On the analysis of the macroscopic coefficients of stochastically inhomogbneous elastic media: PMM vol.38, n≗5, 1974. pp. 915–920

Abstract

We study the application of the methods developed in the electrodynamics of stochastic media and adapted to the quantum field theory [1, 2], to the equations of the dynamics of an elastic medium. The method in question does not impose the usual restrictions of smallness on the magnitude of the fluctuations of the elastic moduli. Separation of the singular part of the Green tensor [1, 3, 4] and the introduction of new field variables, is equivalent to the summation of the quasi-static parts of the elastic moduli. Application of the discontinuous WeberSchafheitlin integrals [5, 6] makes possible the accurate computation of the Born approximation. This brings to light various effects characteristic for the media possessing a structure [7]; when the dispersive properties can become discontinuous, resonant phenomena arise at the wavelengths comparable with the dimension of the structure. For a strongly isotropic medium [8] and an exponential correlation function, we give explicit expressions for the macroscopic elastic coefficients and for the eigenvalues of the operators. The methods of computing the static macroscopic coefficients were developed and used by many authors (see e.g. [3, 4, 9, 10]). Under certain assumptions made about the mean stress-strain state and the texture of the medium, the operator relations connecting the average fields became, generally speaking, algebraic. When the dynamic effective parameters are computed, the relations appearing also have a non-local character and this complicates the dispersion equations considerably. In [11–14] the authors had computed, for the Born approximation, the macroscopic coefficients for the case of long and short waves when the spatial dispersion could be neglected.