不用Kirchhoff－Love假设的三维弹性板二级近似理论及其边界条件

The Second Order Approximation Theory of Three Dimensional Elastic Plates and lts Boundary Conditions Without Using Kirchhoff-Love Assumptions

前文￣［１］给出了不用Ｋｉｒｃｈｈｏｆｆ－Ｌｏｖｅ假设的三维弹性板的一级近似理论及其边界条件。这个理论有６个微分方程求解６个待定平面函数，即ｕ＿０，ｕ＿α，Ａ＿（０），Ｓ＿（２）＿α，其中有３个方程为一组求解３个待定平面函数ｕ＿０，Ｓ＿（２）＿α，而另一组３个方程求解另外３个待定平面函数ｕ＿α，Ａ＿（０）。它们的边界条件和这些方程一样，可以从本问题的广义变分原理的泛函变分的驻值条件求得，当板厚ｈ和板宽α之比ｈ／α很小时，这种解接近于经典薄板解，当ｈ／α值较大时（如ｈ／α≈０．３），这种解和经典薄板解，就有较大差别．但这种差别在ｈ／α值的什么范围内是合理的这一问题，并不清楚，为了解决这一问题，我们必须研究本问题的二级近似理论。本文是前文的继续，我们将用本问题的广义变分原理的泛函变分驻值条件，导出９个微分方程和有关边界条件，用以求解９个二级近似解的待定平面函数ｕ＿０，ｕ＿α，Ａ＿（０），Ａ＿（１），Ｓ＿（２）＿α，Ｓ＿（３）＿α，把二级近似理论解和一级近似理论以及经典理论的解相比较，就能明确一级近似理论的适用范围，这里必须指出，二级近似理论也可以分成两组方程求解，求解过程也并不过分复杂。有关符号和前文相同，这里?

The first order approximation theory of three diinensional elastic plates and its boundary conditions presented in the previous paper￣[1]establishes six differn-tial equations for. the solutions of six undetermined functions u￣0, u￣α,A_(0) and S_(2)_α defined in the x, y plane. They can be divided into two groups, each constitutes three equations, to calculate u￣0, s_(2)_αand u_α, A_(0) respectively. Their boundary conditions as well as these equations are derived from the stationary conditions of variations of a functional for this problem based on the generalized variatio-nal principle. The solutions given by this theory are close to tkose given by the classical theory of thin plates as the ratio of thickness h to width a is small.For large ratio, say h/α=0.3, a considerable differece arises between the two theories. It has not been made clear that in what range of the ratio such a dif-ference is reasonable to give more precise solutions. In order to solve this problem,we must study the second order approximation theory. In this paper following the preyious one, we shall establish the second order approxitnation theory by apply-ing the stationary condition of variations of a functional for this problem based on the generlized variational principle to derive nine differential cquations and the related boundary conditions. which are used to calculate nineundetermined functions u￣0,u_α, A_(0), A_(1), S_(2)α and S_(3)α And the range of the validity of the first order approximation theory can be found out by comparing the second order theory with the first order theory and the classical theory. It should be pointed out here that the cquations of the second order theory can also be divided into two groups to be solved separately, and the procedure of the solution is not too complicate to perform as well. Here, we will use the same notations adopted in the previous paper, and not repeat their definitions.