Point-instabilities, point-coercivity (meta-stability), and<br/><br/>point-calculus

Tuesday, July 22, 2008 - 2:45pm - 2:55pm
EE/CS 3-180
Evan Hohlfeld (University of California, Berkeley)
For general non-linear elliptic PDEs, e.g. non-linear rubber
elasticity, linear stability analysis is false. This is because of the
possibility of point-instabilities. A point-instability is a non-linear
instability with zero amplitude threshold that occurs while linear
stability still holds. Examples include cavitation, fracture, and the formation
of a crease, a self-contacting fold in an otherwise free surface.
Each of which represents a kind of topological change. For any such
PDE, a point-instability occurs whenever a certain auxiliary
scale-invariant problem has a non-trivial solution. E.g. when sufficient strain
is applied at infinity in a rubber (half-)space to support a
single, isolated crease, crack, cavity, etc. Owing to scale-invariance,
when one such solution exists, an infinite number or geometrically
similar solutions also exist, so the appearance of one particular
solution is the spontaneous breaking of scale-invariance. We then identify
this (half-)space with a point in a general domain. The condition
that no such solutions exist is called point-coercivity, and can be
formulated as non-linear eigenvalue problem that predicts the critical
stress for fracture, etc. And when point-coercivity fails for a system,
the system is susceptible to the nucleation and self-similar growth of
some kind of topological defect. Viewing fracture, etc. as symmetry breaking
processes explains their macroscopic robustness.

Point-coercivity is similar to, but more general than,
quasi-convexity, as it can be formulated for any elliptic PDE, not just
Euler-Lagrange systems (i.e. for out-of-equilibrium systems, and so defining
meta-stability in a general sense). Indeed, these are just two
examples of a host of point-conditions, the study of which might be
called point-calculus. Time allowing, I will show that for almost any
elliptic PDE, linear- and point-instabilities exhaust the possible kinds
of instabilities. The lessons learned from elliptic systems will
be just as valid for parabolic and hyperbolic systems since the underlying
reason linear analysis breaks down – taking certain limits in the
wrong order holds for these systems as well.
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