Theoretical Models of Shape Formation: Analysis, Geometry and Energy Scaling Laws

最近，人们一直对生长诱导的形态发生（即形状形成），特别是低维结构，如细丝、层和它们的组合，在生物系统及其人工模拟中经常出现的兴趣。形态发生的物理基础可以用一个简单的原理来表示：一个物体的不同生长导致残余应变，通常导致其形状的变化。最终，生长模式预计将依次由这些菌株调节，因此这一原则很可能是生物组织物理自组织的基础。这些课题是生物学、化学和物理学的交叉点，同时也涉及工程设计等方面的实际问题。残余应力层存在于科学技术的各种场合；从原子级薄的石墨烯（厚度为1纳米，横向跨度为几厘米）到地壳（厚度为10公里，横向跨度为数千公里）。在日常尺度上，人们已经做了很多工作，试图理解这些层在被驱动时的机制，比如在生长的叶子、膨胀或收缩的凝胶片、塑料拉伸片等中。对控制平衡和这种结构演化的规律的理解有许多潜在的应用。由于结构的生长模式和材料中的残余应变之间的相互作用，研究者研究了与这些低维结构形状发展相关的数学问题。学生们在项目的过程中得到训练。关于形状发展的问题从根本上也具有深刻的几何和分析特征。事实上，它们可以被看作是微分几何经典主题的一种变体，即在可能不同维度的空间中嵌入具有给定度量的形状。在这个项目中，研究者的目的不仅是说明这种嵌入可能（或不）的条件，而且是：1)建设性地确定通过最小化测量强制度量和变形形状实现的度量之间的总体差异的能量而产生的形状，2)根据适当的力学理论确定上述形状，3)研究在细长结构中自然产生的尺度分离，并诱导与生长规律相关的约束。

Recently there has been sustained interest in growth-induced morphogenesis (i.e., shape formation), particularly of low-dimensional structures such as filaments, laminae, and their assemblies, which arise routinely in biological systems and their artificial mimics. The physical basis for morphogenesis can be presented in terms of a simple principle: differential growth in a body leads to residual strains that generically result in changes of its shape. Eventually, the growth patterns are expected to be, in turn, regulated by these strains, so that this principle might well be the basis for the physical self-organization of biological tissues. Such topics lie at the interface of biology, chemistry and physics, with practical questions of engineering design and others. Residually stressed laminae are present in science and technology in a variety of situations; from atomically thin grapheme (of thickness 1 nm, with a lateral span of a few cm), to the earth's crust (of thickness 10 km, which spans thousands of km laterally). On the everyday scale, there has been much work on trying to understand the mechanics of these laminae when they are actuated, as in a growing leaf, a swelling or shrinking sheet of gel, a plastically strained sheet, etc. Understanding of the laws governing the equilibria and the evolution of such structures has many potential applications. The investigator studies mathematical problems related to the development of the shapes of these low-dimensional structures due to the interplay between growth patterns of the structures and residual strains in the material. Students are trained in the course of the project. Questions about the development of shapes fundamentally have also a deeply geometric and analytical character. Indeed, they may be seen as a variation on a classical theme in differential geometry -- that of embedding a shape with a given metric in a space of possibly different dimension. In this project the investigator aims not only to state the conditions when this embedding might be done (or not), but also to: 1) constructively determine the shapes resulting from minimizing the energy that measures the overall discrepancy between the imposed metric and the metric realized by the deformed shape, 2) determine the shapes as above in terms of an appropriate mechanical theory, and 3) investigate the separation of scales that arises naturally in slender structures and induces the constraints associated with the prescription of growth laws.