Exact solution of the boundary value problem for strain and stress fields in a thick tube made of physically non-linear elasto-viscoplastic material under given internal and external pressures

We construct the analytic solution of the quasi-static boundary value problem for a hollow cylinder (a thick tube) under given pressure similar to the Lame problem in the elasticity theory but for a cylinder made of physically non-linear elasto-viscoplastic material governed by the Rabotnov constitutive equation with an arbitrary material functions. We assume that pressure values preset on an internal and external surfaces of the thick tube depend on time but change slowly enough to neglect inertia terms in the equilibrium equations. We also suppose that a material is homogeneous, isotropic and incompressible and a plain strain state is realized, i. e. the tube is long enough or zero axial displacements are given on the edge cross sections of the tube. We derive explicit closed form expressions for displacement, strain and stress fields via the unknown function of time and obtain functional equation to determine this function implying radii of the tube, pressure values dependence on time and material functions of the Rabotnov constitutive equation are given. It follows from the exact solution of the boundary value problem that this unknown function of time can be simply measured in pressure tests of tubular specimen. This observation allows to use the solution constructed for identification of the Rabotnov constitutive equation.