Purpose. The goal of the work is to obtain an exact solution to the problem of the stress-strain state of a long thick-walled bimetallic cylinder within the framework of the classical theory of elastic materials. Then, using the results obtained with this formulation, it is necessary to propose simpler engineering approaches and explore the possibilities of using asymptotic formulas for express analyzes at the design stage of such structural elements. Research methods. For the main body of the cylinder, the classical equations of the theory of elasticity in displacements are used. For the outer coating (sputtering), the shell theory equations are written based on the Kirchhoff-Love hypotheses. The complex integral Fourier transform and the Failon method are used to approximately find the original stresses and displacements. Asymptotic representations of cylindrical Bessel functions for large values of the argument and representations of improper integrals in the form of combinations of elementary and special tabulated functions are also used. Results. A mathematical model has been constructed to analyze the stress-strain state of a bimetallic cylinder with a thin outer layer of a material different from that of the inner layer. Various boundary conditions are recorded on the inner surface of the cylinder, describing the transmission from a rigid body of either specified forces or specified displacements. For all considered options, using the method of integral transformations, the results were obtained in the form of improper integrals, for the calculation of which a special method was used, aimed at calculating integrals with highly oscillating functions. Examples of specific graphs of changes in the components of the stress-strain state in the cylinder material are given. Depending on the conditions on the inner surface of the cylinder, simpler models are proposed to describe the main body, which are based, depending on the nature of the description of the interaction of the rigid body and the cylinder, on one equation of the theory of elasticity. With this approach, in some important cases it was possible to obtain improper inversion integrals using the asymptotic approach in closed form as a combination of elementary and special tabulated functions. Comparison with the exact approach allowed us to prove the possibility of using approximate models. Scientific novelty. A model of the behavior of a bimetallic cylinder as a body is constructed, the main layer of which is described by the equations of the theory of elasticity, and the theory of shells is used for the outer coating. Various methods of describing the transfer of forces and displacements from a rigid body to the inner surface of the cylinder are considered. The possibility of using the asymptotic approach to obtain relatively simple formulas for carrying out preliminary calculations at the design stage of such structural elements is shown.
Read full abstract