On the finite displacement problem of a hollow cylinder under internal and external pressures

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 ⁽²⁾_RR and E ⁽²⁾_ϕϕ nor the sum of the stress components E ⁽¹⁾_RR and E ⁽¹⁾_ΦΦ maintains contant throughout the cylinder. The latter effect, which is absent from classical elasticity, bears respensibility for the presence of the E ⁽¹⁾_ZZ . Moreover, there exhibits a linear relation between E ⁽¹⁾_ZZ and (E ⁽²⁾_RR +E ⁽¹⁾_OO ), with the proportionality coefficients depending only on the material of the cylinder.