Homogenization of Surface Energy and Elasticity for Highly Rough Surfaces

Abstract

Surface energy plays a central role in several phenomena pertaining to nearly all aspects of materials science. This includes phenomena such as self-assembly, catalysis, fracture, void growth, and microstructural evolution among others. In particular, due to the large surface-to-volume ratio, the impact of surface energy on the physical response of nanostructures is nothing short of dramatic. How does the roughness of a surface renormalize the surface energy and associated quantities such as surface stress and surface elasticity? In this work, we attempt to address this question by using a multi-scale asymptotic homogenization approach. In particular, the novelty of our work is that we consider highly rough surfaces, reminiscent of experimental observations, as opposed to gentle roughness that is often treated by using a perturbation approach. We find that softening of a rough surface is significantly underestimated by conventional approaches. In addition, our approach naturally permits the consideration of bending resistance of a surface, consistent with the Steigmann–Ogden theory, in sharp contrast to the surfaces in the Gurtin–Murdoch surface elasticity theory that do not offer flexural resistance.