Axisymmetric Problem of the Elasticity Theory for the Radially Inhomogeneous Cylinder with a Fixed Lateral Surface

By the method of the asymptotic integration of the equations of elasticity theory, the axisymmetric problem of elasticity theory is studied for a radially inhomogeneous cylinder of small thickness. It is considered that the elasticity moduli are arbitrary positive continuous functions of the radius of the cylinder. It is also assumed that the lateral surface of the cylinder is fixed, and stresses are imposed at the end faces of the cylinder, which leave the cylinder in equilibrium. The analysis is carried out when the cylinder thickness tends to zero. It is shown that solutions corresponding to the first and second iterative processes that determine the internal stress-strain state of the radially inhomogeneous cylinder with a fixed surface do not exist. The third iterative process defines solutions that have the boundary layer character equivalent to the Saint-Venant end effect on the theory of inhomogeneous plates. The stresses determined by the third iterative process are localized at the ends of the cylinder and decrease exponentially with distance from the ends. The asymptotic integration method is used to study the problem of torsion of the radially inhomogeneous cylinder of small thickness. The nature of the stress-strain state is analyzed.