TY - JOUR

T1 - A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence

AU - Friesecke, Gero

AU - James, Richard D.

AU - Müller, Stefan

PY - 2006/5

Y1 - 2006/5

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=33644602781&partnerID=8YFLogxK

U2 - 10.1007/s00205-005-0400-7

DO - 10.1007/s00205-005-0400-7

M3 - Article

AN - SCOPUS:33644602781

SN - 0003-9527

VL - 180

SP - 183

EP - 236

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

IS - 2

ER -