Advances of the Shear Deformation Theory for Analyzing the Dynamics of Laminated Composite Plates: An Overview

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ADVANCES OF THE SHEAR DEFORMATION THEORY FOR ANALYZING THE DYNAMICS OF LAMINATED COMPOSITE PLATES: AN OVERVIEW

S. P. Parida and P. Ch. Jena*

Keywords: laminated plate, vibration, modified shear theory An attempt has been made to perform a comparative study on advances in the shear deformation theory for analyzing the statics and dynamics of different plates with the use of numerical examples. Initially, shape functions for the displacement field across the plate thickness are compared, and then, the stresses, deflections, and natural frequencies of laminated composite plates are also compared.

1. Introduction Composite laminates are widely preferred structures due to their high stiffness and strength-to-weight ratio. Besides, these structures have a high fatigue strength and good corrosion resistance. To investigate the structural and mechanical properties of these types of materials, studies on their static and dynamic behavior are needed. They include the analysis of inplane/out-of-plane stress vs. strain, load vs. displacement, buckling vs. postbuckling, and free and forced vibration relations. The initial development of beam and plate theories started in the beginning of the 17th century. Daniel Bernoulli and Leonhard Euler were the first who proposed a beam theory including all kinematic and static assumptions. This theory later became known as the Euler–Bernoulli beam theory and paved a way for the development of various theories for plates and shells. In the middle of the 18th century, Kirchhoff [1] formulated a theory purely devoted to plates. Chladni experimentally verified the theory, and S. Germain was the first to propose an equation for vibrations of plates, as reported in [2].

Department of Production Engineering, VSS University of Technology, Burla, India, 768018 * Corresponding author; tel.: +918895100075; e-mail: [email protected]

Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 56, No. 4, pp. 675-714, July-August, 2020. Original article submitted December 10, 2019; revision submitted February 17, 2020. 0191-5665/20/5604-0455 © 2020 Springer Science+Business Media, LLC

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Kirchhoff reduced the 3-D equations of motion of plates to 2-D ones, derived using Navier and Poisson power series and introducing the assumption that the normal to plate midplane remains unchanged after bending of plates. The central deflection w of a plate under the action of a uniform surface load q is determined by the equation



∆∆w =

q Eh3 , D= , D 12(1 − γ 2 )

where D is the Laplace operator. Later, Strutt [3] approximated the results of boundary value problems using a direct method. In 1908, Ritz [4] reformulated an approximation technique to obtain a generalized solution, which is currently known as the Rayleigh–Ritz method. An analogical approach for shells was suggested by Love [5] and is known as the Kirchhoff–Love shell theory. For the first time, the shear deformation effect was introduced into deformation equation by Timoshenko [6]. He used it for beams and reformulated

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