Calculus of variations on pre-strained elastic structures

用变分方法研究材料的弹性性质已成为固体弹性体数学分析中的一个重要领域。预应变材料研究的最新进展给该领域提出了新的悬而未决的问题。预应变可能是由一系列原因引起的：不均匀生长、塑性变形、温度或溶剂吸收导致的膨胀或收缩。基本模型称为“三维预应变弹性”，是经典的非线性变分弹性理论的推广，适用于考虑预应变。从三维模型严格推导出二维壳模型并研究其性质是该项目的主要目标之一。在这种背景下，研究结构的几何性质，例如二维曲面的各种等距空间和无穷小等距空间的性质，就成为研究的前沿。为了达到完全的理解，必须研究某些具有几何约束的弱可微映射类，即Sobolev无穷小等距和等距浸入类，以及Monge-Ampere型方程的Sobolev解类。该奖项将支持对涉及试图形成理想但不可实现的形状的结构的物理或生物现象的系统研究。这种被广泛称为预应变结构的情况经常发生，例如在生长组织中，即那些生长的结构，例如叶子或肿瘤，或在工程凝胶中，这些结构以不均匀的密度制造，然后被迫按密度分布成比例地拉伸或压缩。在最近的实验中观察到了这类材料令人费解的方面，因此更好地理解它们是相当有兴趣的。理论理解的主要框架是弹性数学理论，其目标是根据一些共同的数学原理解释各种明显不同的现象。对预应变及其对弹性体行为的影响的基本了解可以导致诸如制造智能延展性材料或更好地理解正在生长的生物组织的弹性和形态的应用。该奖项还将促进跨学科的互动，研究将与应用数学、材料科学和生物力学社区密切联系起来进行。一些研究生将通过参与这个项目来接受数学研究方面的培训。

Studying the elasticity properties of materials through methods of calculus of variations has become a field of its own in the mathematical analysis of solid elastic bodies. Recent developments in the study of pre-strained materials has put forward new unanswered questions in this field. The pre-straining might arise from a range of causes: inhomogeneous growth, plastic deformation, swelling or shrinkage driven by temperature or by solvent absorption. The basic model, called ``three dimensional pre-strained elasticity'', is a generalization of the classical variational nonlinear elasticity theory, adapted to take into account the pre-straining. Rigorously deriving 2-dimensional shell models from the 3 dimensional model and studying their properties is among the main objectives of the project. In this context, the study of geometric properties of structures, e.g. the properties of various spaces of isometries and infinitesimal isometries of 2 dimensional surfaces, comes to the front line of the research. Some classes of weakly differentiable mappings with geometric constraints, namely the classes of Sobolev infinitesimal isometries and isometric immersions and of Sobolev solutions to Monge-Ampere type equations, must be studied in order to achieve a full understanding.The award will support the methodical study of physical or biological phenomena involving structures that try to form an ideal yet unrealizable shape. Such situations, known broadly as pre-strained structures, occur quite often, e.g. in growing tissues, i.e. those structures that grow, such as leaves or tumors, or in engineered gels, which are manufactured with inhomogeneous density and then forced to stretch or compress proportionally to the distribution of the density. Puzzling aspects of such materials have been observed in recent experiments, and their better understanding is therefore of considerable interest. The main framework for a theoretical understanding is the mathematical theory of elasticity, whose goal is to explain various, apparently different, phenomena in terms of some shared mathematical principles. The basic understanding of the pre-straining and its effects on the behavior of the elastic bodies could lead to applications such as manufacturing of intelligently malleable materials or a better understanding of elasticity and morphology of growing biological tissues. The award will also promote interdisciplinary interactions and the research will be carried on in close connection with the applied mathematics, materials science and the biomechanics communities. Some graduate students will be trained in doing mathematical research through involvement in this project.