Proper formulation of viscous dissipation for nonlinear waves in solids

Area of Science:

• Solid mechanics
• Acoustics
• Materials science

Background:

• Modeling nonlinear viscous dissipative motions in solids commonly involves adding linear terms of the material time derivative of the Lagrangian strain tensor (Ė) to the elastic stress tensor (σ).
• This approach, prevalent for decades, is derived from third or fourth-order expansions of strain energy density.

Purpose of the Study:

• To identify and correct fundamental physical inaccuracies in the standard modeling of nonlinear viscous dissipative motions in solids.
• To demonstrate the consequences of these inaccuracies on phenomena like nonlinear shear wave propagation.

Main Methods:

• Theoretical analysis of the symmetry properties of the elastic stress tensor (σ) and the material time derivative of the Lagrangian strain tensor (Ė).
• Examination of frame-invariance properties of σ and Ė under observer transformations.
• Application of the corrected model to analyze nonlinear shear wave propagation, specifically focusing on kink formation in soft solids.

Main Results:

• The standard practice is physically flawed due to the asymmetry of σ versus the symmetry of Ė, leading to unphysical viscous stresses.
• The non-frame-invariance of σ compared to Ė violates the principle of material frame indifference.
• These flaws have significant implications for modeling nonlinear shear waves, particularly in soft, nearly incompressible solids.

Conclusions:

• The conventional method for modeling nonlinear viscous dissipative solids is fundamentally incorrect and requires revision.
• Correcting these inaccuracies is crucial for accurate predictions of wave propagation and material behavior in soft solids.