Keywords: Reissner-Mindlin plates, dynamic analysis, breathing frequency, natural frequencies.

The shear deformation effect on the plate bending behavior is a necessary path in the dynamic formulation to obtain correct responses for wavelengths approaching the plate thickness. These are important in modes of vibration of high order or in sharp transients. Reissner [1] presented a generalization of the classical bending model with respect to the shear deformation effect and applied it to static problems. Furthermore, the linearly weighted average effect of the normal stress component in the thickness direction was also considered by him, which is related to the pressure of the distributed loads on the plate surface. A similar bending model to study vibration problems including the shear deformation effect was presented by Mindlin [2]. The rotatory inertia and shear deformation effects were included in his model in the same manner they were considered in the Timoshenko one-dimensional theory of bars [3]. Mindlin proposed an equation to obtain the parameter related to the shear deformation according to Poisson's ratio from the relations of the analysis of propagation of straight-crested waves into an infinite domain using three-dimensional elasticity.

Most of dynamic analyses of plates using the boundary element method (BEM) have been performed with or without the shear deformation effect using the static solution or the frequency domain solution beyond special BEM formulations to analyze specific problems [4]. The aim of this paper is the harmonic analysis of plates with the Reissner bending model and the elastodynamic fundamental solution for the BEM presented in [5]. The linearly weighted average effect of the normal stress component in the thickness direction is considered in the direct boundary integral equation (DBIE) as an additional domain integral when distributed loads are considered but it can be cut off to assess results with the Mindlin model. The field decomposition was employed in the solution and the vector of plate rotations was written in terms of its scalar and vector potentials. The shear deformation effect can be decoupled in the formulation and a solution according to classical plate theory was reached using an irrotational approach for the rotation field. The DBIE obtained for the classical bending model is analogous to that employed in static analyses with an additional degree of freedom for the tangential boundary rotation [6].

The frequency response was considered to map the behavior of plates in the range starting from a thin plate to a block. Natural frequencies were obtained with flexural vibration modes as well as those with thickness-twist vibration modes, which are related to the scalar and the vector potential field, respectively [7]. The results were compared to available solutions in the literature considering the three-dimensional elasticity theory, the Mindlin and the classical bending models.

1

Reissner, E., The effect of Transverse Shear Deformation on the Bending of Elastic Plates, Journal of Applied Mechanics, A69-A76, 1945.

2

Mindlin, R.D., Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates, Journal of Applied Mechanics, March, 31-38, 1951.

3

Timoshenko S, Young, D.H., Weaver Jr., W., Vibration Problems in Engineering, 4th ed., New York, John Wiley & Sons; 1974.

4

Providakis, C. P., Beskos, D.E., Dynamic analysis of plates by the boundary elements, Appl. Mech. Rev., vol 52, no 7, ASME, 1999. doi:10.1115/1.3098936

5

Palermo Jr., L., On the fundamental solution to perform the dynamic analysis of Reissner-Mindlin's plates, Boundary Element Technology XV, Editors C.A. Brebbia, R.E. Dippery, 2003.

6

Palermo Jr., L, Plate bending analysis using the classical or the Reissner-Mindlin models, Engineering Analysis with Boundary Elements, v. 27, p. 603-609, Elsevier Publications, U.K., 2003.

7

Mindlin, R.D., Schacknow A., Deresiewicz, H., Flexural vibrations of rectangular plates, Journal of Applied Mechanics, 23, 430-436, 1956.

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