- Book Chapter
- 10.1093/oso/9780198502463.003.0025
Homogenization of multi-dimensional structures
- Nov 26, 1998
- Andrea Braides + 1 more +1
In this chapter we describe some homogenization processes that may model the macroscopic properties of media whose microscopic behaviour takes lower dimensional or multi-dimensional structures into account. We will show that the method for homogenization described in Chapter 14 can be adapted to the asymptotic analysis of these functionals. In addition to this study, we also have to face the problem of the choice of a proper description of the small-scale functionals themselves, whose natural domain is not the usual Sobolev-space setting. In Sections 24.1 and 24.2 we deal with energies defined on lower-dimensional structures whose homogenization gives rise to integral functionals defined on full dimensional domains. We are going to describe two different and, to some extent, complementary approaches to this problem. We remark that while the limit analysis will be performed in detail, the description of the energies at length scale e is only outlined, leaving some details to the reader. In Section 24.3 we consider energies which describe structures as rods, plates or thin films, where one or more dimensions are negligible with respect to the others. These lower-dimensional energies are obtained by a ┌-limit procedure from (normalized) energies defined on full-dimensional domains contracting to lower-dimensional ones. We treat the case when the full dimensional structures possess in addition a (periodic) cellular structure. For the sake of illustration, we only deal with the simplified case when the period and the ‘thickness ‘ of the structure are on a comparable scale.
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