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Analytical solutions of the cylindrical bending problem for the relaxed micromorphic continuum and other generalized continua

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Abstract

We consider the cylindrical bending problem for an infinite plate as modeled 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 exhibits 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 (Neff et al. in Acta Mechanica 211(3–4):237–249, 2010) 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.

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Notes

  1. Here are reported the macroscopic 3D Poisson’s ratio \(\nu _{ \hbox {macro}} = \frac{\lambda _{\hbox { macro}}}{2\left( \lambda _{\hbox { macro}} + \mu _{\hbox { macro}}\right) }\), the Young modulus

    \(E_{ \hbox {macro}} = \frac{\mu _{\hbox { macro}} \left( 3\lambda _{\hbox { macro}} + 2 \mu _{\hbox { macro}}\right) }{\lambda _{\hbox { macro}} + \mu _{\hbox { macro}}} = 2\mu _{\hbox { macro}}\left( 1+\nu _{\hbox { macro}}\right) \), and the bulk modulus \(\kappa _{\hbox { macro}} = \frac{2\mu _{\hbox { macro}} + 3\lambda _{\hbox { macro}}}{3}\).

  2. Note that under the plane stress hypothesis the first Lamé parameter becomes \({\widehat{\lambda }}_{\hbox { macro}} = \frac{2 \, \lambda _{\hbox { macro}} \, \mu _{\hbox { macro}}}{\lambda _{\hbox { macro}} + 2 \mu _{\hbox { macro}}}\) , while the shear modulus \({\widehat{\mu }}_{\hbox { macro}} = \mu _{\hbox { macro}}\) , the Young modulus \({\widehat{E}} = E = \frac{\mu _{\hbox { macro}}(3\lambda _{\hbox { macro}} + 2\mu _{\hbox { macro}})}{\lambda _{\hbox { macro}} + \mu _{\hbox { macro}}}\) , and the Poisson’s ratio \({\widehat{\nu }} = \mu _{\hbox { macro}} = \frac{\lambda _{\hbox { macro}}}{2\lambda _{ \hbox {macro}}+2\mu _{\hbox { macro}}}\) do not change. It is also reported here the more used bending stiffness expression \({\widehat{\lambda }}_{\hbox { macro}} + 2\mu _{\hbox { macro}} = \frac{E_{\hbox { macro}}}{1- \nu _{\hbox { macro}}^2}\) .

  3. Are here reported the 3D Poisson’s ratio \(\nu _{ \hbox {macro}} = \frac{\lambda _{\hbox { macro}}}{2\left( \lambda _{\hbox { macro}} + \mu _{\hbox { macro}}\right) }\), the 3D Young modulus \(E_{ \hbox {macro}} = \frac{\mu _{\hbox { macro}} \left( 3\lambda _{\hbox { macro}} + 2 \mu _{\hbox { macro}}\right) }{\lambda _{\hbox { macro}} + \mu _{\hbox { macro}}}\), and the micro and the mesoexpression of the Poisson’s ratio in plane stress \(\nu _{ \hbox {micro}} = \frac{\lambda _{\hbox { micro}}}{2\left( \lambda _{\hbox { micro}} + \mu _{\hbox { micro}}\right) }\) and the \(\nu _e = \frac{\lambda _e}{2\left( \lambda _e + \mu _e\right) }\), respectively.

  4. \(\text {sech}(x) := \frac{1}{\text {cosh}(x)} = \frac{2}{e^{x}+e^{-x}}\).

  5. Where \(\kappa _e = \frac{2\mu _e + 3\lambda _e}{3}\) and \(\kappa _{\hbox { micro}} = \frac{2\mu _{\hbox { micro}} + 3\lambda _{\hbox { micro}}}{3}\) are the meso- and the micro-scale 3D bulk modulus.

  6. The equivalent formulation in terms of a rotation vector \(\vartheta :=\hbox {axl} ({\varvec{A}}) \in {\mathbb {R}}^3\) is given in appendix D of [42].

  7. Where \(\kappa _e = \frac{2\mu _e + 3\lambda _e}{3}\) and \(\kappa _{\hbox { micro}} = \frac{2\mu _{\hbox { micro}} + 3\lambda _{\hbox { micro}}}{3}\) are he meso- and the micro-scale 3D bulk modulus.

  8. Note that \(\left\Vert \hbox {Curl} \left( \omega \varvec{\mathbb {1}}\right) \right\Vert ^2_{{\mathbb {R}}^{3\times 3}} = \left\Vert \hbox {Anti} \left( \varvec{\hbox {D}} \omega \right) \right\Vert ^2_{{\mathbb {R}}^{3\times 3}} = 2 \left\Vert \hbox {axl} \, \left( \hbox {Anti} \left( \varvec{\hbox {D}} \omega \right) \right) \right\Vert ^2_{{\mathbb {R}}^3} = 2 \left\Vert \varvec{\hbox {D}} \omega \right\Vert ^2_{{\mathbb {R}}^3}\)

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Acknowledgements

Angela Madeo and Gianluca Rizzi acknowledge funding from the French Research Agency ANR, “METASMART” (ANR-17CE08-0006). Angela Madeo and Gianluca Rizzi acknowledge support from IDEXLYON in the framework of the “Programme Investissement d’Avenir” ANR-16-IDEX-0005. Patrizio Neff acknowledges support in the framework of the DFG-Priority Programme 2256 “Variational Methods for Predicting Complex Phenomena in Engineering Structures and Materials," Neff 902/10-1, Project-No. 440935806.

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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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Communicated by Marcus Aßmus, Victor A. Eremeyev and Andreas Öchsner.

Dedicated to Holm Altenbach on the occasion of his 65th birthday

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Rizzi, G., Hütter, G., Madeo, A. et al. Analytical solutions of the cylindrical bending problem for the relaxed micromorphic continuum and other generalized continua. Continuum Mech. Thermodyn. 33, 1505–1539 (2021). https://doi.org/10.1007/s00161-021-00984-7

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