Calculus of variations on pre-strained elastic structures

University Of Pittsburgh, Pittsburgh PA

Investigators

Abstract

Studying the elasticity properties of materials through methods of calculus of variations has become a field of its own in the mathematical analysis of solid elastic bodies. Recent developments in the study of pre-strained materials has put forward new unanswered questions in this field. The pre-straining might arise from a range of causes: inhomogeneous growth, plastic deformation, swelling or shrinkage driven by temperature or by solvent absorption. The basic model, called ``three dimensional pre-strained elasticity'', is a generalization of the classical variational nonlinear elasticity theory, adapted to take into account the pre-straining. Rigorously deriving 2-dimensional shell models from the 3 dimensional model and studying their properties is among the main objectives of the project. In this context, the study of geometric properties of structures, e.g. the properties of various spaces of isometries and infinitesimal isometries of 2 dimensional surfaces, comes to the front line of the research. Some classes of weakly differentiable mappings with geometric constraints, namely the classes of Sobolev infinitesimal isometries and isometric immersions and of Sobolev solutions to Monge-Ampere type equations, must be studied in order to achieve a full understanding. The award will support the methodical study of physical or biological phenomena involving structures that try to form an ideal yet unrealizable shape. Such situations, known broadly as pre-strained structures, occur quite often, e.g. in growing tissues, i.e. those structures that grow, such as leaves or tumors, or in engineered gels, which are manufactured with inhomogeneous density and then forced to stretch or compress proportionally to the distribution of the density. Puzzling aspects of such materials have been observed in recent experiments, and their better understanding is therefore of considerable interest. The main framework for a theoretical understanding is the mathematical theory of elasticity, whose goal is to explain various, apparently different, phenomena in terms of some shared mathematical principles. The basic understanding of the pre-straining and its effects on the behavior of the elastic bodies could lead to applications such as manufacturing of intelligently malleable materials or a better understanding of elasticity and morphology of growing biological tissues. The award will also promote interdisciplinary interactions and the research will be carried on in close connection with the applied mathematics, materials science and the biomechanics communities. Some graduate students will be trained in doing mathematical research through involvement in this project.