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Analytical solutions of the simple shear problem for micromorphic models and other generalized continua

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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.

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Notes

  1. The following energy expression has been used in [41]: \(W \left( \varvec{\nabla u}, \varvec{P}, \varvec{\nabla P}\right) = \mu \left\Vert \hbox {sym} \, \varvec{\nabla u} \right\Vert ^2 + \lambda /2 \, \hbox {tr}^2 \left( \varvec{\nabla u} \right) + \alpha \, \mu \left\Vert \varvec{\nabla u} - \varvec{P} \right\Vert ^2 + \alpha \, \lambda /2 \, \hbox {tr}^2 \left( \varvec{\nabla u} - \varvec{P} \right) +\mu \, L_{c}^2/2 \, \left\Vert \varvec{\nabla P} \right\Vert ^2 \). This formulations is not reconcilable with the relaxed micromorphic model even if we neglect the curvature part.

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Acknowledgements

AM acknowledge funding from the French Research Agency ANR, “METASMART” (ANR-17CE08-0006). AM and GR acknowledges support from IDEXLYON in the framework of the “Programme Investissement d’Avenir” ANR-16-IDEX-0005.

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Authors and Affiliations

  1. GEOMAS, INSA-Lyon, Université de Lyon, 20 avenue Albert Einstein, 69621, Villeurbanne cedex, France

    Gianluca Rizzi & Angela Madeo

  2. TU Bergakademie Freiberg, Institute of Mechanics and Fluid Dynamics, Lampadiusstr. 4, 09596, Freiberg, Germany

    Geralf Hütter

  3. Head of Chair for Nonlinear Analysis and Modelling, Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann-Straße 9, 45127, Essen, Germany

    Patrizio Neff

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  1. Gianluca Rizzi
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Correspondence to Gianluca Rizzi.

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Rizzi, G., Hütter, G., Madeo, A. et al. Analytical solutions of the simple shear problem for micromorphic models and other generalized continua. Arch Appl Mech 91, 2237–2254 (2021). https://doi.org/10.1007/s00419-021-01881-w

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