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Derivation of a plate theory for incompressible materials. (English) Zbl 1112.74037
Summary: We derive a two-dimensional model for elastic plates as a \(\varGamma\)-limit of three-dimensional nonlinear elasticity with the constraint of incompressibility. The energy density of the reduced problem describes plate bending, and is determined from the elastic moduli at the identity of the energy density of the three-dimensional problem. Without the constraint of incompressibility, \(\varGamma\)-convergence to a plate theory was first derived by G. Friesecke, R. James and S. Müller [Arch. Ration. Mech. Anal. 180, 183–236 (2006)]. The main difficulty in the present result is the construction of a recovery sequence which satisfies pointwise the nonlinear constraint of incompressibility.

MSC:
74K20 Plates
74B20 Nonlinear elasticity
74G10 Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of equilibrium problems in solid mechanics
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