A theory of magneto-elastic nanorods obtained through rigorous dimension reduction

TL;DR

This paper develops a rigorous one-dimensional model for magneto-elastic nanorods from a two-dimensional theory, incorporating microstructure effects, and validates it through numerical simulations showing excellent agreement.

Contribution

It introduces a novel dimension reduction approach for magneto-elastic materials, capturing microstructure effects and interaction energies in a rod model.

Findings

Magnetically-induced buckling depends on rod length and field intensity.

The one-dimensional model accurately predicts behavior compared to the two-dimensional theory.

Size effects are incorporated via second gradient regularization.

Abstract

Starting from a two-dimensional theory of magneto-elasticity for fiber-reinforced magnetic elastomers we carry out a rigorous dimension reduction to derive a rod model that describes a thin magneto-elastic strip undergoing planar deformations. The main features of the theory are the following: a magneto-elastic interaction energy that manifests itself through a distributed torque; a penalization term that prevent local interpenetration of matter; a regularization that depends on the second gradient of the deformation and models microstructure-induced size effects. As an application, we study a problem involving magnetically-induced buckling and we show that the intensity of the field at the onset of the instability increases if the length of the rod is decreased. Finally, we assess the accuracy of the deduced model by performing numerical simulations where we compare the two-dimensional and the one-dimensional theory in some special cases and we observe excellent agreement.