The modified indeterminate couple stress model: Why Yang et al.'s arguments motivating a symmetric couple stress tensor contain a gap and why the couple stress tensor may be chosen symmetric nevertheless

We rigorously determine the scale-independent short range elastic parameters in the relaxed micromorphic generalized continuum model for a given periodic microstructure. This is done using both classical periodic homogenization and a new procedure involving the concept of apparent material stiffness of a unit-cell under affine Dirichlet boundary conditions and Neumann's principle on the overall representation of anisotropy. We explain our idea of "maximal" stiffness of the unit-cell and use state of the art first order numerical homogenization methods to obtain the needed parameters for a given tetragonal unit-cell. These results are used in the accompanying paper [16] to describe the wave propagation including band-gaps in the same tetragonal metamaterial.

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“…Letting finally C c → +∞, we obtain Toupin's indeterminate couple stress model [30,48,55,62] with energy…”

Section: Comparison To Eringen-mindlin Micromorphic Modelsmentioning

confidence: 99%

Identification of Scale-Independent Material Parameters in the Relaxed Micromorphic Model Through Model-Adapted First Order Homogenization

Neff

Eidel

d’Agostino

et al. 2019

J Elast

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103

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“…see [58,59]. Based on these transformation laws, we may investigate the anisotropy of the rotational coupling with the representation in terms of the second order tensor C c .…”

Section: Determination Of the Fourth Order Tensors C C In Terms Of C Cmentioning

confidence: 99%

Transparent anisotropy for the relaxed micromorphic model: Macroscopic consistency conditions and long wave length asymptotics

Barbagallo

Madeo

d’Agostino

et al. 2017

International Journal of Solids and Structures

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106

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In this paper, we study the anisotropy classes of the fourth order elastic tensors of the relaxed micromorphic model, also introducing their second order counterpart by using a Voigt-type vector notation. In strong contrast with the usual micromorphic theories, in our relaxed micromorphic model only classical elasticity-tensors with at most 21 independent components are studied together with rotational coupling tensors with at most 6 independent components. We show that in the limit case Lc → 0 (which corresponds to considering very large specimens of a microstructured metamaterial) the meso-and microcoefficients of the relaxed model can be put in direct relation with the macroscopic stiffness of the medium via a fundamental homogenization formula. We also show that a similar homogenization formula is not possible in the case of the standard Mindlin-Eringen-format of the anisotropic micromorphic model. Our results allow us to forecast the successful short term application of the relaxed micromorphic model to the characterization of anisotropic mechanical metamaterials.

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“…Such a model is still well-posed [45] leading to existence and uniqueness results with only one additional material length scale parameter, while it is not pointwise uniformly positive definite. The initial motivation of Yang et al [88] for using the modified couple stress model is based on incomplete arguments [61], even if their conclusions concerning a symmetric couple stress tensor may be kept in some particular phenomenological cases [61].…”

Section: The Indeterminate Couple Stress Modelmentioning

confidence: 99%

A new view on boundary conditions in the Grioli–Koiter–Mindlin–Toupin indeterminate couple stress model

Madeo

Ghiba

Neff

et al. 2016

European Journal of Mechanics - A/Solids

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In this paper we consider the Grioli-Koiter-Mindlin-Toupin linear isotropic indeterminate couple stress model. Our main aim is to show that, up to now, the boundary conditions have not been completely understood for this model. As it turns out, and to our own surprise, restricting the well known boundary conditions stemming from the strain gradient or second gradient models to the particular case of the indeterminate couple stress model, does not always reduce to the Grioli-Koiter-Mindlin-Toupin set of accepted boundary conditions. We present, therefore, a proof of the fact that when specific "mixed" kinematical and traction boundary conditions are assigned on the boundary, no "a priori" equivalence can be established between Mindlin's and our approach.

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The modified indeterminate couple stress model: Why Yang et al.'s arguments motivating a symmetric couple stress tensor contain a gap and why the couple stress tensor may be chosen symmetric nevertheless

Cited by 48 publications

References 49 publications

Identification of Scale-Independent Material Parameters in the Relaxed Micromorphic Model Through Model-Adapted First Order Homogenization

Identification of Scale-Independent Material Parameters in the Relaxed Micromorphic Model Through Model-Adapted First Order Homogenization

Transparent anisotropy for the relaxed micromorphic model: Macroscopic consistency conditions and long wave length asymptotics

A new view on boundary conditions in the Grioli–Koiter–Mindlin–Toupin indeterminate couple stress model

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