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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.