A Multiscale Geometric PDE Framework Coupling Macroscopic Curvature and Microscopic Grain Boundary Motion: An Application to Snow Deformation and Sintering

hiroshi matsuda , Shin'ichi HOMMA

The motion of grain boundaries in polycrystalline materials and porous media has conventionally been modeled by local geometric evolution equations, such as Mean Curvature Flow (MCF). However, existing models primarily depend on local interfacial geometry and generally do not account for the influence of macroscopic curvature fields induced by the deformation of the bulk continuum. In this study, we propose a multiscale geometric continuum framework in which the macroscopic Gaussian curvature field modulates the evolution velocity of the microscopic MCF through a variationally motivated geometric driving term.
In our framework, for analytical clarity, the macroscopic continuum is restricted to a two-dimensional manifold. Its deviation from a stress-free state due to non-uniform deformation is described as the emergence of a nonzero Gaussian curvature field. Simultaneously, at the microscopic scale, the thermodynamic interface growth vector governed by the Laplace-Beltrami operator is formulated. This model integrates the dynamics of two distinct scales---the accumulation of strain energy due to macroscopic curvature and the microscopic topological changes (bonding and rupture of the grain boundary network)---into a single geometric evolution equation.
We discuss the mathematical well-posedness of the proposed coupled system, indicating local-in-time existence under bounded curvature assumptions (|K| < K0). As an illustrative application, the framework is applied to the macroscopic deformation and microscopic sintering process of snow, a porous material undergoing structural changes near its melting point. This geometric approach provides an analytical foundation for examining the mechanisms of fracture and topology changes in materials undergoing phase changes.