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

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.

arXiv comment50 pages, 28 figures (38 pictures)

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Cited by in corpus (8)

• Size-effects of metamaterial beams subjected to pure bending: on boundary conditions and parameter identification in the relaxed micromorphic model
• Analytical solutions of the cylindrical bending problem for the relaxed micromorphic continuum and other generalized continua (including full derivations)
• Primal and mixed finite element formulations for the relaxed micromorphic model
• Lagrange and $H(\mathrm{curl},\mathcal{B})$ based Finite Element formulations for the relaxed micromorphic model
• A computational approach to identify the material parameters of the relaxed micromorphic model
• Green's functions for the isotropic planar relaxed micromorphic model -- concentrated force and concentrated couple
• Micromorphic FE${}^2$ Simulation of Plastic Deformations of Foam Structures
• Analytical solutions of the simple shear problem for certain types of micromorphic continuum models -- including full derivations