A non-classical Kirchhoff plate model incorporating microstructure, surface energy and foundation effects

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• 作者：X. -L. Gao ; G. Y. Zhang
• 关键词：Kirchhoff plate ; Size effect ; Couple stress theory ; Surface elasticity ; Hamilton’s principle ; Winkler foundation ; Pasternak foundation ; Plate theory ; Free vibration ; Natural frequency
• 刊名：Continuum Mechanics and Thermodynamics
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：28
• 期：1-2
• 页码：195-213
• 全文大小：1,679 KB
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• 作者单位：X. -L. Gao (1)
  G. Y. Zhang (1)
  
  1. Department of Mechanical Engineering, Southern Methodist University, P. O. Box 750337, Dallas, TX, 75275-0337, USA
  
• 刊物类别：Engineering
• 刊物主题：Theoretical and Applied Mechanics
  Engineering Thermodynamics and Transport Phenomena
  Mechanics, Fluids and Thermodynamics
  Structural Materials
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-0959

文摘

A new non-classical Kirchhoff plate model is developed using a modified couple stress theory, a surface elasticity theory and a two-parameter elastic foundation model. A variational formulation based on Hamilton’s principle is employed, which leads to the simultaneous determination of the equations of motion and the complete boundary conditions and provides a unified treatment of the microstructure, surface energy and foundation effects. The new plate model contains a material length scale parameter to account for the microstructure effect, three surface elastic constants to describe the surface energy effect, and two foundation moduli to represent the foundation effect. The current non-classical plate model reduces to its classical elasticity-based counterpart when the microstructure, surface energy and foundation effects are all suppressed. In addition, the newly developed plate model includes the models considering the microstructure dependence or the surface energy effect or the foundation influence alone as special cases and recovers the Bernoulli–Euler beam model incorporating the microstructure, surface energy and foundation effects. To illustrate the new model, the static bending and free vibration problems of a simply supported rectangular plate are analytically solved by directly applying the general formulas derived. For the static bending problem, the numerical results reveal that the deflection of the simply supported plate with or without the elastic foundation predicted by the current model is smaller than that predicted by the classical model. Also, it is observed that the difference in the deflection predicted by the new and classical plate models is very large when the plate thickness is sufficiently small, but it is diminishing with the increase of the plate thickness. For the free vibration problem, it is found that the natural frequency predicted by the new plate model with or without the elastic foundation is higher than that predicted by the classical plate model, and the difference is significant for very thin plates. These predicted trends of the size effect at the micron scale agree with those observed experimentally. In addition, it is shown both analytically and numerically that the presence of the elastic foundation reduces the plate deflection and increases the plate natural frequency, as expected. Keywords Kirchhoff plate Size effect Couple stress theory Surface elasticity Hamilton’s principle Winkler foundation Pasternak foundation Plate theory Free vibration Natural frequency