Rotational elasticity

TL;DR

This paper develops a nonlinear rotational elasticity model based on Cosserat theory, constructs explicit plane wave solutions for rotations, and links a special case to massless Dirac equations, advancing understanding of rotational wave phenomena.

Contribution

It introduces a nonlinear rotational elasticity framework with explicit traveling wave solutions and connects a special case to Dirac equations, expanding theoretical insights.

Findings

Explicit plane wave solutions for rotational waves

Nonlinear Euler-Lagrange equations in rotational elasticity

Special case reduces to massless Dirac equations

Abstract

We consider an infinite 3-dimensional elastic continuum whose material points experience no displacements, only rotations. This framework is a special case of the Cosserat theory of elasticity. Rotations of material points are described mathematically by attaching to each geometric point an orthonormal basis which gives a field of orthonormal bases called the coframe. As the dynamical variables (unknowns) of our theory we choose the coframe and a density. We write down the general dynamic variational functional for our rotational theory of elasticity, assuming our material to be physically linear but the kinematic model geometrically nonlinear. Allowing geometric nonlinearity is natural when dealing with rotations because rotations in dimension 3 are inherently nonlinear (rotations about different axes do not commute) and because there is no reason to exclude from our study large rotations such as full turns. The main result of the paper is an explicit construction of a class of time-dependent solutions which we call plane wave solutions; these are travelling waves of rotations. The existence of such explicit closed form solutions is a nontrivial fact given that our system of Euler-Lagrange equations is highly nonlinear. In the last section we consider a special case of our rotational theory of elasticity which in the stationary setting (harmonic time dependence and arbitrary dependence on spatial coordinates) turns out to be equivalent to a pair of massless Dirac equations.