Abstract
Asymptotic methods play an important role in solving three-dimensional elasticity problems. The method of asymptotic integration of three-dimensional equations of elasticity theory takes an important place in solving the problems of the limited transition from three-dimensional problems to two-dimensional for elastic membranes. Based on the method of asymptotic integration of equations of the elasticity theory, the axisymmetric problem of elasticity theory for radially non-homogeneous cylinder of small thickness is explored. The case when elasticity modules change by the radius according to the linear law is considered. It is expected that the lateral part of the cylinder is free from stresses and boundary conditions, leaving a cylinder in equilibrium, are assigned at the ends of a cylinder. The stated boundary-value problem is reduced to the spectral problem. The behavior of solutions to the spectral problem both in the inner part of a cylinder, and near the ends of a cylinder if the parameter of thinness of cylinder’s walls tends to zero, is studied. Three groups of solutions were obtained and the nature of the constructed homogeneous solutions was explained. The solution corresponding to the first iterative process determines the penetrating stressed-strained state of a cylinder. The solution corresponding to the second iterative process represents edge effects in the applied theory of shells. The third iterative process determines the solution which has the character of a boundary layer. The solution corresponding to the first and second iterative processes determines the internal stressed-strained state of the cylinder. In the first term of asymptotics, they can be regarded as a solution on the applied theory of shells. It was shown that the stressed-strained state, similar to the case of a homogeneous cylinder of small thickness, consists of three types: penetrating stressed state, simple edge effect and a boundary layer. The problem of meeting the boundary conditions on the ends of a radially non-homogeneous cylinder using the Lagrangian variation principle was considered.
Highlights
The studies of non-homogeneous shells take one of the special places in the theory of shells
The aim of this study is to reveal the features and to construct effective methods for calculating the stressed-strained state of a radially inhomogeneous cylinder
The following tasks have been set: – to construct homogeneous solutions using the method of asymptotic integration of equations of the elasticity theory, based on three iterative processes; – to construct asymptotic formulas for displacements and stresses; – to analyze the stressed-strained states, corresponding to various types of homogenous solutions; – meeting the boundary conditions at the ends of a cy linder
Summary
The studies of non-homogeneous shells take one of the special places in the theory of shells. Many issues related to the study of stressed-strained state for non-homogeneous shells can be properly solved only within the framework of the elasticity theory. This is especially important when researching non-stationary and. Paper [3] developed a ge neral theory of a transversely-isotropic cylinder of the small thickness, which includes the methods for constructing heterogeneous and homogeneous solutions, which make it possible to reveal the characteristics of stressed-strained state of an anisotropic cylindrical shell. In papers [2, 3], using the method of homogenous solutions, an axisymmetric dynamic problem of the elasticity theory for an isotropic and transversely-isotropic hollow cylinder of small thickness was explored. In article [9], the influence of non-homogeneity of the material on the stressed-strained state of a cylinder was explored
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