Rotational elasticity and couplings to linear elasticity

TL;DR

This paper introduces a nonlinear rotational elasticity model using orthogonal matrices, analyzes wave propagation including transversal and longitudinal waves, and explores coupling with linear elasticity, revealing conditions for auxetic behavior.

Contribution

It presents a new nonlinear rotational elasticity framework, identifies wave solutions, and couples rotational and linear elastic waves, expanding understanding of elastic wave dynamics.

Findings

Two classes of rotational elastic waves identified

Transversal wave velocity can exceed longitudinal velocity

Model can describe auxetic materials under certain parameters

Abstract

It is the aim of the paper to present a new point of view on rotational elasticity in a nonlinear setting using orthogonal matrices. The proposed model, in the linear approximation, can be compared to the well known equilibrium equations of static linear elasticity. An appropriate kinetic energy is identified and we present a dynamical model of rotational elasticity. The propagation of elastic waves in such a medium is studied and we find two classes of waves, transversal rotational waves and longitudinal rotational waves, both of which are solutions of the nonlinear partial differential equations. For certain parameter choices, the transversal wave velocity can be greater than the longitudinal wave velocity. Moreover, parameter ranges are identified where the model describes an auxetic material. However, in all cases the potential energy functional is positive definite. Finally, we couple the rotational waves to linear elastic waves to study the behaviour of the coupled system. We find wave like solutions to the coupled equations and can visualise our results with the help of suitable figures.