On plate theories and Saint-Venant's principle
On plate theories and Saint-Venant's principle
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DOI：
10.1016/0020-7683(85)90052-6
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发表时间：
1985
影响因子：
3.6
通讯作者：
R. D. Gregory;F. Wan
中科院分区：
文献类型：
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作者：
R. D. Gregory;F. Wan
众所周知，经典的Germain-Kirchhoff板理论是各向同性薄平板线性弹性静力学在小厚度参数下的外（渐近展开或内）解的主导项。这个主导项（或实际的）外部解决方案本身不能满足任意规定的可接受的边缘数据。另一方面，互补的内部（渐近展开或边界层）的解决方案是由一系列的边界值问题，这是几乎一样难以解决的原始问题。对于应力边界数据，圣维南原理已被调用，以产生一组应力边界条件的经典板理论，以及高阶项的外部扩展（给各种厚板理论），没有任何参考的内部解决方案。文献中对一般形状板的位移边界条件和其它类型的边界条件的推导都是不成功的，本文应用作者提出的一种通用方法，推导出任意给定的可容许边界条件的正确边界条件（没有内（或边界层）解的显式解），包括位移边界数据的情况。我们的一般结果还表明，圣维南原理仅适用于首项外解，即经典板理论，才是严格正确的。
It is generally known (hat the classical Germain-Kirchhoff plate theory is the leading term of the outer (asymptotic expansion or interior) solution in a small thickness parameter for the linear elastostatics of thin, flat, isotropic bodies. This leading term (or the actual) outer solution alone cannot satisfy arbitrarily prescribed admissible edge-data. On the other hand, the complementary inner (asymptotic expansion or boundary layer) solution is determined by a sequence of boundary value problems which are nearly as difficult to solve as the original problem. For stress edge-data, Saint-Venant's principle has been invoked to generate a set of stress boundary conditions for the classical plate theory as well as for higher order terms in the outer expansion (giving various thick plate theories) without any reference to the inner solution. Attempts to derive the corresponding boundary conditions for displacement and other types of edge-data in the literature for general shape plates have not been successful.The present study applies a general method developed by the authors to derive the correct set of boundary conditions for arbitrarily prescribed admissible edge-data (without an explicit solution of the inner (or boundary layer) solution) for a number of special cases of general interest, including cases with displacement edge-data. Our general results also show that, to be strictly correct, Saint-Venant's principle should be applied only to the leading term outer solution, i.e. the classical plate theory.