Homogenization, linearization and dimension reduction in elasticity with variational methods
Stefan Neukamm · mediaTUM – the media and publications repository of the Technical University Munich (Technical University Munich) · 2010
The objective of this thesis is the derivation of effective theories for thin elastic bodies with periodic microstructure. The main result is the rigorous, ansatz free derivation of a homogenized Cosserat theory for inextensible rods from nonlinear three-dimensional elasticity. The approach is based on the variational point of view and the derivation is expressed in the language of Gamma-convergence employing periodic unfolding methods. A peculiarity of thin elastic objects is their capability to undergo large deformations at low energy. Mathematically this phenomenon is captured by studying an appropriate scaling of the energy and leads to a linearization effect in the limiting process. In this thesis we develop new general methods for multiscale problems that simultaneously involve homogenization, dimension reduction and linearization.