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Scaling laws for non-Euclidean plates and the W^{2,2} isometric immersions of Riemannian metrics
Lewicka, Marta; Pakzad, Reza (2009)
 

Title 
Scaling laws for non-Euclidean plates and the W^{2,2} isometric immersions of Riemannian metrics

Issue Date
2009-07-09

Type
Article

Abstract
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 $\Gamma$-convergence under the proper scaling. As a corollary, we obtain new necessary and sufficient conditions for existence of a $W^{2,2}$ isometric immersion of a given $2$d metric into $\mathbb R^3$.

Appears in Collection(s)
Preprints [18]

Suggested Citation
Lewicka, Marta; Pakzad, Reza. (2009). Scaling laws for non-Euclidean plates and the W^{2,2} isometric immersions of Riemannian metrics. Retrieved from the University of Minnesota Digital Conservancy, http://purl.umn.edu/51946.


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