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Dynamic stability of a porous rectangular plate
Author(s) -
Debowski Daniel,
Magnucki Krzysztof
Publication year - 2006
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200610088
Subject(s) - isotropy , mathematical analysis , galerkin method , mathematics , bending of plates , plate theory , boundary value problem , porous medium , partial differential equation , dimensionless quantity , geometry , equations of motion , differential equation , ordinary differential equation , displacement field , plane stress , linear elasticity , porosity , mechanics , physics , classical mechanics , finite element method , materials science , composite material , optics , bending , thermodynamics
The study is devoted to a axial compressed porous‐cellular rectangular plate. Mechanical properties of the plate vary across is its thickness which is defined by the non‐linear function with dimensionless variable and coefficient of porosity. The material model used in the current paper is as described by Magnucki, Stasiewicz papers. The middle plane of the plate is the symmetry plane. First of all, a displacement field of any cross section of the plane was defined. The geometric and physical (according to Hook's law) relationships are linear. Afterwards, the components of strain and stress states in the plate were found. The Hamilton's principle to the problem of dynamic stability is used. This principle was allowed to formulate a system of five differential equations of dynamic stability of the plate satisfying boundary conditions. This basic system of differential equations was approximately solved with the use of Galerkin's method. The forms of unknown functions were assumed and the system of equations was reduced to a single ordinary differential equation of motion. The critical load determined used numerically processed was solved. Results of solution shown in the Figures for a family of isotropic porous‐cellular plates. The effect of porosity on the critical loads is presented. In the particular case of a rectangular plate made of an isotropic homogeneous material, the elasticity coefficients do not depend on the coordinate (thickness direction), giving a classical plate. The results obtained for porous plates are compared to a homogeneous isotropic rectangular plate. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)