The basic problem of satisfaction of boundary conditions is considered when the generalized stress vector is given on the surfaces of elastic plates and shells. This problem has so far remained open for both refined theories in a wide sense and hierarchic type models. In the linear case, it was formulated by I. N. Vekua for hierarchic models. In the nonlinear case, bending and compression-expansion processes do not split and in this context the exact structure is presented for the system of differential equations of von Kármán–Mindlin–Reisner (KMR) type, constructed without using a variety of ad hoc assumptions since one of the two relations of this system in the classical form is the compatibility condition, but not the equilibrium equation. In this paper, a unity mathematical theory is elaborated in both linear and nonlinear cases for anisotropic inhomogeneous elastic thin-walled structures. The theory approximately satisfies the corresponding system of partial differential equations and the boundary conditions on the surfaces of such structures. The problem is investigated and solved for hierarchic models too. The obtained results broaden the sphere of applications of complex analysis methods. The classical theory of finding a general solution of partial differential equations of complex analysis, which in the linear case was thoroughly developed in the works of Goursat, Weyl, Walsh, Bergman, Kolosov, Muskhelishvili, Bers, Vekua and others, is extended to the solution of basic nonlinear differential equations containing the nonlinear summand, which is a composition of Laplace and Monge–Ampére operators.

中文翻译：

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薄壁结构的理论与实践

给出弹性板壳表面的广义应力向量时，考虑了满足边界条件的基本问题。到目前为止，这个问题对于广义上的精炼理论和分层类型模型来说仍然是开放的。在线性情况下，它是由 IN Vekua 为分层模型制定的。在非线性情况下，弯曲和压缩-膨胀过程不会分裂，在这种情况下，提供了 von Kármán-Mindlin-Reisner (KMR) 类型的微分方程系统的确切结构，该系统不使用各种临时假设构建因为这个系统在经典形式中的两个关系之一是相容性条件，而不是平衡方程。在本文中，在各向异性非均匀弹性薄壁结构的线性和非线性情况下，阐述了统一数学理论。该理论近似满足相应的偏微分方程组和此类结构表面的边界条件。该问题也针对分层模型进行了调查和解决。获得的结果拓宽了复杂分析方法的应用范围。Goursat、Weyl、Walsh、Bergman、Kolosov、Muskhelishvili、Bers、Vekua 等人的著作在线性情况下彻底发展了寻找复分析偏微分方程通解的经典理论，扩展到包含非线性被加数的基本非线性微分方程的解，