Analytical solutions of the cylindrical bending problem for the relaxed micromorphic continuum and other generalized continua (including full derivations)

TL;DR

This paper derives analytical solutions for cylindrical bending in various generalized continuum models, demonstrating that the relaxed micromorphic model predicts physically realistic, bounded bending stiffness for thin specimens, unlike other models.

Contribution

It provides the first full analytical solutions for the relaxed micromorphic continuum in cylindrical bending, highlighting its advantages over classical models.

Findings

Relaxed micromorphic model predicts bounded bending stiffness for thin specimens.

Classical micromorphic and gradient models show unbounded stiffness.

The solutions help identify relevant material parameters.

Abstract

We consider the cylindrical bending problem for an infinite plate as modelled with a family of generalized continuum models, including the micromorphic approach. The models allow to describe length scale effects in the sense that thinner specimens are comparatively stiffer. We provide the analytical solution for each case and exhibit the predicted bending stiffness. The relaxed micromorphic continuum shows bounded bending stiffness for arbitrary thin specimens, while classical micromorphic continuum or gradient elasticity as well as Cosserat models [35] exhibit unphysical unbounded bending stiffness for arbitrary thin specimens. This finding highlights the advantage of using the relaxed micromorphic model, which has a definite limit stiffness for small samples and which aids in identifying the relevant material parameters.