Equivalence of the Refined Theory and the Decomposed Theorem of an Elastic Plate
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摘要: 将Cheng氏精化理论和Gregory分解定理联系起来，获得了两者的等价性(Cheng利用算子矩阵行列式求解多元偏微方程组的方法，得到了一个方程，他认为这个方程的解是3个微分方程的解的和，没有证明这种分解的合理性)．从Papkovich-Neuber通解出发给出一个完整的精化理论的证明．首先将板内的位移利用中面上位移及其沿板厚方向的梯度表示出来，并获得板内应力张量．再利用附录中给出的定理，由边界条件和Lur'e算子方法获得精化理论．最后利用基本的数学工具分别证明了，Cheng氏精化理论中的3个方程分别与Gregory分解定理的三个应力状态的等价性．即：Cheng氏精化理论的双调和方程、剪切方程、超越方程与Gregory分解定理的内应力状态、剪切应力状态、Papkovich-Fadle应力状态一一等价．
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关键词:
- 弹性板 /
- 各向同性 /
- 精化理论 /
- 分解定理 /
- Papkovich-Neuber通解
Abstract: A connection between Cheng's refined theory and Gregory's decomposed theorem is analyzed.The equivalence of the refined theory and the decomposed theorem is given.Using operator matrix determinant of partial differential equation,Cheng gained one equation,and he substituted the sum of the general integrals of three differential equations for the equation's solution.But he didn't prove the rationality of substitute.There,a whole proof for the refined theory from Papkovich-Neuber solution was given.At first expressions were obtained for all the displacements and stress components in term of the mid-plane displacement and its derivatives.Using Lur'e method and the theorem of appendix,the refined theory was given.At last,using basic mathematic method,the equivalence between Cheng's refined theory and Gregory's decomposed theorem was proved,i.e.,Cheng's bi-harmonic equation,shear equation and transcendental equation are equivalent to Gregory's interior state,shear state and Papkovich-Fadle state,respectively. -
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