Analytical solutions for long cylindrical shells under radial deformations based on the isotropic relaxed micromorphic continuum

TL;DR

This paper derives an exact analytical solution for the elastostatic response of long cylindrical shells made of microstructured materials using the relaxed micromorphic continuum model, highlighting microstructural effects on deformation.

Contribution

It introduces a closed-form solution for cylindrical shells in the relaxed micromorphic model, accounting for microstructural effects and providing a benchmark for numerical validation.

Findings

Microstructural effects significantly influence displacement fields.

The solution reveals deviations from classical elasticity predictions.

Material parameters and characteristic length impact the response.

Abstract

This study presents a closed-form analytical solution for the elastostatic response of long cylindrical shells composed of microstructured materials within the framework of the isotropic relaxed micromorphic continuum. The formulation accounts for microstructural effects by introducing an independent micro-distortion tensor field in addition to the classical displacement field. Under the assumptions of axisymmetric deformation and plane strain conditions, the governing equilibrium equations reduce to a coupled system of ordinary differential equations in the radial coordinate. By introducing suitable auxiliary variables, the system is reformulated into a non-homogeneous modified Bessel equation, which admits an exact analytical solution. Explicit expressions are derived for the radial displacement field and the non-zero components of the micro-distortion tensor. Numerical examples are presented to illustrate the influence of material parameters and the characteristic length on the displacement. The results demonstrate that the relaxed micromorphic model predicts deviations from classical elasticity where microstructural effects are more pronounced. The obtained solution provides valuable physical insight into the mechanics of cylindrical shells and serves as a benchmark for validating numerical implementations of relaxed micromorphic models.