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

Rizzi, Gianluca; Hütter, Geralf; Khan, Hassam; Ghiba, Ionel Dumitrel; Madeo, Angela; Neff, Patrizio

doi:10.1007/s00161-021-00984-7

Cited by 21 publications

(34 citation statements)

References 89 publications

(151 reference statements)

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“…The strain energy for the isotropic Cosserat continuum in dislocation tensor format (curvature energy expressed in terms of Curl A) can be written as [3,8,13,14,18,21,25,28,29]…”

Section: Uniaxial Extension Problem For the Isotropic Cosserat Continuummentioning

confidence: 99%

“…In this paper we continue our investigation of analytical solutions for the isotropic relaxed micromorphic model (and other isotropic generalized continuum models). It follows our recent exposition of analytical solutions for the simple shear [28], bending [25], and torsion problem [13,27]. Here, we consider the uniaxial extension problem, which, in classical isotropic linear elasticity, allows to determine the size-independent longitudinal modulus M macro = λ macro + 2μ macro .…”

Section: Introductionmentioning

confidence: 99%

“…In that case, the inhomogeneous response is triggered by the boundary conditions for the additional kinematic fields which are applied at the upper and lower surface. We refer the reader to the introduction of [25,27,28,32] concerning the relevance of the scientific question as well as its importance for the determination of material parameters for generalized continua [33]. Indeed, the obtained analytical formulas can be used to determine size-dependent and size-independent material parameters.…”

Section: Introductionmentioning

confidence: 99%

“…Indeed, the obtained analytical formulas can be used to determine size-dependent and size-independent material parameters. The notation follows that of [25,27,28]. We recapitulate shortly.…”

Section: Introductionmentioning

confidence: 99%

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Analytical solution of the uniaxial extension problem for the relaxed micromorphic continuum and other generalized continua (including full derivations)

et al. 2021

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“…The strain energy for the isotropic Cosserat continuum in dislocation tensor format (curvature energy expressed in terms of Curl A) can be written as [3,8,13,14,18,21,25,28,29]…”

Section: Uniaxial Extension Problem For the Isotropic Cosserat Continuummentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Analytical solution of the uniaxial extension problem for the relaxed micromorphic continuum and other generalized continua (including full derivations)

et al. 2021

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“…This is achieved via homogenization of the material parameters and the introduction of the characteristic length L c [19,29], which determines the influence of the dislocation density in the free energy functional. Specific analytical solutions to the full isotropic relaxed micromorphic model are presented in [35][36][37].…”

Section: Introductionmentioning

confidence: 99%

A hybrid $$ H ^1\times H (\mathrm {curl})$$ finite element formulation for a relaxed micromorphic continuum model of antiplane shear

et al. 2021

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One approach for the simulation of metamaterials is to extend an associated continuum theory concerning its kinematic equations, and the relaxed micromorphic continuum represents such a model. It incorporates the Curl of the nonsymmetric microdistortion in the free energy function. This suggests the existence of solutions not belonging to $$ H ^1$$ H 1 , such that standard nodal $$ H ^1$$ H 1 -finite elements yield unsatisfactory convergence rates and might be incapable of finding the exact solution. Our approach is to use base functions stemming from both Hilbert spaces $$ H ^1$$ H 1 and $$ H (\mathrm {curl})$$ H ( curl ) , demonstrating the central role of such combinations for this class of problems. For simplicity, a reduced two-dimensional relaxed micromorphic continuum describing antiplane shear is introduced, preserving the main computational traits of the three-dimensional version. This model is then used for the formulation and a multi step investigation of a viable finite element solution, encompassing examinations of existence and uniqueness of both standard and mixed formulations and their respective convergence rates.

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On the identification of material parameters in the relaxed micromorphic continuum

Sarhil,

Scheunemann,

Schröder

et al. 2023

Proc Appl Math and Mech

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For modeling materials with size‐effects, the classical Cauchy linear elasticity theory and first‐order homogenization procedures are insufficient. Enriched continua are able to capture such phenomena in principle. In this work, we model the size‐effects of metamaterial beams via the relaxed micromorphic continuum. The material parameters of the relaxed micromorphic model are evaluated. We present a simple procedure for the determination of the material parameters based on the fact that the model operates between two well‐defined scales that are the classical continuum scales of macroscopic linear elasticity, whose stiffness is given by a known macroscopic elasticity tensor, and a microscopic scale with a microscopic elasticity tensor yet to be identified. The size‐effects of metamaterial cantilevers under traction load are captured successfully.

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

Abstract: We derive analytical solutions for the uniaxial extension problem for the relaxed micromorphic continuum and other generalized continua. These solutions may help in the identification of material parameters of generalized continua which are able to disclose size-effects.

Cited by 21 publications

References 89 publications

Analytical solution of the uniaxial extension problem for the relaxed micromorphic continuum and other generalized continua (including full derivations)

Analytical solution of the uniaxial extension problem for the relaxed micromorphic continuum and other generalized continua (including full derivations)

A hybrid $$ H ^1\times H (\mathrm {curl})$$ finite element formulation for a relaxed micromorphic continuum model of antiplane shear

On the identification of material parameters in the relaxed micromorphic continuum

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