engleski

Homogenization of the elastic plate equation

The main goal of this thesis is to study homogenization of the Kirchhoff-Love model for pure bending of a thin symmetric elastic plate, which is described by the fourth order elliptic equation. Homogenization theory is one of the most successful approaches for dealing with optimal design problems (in conductivity or linearized elasticity), which consists of arranging given materials such that obtained body satisfies some optimality criteria, typically expressed mathematically as the minimization of some (integral) functional under some (PDE) constraints. The key role in homogenization theory has H-convergence. After a brief introduction, in Chapter 1 we prove a number of properties of H- convergence, such as locality, independence of boundary conditions, metrizability of H-topology, convergence of energies and a corrector result. We also discuss smooth dependence of H-limit on a parameter and calculate the H-limit of a periodic sequence of tensors. Moreover, we give special emphasis to calculating the first correction in the small-amplitude homogenization limit of a sequence of periodic tensors. Using this newly developed theory, in Chapter 2 we put our focus on the composite elastic plate. We show the local character of the set of all possible composites, also called the G-closure, and prove that the set of composites obtained by periodic homogenization is dense in that set. Additionally, we derive explicit expressions for elastic coefficients of composite plate obtained by mixing two materials in thin layers (known as laminated material), and for mixing two materials in the low-contrast regime. Moreover, we derive optimal bounds on the effective energy of a composite material, known as Hashin-Shtrikman bounds. In the case of two-phase isotropic materials, explicit optimal Hashin- Shtrikman bounds are calculated. We show that an analogous results can be derived for the complementary energy of a composite material.

Kirchhoff-Love model of elastic plate, composite material, G-closure, Hashin-Shtrikman bounds, homogenization, H-convergence, laminated material, small-amplitude homogenization

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