Calculus of variations on pre-strained elastic structures

用变分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独立领域。预应变材料研究的最新进展在这一领域提出了新的未回答的问题。预应变可能由一系列原因引起：不均匀生长、塑性变形、由温度或溶剂吸收驱动的溶胀或收缩。 基本模型称为"三维预应变弹性“，是经典变分非线性弹性理论的推广，适用于考虑预应变。该项目的主要目标之一是从3维模型严格推导出2维壳模型并研究其特性。在这种背景下，结构的几何性质的研究，例如各种空间的等距和无穷小等距的二维表面的性质，来到了研究的前沿。一些类的弱可微映射的几何约束，即类的Sobolev无穷小等距和等距浸入和Sobolev解决方案的Monge-Ampere型方程，必须进行研究，以达到一个完整的理解。该奖项将支持的物理或生物现象，涉及试图形成一个理想的，但无法实现的形状结构的有条不紊的研究。这样的情况，广泛地称为预应变结构，经常发生在例如生长组织中，即生长的那些结构，例如叶子或肿瘤，或者在工程化凝胶中，其被制造成具有不均匀的密度，然后被迫与密度分布成比例地拉伸或压缩。在最近的实验中已经观察到这些材料的令人困惑的方面，因此更好地理解它们具有相当大的兴趣。理论理解的主要框架是弹性的数学理论，其目标是用一些共有的数学原理来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理解可以导致诸如智能可延展材料的制造或更好地理解生长生物组织的弹性和形态等应用。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。一些研究生将通过参与这个项目进行数学研究的培训。

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.