On problematic case of product approximation in Backus average

TL;DR

This paper investigates the limitations of the product approximation in the Backus average, especially for materials with negative Poisson's ratio, and explores its implications for seismic wave modeling.

Contribution

It identifies a problematic case of the product approximation related to auxetic materials and derives stability conditions for various symmetry classes in the context of seismic modeling.

Findings

Negative Poisson's ratio causes inaccuracies in the product approximation.

Averaging cubic layers can produce tetragonal symmetry.

Auxetic materials are more common than previously thought.

Abstract

Elastic anisotropy might be a combined effect of the intrinsic anisotropy and the anisotropy induced by thin-layering. The Backus average, a useful mathematical tool, allows us to describe such an effect quantitatively. The results are meaningful only if the underlying physical assumptions are obeyed, such as static equilibrium of the material. We focus on the only mathematical assumption of the Backus average, namely, product approximation. It states that the average of the product of a varying function with nearly-constant function is approximately equal to the product of the averages of those functions. We discuss particular, problematic case for which the aforementioned assumption is inaccurate. Further, we focus on the seismological context. We examine numerically if the inaccuracy affects the wave propagation in a homogenous medium -- obtained using the Backus average -- equivalent to thin layers. We take into consideration various material symmetries, including orthotropic, cubic, and others. We show that the problematic case of product approximation is strictly related to the negative Poisson's ratio of constituent layers. Therefore, we discuss the laboratory and well-log cases in which such a ratio has been noticed. Upon thorough literature review, it occurs that examples of so-called auxetic materials (media that have negative Poisson's ratio) are not extremely rare exceptions as thought previously. The investigation and derivation of Poisson's ratio for materials exhibiting symmetry classes up to monoclinic become a significant part of this paper. Except for the main objectives, we also show that the averaging of cubic layers results in an equivalent medium with tetragonal (not cubic) symmetry. Additionally, we present concise formulations of stability conditions for low symmetry classes, such as trigonal, orthotropic, and monoclinic.