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

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

doi:10.1007/s00419-021-02064-3

Cited by 11 publications

(10 citation statements)

References 34 publications

(49 reference statements)

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“…8-10-11-12-13-14). These solutions were derived in [66,67,69]. For the limit L c → 0, all the models converge to a classical Cauchy material with stiffness C macro (see Fig.…”

Section: Comparison Of Analytical Solutions For the Different Boundar...mentioning

confidence: 99%

“…The general expression of the strain energy for the isotropic relaxed micromorphic continuum is [66,67,69] W (Du, P , Curl P ) = µ e sym (Du − P ) where (µ e ,λ e ), (µ micro ,λ micro ), µ c , L c > 0, and (a 1 ,a 2 ,a 3 ) are the parameters related to the meso-scale, the parameters related to the micro-scale, the Cosserat couple modulus, the characteristic length, and the three general isotropic curvature parameters, respectively. This energy expression represents the most general isotropic form possible for the relaxed micromorphic model.…”

Section: Relaxed Micromorphic Modelmentioning

confidence: 99%

“…The strain energy density for the isotropic second gradient model with simplified curvature [3,10,48,66,68,69] is a 3 tr (∆ (Du)) 1 = 0 .…”

Section: Second Gradient Modelmentioning

confidence: 99%

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The consistent coupling boundary condition for the classical micromorphic model: existence, uniqueness and interpretation of parameters

d’Agostino¹,

Rizzi²,

Khan³

et al. 2021

Preprint

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We consider the classical Mindlin-Eringen linear micromorphic model with a new strictly weaker set of displacement boundary conditions. The new consistent coupling condition aims at minimizing spurious influences from arbitrary boundary prescription for the additional microdistortion field P . In effect, P is now only required to match the tangential derivative of the classical displacement u which is known at the Dirichlet-part of the boundary.We derive the full boundary condition, in adding the missing Neumann condition on the Dirichletpart. We show existence and uniqueness of the static problem for this weaker boundary condition. These results are based on new coercive inequalities for incompatible tensor fields with prescribed tangential part.Finally, we show that compared to classical Dirichlet conditions on u and P , the new boundary condition modifies the interpretation of the constitutive parameters.

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“…8-10-11-12-13-14). These solutions were derived in [66,67,69]. For the limit L c → 0, all the models converge to a classical Cauchy material with stiffness C macro (see Fig.…”

Section: Comparison Of Analytical Solutions For the Different Boundar...mentioning

confidence: 99%

Section: Relaxed Micromorphic Modelmentioning

confidence: 99%

See 1 more Smart Citation

The consistent coupling boundary condition for the classical micromorphic model: existence, uniqueness and interpretation of parameters

d’Agostino¹,

Rizzi²,

Khan³

et al. 2021

Preprint

Self Cite

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show abstract

“…This behavior is depicted in Figure 1. Furthermore, it has been shown that the relaxed micromorphic model indeed captures the size-effects of bending, tension/compression, torsion, and shear problems, see [2][3][4][5].…”

Section: Introductionmentioning

confidence: 99%

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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“…It is beneficial to verify the computations by analytical solutions. The analytical solutions for some example problems in the generalized continua are presented, for example, in previous studies [57][58][59][60][61].…”

Section: Introductionmentioning

confidence: 99%

A benchmark strain gradient elasticity solution in two-dimensions for verifying computational approaches by means of the finite element method

Shekarchizadeh

Abali

Bersani

2022

Mathematics and Mechanics of Solids

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In elasticity, microstructure-related deviations may be modeled by strain gradient elasticity. For so-called metamaterials, different implementations are possible for solving strain gradient elasticity problems numerically. Analytical solutions of simple problems are used to verify the numerical approach. We demonstrate such a case in a two-dimensional continuum as a benchmark case for computations. As strain gradient enforces higher regularity conditions in displacements, in the finite element method (FEM), the use of standard elements is often seen as inadequate. For such piecewise or elementwise continuous elements, we examine a possible remedy to correctly simulate strain gradient elasticity problems by implementing two techniques. First, we enforce continuity of displacement gradient across elements; second, we employ a mixed finite element method where displacement and its gradient are solved both as unknowns. The results show the pros and cons of each numerical technique. All methods converge monotonically, but the mixed method is more reliable than the other one.

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

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 11 publications

References 34 publications

The consistent coupling boundary condition for the classical micromorphic model: existence, uniqueness and interpretation of parameters

The consistent coupling boundary condition for the classical micromorphic model: existence, uniqueness and interpretation of parameters

On the identification of material parameters in the relaxed micromorphic continuum

A benchmark strain gradient elasticity solution in two-dimensions for verifying computational approaches by means of the finite element method

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