Effective Theories and Energy Minimizing Configurations for Heterogeneous Layers

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Thin elastic sheets have been the subject of intense research over the last years, both from the analytical and from the experimental point of view. Seminal work of Friesecke, James and Müller showed that classical plate theories can be obtained from three-dimensional nonlinear elasticity rigorously in terms of a variational limit process. From the physical perspective, a method for a self-organized fabrication of nano-scrolls from atomistically thin heterogeneous layers with internal stress has been reported by Schmidt and Eberl. A first aim of the proposed project is to derive effective theories for heterogeneous elastic layers with internal stress mathematically rigorously. As for atomistically thin films it is unclear if a pure continuum description can describe the anticipated effects sufficiently accurately, these theories should in particular be obtained from fundamental interatomic interactions. We expect new terms that reflect the underlying discrete lattice structure of the materials involved. In a second step, these theories are to be examined for their energy minimizing configurations in order to obtain a thorough understanding for the stress induced geometry of such objects. While this question could be answered in a special case in a previous contribution of ours, for the plate theories under investigation we expect to see novel results. The results of this project cover the range from fundamental questions of justifying continuum mechanical methods through suitable passages from atomistic to continuum to obtaining a detailed understanding of a current physical technique on self-organization in nanotechnology.