Derivation of Elastic Wave Equation from New Motion Description

TL;DR

This paper derives a new elastic wave equation by modeling material elements as rigid bodies, linking wave types to translational and rotational motions, and challenging traditional assumptions in continuum mechanics.

Contribution

It introduces a novel derivation of the elastic wave equation based on rigid body motion of material elements, modifying classical constitutive relations.

Findings

Longitudinal and transverse waves correspond to translation and rotation.

Reciprocity of shear stress and strain is not necessary.

Local rigid body rotation influences stress.

Abstract

In classical mechanics, the motion of an object is described with Newton's three laws of motion, which means that the motion of the material elements composing a continuum can be described with the particle model. However, this viewpoint is not objective, since the existence of transverse wave cannot be predicted by the theory of elasticity based on the particle model. In this paper, the material element of an elastomer is regarded as a rigid body, and the traditional elastic wave equation is derived based on it. In the derivation, the constitutive relations and strain-displacement relations are correspondingly modified. The study reveals that the longitudinal and transverse waves in elastomer correspond to the translational and rotational motion of the material element, respectively. Besides, the reciprocity of shear stress and shear strain is no longer requisite in continuum mechanics, and the local rigid body rotation contributes stress.