LAUSR

We propose results that relate the following two contexts: (i) Given a Riemann metric G on $$\Omega ^1=\omega \times (-\frac{1}{2}, \frac{1}{2})$$ Ω 1 = ω × ( - 1 2 , 1 2 ) , we find the infimum of the averaged pointwise deficit of an immersion from attaining the orientation-preserving isometric immersion of $$G_{\mid \Omega ^h}$$ G ∣ Ω h on $$\Omega ^h=\omega \times (-\frac{h}{2}, \frac{h}{2})$$ Ω h = ω × ( - h 2 , h 2 ) , over all weakly regular immersions. This deficit is measured by the non-Euclidean energies $${\mathcal {E}}^h$$ E h , which can be seen as modifications of the classical nonlinear three-dimensional elasticity. (ii) We complete the scaling analysis of $${\mathcal {E}}^h$$ E h in the context of dimension reduction as $$h\rightarrow 0$$ h → 0 , and the derivation of $$\Gamma $$ Γ -limits of the scaled energies $$h^{-2n}{\mathcal {E}}^h$$ h - 2 n E h for all $$n\geqq 1$$ n ≧ 1 . We show the energy quantisation, in the sense that the even powers 2 n of h are indeed the only possible ones (all of them are also attained). For each n , we identify conditions for the validity of the scaling $$h^{2n}$$ h 2 n , in terms of the vanishing of Riemann curvatures of G up to appropriate orders, and in terms of the matched isometry expansions. We also establish the asymptotic behaviour of the minimizing immersions as $$h\rightarrow 0$$ h → 0 .