University of Minnesota
University of Minnesota
Mathematics of Materials and Macromolecules: Multiple Scales, Disorder, and Singularities, September 2004 - June 2005

Professor Stuart S. Antman
(Department of Mathematics, University of Maryland)

will be visiting the IMA and the School of Mathematics of the University of Minnesota March 20-April 1. He will be speaking on Geometric obstructions in the nonlinear equations from solid mechanics on Wednesday, March 30, 3:35, Vincent Hall 301 and Incompressibility in the Mathematics colloquium at 3:30 on Thursday, March 31, in Vincent Hall 16.

Poster:   pdf

Geometric obstructions in the nonlinear equations from solid mechanics

Wednesday, March 30, 3:35, Vincent Hall 301

Abstract: Many of the difficulties presented by the nonlinear partial differential equations from solid mechanics are inherently geometrical, reflecting that the equations must (i) describe one-to-one deformations of regions of Euclidean space, and (ii) meet certain invariance requirements, which complicate the geometrical description. This lecture treats geometrically exact problems governed by quasilinear parabolic-hyperbolic systems in which there is but one independent spatial variable. The main emphasis is on how standard methods of nonlinear analysis, like the Faedo-Galerkin method, must be significantly modified to accommodate the intrinsic difficulties of solid mechanics.


Thursday, March 31, 3:30, Vincent Hall 16

Abstract: A material body is incompressible if every deformation of it locally preserves its volume, in particular, if the Jacobian determinant of every continuously differentiable deformation of it is identically 1. Since the nonlinear PDEs of evolution for such 3-dimensional bodies have largely resisted analysis, it is useful to have effective theories for slender bodies governed by equations with but one independent spatial variable. This lecture shows that the actual construction of one such very attractive theory requires the solutions of a sequence of first-order PDEs (by the method of characteristics). Although the resulting equations are more complicated than those for bodies not subject to the constraint of incompressibility, they have novel regularity properties not enjoyed by the latter. The governing equations for an elastic body can be characterized by Hamilton's Principle. The ODEs governing travelling waves for these equations can also be characterized by Hamilton's Principle, but the kinetic and potential energies for these ODEs do not correspond to those of the PDEs. These ODEs admit periodic travelling waves with wave speeds that are are supersonic with respect to some modes of motion and subsonic with respect to others.


IMA Tutorial/Workshop: New Paradigms in Computation, March 28-30, 2005