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

d’Agostino, Marco Valerio; Rizzi, Gianluca; Khan, Hassam; Lewintan, Peter; Madeo, Angela; Neff, Patrizio

doi:10.1007/s00161-022-01126-3

Cited by 11 publications

(3 citation statements)

References 75 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…An additional dependence between the displacement field and the micro-distortion field on the boundary was proposed in [81] and subsequently considered in [26,85,86,105]. This so-called consistent coupling condition is defined by…”

Section: The Relaxed Micromorphic Modelmentioning

confidence: 99%

Size-effects of metamaterial beams subjected to pure bending: on boundary conditions and parameter identification in the relaxed micromorphic model

Sarhil

Scheunemann²,

Schröder

et al. 2023

Comput Mech

Self Cite

View full text Add to dashboard Cite

In this paper we model the size-effects of metamaterial beams under bending with the aid of the relaxed micromorphic continuum. We analyze first the size-dependent bending stiffness of heterogeneous fully discretized metamaterial beams subjected to pure bending loads. Two equivalent loading schemes are introduced which lead to a constant moment along the beam length with no shear force. The relaxed micromorphic model is employed then to retrieve the size-effects. We present a procedure for the determination of the material parameters of the relaxed micromorphic model based on the fact that the model operates between two well-defined scales. These scales are given by linear elasticity with micro and macro elasticity tensors which bound the relaxed micromorphic continuum from above and below, respectively. The micro elasticity tensor is specified as the maximum possible stiffness that is exhibited by the assumed metamaterial while the macro elasticity tensor is given by standard periodic first-order homogenization. For the identification of the micro elasticity tensor, two different approaches are shown which rely on affine and non-affine Dirichlet boundary conditions of candidate unit cell variants with the possible stiffest response. The consistent coupling condition is shown to allow the model to act on the whole intended range between macro and micro elasticity tensors for both loading cases. We fit the relaxed micromorphic model against the fully resolved metamaterial solution by controlling the curvature magnitude after linking it with the specimen’s size. The obtained parameters of the relaxed micromorphic model are tested for two additional loading scenarios.

show abstract

Section: The Relaxed Micromorphic Modelmentioning

confidence: 99%

Size-effects of metamaterial beams subjected to pure bending: on boundary conditions and parameter identification in the relaxed micromorphic model

Sarhil

Scheunemann²,

Schröder

et al. 2023

Comput Mech

Self Cite

View full text Add to dashboard Cite

show abstract

“…This is the reason for which the crucial homogenization formula (2) for 𝐿 c → 0 can be obtained. Unlike for the linear Cosserat (micropolar) model, the relaxed micromorphic model remains operative and well posed (Neff et al, 2014;d'Agostino et al, 2022;Ghiba et al, 2015) also for zero Cosserat couple modulus 𝜇 c ≡ 0 (C c ≡ 0), in which case the force stress tensor remains symmetric. The well-posedness is established using novel generalized Korn's inequalities for incompatible tensor fields Neff, 2022, 2021b;Lewintan et al, 2021;Lewintan and Neff, 2021a,a;Neff et al, 2015bNeff et al, , 2012, whereby sharp criteria for the validity of such coercivity estimates were given in the recent works (Gmeineder and Spector, 2021;Gmeineder et al, 2023b,a).…”

Section: Introductionmentioning

confidence: 99%

Green’s functions for the isotropic planar relaxed micromorphic model — Concentrated force and concentrated couple

Gourgiotis,

Rizzi,

Lewintan

et al. 2024

International Journal of Solids and Structures

Self Cite

View full text Add to dashboard Cite

“…1. For the Dirichlet boundary, the relaxed micromorphic theory introduces the so called consistent coupling condition [2]…”

mentioning

confidence: 99%

A quadratic finite element for the relaxed micromorphic model

Sky

Muench²,

Neff

2023

Proc Appl Math and Mech

Self Cite

View full text Add to dashboard Cite

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.

show abstract

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

Cited by 11 publications

References 75 publications

Size-effects of metamaterial beams subjected to pure bending: on boundary conditions and parameter identification in the relaxed micromorphic model

Size-effects of metamaterial beams subjected to pure bending: on boundary conditions and parameter identification in the relaxed micromorphic model

Green’s functions for the isotropic planar relaxed micromorphic model — Concentrated force and concentrated couple

A quadratic finite element for the relaxed micromorphic model

Contact Info

Product

Resources

About