ABSTRACT:

This paper presents a critical analysis of three microstructural modellings, namely, the Double gradient theory the Cosserat theory, and the homogenization up to the third order. The basic assumptions for obtaining a macroscopic description of heterogeneous media are first reviewed. Then the interpretation of averaged stresses used in these approaches is addressed, and the question of boundary conditions is evoked. This analysis leads us to define the validity domain of the homogenized and double gradient microstructural description, and to prove that microstructural effects cannot be modelled by the Cosserat mechanics.

RÉSUMÉ:

On presente dans cet article une analyse critique des trois modèles microstructuraux obtenus par les theories du Double gradient, de Cosserat et de l'homogeneisation jusqu'au troisième ordre. En premier lieu on expose les hypothèses fondamentales pour obtenir une description macroscopique d'un milieu heterogène. Puis on s'attache à l'interpretation des moyennes de contrainte utilisees dans ces approches ainsi qu'à la question des conditions aux limites. Cette analyse nous conduit à definir Ie domaine de validite des descriptions homogeneisee et double gradient, et à demontrer que les effets microstructuraux ne peuvent pas être modelises par la mecanique de Cosserat.

ZUSAMMENFASSUNG:

Dieser Artikel stellt eine kritische Analyse ueber drei mikrostrukturelen Modellen vor, die durch die Theorie des "Double gradient", die von Cosserat und die "Homogenization" bis zur dritten Ordnung erreichr. Zuerst werden die Grundhypothese dargelegt, um eine makroskopische Beschreibung eines heterogenen Materiales zu bekommen. Dann werden die in diesen Theorien benutzten durchschnittlichen Spannungen erklart und die Grenzbedigungen studiert. Die Untersuchung ermöglicht die Bestimmung des Geltungsbereichs der Beschreibungen der "Homogenization" und des "Double gradient" und den Beweis, daß die mikrostrukturelen Wirkungen durch dieMechanik von Cosserat dargestellt nicht werden können.

1 INTRODUCTION

The problems in rocks mechanics are usually dealt with using the mechanics of continuous media. However rocks are heterogeneous media and some phenomena (localization of the deformations, surface instabilities) are due to the presence of a microstructure. In order to describe these phenomena, microstructural modellings such as Homogenization at superior orders, Double gradient, or Cosserat theories were proposed by various authors. In this paper we focus on the physical meaning of these approaches, and for simplification purpose, we only study small deformations and Consider rocks as elastic composites. The definition (and existence) of a macroscopic stress tensor is examined, problems related to the expression of boundary conditions are studied, and the validity of the microstructural descriptions is analyzed. The common point of the various modellings is to try to obtain a "continuum" description including the microstructural effects. It is then useful to review the basic assumptions for obtaining such a macroscopic description for heterogeneous media. Obviously, there is no way to obtain such a description if the studied phenomenon essentially varies at the heterogeneity level. Then, a basic condition is that the phenomenon presents a characteristic size of evolution, L, much larger than that of the heterogeneities, I. Thus, the search for a homogenized behavior makes sense only if the condition of scale separation is fulfilled. Actually, this condition leads to two requirements which result in a "local invariance": - the first concerns the material, which must be sufficiently regular so that one can define a an elementary representative volume (VER).

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