Dimension Reduction and Singular Limits of Prestrained Structures

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This project is directed at the study of properties and behavior of "prestrained elastica". Elastica (elastic materials) are the solid materials which return to their original shape and size after forces applied to them are removed. If an elastic body is appropriately processed mechanically (e.g. rolled), thermally (cooled non-uniformly during heat treatment), chemically ("nitrided" through surface absorption of nitrogen) or exposed to inhomogeneous growth, stresses and strains may develop in the body at equilibrium, leading to the material that has been prestrained (strained in advance). A characteristic which singles out the quality of prestraining in a body is that even in the absence of exterior forces the body assumes a shape that is radically different from the same body without strains. This phenomenon has been observed in different contexts: from growing leaves, through liquid crystals (used in various displays), to polymer gels; it is especially significant for materials formed as thin sheets. With the advancement of wide manufacturing and use of the novel materials in the thin film shape (molecular thin films, nanotubes, perforated domains, engineered gels), it becomes especially important to gain a theoretical insight on how to relate the prestrain with the elastic energy stored in the body. Attaining such a theoretical insight is the overarching objective of this project. One of applications of this research is controlling the structural properties of the desired final product through fine-tuning its manufacturing conditions. This project includes opportunities for the research training of students at various educational levels and disseminating obtained results to research communities of mathematicians and engineers. The project has theoretical as well as applied aspects, representing contributions to mathematical analysis, differential geometry, calculus of variations, materials science, and engineering design. Specifically, the general dimension reduction classification of prestrained elastic materials, with the prestrain given through a Riemannian metric of arbitrarily large (or arbitrarily small) curvature, as well as distributed across the mid-surface and the thickness of the film, will be developed. Further, the curvature constraints obtained in the process of dimensionally reducing the energy of a prestrained thin film and corresponding to different energy and regularity of deformations regimes, will be studied. This includes the rigidity and flexibility of Holder-continuous solutions to the quadratic Monge-Ampere system in arbitrary spatial dimensions. Finally, the project will involve the time-dependent and discrete versions of the stationary continuum problems mentioned above, including the random environment setting.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.