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A justification of nonlinear properly invariant plate theories. (English) Zbl 0789.73039
The paper represents a reformulation and a precise formalization of the asymptotic approach to the three-dimensional nonlinear elasticity. In the formal asymptotic analysis two assumptions are made: (i) the applied loads admit an expansion in power series in the thickness \(\varepsilon\), after a reparametrization of an \(\varepsilon\)-independent domain is introduced, (ii) the deformations of the plate, reparametrized to the \(\varepsilon\)-independent domain, can be formally expanded in powers of \(\varepsilon\). It is shown, that the leading terms in an asymptotic expansion of the three-dimensional nonlinear theory describe the nonlinear membrane theory and the nonlinear inextensional theory. These models, contrary to the von Kármán model of plates, inherit the full invariance properties of the three-dimensional theory. The relation to the model of Cosserat’s directed media and the possible generalization of the here considered Saint Venant-Kirchhoff materials are discussed.

MSC:
74K20 Plates
74B20 Nonlinear elasticity
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