Homogenization of Elastic Media with Gaseous Inclusions

Abstract

We study the asymptotic behavior of a system modeling a composite material made of an elastic periodically perforated support, with period ε > 0, and a perfect gas placed in each of these perforations, as ε goes to zero. The model we use is linear corresponding to deformations around a reference configuration. We apply both two-scale asymptotic expansion and two-scale convergence methods in order to identify the limit behaviors as ε goes to 0. We state that in the limit, we get a two-scale linear elasticity-like boundary value problem. From this problem, we identify the corresponding homogenized and periodic cell equations which allows us to find the first corrector term. The analysis is performed both in the case of an incompressible and compressible material. We derive some mechanical properties of the limit materials by studying the homogenized coefficients. Finally, we numerically calculate the homogenized coefficients in the incompressible case, for different types of elastic materials.

Multiscale modelling of sound propagation through the lung parenchyma

“…Let u ε be the (small) displacement field of the elastic media. Linearizing around an equilibrum state of atmospheric pressure p a and volume B k ε , the equation governing the pressure inside the bubbles is [2]:…”

“…Thanks to the two-scale convergences (1.11) we can pass to the limit in most terms of the weak formulation (1.9). We refer to [2] for details on the specific treatment of the term containing a non-local product of integrals in (1.9). In the end, we obtain that u, u 1 and p are solutions of the variational formulation, for all v ∈ V and…”

“…Existence and uniqueness of the limit pressure p can be proved using a two-scale inf-sup condition (Lemma 3.6 in [2]). …”

“…In previous attempts to derive a law at the macroscopic scale fitting experimental results, the alveolar wall was modeled as an homogeneous elastic or viscoelastic medium [2,28,35]. Here, we change this point of view and try to investigate, theoretically and numerically, some macroscopic effects of the heterogeneity of the alveolar material at the micro-scale.…”

“…The effective coefficients describing this homogenized medium are recovered from the given periodic micro-scale structure. Note that the homogenization of a similar model, in the static case, was considered in [2]. As is well-known [40], the homogenization of heterogeneous viscoelastic composites, by the interaction of temporal and spatial variations of the coefficients in the differential equations of the model, gives rise to new memory effects.…”