The project is motivated by several questions in two distinct areas of materials science. The first part of the project undertakes the development of a general theory that allows for efficient characterizations of the yield set of a polycrystal by means of suitable variational principles in L-infinity. A new variational formalism leads naturally to several problems in the emerging area of calculus of variations in L-infinity for the case where the functionals to be optimized are the essential supremum of some expression involving divergence-free fields. Here the natural approach is to minimize the maximum of certain relevant quantities instead of their average. The investigator studies selected issues in this direction, with an emphasis on those that can have an impact on our understanding of plasticity. Among these, the study of the systems of partial differential equations that arise as Aronsson equations associated to the new variational principles in L-infinity is of particular interest. The second part of the project deals with the derivation of nonlinear membrane theories from three-dimensional elasticity, seeking to better understand the formation of interfaces in thin films of martensitic materials. The project sheds light on the structure of the minimizers for a new effective thin film energy, and indicates appropriate minimizing sequences.
The project aims to set on rigorous mathematical grounds some of the traditional models used by engineers and material scientists and, on the other hand, to explore new variational principles in the case where one wants to know how big some quantity can be at its maximum value, not merely on average. This question arises in many areas of science and engineering and is of fundamental importance in such applications as elasticity, image processing, damage and fracture mechanics, and plasticity. The investigator studies this question in the first part of the project. In the second part he deals with the derivation of nonlinear membrane theories from three-dimensional elasticity. This is motivated in part by the need, originating from important technological applications, to better understand and predict the formation of interfaces in thin films of martensitic materials. The project sheds light on the structure of the minimizers for a new effective thin film energy recently proposed by the investigator, and indicates appropriate minimizing sequences that are expected to provide some insight into the design of technologically useful active material structures, more complex than those suggested by existing thin film theories.
