

Preprint 127/2006
Stability of slender bodies under compression and validity of the von Kármán theory
Myriam Lecumberry and Stefan Müller
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Submission date: 12. Nov. 2006 (revised version: March 2007)
Pages: 70
published in: Archive for rational mechanics and analysis, 193 (2009) 2, p. 255-310
DOI number (of the published article): 10.1007/s00205-009-0232-y
Bibtex
MSC-Numbers: 74K20, 49J40, 35Q72, 74G60
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Abstract:
SSince their formulation almost 100 years ago the
von Kármán (vK) plate equations have been frequently
used both by engineers and by analysts
to study thin elastic bodies, in particular their stability
behaviour under applied loads. At the same
time the derivation of these equations
met some harsh criticism and their precise
mathematical status has been unclear until very recently.
Following up on a recent variational derivation of the vK theory
by Friesecke, James and Müller
from three dimensional nonlinear elasticity we study
the predictions and the validity of the vK equation in the presence
of in-plane compressive forces. The first main result
is a stability alternative: either the load leads to
in instability already in the nonlinear bending theory of
plates (Kirchhoff-Love theory) or it leads to an instability
in a geometrically linear KL theory ('linearized instability')
or vK theory is valid.
The second main result states that under suitable conditions
the critical loads for
nonlinear stability and linearized instability coincide.
The third main result asserts this critical load also
agrees with the load beyond which the infimum of the vK
functional is .
The main ingredients are a sharp rigidity estimate for maps
with low elastic energy and a study of the properties of isometric
immersions from a set in
to
and
their geometrically linear counterparts.