Normal modes and acoustic properties of an elastic solid with line defects

Abstract

The normal modes of a continuum solid endowed with a random distribution of line defects that behave like elastic strings are described. These strings interact with elastic waves in the bulk, generating wave dispersion and attenuation. Explicit formulas are provided that relate these properties to the density of string states. For a density of states that mimics the boson peak (BP) in amorphous materials, the attenuation as a function of frequency omega behaves as omega(4) for low frequencies, and, as frequency increases, crosses over to omega(2) near the BP, and then to linear in omega. An Ioffe-Regel criterion is satisfied at the BP. Dispersion is negative in the frequency range where attenuation is quartic and quadratic in frequency, with effective velocity reaching a minimum near the BP. Continuum mechanics can thus be applied to both crystalline materials and their amorphous counterparts at similar length scales. The possibility of linking this model with the microstructure of amorphous materials is discussed.