Abstract: Using the method of successive approximations we find of this boundary-value problem the first-and second-order solutions. And then we obtain the formulae in the second approximation for the displacement, strain, and stress fields. Also, our results show that after deformation (i) a cross-section of the cylinder must be displaced into a plane section perpendicular to the central axis of the cylinder; and (ii) neither the sum of the strain components E_RR⁽²⁾ and E_φφ⁽²⁾ nor the sum of the stress components ∑_RR⁽²⁾ and ∑_φφ⁽²⁾ maintains contant throughout the cylinder. The latter effect, which is absent from classical elasticity, bears responsibility for the presence of the ∑_zz⁽²⁾ Moreover, there exhibits a linear relation between ∑_zz⁽²⁾ and (∑_RR⁽²⁾+∑_φφ⁽²⁾), with the proportionality coefficients depending only on the material of the cylinder.