Proper formulation of viscous dissipation for nonlinear waves in solids

To model nonlinear viscous dissipative motions in solids, acoustical physicists usually add terms linear in (E) over dot, 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 epsilon = epsilon(tr E, tr E-2, tr 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 (E) over dot is symmetric, so that motions for which sigma + sigma(T) not equal 0 will give rise to elastic stresses that have no viscous pendant. Another reason is that (E) over dot 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. (C) 2013 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4776178]