Dimension Reduction and Singular Limits of Prestrained Structures

本项目旨在研究“预应变弹性体”的性能和行为。弹性材料(弹性材料)是施加在其上的力被移除后恢复到其原始形状和大小的固体材料。如果对弹性体进行适当的机械处理(例如轧制)、热处理(在热处理过程中不均匀冷却)、化学处理(通过表面氮气吸收进行氮化)或暴露在非均匀生长中，则应力和应变可能在平衡状态下在体内发展，从而导致材料已经预应变(预先应变)。身体预紧的一个特征是，即使在没有外力的情况下，身体也会呈现出与没有张力的身体完全不同的形状。这种现象在不同的环境中都可以观察到：从生长的叶子，到液晶(用于各种显示器)，再到聚合物凝胶；这对形成薄片的材料尤其重要。随着薄膜形状的新型材料(分子薄膜、纳米管、穿孔域、工程凝胶)的广泛制造和使用，如何从理论上了解预应变与体内储存的弹性能量之间的关系变得尤为重要。达到这样的理论洞察力是这个项目的首要目标。这项研究的应用之一是通过微调其制造条件来控制所需最终产品的结构特性。该项目包括对不同教育水平的学生进行研究培训的机会，并将所取得的成果传播给数学家和工程师的研究界。该项目既有理论方面的，也有应用方面的，代表了对数学分析、微分几何、变分、材料科学和工程设计的贡献。具体地说，将发展预应变弹性材料的一般降维分类，其中预应变通过任意大(或任意小)曲率的黎曼度量给出，并且分布在膜的中表面和厚度上。此外，还将研究在量纲降低预应变薄膜能量的过程中获得的对应于不同能量和变形区域规律性的曲率约束。这包括任意空间维度上二次Monge-Ampere系统Holder连续解的刚性和灵活性。最后，该项目将涉及上述静止连续体问题的时间依赖和离散版本，包括随机环境设置。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project is directed at the study of properties and behavior of "prestrained elastica". Elastica (elastic materials) are the solid materials which return to their original shape and size after forces applied to them are removed. If an elastic body is appropriately processed mechanically (e.g. rolled), thermally (cooled non-uniformly during heat treatment), chemically ("nitrided" through surface absorption of nitrogen) or exposed to inhomogeneous growth, stresses and strains may develop in the body at equilibrium, leading to the material that has been prestrained (strained in advance). A characteristic which singles out the quality of prestraining in a body is that even in the absence of exterior forces the body assumes a shape that is radically different from the same body without strains. This phenomenon has been observed in different contexts: from growing leaves, through liquid crystals (used in various displays), to polymer gels; it is especially significant for materials formed as thin sheets. With the advancement of wide manufacturing and use of the novel materials in the thin film shape (molecular thin films, nanotubes, perforated domains, engineered gels), it becomes especially important to gain a theoretical insight on how to relate the prestrain with the elastic energy stored in the body. Attaining such a theoretical insight is the overarching objective of this project. One of applications of this research is controlling the structural properties of the desired final product through fine-tuning its manufacturing conditions. This project includes opportunities for the research training of students at various educational levels and disseminating obtained results to research communities of mathematicians and engineers. The project has theoretical as well as applied aspects, representing contributions to mathematical analysis, differential geometry, calculus of variations, materials science, and engineering design. Specifically, the general dimension reduction classification of prestrained elastic materials, with the prestrain given through a Riemannian metric of arbitrarily large (or arbitrarily small) curvature, as well as distributed across the mid-surface and the thickness of the film, will be developed. Further, the curvature constraints obtained in the process of dimensionally reducing the energy of a prestrained thin film and corresponding to different energy and regularity of deformations regimes, will be studied. This includes the rigidity and flexibility of Holder-continuous solutions to the quadratic Monge-Ampere system in arbitrary spatial dimensions. Finally, the project will involve the time-dependent and discrete versions of the stationary continuum problems mentioned above, including the random environment setting.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.