Effective Theories and Energy Minimizing Configurations for Heterogeneous Layers

在过去的几年里，从分析和实验的角度来看，薄弹性片一直是激烈研究的主题。Friesecke， James和m<e:1> ller的开创性工作表明，经典板理论可以从三维非线性弹性中严格地以变分极限过程获得。从物理角度来看，Schmidt和Eberl报道了一种由具有内应力的原子薄非均质层自组织制造纳米卷轴的方法。提出的项目的第一个目标是推导出具有内应力的非均质弹性层的有效理论。至于原子薄膜，尚不清楚纯连续体描述是否能够足够准确地描述预期的效应，这些理论尤其应该从基本的原子相互作用中获得。我们期望新的术语能够反映所涉及的材料的潜在离散晶格结构。在第二步中，这些理论将被检查其能量最小化配置，以获得对这些物体的应力诱导几何的彻底理解。虽然这个问题可以在我们之前的贡献中在一个特殊情况下得到回答，但对于正在研究的板块理论，我们希望看到新的结果。这个项目的结果涵盖了从证明连续体力学方法的基本问题，通过从原子到连续体的适当通道，获得对纳米技术中自组织的当前物理技术的详细理解。

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.