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 引用本文: 段铁城, 李录贤. 厚板的高阶剪切变形理论研究[J]. 力学学报, 2016, 48(5): 1096-1113.
Duan Tiecheng, Li Luxian. STUDY ON HIGHER-ORDER SHEAR DEFORMATION THEORIES OF THICK-PLATE[J]. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(5): 1096-1113.
 Citation: Duan Tiecheng, Li Luxian. STUDY ON HIGHER-ORDER SHEAR DEFORMATION THEORIES OF THICK-PLATE[J]. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(5): 1096-1113.

## STUDY ON HIGHER-ORDER SHEAR DEFORMATION THEORIES OF THICK-PLATE

• 摘要: 已有多种厚板理论和高阶剪切变形模型，但仍需要进一步研究以更加完善.首先根据平均转角及上下表面剪应力自由这两个条件，提出了具有统一高阶剪切变形模型的中面位移模式，并将之表示为正交分解形式.根据正交特性，定义了板的广义应力；运用板问题应变能密度表示的等价性，提出了与广义应力功共轭的广义应变表示形式，建立了板的本构关系.证明了不同转角定义时虚功原理板理论表示的客观性，以及与三维弹性理论表示的等价性.运用虚功原理，建立了变分自洽的高阶厚板理论和变分渐近的低阶厚板理论，推导了相应的平衡方程及边界条件，分析了与已有板理论的异同.以广义应力形式建立了厚板理论的平衡方程，厘清了不同转角表示时板理论间的关系、低阶厚板理论与高阶厚板理论间的关系以及剪切系数计算等若干基本问题.对圣维南扭转问题的求解证明了该理论的正确性.

Abstract: It is still necessary to study the thick plate theory and higher-order shear deformation models with a lot of published work. Starting with the definition of average rotation and the free shear stress condition at the bottom and top surfaces, the displacements on the neutral plane are suggested with a unified higher-order shear deformation model, and then expressed in the orthogonal form. On this basis, the generalized stresses are defined, then the generalized strains are obtained in light of the work conjugate, and the constitutive relations are established for the plate theory. The objectivity of the principle of virtual work in the plate theory is proved for different definitions of rotation, as well as the identity to three-dimensional elasticity theory. Based on the principle of virtual work, the variationally consistent higherorder plate theory and the variationally asymptotic lower-order plate theory are respectively established by deriving the corresponding equilibrium equations and boundary conditions, and then compared with the existing plate theories. The current work originally presents the equilibrium equations of the plate theory in terms of the generalized stresses, and clarifies some fundamental problems such as the relations of different definitions of rotation, the relation between the higher-order plate theory and the lower-order plate theory, and the evaluation of the shear factor. The current plate theory is finally validated by solving the Saint-Venant torsion problem.

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