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

形状形成的理论模型：分析、几何和能量缩放定律

• 批准号: 1406730
• 负责人: Marta Lewicka
• 金额: $ 16.9万
• 依托单位: University of Pittsburgh
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1406730&HistoricalAwards=false
• 关键词: Theoretical; Models; Shape; Formation; Analysis

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.

最近，人们对生长引起的形态发生（即形状形成）持续持续兴趣，特别是对诸如细丝，薄片及其组件等低维结构，这些结构在生物系统及其人工模仿中通常会产生。 形态发生的物理基础可以从一个简单的原理来表示：身体的差异生长会导致残留菌株通常会导致其形状变化。 最终，预计生长模式将受这些菌株的调节，因此该原理很可能是生物组织物理自我组织的基础。 这些主题在于生物学，化学和物理学的界面，以及工程设计等的实用问题。 在各种情况下，残留的压力层中存在于科学技术中。从原子上薄的石素（厚度为1 nm，横向跨度为几厘米），到地壳（厚度为10 km），横向跨越数千km）。 在日常规模上，有很多努力试图在被驱动时了解这些薄片的机制，例如在越来越多的叶片，凝胶或收缩的凝胶片，塑料张力张紧的片片等。理解有关等于平衡的法律的理解和此类结构的演变具有许多潜在的应用。 研究人员研究了与这些低维结构形状的发展有关的数学问题，这是由于结构的生长模式与材料中的残留菌株之间的相互作用。 在项目过程中，学生接受了培训。从根本上讲，有关形状发展的问题也具有深刻的几何和分析特征。 确实，它们可以被视为差异几何形状中经典主题的变体 - 将给定指标嵌入形状可能不同的空间。 在这个项目中，研究者不仅旨在说明可能完成此嵌入的条件（或不进行），而且要：1）建设性地确定最小化的能量所产生的形状，从而最大程度地减少了衡量所施加的度量的总体差异和由变形形状所实现的指标和指标之间的整体差异，2）在适当的机制上确定型号，并在适当的机制上确定型号，并确定3）的范围。与增长法处方有关的约束。