Acta Mechanica

, Volume 228, Issue 10, pp 3403–3419

# An elastic plate bending equation of second-order accuracy

• Petr Tovstik
• Tatiana Tovstik
Original Paper

## Abstract

A study is carried out of a thin plate of constant thickness made of linearly elastic material which is transversally isotropic and heterogeneous in the thickness direction. Asymptotic expansions in powers of the relative plate thickness are constructed, and the bending equation of second-order accuracy (the SA model) is delivered. The results of the SA model are compared with the Kirchhoff–Love classical model and with the Timoshenko–Reissner (TR) model, as well as with the exact solution. To this end, some problems for a functionally gradient plate bending, and for a multi-layer plate bending and free vibration are solved and analysed. The range of plate heterogeneity, for which the error of the approximate models is small, is established. The TR model and the SA model are proved to yields results close to each other and the exact results for a very broad range of heterogeneity. That is why the generalized TR model for one-layered homogeneous transversely isotropic plate is proposed. Parameters of this model are chosen so that the results are close to the exact results and the results by the SA model. For the Navier boundary conditions, the analytical solution of 3D problems for a rectangular heterogeneous plate is constructed.

## Notes

### Acknowledgements

The work is supported by Russian Foundation for Basic Research, Grants 16-01-00580-a and 16-51-52025 MNT-a.

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