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Vibration of Mindlin plates. Programming the p-version Ritz method. (English) Zbl 0940.74002

Amsterdam: Elsevier. 212 p. (1998).
The monograph is an excellent summary of many contributions of the authors on the vibrations of Mindlin plates. The Kirchhoff theory of thin plates is based on the assumption that the plate deflection is smaller than its thickness. When the deflection is comparable to the thickness, the strains and rotations of the middle surface are no longer negligible. Hence Mindlin, no doubt inspired by Timoshenko beam theory, included shear deformation and rotatory inertia in the equations of motion, thus establishing a thick plate theory. Like in Timoshenko beam theory, in the Mindlin theory the frequencies of higher modes are more accurately represented than in more elementary theories which yield unrealistically high values. From this one can conclude that shear deformation and rotatory inertia terms act as energy dissipation functions.
In this monograph, the authors use Ritz method to develop four computer codes which cover a variety of geometric configurations. Two codes are written in polar coordinates, they address axisymmetric and nonaxisymmetric plate problems. The third code is written in rectangular Cartesian coordinates, it can be used to solve problems of plates whose shape can be characterized by a polynomial function. Finally, the fourth program is written in skewed Cartesian coordinates to accomodate the solution of plates having parallel oblique edges.

MSC:

74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids
74H45 Vibrations in dynamical problems in solid mechanics
74H15 Numerical approximation of solutions of dynamical problems in solid mechanics
74K20 Plates

Software:

Mathematica
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