Plane Vibrations and Stability of Elastic Plates

[摘要] We consider plane harmonic incremental displacements superimposed on a finite deformation, corresponding to pure homogeneous strain, of a plate with rectangular cross-section, for which motions normal to the plane of this cross- section are neglected. We assume that the plate is isotropic and hyperelastic, with the underlying finite deformation satisfying the strong-ellipticity condition, and consider both incompressible and compressible materials. Having set up the governing equations of finite and incremental elasticity we consider the incompressible case and study the problem with mixed traction-displacement boundary conditions defined on two opposing faces and pure traction boundary conditions defined on the other sides. It is found that, in general, nine distinct cases may occur and frequency equations are derived for each case in turn, with two separate modes possible in most cases. Necessary conditions for the existence of nontrivial solutions are obtained for both the static and the dynamic problems and, in order to study the frequency equations numerically, we consider the restriction to equibiaxial underlying deformations as well as looking at the full problem for two particular strain-energy functions. The corresponding problem for compressible materials is then considered and frequency equations, similar to those in the incompressible case, are derived. Unfortunately, due to the more complicated algebra, fewer explicit results can be obtained in this case, as compared to the incompressible case, but some necessary existence conditions are derived. Numerical results are obtained in the static case for three particular strain-energy functions and one of these stain-energy functions is then used to illustrate the full dynamic problem. Finally we consider the related problem of an infinite layer, of finite thickness, with traction boundary conditions applied on the surfaces, and show how the results for the finite plate can be applied to this problem.

[发布机构] University:University of Glasgow

[关键词] Mechanical engineering