Investigation of the stress-strain state of a hollow cylinder with a coating based on the gradient model of thermoelasticity

Abstract

The study of the stress-strain state of a thermoelastic hollow cylinder with a homogeneous coating is carried out taking into account the scale effects. Aifantis' one-parameter gradient model is used to account for scale effects. Equilibrium equations and boundary conditions for a composite hollow thermoelastic cylinder are obtained on the basis of the Lagrange variational principle. In comparison with the classical formulation of the problem, additional boundary conditions and conjugation conditions are set for moment stresses and displacement gradients. The dimensionlessness of the task of thermoelasticity has been carried out. Solving the problem of uncoupled thermoelasticity begins with finding the radial temperature distribution of a layered cylinder on the basis of solving the problem of heat conduction in the classical formulation. The solution of the problem in displacements is presented as a sum of solutions in the classical formulation of the problem and additional boundary layer terms found on the basis of the asymptotic properties of the modified Bessel functions. Simplified analytical expressions are obtained for finding radial displacements, radial and circumferential Cauchy stresses, nonzero components of the tensor of moment and total stresses. On specific examples, calculations of the radial distribution of displacements and stresses of a composite cylinder in the case of both mechanical and thermal loading are carried out. The limits of applicability of the asymptotic solution of the problem are investigated. The difference between the radial distribution of displacements and stresses found on the basis of solutions to the problem in the classical formulation and in the gradient formulation is shown. It was found that the Cauchy radial stresses experience a jump at the boundary of the cylinder and the coating, which is explained by the continuity of radial displacements and their first derivatives. The components of the moment stress tensor either take on peak values or experience a jump at the interface. The moment stresses are proportional to the square of the gradient parameter, at small values of which they have values that are much less than the values of the total stresses. With an increase in the dimensionless scale parameter, the values of radial displacements and total circumferential stresses decrease, but moment stresses increase.