### Abstract

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 ∼ h ^{β} , where h is the thickness of the plate. This is in turn related to the strength of the applied force ∼ h ^{α} . Membrane theory, derived earlier by Le Dret and Raoult, corresponds to α=β=0, nonlinear bending theory to α=β=2, von Kármán theory to α=3, β=4 and linearized vK theory to α>3. Intermediate values of α lead to certain theories with constraints. A key ingredient in the proof is a generalization to higher derivatives of our rigidity result [29] which states that for maps v:(0,1)^{3}→ ^{3}, the L ^{2} distance of ∇. v from a single rotation is bounded by a multiple of the L ^{2} distance from the set SO(3) of all rotations.

Original language | English (US) |
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Pages (from-to) | 183-236 |

Number of pages | 54 |

Journal | Archive For Rational Mechanics And Analysis |

Volume | 180 |

Issue number | 2 |

DOIs | |

State | Published - May 1 2006 |

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## Cite this

*Archive For Rational Mechanics And Analysis*,

*180*(2), 183-236. https://doi.org/10.1007/s00205-005-0400-7