A Riemannian approach to reduced plate, shell, and rod theories

2012/01/31 by Raz Kupferman, Jake P. Solomon · 48 citations
Engineering · Mathematics · Physics and Astronomy · #Advanced Materials and Mechanics #Classical mechanics #Composite Structure Analysis and Optimization #Composite material #Geometry #Materials science #Mathematics #Physics #Shell (structure) #Structural Analysis and Optimization #Theoretical physics #cond-mat.soft #math.DG #math.FA #msc:53C42 #msc:74B20 #msc:74K10 #msc:74K20 #msc:74K25

paper · pdf · doi:10.1016/j.jfa.2013.09.003

published in Journal of Functional Analysis 266(5), 2989-3039 (Elsevier BV) · 61 pages, added references, fixed typos

arxiv created 2013/04/19 · openalex publication_date 2013/09/18 · arxiv updated 2014/09/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We derive a dimensionally-reduced limit theory for an n-dimensional nonlinear elastic body that is slender along k dimensions. The starting point is to view an elastic body as an n-dimensional Riemannian manifold together with a not necessarily isometric W^1,2-immersion in n-dimensional Euclidean space. The equilibrium configuration is the immersion that minimizes the average discrepancy between the induced and intrinsic metrics. The dimensionally reduced limit theory views the elastic body as a k-dimensional Riemannian manifold along with an isometric W^2,2-immersion in n-dimensional Euclidean space and linear data in the normal directions. The equilibrium configuration minimizes a functional depending on the average covariant derivatives of the linear data. The dimensionally-reduced limit is obtained using a Γ-convergence approach. The limit includes as particular cases plate, shell, and rod theories. It applies equally to "standard" elasticity and to "incompatible" elasticity, thus including as particular cases so-called non-Euclidean plate, shell, and rod theories.

Citations

• Über das Gleichgewicht und die Bewegung einer elastischen Scheibe.
• Differential Forms in Algebraic Topology
• Riemann–Cartan Geometry of Nonlinear Dislocation Mechanics
• On the geometric structures of simple bodies, a mathematical foundation for the theory of continuous distributions of dislocations
• A theorem on geometric rigidity and the derivation of nonlinear plate theory from three‐dimensional elasticity
• Continuous distributions of dislocations: a new application of the methods of non-Riemannian geometry
• Scaling laws for non-Euclidean plates and the W^2,2 isometric immersions of Riemannian metrics

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• Point-like bounding chains in open Gromov–Witten theory
• Dimension reduction for thin films with transversally varying prestrain: the oscillatory and the non-oscillatory case