In this paper, we substantiate the claim implicitly made in previous works that the relaxed micromorphic model is the only linear, isotropic, reversibly elastic, nonlocal generalized continuum model able to describe complete band-gaps on a phenomenological level. To this end, we recapitulate the response of the standard Mindlin–Eringen micromorphic model with the full micro-distortion gradient ∇P, the relaxed micromorphic model depending only on the Curl P of the micro-distortion P, and a variant of the standard micromorphic model, in which the curvature depends only on the divergence Div P of the micro distortion. The Div-model has size-effects, but the dispersion analysis for plane waves shows the incapability of that model to even produce a partial band gap. Combining the curvature to depend quadratically on Div P and Curl P shows that such a model is similar to the standard Mindlin–Eringen model, which can eventually show only a partial band gap.
Accepted:
Published online:
Angela Madeo 1, 2; Patrizio Neff 3; Marco Valerio d'Agostino 1; Gabriele Barbagallo 4
@article{CRMECA_2016__344_11-12_784_0,
author = {Angela Madeo and Patrizio Neff and Marco Valerio d'Agostino and Gabriele Barbagallo},
title = {Complete band gaps including non-local effects occur only in the relaxed micromorphic model},
journal = {Comptes Rendus. M\'ecanique},
pages = {784--796},
publisher = {Elsevier},
volume = {344},
number = {11-12},
year = {2016},
doi = {10.1016/j.crme.2016.07.002},
language = {en},
}
TY - JOUR AU - Angela Madeo AU - Patrizio Neff AU - Marco Valerio d'Agostino AU - Gabriele Barbagallo TI - Complete band gaps including non-local effects occur only in the relaxed micromorphic model JO - Comptes Rendus. Mécanique PY - 2016 SP - 784 EP - 796 VL - 344 IS - 11-12 PB - Elsevier DO - 10.1016/j.crme.2016.07.002 LA - en ID - CRMECA_2016__344_11-12_784_0 ER -
%0 Journal Article %A Angela Madeo %A Patrizio Neff %A Marco Valerio d'Agostino %A Gabriele Barbagallo %T Complete band gaps including non-local effects occur only in the relaxed micromorphic model %J Comptes Rendus. Mécanique %D 2016 %P 784-796 %V 344 %N 11-12 %I Elsevier %R 10.1016/j.crme.2016.07.002 %G en %F CRMECA_2016__344_11-12_784_0
Angela Madeo; Patrizio Neff; Marco Valerio d'Agostino; Gabriele Barbagallo. Complete band gaps including non-local effects occur only in the relaxed micromorphic model. Comptes Rendus. Mécanique, Volume 344 (2016) no. 11-12, pp. 784-796. doi : 10.1016/j.crme.2016.07.002. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2016.07.002/
[1] Nonlinear theory of simple micro-elastic solids – I, Int. J. Eng. Sci., Volume 2 (1964) no. 2, pp. 189-203
[2] Mechanics of micromorphic materials, Applied Mechanics, Springer Berlin Heidelberg, Berlin, Heidelberg, 1966, pp. 131-138
[3] Microcontinuum Field Theories, Springer-Verlag, New York, 1999
[4] Micro-structure in linear elasticity, Arch. Ration. Mech. Anal., Volume 16 (1964) no. 1, pp. 51-78
[5] The method of virtual power in continuum mechanics. Part 2: microstructure, SIAM J. Appl. Math., Volume 25 (1973) no. 3, pp. 556-575
[6] Transient computational homogenization for heterogeneous materials under dynamic excitation, J. Mech. Phys. Solids, Volume 61 (2013) no. 11, pp. 2125-2146
[7] Homogenization of locally resonant acoustic metamaterials towards an emergent enriched continuum, Comput. Mech., Volume 57 (2016) no. 3, pp. 423-435
[8] The relaxed linear micromorphic continuum: well-posedness of the static problem and relations to the gauge theory of dislocations, Q. J. Mech. Appl. Math., Volume 68 (2014) no. 1, pp. 53-84
[9] A unifying perspective: the relaxed linear micromorphic continuum, Contin. Mech. Thermodyn., Volume 26 (2014) no. 5, pp. 639-681
[10] Band gaps in the relaxed linear micromorphic continuum, Z. Angew. Math. Mech., Volume 95 (2014) no. 9, pp. 880-887
[11] Wave propagation in relaxed micromorphic continua: modeling metamaterials with frequency band-gaps, Contin. Mech. Thermodyn., Volume 27 (2015) no. 4–5, pp. 551-570
[12] The relaxed linear micromorphic continuum: existence, uniqueness and continuous dependence in dynamics, Math. Mech. Solids, Volume 20 (2014) no. 10, pp. 1171-1197
[13] Transparent anisotropy for the relaxed micromorphic model: macroscopic consistency conditions and long wave length asymptotics, 2016 | arXiv
[14] A geometrically exact micromorphic model for elastic metallic foams accounting for affine microstructure. Modelling, existence of minimizers, identification of moduli and computational results, J. Elast., Volume 87 (2007) no. 2–3, pp. 239-276
[15] On the validity of Friedrichs' inequalities, Math. Scand., Volume 54 (1984), pp. 17-26
[16] Rotational invariance conditions in elasticity, gradient elasticity and its connection to isotropy, 2016 | arXiv
[17] Determining material constants in micromorphic theory through phonon dispersion relations, Int. J. Eng. Sci., Volume 41 (2003) no. 8, pp. 871-886
[18] Connecting molecular dynamics to micromorphic theory. (I). Instantaneous and averaged mechanical variables, Phys. A, Stat. Mech. Appl., Volume 322 (2003), pp. 359-376
[19] Atomistic viewpoint of the applicability of microcontinuum theories, Int. J. Solids Struct., Volume 41 (2004) no. 8, pp. 2085-2097
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