On problems in homogenization and two-scale convergence

Stelzig, Philipp Emanuel (2012) On problems in homogenization and two-scale convergence. PhD thesis, University of Trento.

PDF - Doctoral Thesis


This thesis addresses two problems from the theory of periodic homogenization and the related notion of two-scale convergence. Its main focus rests on the derivation of equivalent transmission conditions for the interaction of two adjacent bodies which are connected by a thin layer of interface material being perforated by identically shaped voids. Herein, the voids recur periodically in interface direction and shall in size be of the same order as the interface thickness. Moreover, the constitutive properties of the material occupying the bodies adjacent to the interface are assumed to be described by some convex energy densities of quadratic growth. In contrast, the interface material is supposed to show "extremal" constitutive behavior. More precisely, it is assumed that the constitutive relations of the interface material are once more characterized by a convex energy density of quadratic growth which, however, scales with the inverse thickness of the interface layer. In accordance to recent works on homogeneous interfaces in an analogous constitutive setting, it is found that in the limit of vanishing interface thickness and void size the interaction of the bodies adjacent to the interface can asymptotically be described by equivalent transmission conditions on the flattened interface. These transmission conditions are such that they penalize in-plane gradients in the interface. Cell formulas defining the homogenized transmission conditions on the flattened interface are derived, using a combination of Gamma-convergence methods and suitable adaptions of the periodic unfolding method in rescaled (perforated) thin domains. Depending on whether the voids touch the periodicity cells' faces, different periodic unfolding operators can be applied. The thesis' final chapter contains the results of a recent collaboration with Stefan Neukamm on the effects on two-scale convergence caused by a translation of the coordinate frame. It is observed that given a vanishing sequence of microscale parameters, a once two-scale convergent sequence is in general no longer two-scale convergent when described in a translated coordinate frame, even though it remains two-scale convergent along suitable subsequences. Yet, all two-scale cluster points of the translated sequence are indeed revealed as translates of the original two-scale limit. In fact, these two-scale cluster points are not only translated in the macroscopic variable, but also in the microscopic variable by microtranslations which belong to a set that is determined solely by the translation of the coordinate frame and the vanishing sequence of microscale parameters. This result is then applied to a novel homogenization problem that involves two different coordinate frames and yields a limiting behavior governed by emerging microscopic translations. Finally, in addition to these results the thesis also indicates a possible extension of the periodic unfolding method to so-called non-translatory periodic microstructures.

Item Type:Doctoral Thesis (PhD)
Doctoral School:Mathematics
Subjects:Area 01 - Scienze matematiche e informatiche > MAT/05 ANALISI MATEMATICA
Area 01 - Scienze matematiche e informatiche > MAT/07 FISICA MATEMATICA
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