Size-dependent normal mode vibrations of first strain gradient elastic medium

Abstract

This study investigates the size-dependence of normal mode vibrations of an isotropic first strain gradient elasticity of Cauchy type. The elastic medium has a two-dimensional rectangular shape with the dimensions of L1 = 1.1ξ and L2 = 0.9ξ, where ξ is a scaling factor. Lagrangian density includes two kinds of length parameters, d and g, and which control the phonon dispersion relations. Numerical analysis conucted by the Ritz method revealed that the vibration frequencies follow the classical scaling law when the size of the medium is sufficiently large. On the other hand, however, notable deviations from the scaling law are observed below ξ < 5, irrespective to the symmetry of vibration. These results demonstrate that phonon dispersion is responsible for the emergence of size-effect in the normal mode vibrations.