Proper formulation of viscous dissipation for nonlinear waves in solids

TL;DR

This paper critiques the common practice of adding linear viscous terms to elastic stress tensors in nonlinear wave modeling, highlighting physical inconsistencies and proposing corrections for accurate representation.

Contribution

It identifies fundamental issues with traditional viscous dissipation formulations in nonlinear solid wave models and offers a corrected, physically consistent approach.

Findings

Traditional viscous models violate symmetry and frame-invariance principles.

Incorrect formulations can lead to unphysical predictions in shear wave propagation.

Proposed corrections improve the physical accuracy of nonlinear wave simulations.

Abstract

In order to model nonlinear viscous dissipative motions in solids, acoustical physicists usually add terms linear in dot{E}, the material time derivative of the Lagrangian strain tensor E, to the elastic stress tensor sigma derived from the expansion to the third- (sometimes fourth-) order of the strain energy density e=e(trace(E), trace(E^2), trace(E^3)). Here, it is shown that this practice, which has been widely used in the past three decades or so, is physically wrong for at least two reasons, and that it should be corrected. One reason is that the elastic stress tensor sigma is not symmetric while dot{E) is symmetric, so that motions for which sigma + sigma^T <> 0 will give rise to elastic stresses which have no viscous pendant. Another reason is that dot{E} is frame-invariant, while sigma is not, so that an observer transformation would alter the elastic part of the total stress differently than it would alter the dissipative part, thereby violating the fundamental principle of material frame-indifference. These problems can have serious consequences for nonlinear shear wave propagation in soft solids, as seen here with an example of a kink in almost incompressible soft solids.