1,504 research outputs found
A note on convergence of low energy critical points of nonlinear elasticity functionals, for thin shells of arbitrary geometry
We prove that the critical points of the 3d nonlinear elasticity functional
on shells of small thickness and around the mid-surface of arbitrary
geometry, converge as to the critical points of the von K\'arm\'an
functional on , recently derived in \cite{lemopa1}. This result extends the
statement in \cite{MuPa}, derived for the case of plates when
. We further prove the same convergence result for the
weak solutions to the static equilibrium equations (formally the Euler-
Lagrange equations associated to the elasticity functional). The convergences
hold provided the elastic energy of the 3d deformations scale like and
the external body forces scale like .Comment: 15 page
Scaling laws for non-Euclidean plates and the isometric immersions of Riemannian metrics
This paper concerns the elastic structures which exhibit non-zero strain at
free equilibria. Many growing tissues (leaves, flowers or marine invertebrates)
attain complicated configurations during their free growth. Our study departs
from the 3d incompatible elasticity theory, conjectured to explain the
mechanism for the spontaneous formation of non-Euclidean metrics.
Recall that a smooth Riemannian metric on a simply connected domain can be
realized as the pull-back metric of an orientation preserving deformation if
and only if the associated Riemann curvature tensor vanishes identically. When
this condition fails, one seeks a deformation yielding the closest metric
realization. We set up a variational formulation of this problem by introducing
the non-Euclidean version of the nonlinear elasticity functional, and establish
its -convergence under the proper scaling. As a corollary, we obtain
new necessary and sufficient conditions for existence of a isometric
immersion of a given 2d metric into .Comment: 18 pages, 1 figur
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