Derivation and verification of a lattice model for bending vibration of a plate

• Published in: ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik
• Main Authors: van Vliet, Renate, Metrikine, Andrei V.
• Format: Article in Journal/Newspaper
• Language: English
• Published: Wiley 2017
• Subjects: Ice Sheet
• Online Access: http://dx.doi.org/10.1002/zamm.201700024; https://api.wiley.com/onlinelibrary/tdm/v1/articles/10.1002%2Fzamm.201700024; https://onlinelibrary.wiley.com/doi/pdf/10.1002/zamm.201700024

Abstract Lattice models are successfully used in modelling of fracture of brittle materials. To date, most of the lattice multi‐dimensional (2D and 3D) models known to the authors describe either in‐plane or three‐dimensional mechanics of the materials. Only a few lattice models are available in the literature for the description of the out‐of‐plane mechanics of plates. However, the parameters of those lattice models have not been linked to those of the classical plate models, such as Mindlin‐Reissner plate theory, based on the classical continuum theory. Nor has the dynamic behaviour of the out‐of‐plane lattice models been compared to that of the classical plates. This gap is closed in this paper by means of developing a lattice model that reproduces the out‐of‐plane dynamics of a shear‐deformable plate in the low frequency band. The developed model can be applied in various fields of engineering. In this paper, however, it is discussed taking example of an ice sheet. The linear dynamics of the model is focused upon in this paper in order to show its consistency with a continuum plate theory in the low frequency band and underline the differences emerging at higher frequencies. The developed model is composed of masses and springs whose morphology and properties were derived to match the out‐of‐plane deformations of thick plates as described by the Mindlin‐Reissner theory. Bending, shear and torsion are taken into account. The eigenfrequencies and the steady‐state response of the model to a sinusoidal in time point load are computed and compared to those of a corresponding continuum plate. It is proven that the developed lattice predicts the same dynamic behaviour as the corresponding continuum plate at relatively low frequencies. At higher frequencies deviations occur. These are discussed in this paper in terms of the dispersion, anisotropy and specific boundary effects of the lattice model.