A harmonic/anharmonic energy partition method for lattice statics computations

Abstract

A method of lattice statics analysis is developed. Consideration of anharmonic effects is restricted to finite regions surrounding lattice defects. All displacements of the crystal are expressed as the effect of unknown forces applied to a perfect harmonic lattice of infinite extent. Displacements are related to the unknown applied forces by means of the Green function of the perfect harmonic lattice, so that equilibrating forces need only be applied to the anharmonic region. The unknown forces are determined so as to maximize the complementary energy of the crystal, which yields a lower bound to the potential energy. The method does not require the explicit enforcement of equilibrium or compatibility conditions across the boundary between the harmonic and anharmonic regions. The performance of the method is assessed on the basis of selected numerical examples. The rate of convergence of the method with increasing domain size is found to be cubic. This is one or two orders of magnitude faster than rigid boundary methods based on the harmonic and continuum solutions, respectively.