Soliton-like solutions based on geometrically nonlinear Cosserat micropolar elasticity

TL;DR

This paper develops a geometrically nonlinear Cosserat micropolar elasticity model that admits soliton solutions, extending previous linear theories by incorporating full rotations and interaction terms, and reducing the equations to a Sine-Gordon type.

Contribution

It introduces a nonlinear Cosserat model with all interaction terms and derives soliton solutions via reduction to a Sine-Gordon equation, a novel approach in micropolar elasticity.

Findings

Derivation of a nonlinear Cosserat model with full rotational effects.

Reduction of the governing equations to a Sine-Gordon type equation.

Existence of soliton solutions within this nonlinear micropolar framework.

Abstract

The Cosserat model generalises an elastic material taking into account the possible microstructure of the elements of the material continuum. In particular, within the Cosserat model the structured material point is rigid and can only experience microrotation, which is also known as micropolar elasticity. We present the geometrically nonlinear theory taking into account all possible interaction terms between the elastic and microelastic structure. This is achieved by considering the irreducible pieces of the deformation gradient and of the dislocation curvature tensor. In addition we also consider the so-called Cosserat coupling term. In this setting we seek soliton type solutions assuming small elastic displacements, however, we allow the material points to experience full rotations which are not assumed to be small. By choosing a particular ansatz we are able to reduce the system of equations to a Sine-Gordon type equation which is known to have soliton solutions.