Analytical solutions of the simple shear problem for certain types of micromorphic continuum models -- including full derivations

TL;DR

This paper analytically solves the simple shear problem for various micromorphic continuum models, providing insights into their stability, modeling limits, and the length-scale dependence of shear stiffness.

Contribution

It offers full analytical solutions for the simple shear problem across multiple generalized continuum models, highlighting the role of shear stiffness and parameter interpretation.

Findings

Shear stiffness $*\mu^{*}$ is length-scale dependent.

Limit cases help interpret material parameters.

Analytical solutions inform stability and modeling limits.

Abstract

To draw conclusions as regards the stability and modelling limits of the investigated continuum, we consider a family of infinitesimal isotropic generalized continuum models (Mindlin-Eringen micromorphic, relaxed micromorphic continuum, Cosserat, micropolar, microstretch, microstrain, microvoid, indeterminate couple stress, second gradient elasticity, etc.) and solve analytically the simple shear problem of an infinite stripe. A qualitative measure characterizing the different generalized continuum moduli is given by the shear stiffness $\mu^{*}$. This stiffness is in general length-scale dependent. Interesting limit cases are highlighted, which allow to interpret some of the appearing material parameter of the investigated continua.