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

最近，人们对生长诱导的形态发生(即形状形成)一直感兴趣，特别是低维结构，如细丝、片层及其组件，这些结构经常出现在生物系统及其人工模拟物中。形态发生的物理基础可以用一个简单的原理来表示：身体的不同生长导致残余应变，而残余应变通常导致其形状的变化。最终，生长模式有望反过来受到这些菌株的调节，因此这一原理很可能成为生物组织物理自组织的基础。这样的主题存在于生物、化学和物理的交界处，以及工程设计和其他实际问题。剩余应力薄层在科学和技术中存在于各种情况；从原子薄的石墨层(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)。在日常的尺度上，已经有很多工作试图理解这些薄层在它们被驱动时的机制，例如在生长的叶子、膨胀或收缩的凝胶片、塑性应变的薄片等中。理解控制这些结构的平衡和演化的规律有许多潜在的应用。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，研究人员研究了与这些低维结构的形状发展相关的数学问题。学生们在这个项目的过程中接受了培训。关于形状发展的问题从根本上也具有深刻的几何性和解析性。事实上，它们可能被视为微分几何中一个经典主题的变体--将具有给定度量的形状嵌入到可能不同维度的空间中。在这个项目中，研究人员的目标不仅是陈述这种嵌入可能(或不可能)的条件，而且还包括：1)建设性地确定通过最小化能量而产生的形状，该能量测量施加的度量和由变形的形状实现的度量之间的总体差异；2)根据适当的力学理论确定如上所述的形状；以及3)研究细长结构中自然产生的鳞片的分离，并归纳出与生长定律规定相关联的约束。

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.