity: Elucidating anomalous size-effects and emergent fractional nonlocal continuum behavior

Extensive experiments over the decades unequivocally point to a pronounced scale-dependency of plastic deformation in metals. This observation is fairly general, and broadly speaking, strengthening against deformation is observed with the decrease in the size of a relevant geometrical feature of the material, e.g., the thickness of a thin film. The classical theory of plasticity is size-independent, and this has spurred extensive research into an appropriate continuum theory to elucidate the observed size effects. This pursuit has led to the emergence of strain gradient plasticity, along with its numerous variants, as the paradigm of choice. Recognizing the constrained shear of a thin metallic film as the model problem to understand the observed size-effect, all conventional (and reasonable candidate) theories of strain gradient plasticity predict a scaling of yield strength that inversely varies with the film thickness $\sim h^{-1}$. Experimental findings indicate a considerably diminished scaling, the magnitude of which can exhibit significant variation based on processing conditions or even the mode of deformation. As an example, the scaling exponent as low as $-0.2$ has been observed for as-deposited copper thin films. Two perspectives have been posited to explain this perplexing anomaly. Kuroda and Needleman (2019) argue that the conventional boundary conditions used in strain gradient plasticity theory are not meaningful for the canonical constrained thin film problem and propose a physically motivated alternative. Dahlberg and Ortiz (2019) contend that the intrinsic differential calculus structure of all strain gradient plasticity theories will invariably lead to the incorrect (or rather inadequate) explanation of the size-scaling. They propose a fractional strain gradient plasticity framework where the fractional derivative order is a material property that correlates with the scaling exponent. In this work, we present an alternative approach that complements the existing explanations. We create a statistical mechanics model for interacting microscopic units that deform and yield with the rules of classical plasticity, and plastic yielding is treated as a phase transition. We coarse-grain the model to precisely elucidate the microscopic interactions that can lead to the emergent size-effects observed experimentally. Specifically, we find that depending on the nature of the long-range microscopic interactions, the emergent coarse-grained theory can be of fractional differential type or alternatively a form of integral nonlocal model. Our theory, therefore, provides a partial (and microscopic) justification for the fractional derivative model and makes clear the precise microscopic interactions that must be operative for a continuum plasticity theory to be a valid phenomenological descriptor for capturing the correct size-scale dependency.