Stress singularities along a cycloid rough surface

Abstract

The stress concentration along a rough surface is of importance for understanding the nucleation of misfit dislocations and cracks in heteroepitaxial thin films and for general flaw initiation at material surfaces exposed to environmental corrosion. In order to seek a basic understanding on this issue, a cycloid wavy surface subject to a uniform bulk stress is adopted as a model problem. The elastic stress and displacement fields are determined using Muskhelishvili's conformal mapping method. It is shown that a cusped cycloid surface generates a crack-like singular stress field within a thin surface layer; under uniform tension this singularity is found to have identical strength to a row of periodic parallel cracks. The path-independent J-integral requires that the average surface strain energy density be identical to that in the bulk, implying that any stress magnification along a rough surface must be compensated by unloading along some complementary portions of the surface. Along the cusped cycloid surface, the strain energy distribution becomes Dirac singular at the cusp tips, the rest of the surface being completely stress-free. Thus the effect of cycloid cusps is to redistribute and concentrate all the surface strain energy per wavelength at a single (cusp) point, and because of that we can claim that the cycloid surface is the most efficient stress concentrator at a fixed wavelength. Even at a moderate bulk stress level this concentration may be sufficient to cause failure or nucleation of defects.The full evolution of a rough surface under stress and other corrosion mechanisms must be solved by numerical methods. Our analytic solutions for a cycloid surface are of significant value for guiding numerical computations and gaining insights into essential features of the evolution process. From a global point of view, we show that a cusped cycloid surface becomes energetically favorable once the surface wavelength exceeds a critical value determined from the competition between the strain energy and the surface energy. From a local point of view, analysis of surface diffusion behaviors along an almost cusped cycloid surface indicates that the cusps are stable once they develop. The critical condition for formation of a cusped cycloid corresponds to the Griffith energy balance being exactly satisfied at the cusp tips while the chemical potential remains nearly constant along the rest of the surface. This implies spontaneous Griffith brittle fracture at tension cusps if plastic relaxation is not present to relieve the stress singularity.