Establishment of strain gradient constitutive relations by homogenization

Jia Li

doi:10.1016/j.crme.2011.02.002

Establishment of strain gradient constitutive relations by homogenization

Jia Li ¹

¹ LPMTM, CNRS UPR 9001, Université Paris XIII, 99, avenue Jean-Baptiste-Clément, 93430 Villetaneuse, France

Comptes Rendus. Mécanique, Volume 339 (2011) no. 4, pp. 235-244.

Résumé
Abstract

Dans cette Note, nous décrivons une méthodologie dʼhomogénéisation dans le but dʼétablir des lois de comportement au gradient de déformation pour les matériaux hétérogènes. La méthodologie présentée comporte deux étapes. La première étape est la construction de la densité moyenne dʼénergie de déformation pour un Volume Élémentaire Représentatif (VER) en utilisant une technique dʼhomogénéisation. La deuxième est de transformer cette densité moyenne obtenue pour un VER en celle valable pour un milieu continu. Cette transformation permet de garantir que les modules effectifs généralisés ne dépendent pas de la taille du VER mais seulement des tailles intrinsèques des constituants. Elle garantit également que les modules du second ordre sʼannulent si le contraste entre les propriétés des constituants disparaît. Ces aspects sont illustrés ensuite par un exemple simple.

In this Note, we describe a homogenization methodology with the aim of establishing strain gradient constitutive relations for heterogeneous materials. The presented methodology includes two main steps. The first one is the construction of the average strain-energy density for a well-chosen Representative Volume Element (RVE) by using a homogenization technique. The second one is the transformation of the obtained average strain-energy density to that of the continuum. This transformation permits to ensure that the effective moduli do not depend on the size of an RVE, but only on the intrinsic size of the components. It also ensures that the moduli related to the strain gradient vanish as the contrast between the components disappears. These features were then illustrated by means of a simple example.

Reçu le : 2010-09-27
Accepté le : 2011-02-14
Publié le : 2011-03-03

DOI : 10.1016/j.crme.2011.02.002

Keywords: Mechanisms, Strain gradient theory, Homogenization, Constitutive equations
Mots clés : Mécanismes, Théorie au gradient de déformation, Homogénéisation, Équations constitutives

Affiliations des auteurs :

Jia Li ¹

¹ LPMTM, CNRS UPR 9001, Université Paris XIII, 99, avenue Jean-Baptiste-Clément, 93430 Villetaneuse, France

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     title = {Establishment of strain gradient constitutive relations by homogenization},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {235--244},
     publisher = {Elsevier},
     volume = {339},
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     year = {2011},
     doi = {10.1016/j.crme.2011.02.002},
     language = {en},
}

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Jia Li. Establishment of strain gradient constitutive relations by homogenization. Comptes Rendus. Mécanique, Volume 339 (2011) no. 4, pp. 235-244. doi : 10.1016/j.crme.2011.02.002. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2011.02.002/

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