Quantitative Immersability of Riemann Metrics and the Infinite Hierarchy of Prestrained Shell Models

Abstract

This paper concerns the variational description of prestrained materials, in the context of dimension reduction for thin films Ω h = ω × (− h 2 , h 2 ). Given a Riemann metric G on Ω 1 , we study the question of what is the infimum of the averaged pointwise deficit of a given immersion from being an orientation-preserving isometric immersion of G |Ω h on Ω h , over all weakly regular immersions. This deficit is measured by the non-Euclidean energies E h , which can be seen as modifications of the classical nonlinear three-dimensional elasticity.Building on our previous results, we complete the scaling analysis of E h and the derivation of Γ-limits of the scaled energies h −2n E h , for all n ≥ 1. We show the energy quantisation, in the sense that the even powers 2n of h are indeed the only possible ones (all of them are also attained). For each n, we identify the equivalent conditions for the validity of the corresponding scaling, in terms of the vanishing of appropriate Riemann curvatures of G up to certain orders, and in terms of the matched isometry expansions. We also establish the asymptotic behaviour of the minimizing immersions as h → 0.Date: December 27, 2018.

Dimension Reduction for Thin Films with Transversally Varying Prestrain: Oscillatory and Nonoscillatory Cases

“…These findings are consistent with and pave the way for the follow-up paper [34], which completes the scaling analysis of E h in the non-oscillatory case, including the derivation of Γ-limits of h −2n E h for all n ≥ 1, and proving the energy quantisation in the sense that the even powers 2n of h are indeed the only possible ones (all of them are also attained).…”

“…We finally mention the paper [34], completed after the submission of the present article, which resolves the scaling analysis of E h together with the derivation of Γ-limits of h −2n E h , for all n ≥ 1. There, we identify equivalent conditions for the validity of the scalings h 2n in terms of vanishing of the Riemann curvatures, for an arbitrary non-oscillatory metric G, up to appropriate orders and in terms of the matched isometry expansions.…”

“…We exhibit connections between the two cases via projections of appropriate curvature forms on the polynomial tensor spaces. We also show the effective energy quantisation in terms of scalings as a power of h and discuss the scaling regimes h 2 (Kirchhoff), h 4 (von Kármán) in the general case, as well as all possible (even powers) regimes for conformal metrics, thus paving the way to the subsequent complete analysis of the non-oscillatory setting in [34]. Thirdly, we prove the coercivity inequalities for the singular limits at h 2 -and h 4 -scaling orders, while disproving the full coercivity of the classical von Kármán energy functional at scaling h 4 .…”

Derivation of a Homogenized Bending–Torsion Theory for Rods with Micro-Heterogeneous Prestrain

“…In the results discussed so far the prestrain (if present) is assumed to be infinitesimally small. In the last decade, dimension reduction for finite prestrain (yet smoothly varying on a macroscopic scale) has been studied in the framework of non-Euclidean elasticity theory [14], e.g., [8,29,31,33,35] for the derivation of non-Euclidean theories for rods and plates. Rods and shells with nontrivially curved reference configuration lead to similar models when being pulled back to a flat reference configuration (cf.…”

“…In general, we expect that the type of limiting model does not only depend on α but also on other properties of the prestrain. Indeed, in a situation without homogenization it is shown in [8,36] that bending and von Karman type plate models arise in the case α = 0 depending on the geometry of the prestrain (see also [37] for related results in the case of rods and [31,32] for recent results for plates and shells beyond the von Karman regime). It is an interesting question if these results are stable with respect to (small) rapidly oscillating perturbations.…”