Primal and mixed finite element formulations for the relaxed micromorphic model

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In this work we discuss the relaxed micromorphic model and implementation details for a full three‐dimensional formulation entailing a quadratic Lagrangian‐Nédélec finite element and appropriate boundary conditions in the discrete setting. The relaxed micromorphic model is a generalized continuum theory with the capacity to capture more complex kinematical behaviour than in the classical Cauchy continua. Such behaviour is commonly found in materials with a pronounced micro‐structure such as porous media and metamaterials. The theory introduces the microdistortion field, encompassing nine additional degrees of freedom for each material point in the continuum, effectively turning each material point into a deformable micro‐body. The uncommon discrete formulation stems from the employment of the Curl operator in the energy functional of the relaxed micromorphic model, thus requiring H(curl)‐conforming finite elements for well‐posedness to be inherited in the discrete setting. The model further introduces the so called consistent coupling condition, which requires some technical considerations in order to be upheld correctly. This work demonstrates the finite element formulation, culminating with a numerical example.

“…Further, for the curl we find the contravariant Piola transformation [8,10] 4: Barycentric mapping of edges from the unit domain to the reference element and onto the physical domain.…”

Section: Nédélec Base Functionsmentioning

confidence: 98%

“…with a Lagrangian and Nédélec basis [6,8,9] is presented in the following. The elements are mapped from the reference element…”

Section: Finite Element Formulationsmentioning

confidence: 99%

“…The linear Nédélec basis of the second type [3,8,10] is defined by the following edge degrees of freedom…”

Section: Nédélec Base Functionsmentioning

confidence: 99%

Section: Nédélec Base Functionsmentioning

confidence: 99%

See 2 more Smart Citations

A quadratic finite element for the relaxed micromorphic model

Sky

Muench²,

Neff

2023

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“…Different finite element formulations of the relaxed micromorphic model for triangular and quadrilateral meshes were introduced for the plane strain case in [5,6], antiplane shear in [7] and 3D case in [8]. The suitable finite element space for the micro-distortion field is known as Nédélec space, see [9,10].…”

Section: Finite Element Formulationmentioning

confidence: 99%

Modeling the size‐effect of metamaterial beams under bending via the relaxed micromorphic continuum

Sarhil

Scheunemann

Schröder

et al. 2023

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Mechanical metamaterials are media with periodic unit cells comprised of tailored geometry in order to accomplish extraordinary bulk mechanical properties which can not be found in conventional materials. However, metamaterials exhibit size-effect phenomena which can be captured by enriched continua. In this work, the relaxed micromorphic model will be used to reproduce the mechanical response of an assumed metamaterial. The solution of the fully discretized metamaterial is used as a reference that the homogeneous relaxed micromorphic model should meet. We discuss the parameters identification and boundary conditions. Our analysis indicates that the relaxed micromorphic model faithfully reproduces the size-effect.

On the identification of material parameters in the relaxed micromorphic continuum

Sarhil,

Scheunemann,

Schröder

et al. 2023

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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.