A refined theory of elastic thick plates for extensional deformation |
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Authors: | Yang Gao Bao-Sheng Zhao |
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Institution: | (1) College of Science, China Agricultural University, P.O. Box 74, Beijing, 100083, People’s Republic of China;(2) School of Mechanical Engineering, University of Science and Technology Liaoning, Anshan, 114044, People’s Republic of China |
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Abstract: | Based on elasticity theory, various two-dimensional (2D) equations and solutions for extensional deformation have been deduced
systematically and directly from the three-dimensional (3D) theory of thick rectangular plates by using the Papkovich–Neuber
solution and the Lur’e method without ad hoc assumptions. These equations and solutions can be used to construct a refined
theory of thick plates for extensional deformation. It is shown that the displacements and stresses of the plate can be represented
by the displacements and transverse normal strain of the midplane. In the case of homogeneous boundary conditions, the exact
solutions for the plate are derived, and the exact equations consist of three governing differential equations: the biharmonic
equation, the shear equation, and the transcendental equation. With the present theory a solution of these can satisfy all
the fundamental equations of 3D elasticity. Moreover, the refined theory of thick plate for bending deformation constructed
by Cheng is improved, and some physical or mathematical explanations and proof are provided to support our justification.
It is important to note that the refined theory is consistent with the decomposition theorem by Gregory. In the case of nonhomogeneous
boundary conditions, the approximate governing differential equations and solutions for the plate are accurate up to the second-order
terms with respect to plate thickness. The correctness of the stress assumptions in the classic plane-stress problems is revised.
In an example it is shown that the exact or accurate solutions may be obtained by applying the refined theory deduced herein. |
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Keywords: | Thick plate Refined theory Extensional deformation The Papkovich– Neuber solution The Lur’ e method |
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