Exact solution for the stress relaxation problem in a thickwalled tube of a nonlinear viscoelastic material obeying the rabotnov constitutive equation

We constructed and studied analytically the exact solution of the quasi-static boundary value problem for a hollow cylinder (a tube) made of physically nonlinear homogeneous isotropic viscoelastic material obeying the Rabotnov constitutive equation with two arbitrary material functions (a creep compliance and a function which governs physical nonlinearity). A time-dependent displacement and zero shear stresses are given on the inside cylinder surface, a pressure (normal stress) is given on outside surface and zero axial displacements and zero shear stresses are preset on the end cross sections of the tube (thus, cylinder stress and plain strain are realized in a tube). We supposed that a material is incompressible and that given boundary displacement and pressure vary slowly enough with time to neglect inertia terms in the equilibrium equations. We obtained explicit expressions for strains and stresses at any point via the ratio of a tube radii, given external pressure and integral operators involving composition of two material functions of the constitutive relation and displacement preset on the internal cylindrical surface. In particular, assuming given boundary displacement and pressure are constant we considered the stress relaxation problem. For arbitrary material functions, we calculated all the hereditary integrals involved in the general representation for the stress field and reduced it to simple algebraic formulas convenient for analysis and use. We studied analytically general properties of the stress relaxation curves at any point of a tube and features of stress distributions along radius. We proved that the first material function (relaxation modulus) of the constitutive equation governs completely stresses dependence on time and the second one (non-linearity function) governs only stresses dependence on radial coordinate and found out that hoop stress and axial stress can change sign and can increase, decrease and be non-monotone with respect to radial coordinate. In case of zero external pressure, we proved that all stress relaxation curves at any point of a tube are proportional to each other (ratio of stresses at different points doesn't depend on time), decrease with time and have no flexure points. We also calculated integral mean values of axial and hoop stresses and discovered that their ratio depends on thickness/radius ratio only and doesn't depend on time and material functions although mean stresses do.