The isotropic relaxed micromorphic model in polar coordinates and its application to an elastostatic axisymmetric extension problem

TL;DR

This paper develops an analytical solution for an elastostatic axisymmetric problem using the isotropic relaxed micromorphic model in polar coordinates, incorporating Bessel functions and exploring classical elasticity limits.

Contribution

It introduces a closed-form analytical solution for the relaxed micromorphic model in polar coordinates applied to axisymmetric problems, including limit cases and numerical analysis.

Findings

Explicit solutions for displacement and microdistortion using Bessel functions

Demonstration of classical elasticity limit within the relaxed micromorphic framework

Numerical results illustrating parameter effects and code calibration potential

Abstract

In this paper, we consider the isotropic relaxed micromorphic model in polar coordinates and use this representation to solve explicitly an elastostatic axisymmetric extension problem involving a linear system of ordinary differential equations. To obtain an analytical solution, modified Bessel functions are utilized and closed-form solutions for the displacement and microdistortion are obtained. We show how certain limit cases (classical linear elasticity), which are naturally included in the relaxed micromorphic model, can be efficiently achieved. Furthermore, numerical results are calculated and the effects of various parameters are examined. The results can be used to calibrate and check corresponding finite element codes.