A Formalism for the Consistent Description of Non-Linear Elasticity of Anisotropic Media

Abstract

The propagation of elastic waves is generally treated under four assumptions: - that the medium is isotropic,- that the medium is homogeneous, - that there is a one-to-one relationship between stress and strain, - that stresses are linearly related to strains (equivalently, that strains are linearly related to stresses). Real media generally violate at least some-and often all-of these assumptions. A valid theoretical description of wave propagation in real media thus depends on the qualitative and quantitative description of the relevant inhomogeneity, anisotropy, and non-linearity: one either has to assume (or show) that the deviation from the assumption can - for the problem at hand - be neglected, or develop a theoretical description that is valid even under the deviation. While the effect of a single deviation from the ideal state is rather well understood, difficulties arise in the combination of several such deviations. Non-linear elasticity of anisotropic (triclinic) rock samples has been reported, e. g. by P. Rasolofosaon and H. Yin at the 6th IWSA in Trondheim (Rasolofosaon and Yin, 1996). Non-linear anisotropic elasticity matters only for non-infinitesimalamplitudes, i. e. , at least in the vicinity of the source. How large this vicinity is depends on the accuracy of observation and interpretation one tries to maintain, on the source intensity, and on the level of non-linearity. This paper is concerned with the last aspect, i. e. , with the meaning of the numbers beyond the fact that they are the results of measurements. As a measure of the non-linearity of the material, one can use the strain level at which the effective stiffness tensor deviates significantly from the zero-strain stiffness tensor. Particularly useful for this evaluation is the eigensystem (six eigenstiffnesses and six eigenstrains) of the stiffness tensor : the eigenstrains provide suitable strain typesfor the calculation of the effective stiffness tensor, and the deviation can be expressed by the relative change of the eigenstiffnesses and by the variation in the direction of the eigenstrains (expressed as vectors in six-dimensional strain space). The suggested procedure is applied to the two materials discussed by Rasolofosaon and Yin (1996). The results allow a heuristic evaluation of the meaning of the reference strain , the square root of the ratio of the norms of the fourth-rank and sixth-rank stiffness tensors. It is stressed that this is not a new theory of non-linearity, but only a different way of viewing the existing theory and results.