Soliton solutions in geometrically nonlinear Cosserat micropolar elasticity with large deformations

TL;DR

This paper derives nonlinear equations for Cosserat micropolar elasticity, constructs soliton solutions, and analyzes wave propagation and velocity relations to understand deformation behaviors in three-dimensional materials.

Contribution

It introduces a novel derivation of coupled nonlinear equations and constructs soliton solutions within the fully nonlinear Cosserat elasticity framework.

Findings

Derivation of coupled nonlinear equations of motion from energy functionals

Construction of soliton solutions using a double sine-Gordon equation

Identification of parameter regimes where wave propagation is inhibited

Abstract

We study the fully nonlinear dynamical Cosserat micropolar elasticity problem in space with three dimensionals with various energy functionals dependent on the microrotation $\overline{R}$ and the deformation gradient tensor $F$ . We derive a set of coupled nonlinear equations of motion from first principles by varying the complete energy functional. We obtain a double sine-Gordon equation and construct soliton solutions. We show how the solutions can determine the overall deformational behaviour and discuss the relations between wave numbers and wave velocities thereby identifying parameter values where the waves cannot propagate.