The increasing complexity of the constructive forms and shell elements structure leads to the need to develop both the theory and methods for solving static and dynamic problems for non-homogeneous (heterogeneous) shells.

By the shell heterogeneity, we mean heterogeneity in a broad sense: these are inclusions in the shell body of the different rigidity elements and, as a special case, these are holes; the material inhomogeneity caused by a change in the stress–strain state under the influence of both static and dynamic loads; stiffeners; and by taking into account the physical nonlinearity and different modulus of the shell material.

This work is devoted to the mathematical model creation of the statics and dynamics for non-homogeneous shells in the above sense and consists of two parts.

In the first part, a mathematical model of statics and dynamics for rectangular shells described by the Kirchhoff–Love kinematic model is constructed. Geometric nonlinearity is taken into account on the basis of T. von Kármán’s geometric model, physical nonlinearity — according to the deformation theory of plasticity, based on the elasticity variable parameters method. Stiffness heterogeneity is taken into account using the Heaviside function. The original equations were obtained from Hamilton’s variational principle. A numerical experiment is performed using the Faedo–Galerkin method in higher approximations. The convergence of this method is investigated.

In the second part, the nonlinear dynamics of axisymmetric elastic variable thickness shells is investigated. In contrast to the first part of this work, where the Faedo–Galerkin method in higher approximations was used to reduce partial differential equations to the Cauchy problem, in this part the Ritz method in higher approximations is employed. For differential operators used in two parts of this work, according to the research of S.G. Mikhlin, the Ritz and Faedo–Galerkin methods are equivalent. The convergence of the applied numerical method is investigated. An approach is proposed for the zones study of oscillation types for variable thickness shells using dynamic modes maps.

构造形式和壳单元结构的日益复杂性导致需要发展用于解决非均质（异质）壳的静态和动态问题的理论和方法。

壳体异质性在广义上是指异质性：这些是壳体中不同刚度元素的夹杂物，在特殊情况下，它们是孔；在静态和动态载荷的影响下，应力-应变状态的变化所引起的材料不均匀性；加劲肋; 并考虑到壳体材料的物理非线性和不同模量。

从上述意义上讲，这项工作致力于为非均质壳体建立静力学和动力学的数学模型，它由两部分组成。

在第一部分中，建立了由Kirchhoff-Love运动学模型描述的矩形壳静力学和动力学数学模型。在T. vonKármán的几何模型（物理非线性）的基础上考虑了几何非线性-根据可塑性变形理论，基于弹性变量参数方法。使用Heaviside函数考虑了刚度异质性。原始方程式是从汉密尔顿的变分原理获得的。使用Faedo-Galerkin方法以更高的近似值进行了数值实验。研究了该方法的收敛性。

在第二部分中，研究了轴对称弹性变厚度壳的非线性动力学。与本工作的第一部分相反，在该部分中，采用较高近似值的Faedo-Galerkin方法将偏微分方程简化为柯西问题，而在这一部分中，采用较高近似值的Ritz方法。根据SG Mikhlin的研究，对于这项工作的两个部分中使用的差分算子，Ritz和Faedo-Galerkin方法是等效的。研究了所应用数值方法的收敛性。提出了一种利用动态模式图研究变厚度壳的振动类型的区域方法。