Equilibrium, Stability and Evolution of Elastic Plates under the Combined Effects of Stress and Surface Diffusion

The particles in an elastic plate are permitted to move by a surface diffusion process subject to the constraint that the total free energy does not increase. The static equilibrium, the quasistatic linear stability, and the quasi-static nonlinear evolution of the surface are examined under different loading conditions: tensile/compressive or #exural. The equilibrium con"gurations are such that the surface value of the chemical potential is constant, and their shapes depend upon the relative magnitude of elastic to surface energies. A linear stability analysis indicates that antisymmetric perturbations to the surface pro"le of a #at plate are most unstable for tensile loading and symmetric perturbations display the greatest instability under #exure. A new model for nonlinear non-equilibrium mechanics of thin plates is described and analysed. The main feature is that the elastic energy at the surface is approximated by that of an equivalent thin plate in a state of uniaxial stress, even as the pro"le changes. Nonlinear evolution of a perturbed #at plate is illustrated by numerical solution. A crevice gradually develops in the plate, eventually leading to rapid rupture and breakage. Scaling analysis near the ultimate rupture indicates a simple spatial and temporal dependence.