Abstract

Using the basic recurrence formulas for Chebyshev polynomials of the second kind, the several additional relationships have been obtained which play an important role in the construction of various variants of the theory of thin bodies. The moments of the tensor functions, as well as the moments of their derivatives and the moments of the repeated derivatives are determined, too. The moments of the \(k\)th order of some expressions with respect to Chebyshev polynomials are found. Representations of the equations of motion with respect to the contravariant components of the stress and couple-stress tensors, the heat flow equation, constitutive relations of the micropolar theory, and the Fourier heat conduction law of the \(s\)th order approximation are given. From them it is easy to get the corresponding relations in the moments with respect to the systems of Chebyshev and Legendre polynomials. As a particular case, the zero and first approximations motion equations in moments with respect to the contravariant components of the stress and couple-stress tensors are written out, and also the systems of equations in the displacements of the zero and first approximations in moments for non isothermal processes for any anisotropic material are given.

中文翻译：

---

关于多层薄体的理论

摘要

使用第二类切比雪夫多项式的基本递推公式，已经获得了几个附加关系，这些关系在构造薄体理论的各种变体中起着重要作用。张量函数的矩，以及它们的导数的矩和重复导数的矩也被确定了。找到了关于 Chebyshev 多项式的一些表达式的\(k\)阶矩。关于应力和应力偶应力张量的逆变分量的运动方程的表示、热流方程、微极理论的本构关系以及\(s\)的傅立叶热传导定律给出了 th 阶近似值。从它们很容易得到关于切比雪夫和勒让德多项式系统的时刻的对应关系。作为一个特殊情况，写出关于应力和偶应力张量的逆变分量的矩的零和一阶近似运动方程，以及零和一阶近似的矩位移方程组给出了任何各向异性材料的非等温过程。