A theorem on geometric rigidity and the derivation of nonlinear plate theory from three‐dimensional elasticity

  title={A theorem on geometric rigidity and the derivation of nonlinear plate theory from three‐dimensional elasticity},
  author={Gero Friesecke and Richard D. James and Stefan M{\"u}ller},
  journal={Communications on Pure and Applied Mathematics},
The energy functional of nonlinear plate theory is a curvature functional for surfaces first proposed on physical grounds by G. Kirchhoff in 1850. We show that it arises as a Γ‐limit of three‐dimensional nonlinear elasticity theory as the thickness of a plate goes to zero. A key ingredient in the proof is a sharp rigidity estimate for maps v : U → ℝn, U ⊂ ℝn. We show that the L2‐distance of ∇v from a single rotation matrix is bounded by a multiple of the L2‐distance from the group SO(n) of all… 
Derivation of a plate theory for incompressible materials
Thin-plate theory for large elastic deformations
Homogenization of the nonlinear bending theory for plates
We carry out the spatially periodic homogenization of nonlinear bending theory for plates. The derivation is rigorous in the sense of $$\Gamma $$Γ-convergence. In contrast to what one naturally would
A Hierarchy of Plate Models Derived from Nonlinear Elasticity by Gamma-Convergence
We derive a hierarchy of plate models from three-dimensional nonlinear elasticity by Γ-convergence. What distinguishes the different limit models is the scaling of the elastic energy per unit volume
Derivation of the homogenized von Kármán plate theory from 3 d elasticity
In this paper we derive, by means of Γ -convergence, the homogenized von Kármán plate model starting from three dimensional nonlinear elasticity. We assume that the thickness of the plate is h, the
Convergence of equilibria of three-dimensional thin elastic beams
  • M. G. MoraS. Muller
  • Mathematics
    Proceedings of the Royal Society of Edinburgh: Section A Mathematics
  • 2008
A convergence result is proved for the equilibrium configurations of a three-dimensional thin elastic beam, as the diameter $h$ of the cross-section tends to zero. More precisely, we show that
Justification of the Nonlinear Kirchhoff-Love Theory of Plates as the Application of a New Singular Inverse Method
AbstractIn the framework of isotropic homogeneous nonlinear elasticity for a St. Venant-Kirchhoff material, we consider a three-dimensional plate of thickness ɛ and periodic in the two other
Geometric rigidity for incompatible fields, and an application to strain-gradient plasticity
In this paper, we show that a strain-gradient plasticity model arises as the Γ -limit of a nonlinear semi-discrete dislocation energy. We restrict our analysis to the case of plane elasticity, so


The membrane shell model in nonlinear elasticity: A variational asymptotic derivation
SummaryWe consider a shell-like three-dimensional nonlinearly hyperelastic body and we let its thickness go to zero. We show, under appropriate hypotheses on the applied loads, that the deformations
Singularities, structures, and scaling in deformed m-dimensional elastic manifolds.
There are marked differences in the forms of energy condensation depending on the embedding dimension, and two distinct behaviors of local energy density falloff away from singular points are observed.
New integral estimates for deformations in terms of their nonlinear strains
AbstractIf u is a bi-Lipschitzian deformation of a bounded Lipschitz domain Ω in ℓn (n≧2), we show that the LP norm (p≧1, p≠n) of a certain “nonlinear strain function” e(u) associated with u
Rigorous Bounds for the Föppl—von Kármán Theory of Isotropically Compressed Plates
The Föppl—von Kármán theory for isotropically compressed thin plates in a geometrically linear setting is studied, and upper and lower bounds on the minimum energy linear in the plate thickness σ are obtained.
Theory of plates
Part A. Linear Plate Theory. 1. Linearly elastic plates. 2. Junctions in linearly elastic multi-structures. 3. Linearly elastic shallow shells in Cartesian coordinates. Part B. Nonlinear Plate
3D-2D asymptotic analysis of an optimal design problem for thin films
Abstract The Gamma-limit of a rescaled version of an optimal material distribution problem for a cylindrical two-phase elastic mixture in a thin three-dimensional domain is explicitly computed. Its
A theory of thin films of martensitic materials with applications to microactuators
The morphology and folding patterns of buckling-driven thin-film blisters
A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces
Let M be a compact Riemannian manifold with a fixed conformal structure. Then we introduce the concept of conformal volume of M in the following manner. For each branched conformal immersion q9 of M