Asymptotic solutions for the Föppl – von Kármán equations governing deflections of thin axisymmetric annular plates. (May 2017)

• Record Type: Journal Article
• Authors: Van Gorder, Robert A.
• Is Part Of: International journal of non-linear mechanics. Volume 91(2017)
• Page Start: 8
• Page End: 21
• Publication Date: 2017-05
• Subjects: Föppl – von Kármán equations -- Deflections of thin flat plates -- Nonlinear PDEs -- Analytical solution; Nonlinear mechanics -- Periodicals; Mécanique non linéaire -- Périodiques; Nonlinear mechanics; Periodicals; 531
• DOI: 10.1016/j.ijnonlinmec.2017.02.004
• Languages: English
• ISSNs: 0020-7462

Abstract

Abstract: We are concerned with the deformation of thin, flat annular plates under a force applied orthogonally to the plane of the plate. This mechanical process can be described via a radial formulation of the Föppl – von Kármán equations, a set of nonlinear partial differential equations describing the deflections of thin flat plates. We are able to obtain analytical solutions for the radial Föppl – von Kármán equations with boundary conditions relevant for clamped, loosely clamped, and free inner and outer. This permits us to study the qualitative behavior of the out-of-plane deflections as well as the Airy stress function for a number of cases. Provided that an appropriate non-dimensionalization is taken, we find that the perturbation solutions are surprisingly valid for a wide variety of parameters, and compare favorably with numerical simulations in all cases (rather than just for small parameters). The results demonstrate that the ratio of the inner to outer radius of the annular plate will strongly influence the properties of the solutions, as will the specific boundary data considered. For instance, one may choose to fix the plate in place with a specific set of boundary conditions, in order to minimize the out-of-plane deflections. Other boundary conditions may result in undesirable behaviors. Highlights: Deformations of flat annular plates studied under Föppl – von Kármán equations. Asymptotic solutions obtained for various boundary condition configurations. Ratio of the inner to outer radius will strongly influence the properties of the solutions. Boundary conditions can be selected to control plate deformations.