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This paper concerns the elastic structures which exhibit non-zero strain atfree equilibria. Many growing tissues (leaves, flowers or marine invertebrates)attain complicated configurations during their free growth. Our study departsfrom the 3d incompatible elasticity theory, conjectured to explain themechanism for the spontaneous formation of non-Euclidean metrics.  Recall that a smooth Riemannian metric on a simply connected domain can berealized as the pull-back metric of an orientation preserving deformation ifand only if the associated Riemann curvature tensor vanishes identically. Whenthis condition fails, one seeks a deformation yielding the closest metricrealization. We set up a variational formulation of this problem by introducingthe non-Euclidean version of the nonlinear elasticity functional, and establishits $\Gamma$-convergence under the proper scaling. As a corollary, we obtainnew necessary and sufficient conditions for existence of a $W^{2,2}$ isometricimmersion of a given 2d metric into $\mathbb R^3$.

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