Abstract

The linear statement of the theory of elasticity of a three-dimensional body was used. Issues related to the elimination of unnecessary functions from the known solutions of the theory of elasticity for isotropic and orthotropic materials were examined. A system of three second order partial differential equations was written down. Linear operators have been introduced, and the original system is transformed so each equation includes only two elastic displacements. It is proved that the solution of the resulting system of equations can be reduced to the integration of one sixth-order partial differential equation. Conditions for the introduced operators were found under the general solution of the system of three equations can be expressed in terms of one displacement function. The equation contains nine coefficients that depend on nine independent elastic constants describing the elastic state of an orthotropic material. It is shown that this equation depends on three variables and therefore, in the general case, cannot be decomposed into three factors. The method of separation of variables has been used and a technique has been developed for computing the solution of a sixth-order equation in an orthotropic prism. The characteristic equation was written down. Different variants of the values of its roots were investigated: complex and real. The analytical expression of displacements, strains and stresses have been obtained through the introduced displacement function.

中文翻译：

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求解六阶偏微分方程的正交各向异性体方程化简及其研究

摘要

使用了三维物体弹性理论的线性陈述。研究了与从各向同性和正交各向异性材料的弹性理论的已知解决方案中消除不必要的函数相关的问题。写出一个由三个二阶偏微分方程组成的系统。引入了线性算子，并对原始系统进行了变换，因此每个方程只包含两个弹性位移。证明了所得方程组的解可以简化为一个六阶偏微分方程的积分。引入算子的条件是在三个方程组的通解下找到的，可以用一个位移函数来表示。该方程包含九个系数，这些系数取决于描述正交各向异性材料的弹性状态的九个独立弹性常数。结果表明，该方程取决于三个变量，因此，在一般情况下，不能分解为三个因素。已经使用了变量分离的方法，并开发了一种技术来计算正交各向异性棱镜中的六阶方程的解。写出特征方程。研究了其根值的不同变体：复杂的和真实的。通过引入的位移函数，得到了位移、应变和应力的解析表达式。在一般情况下，不能分解为三个因素。已经使用了变量分离的方法，并开发了一种技术来计算正交各向异性棱镜中的六阶方程的解。写出特征方程。研究了其根值的不同变体：复杂的和真实的。通过引入的位移函数，得到了位移、应变和应力的解析表达式。在一般情况下，不能分解为三个因素。已经使用了变量分离的方法，并开发了一种技术来计算正交各向异性棱镜中的六阶方程的解。写出特征方程。研究了其根值的不同变体：复杂的和真实的。通过引入的位移函数，得到了位移、应变和应力的解析表达式。