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 onetoone 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 parabolichyperbolic systems in which
there is but one independent spatial variable. The main emphasis is on how standard
methods of nonlinear analysis, like the FaedoGalerkin method, must be significantly
modified to accommodate the intrinsic difficulties of solid mechanics.
Incompressibility
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 3dimensional
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 firstorder 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 2830, 2005
