and Computational Methods

doi: 10.18698/2309-3684-2022-4-330

The general asymptotic theory of thin multilayer shells developed by the authors earlier in Part 1 of this study is applied to cylindrical anisotropic thermoelastic shells. It is shown that for cylindrical shells the general theory is substantially simplified: general two-dimensional averaged thermoelasticity equations for multilayer shells are obtained. These equations are similar to the classical equations of cylindrical shells in the Kirchhoff-Love theory, but they are obtained in a completely different way: on the basis of only an asymptotic analysis of the general three-dimensional equations of the theory of thermoelasticity. No hypotheses regarding the distribution of displacements or stresses over the thickness are used in this theory, which makes it logically consistent. In addition, the developed theory makes it possible to obtain explicit analytical expressions for all 6 components of the stress tensor in cylindrical anisotropic shells. Explicit expressions are obtained for all tensor constants included in these stress formulas. An example of calculating thermal stresses in a cylindrical composite shell with axisymmetric bending due to the combined action of external pressure and one-sided non-stationary heating is given. An example of a layered-fiber 4-layer shell with different angles of helical winding of reinforcing fibers is considered. It is shown that the developed one allows one to study in detail such complex effects as the formation of significant transverse thermal stresses during heating, which significantly exceed the level of interlayer shear stresses, which are traditionally considered the most dangerous for layered composites.

Димитриенко Ю.И., Губарева Е.А., Пичугина А.Е., Белькова К.В., Борин Д.М. Моделирование термонапряжений в композитных оболочках на основе асимптотической теории. Часть 2. Расчет цилиндрических оболочек. Математическое моделирование и численные методы, 2022, № 3, с. 3–30

doi: 10.18698/2309-3684-2018-3-114132

The previously developed general asymptotic theory of thin multilayer shells is used for the case of cylindrical shells. The ratios are presented in explicit analytical form for all six components of the stress tensor in a thin multilayer elastic cylindrical shell, depending on the deformations, curvatures of the middle surface of the shell, as well as their derivatives along the longitudinal coordinates. The obtained formulas make it possible to calculate all the distributions of the components of the stress tensor over the thickness in a cylindrical shell after finding solutions to the two-dimensional problem of the theory of KirchhoffLyav shells. An example is given of the calculation of stresses in a cylindrical composite shell underaxisymmetric bending by pressure. To calculate stresses by these formulas, only a differentiation of displacements is required - a deflection and two displacements of the middle surface of the shell, for which an analytical solution is obtained.

Димитриенко Ю.И., Губарева Е.А., Пичугина А.Е. Моделирование напряжений в тонких композитных цилиндрических оболочках на основе асимптотической теории. Математическое моделирование и численные методы, 2018, № 3, с. 114–132.

doi: 10.18698/2309-3684-2020-4-84110

An asymptotic theory of thermoelasticity of multilayer composite shells is proposed, the derivation of the basic equations of which is based on the asymptotic expansion in terms of a small geometric parameter of three-dimensional thermoelasticity equations. This method was previously developed by the authors for thin composite plates, and in this article it is applied to thin-walled shells of an arbitrary frame. According to the developed method, the original three-dimensional problem of thermoelasticity decomposes into a recurrent successor of one-dimensional local problems of thermoelasticity and an averaged two-dimensional problem of thin shells. For local problems of thermoelasticity, analytical solutions are obtained, which make it possible to close the averaged formulation of the problem of the theory of shells with respect to 5 unknown functions: longitudinal displacements, deflection, and two shear forces. It is shown that the averaged problem for multilayer shells coincides with the classical system of equations for Kirchhoff–Love shells, however, it is more substantiated, since the asymptotic theory does not contain any assumptions regarding the pattern of the distribution of permutations and stresses over thickness. In addition, the asymptotic theory makes it possible to calculate all the stresses in the shell, without solving any additional problems, but only by differentiating the averaged displacements.

Димитриенко Ю.И., Губарева Е.А., Пичугина А.Е. Моделирование термона-пряжений в композитных оболочках на основе асимптотической теории. Часть 1. Общая теория оболочек. Математическое моделирование и численные методы, 2020, № 4, с. 84–110.