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

@article{Friesecke2002ATO, 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}, year={2002}, volume={55} }

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…

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## References

SHOWING 1-10 OF 52 REFERENCES

The membrane shell model in nonlinear elasticity: A variational asymptotic derivation

- Mathematics
- 1996

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.

- EngineeringPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2002

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

- Mathematics
- 1982

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

- MathematicsJ. Nonlinear Sci.
- 2000

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

- Engineering
- 1997

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

- Mathematics
- 1998

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

- Materials Science
- 1999

A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces

- Mathematics
- 1982

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…