Room P3.10, Mathematics Building

David Owen, Carnegie Mellon University
Multiscale Geometrical Changes in Continua: Structured Deformations and Some Questions in Analysis

Classes of injective deformations of a continuous body that are stable under composition and under the taking inverses are central to the description of geometrical changes of a continuous body at both the macroscopic and microscopic levels. Classical, simple, and structured deformations are described and assessed from the point of view of continuum mechanics and of variational problems. The natural way of assigning an energy to a structured deformation, through relaxation of an energy assigned to a smaller class of deformations, leads to various alternative, "variationally friendly" notions of structured deformations. Several of these alternative notions are examined from the point of view of approximation by deformations in a smaller class and of relaxation of an initial energy defined on the smaller class of deformations. Questions are raised as to whether or not different notions of structured deformations lead to different relaxed energies for a structured deformation that satisfies the defining requirements of two or more notions. A recent example of an explicit formula for a relaxed energy is used to illustrate these questions.