Cubic Single Crystal Representations in Classical and Size-dependent Couple Stress Elasticity

Abstract

Beginning with Cosserat theory in the early 20th century, there have been several different formulations for size-dependent elastic response. In this paper, we concentrate on the application of classical Cauchy theory and the recent parsimonious consistent couple stress theory to model a homogeneous linear elastic solid, exemplified by a pure single crystal with cubic structure. The focus is on an examination of elastodynamic response based upon wave velocities from ultrasonic excitation and phonon dispersion curves, along with adiabatic bulk moduli measurements. In particular, we consider in detail elastic parameter estimation within classical elasticity and consistent couple stress theory for four different cubic single crystals (NaCl, KCl, Cu, CuZn). The classical theory requires the estimation of three independent material parameters, while only one additional parameter relating skew-symmetric mean curvature to skew-symmetric couple-stress is needed for the size-dependent consistent couple stress theory. This additional parameter can be defined for cubic crystals in terms of a material length scale, which is found to be on the order of tens of microns for the four materials studied here. Furthermore, a detailed statistical investigation provides strong to very strong evidence that couple stress theory is superior to classical Cauchy elasticity for representing the wave velocities and adiabatic bulk moduli for all four single crystals.