Asymptotic theory of harmonic vibration of multilayer thin elastic plates

The paper considers a harmonic vibrations theory of thin elastic multilayer anisotropic plates proposed by the authors. The theory is based on the asymptotic research into three-dimensional general equations of steady-state vibrations of elastic bodies with a small parameter characterizing the length-to-thickness ratio. No hypothesis about both principles of distributing displacement and a stress over the thickness of a plate is used. An asymptotic solution to the problem of eigen vibrations of a plate is found. Recurrent sequences of local vibration problems are formulated, and their solutions are found in the explicit form. It is shown that the averaged problem of the developed plate vibrations theory proves to be similar to the Kirchhoff-Love plate vibrations theory. This method allows us to calculate all 6 stress tenor components including both transverse normal stresses and stresses of an interlayer shift in case of vibrations of thin elastic plates. The solution to the problem offlexural eigen vibrations of a multilayer plate is illustrated by an example. The authors compare computations performed by the developed method and by the finite-element solution of the three-dimensional problem of eigen vibrations on the basis of ANSYS complex. It is shown that the asymptotic theory allows finding eigen frequencies quite accurately and calculating all 6 stress tensor components in a plate with a pinpoint accuracy.

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