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，薄弹性片材一直是密集研究的主题。Friesecke、James和Müller的早期工作表明，经典板理论可以严格地从三维非线性弹性力学的变分极限过程中得到。从物理角度来看，施密特和埃贝尔已经报道了一种从具有内应力的原子级薄异质层自组织制造纳米卷的方法。该项目的第一个目的是推导出有效的理论与内部应力的非均匀弹性层的数学严格。至于原子薄膜，目前尚不清楚是否可以用纯连续描述足够准确地描述预期的效果，这些理论应该特别是从基本的原子间相互作用中获得。我们期待新的条款，反映了所涉及的材料的基本离散晶格结构。在第二步中，这些理论将被检查其能量最小化的配置，以获得一个彻底的理解，这种对象的应力诱导的几何形状。虽然这个问题可以回答在一个特殊的情况下，在我们以前的贡献，板理论正在调查中，我们希望看到新的结果。该项目的结果涵盖了从通过从原子到连续体的适当通道来证明连续体机械方法的合理性的基本问题，到对纳米技术中自组织的当前物理技术的详细了解。

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.