Abstract

The motion equations of the micropolar theory of elasticity in displacements and rotations represented by the matrix differential tensor-operator for any inhomogeneous anisotropic materials. As a particular case the micropolar isotropic homogeneous materials with a center of symmetry is considered. In this case, the matrix differential tensor-operator of cofactors to the matrix differential tensor-operator of the motion equations is constructed. This constructed operator makes possible to decompose the equations. The equations are obtained separately with respect to the displacement and rotation vectors. Decomposed equations also obtained for a reduced medium. In this case, the equation with respect to the displacement vector is the same as the equation of the classical theory, and the equation with respect to the rotation vector has a similar form. In addition, in the absence of volume loads, the equations of the reduced medium do not depend on the properties of the material. This suggests that these equations can be used to identify the material constants of this medium. The cases under which the static boundary conditions are easily split are revealed. From the decomposed equations of the micropolar theories of elasticity, the corresponding decomposed equations of the static (quasistatic) problem of the theories of single-layer and multi-layer prismatic bodies of constant thickness in displacements and rotations are obtained. From the last systems of equations the equations in the moments of unknown vector functions with respect to any systems of orthogonal polynomials are derived. As a particular case, we obtain a system of equations of the eighth approximation in moments with respect to the system of Legendre polynomials, which decomposes into two systems. One of them is the system with respect to the even order moments of the unknown vector function, and the other system is the system with respect to odd order moments of the same functions. Based on the obtained operator of cofactors to the operator of any of these systems we get a high order (the order of the system depends on the order of approximation) elliptic type equation for each moment of the unknown vector function, which characteristic roots are easily found. Using Vekua’s method for solving such equations [66], we can obtain their analytical solution. Note also that the analytic method with the use of the orthogonal polynomial systems (Legendre and Chebyshev) in constructing the one-layer [2, 3, 7, 10, 15, 17, 18, 20–22, 63, 68, 69] and multilayer [4–6, 13, 60, 61] thin body theory was also applied by other authors. In this direction the authors had published the papers [24–31, 33–37, 41–45, 51–53], and others with the application of Legendre and Chebyshev polynomial systems. These expansions can be successfully used in constructing any thin body theory. Despite this, classical theories are far from perfect, and micropolar theories and theories of another rheology are very far from perfect.

中文翻译：

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微极弹性方程与薄体理论的分解

摘要

微极性弹性理论在位移和旋转方面的运动方程式，适用于任何非均质各向异性材料的矩阵微分张量算符。作为特殊情况，考虑具有对称中心的微极性各向同性均质材料。在这种情况下，构造了运动方程的矩阵微分张量算符的辅因子矩阵微分张量算符。该构造的算子使得分解方程成为可能。关于位移和旋转矢量分别获得方程式。还可以针对还原性介质获得分解方程。在这种情况下，关于位移矢量的方程式与经典理论的方程式相同，并且关于旋转矢量的等式具有类似的形式。另外，在没有体积载荷的情况下，还原介质的方程式不取决于材料的特性。这表明这些方程可用于识别该介质的材料常数。揭示了容易分割静态边界条件的情况。从微极弹性理论的分解方程中，得到了位移和旋转不变厚度的单层和多层棱柱体理论的静态（准静态）问题的相应分解方程。从最后的方程组中，导出关于任何正交多项式系统的未知矢量函数时刻的方程。在特定情况下，相对于勒让德多项式系统，我们获得了一个瞬间近似为八分之一的方程组，该系统分解成两个系统。其中一个是关于未知矢量函数的偶数阶矩的系统，另一个系统是关于相同函数的奇数阶矩的系统。根据获得的辅因子算子对这些系统中任何一个的算子，我们对于未知矢量函数的每个矩都得到一个高阶（系统阶数取决于近似阶数）椭圆型方程，其特征根很容易找到了。使用Vekua的方法求解此类方程[66]，我们可以获得其解析解。另请注意，在构造单层[2、3、7、10、15、17、18、20-22、63、68、69]时，使用正交多项式系统（Legendre和Chebyshev）进行分析的方法其他作者也应用了多层[4-6、13、60、61]薄体理论。在这个方向上，作者发表了论文[24-31、33-37、41-45、51-53]，以及其他有关勒让德勒和切比雪夫多项式系统应用的论文。这些扩展可以成功地用于构建任何薄体理论。尽管如此，古典理论还远非完美，微极性理论和另一种流变学则远非完美。[41–45，51–53]，以及其他在勒让德和切比雪夫多项式系统中的应用。这些扩展可以成功地用于构建任何薄体理论。尽管如此，古典理论还远非完美，微极性理论和另一种流变学则远非完美。[41–45，51–53]，以及其他在勒让德和切比雪夫多项式系统中的应用。这些扩展可以成功地用于构建任何薄体理论。尽管如此，古典理论还远非完美，微极性理论和另一种流变学则远非完美。