Formulating mesodynamics for polycrystalline materials

For materials consisting of polycrystalline grains, we formulate a minimal basis for mesoscale dynamics— mesodynamics —where the mass points are individual mesoscale grains that interact via a short-ranged and pairwise-additive mesopotential. Newton's equations of motion for the grains are augmented by relative-velocity viscous damping between grains, representing sub-grain dissipative processes. We require, at minimum, two things of the mesopotential: i) in compression, it must be consistent with the nonlinear elastic equation of state, and ii) under tension, the mesoscale bonding between two grains must reflect the fact that grains separate at their mutual interface (grain boundary) rather than in the bulk. We then show that a polycrystalline system interacting by this minimal mesopotential fails under tension at a yield strength that decreases inversely with the square root of grain size, with a predicted coefficient that agrees remarkably well with Hall-Petch experimental values.