On von Kármán equations and the buckling of a thin circular elastic plate. (English) Zbl 1321.35230

Janczewska, Joanna; Zgorzelska, Anita; Guze, Hanna

Adv. Nonlinear Stud. 15, No. 3, 613-628 (2015).

Summary: We shall be concerned with the buckling of a thin circular elastic plate simply supported along a boundary, subjected to a radial compressive load uniformly distributed along its boundary. One of the main engineering concerns is to reduce deformations of plate structures. It is well-known that von Kármán equations provide an established model that describes nonlinear deformations of elastic plates. Our approach to study plate deformations is based on bifurcation theory. We will find critical values of the compressive load parameter by reducing von Kármán equations to an operator equation in Hölder spaces with a nonlinear Fredholm map of index zero.
We will prove a sufficient condition for bifurcation by the use of a gradient version of the Crandall-Rabinowitz theorem due to A. Yu. Borisovich and basic notions of representation theory. Moreover, applying the key function method by Yu. I. Sapronov we will investigate the shape of bifurcation branches.

MSC:

35Q74	PDEs in connection with mechanics of deformable solids
35B32	Bifurcations in context of PDEs
74G20	Local existence of solutions (near a given solution) for equilibrium problems in solid mechanics (MSC2010)
74K20	Plates
74B20	Nonlinear elasticity

Keywords:

bifurcation; von Kármán equations; elastic plate; nonlinear Fredholm operator; variational gradient; variational approach; group action

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