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），特别是低维结构，如细丝、薄片及其组装，其通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描述：身体的差异生长导致残余应变，这些应变通常会导致其形状的变化。 最终，生长模式预计将反过来受到这些菌株的调节，因此这一原则很可能成为生物组织物理自组织的基础。 这些主题位于生物学，化学和物理学的接口，与工程设计和其他实际问题。 残余应力纹层存在于科学和技术中的各种情况下;从原子薄的石墨烯（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）。 在日常规模上，已经有很多工作试图了解这些层的力学时，他们被驱动，在一个不断增长的叶子，一个膨胀或收缩片的凝胶，一个塑性应变片等了解的法律管辖的平衡和这种结构的演变有许多潜在的应用。 研究人员研究与这些低维结构的形状发展相关的数学问题，这是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。 学生在项目过程中接受培训。关于形状发展的问题，从根本上说，也具有深刻的几何和分析性质。 事实上，它们可以被看作是微分几何中经典主题的一种变体--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进行（或不进行）的条件，而且是：1）建设性地确定由最小化能量而产生的形状，所述能量测量所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，3）研究细长结构中自然产生的尺度分离，并归纳出与增长律处方有关的约束。

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.