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

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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.