Analytical-based exact-kernel vibration and long-term creep stress and large deformation redistributions of the suddenly pressurized incompressible visco-hyperelastic thick cylinders

Abstract

The long-term creep and short-term damped dynamic redistributions of the large deformations and stresses are investigated here for long, suddenly pressurized, rubber-like visco-hyperelastic thick-walled cylinders, for the first time. The extension of the Mooney–Rivlin strain energy density function to incompressible visco-hyperelastic materials is accomplished by using a hereditary integral based on the Prony series and an exponential relaxation kernel. To extract a more accurate/efficient formulation, the relaxation kernel is processed instead of the common approximate discretization of the time derivative of the stress tensor in the convolution integral. The resulting nonlinear integrodifferential equations whose number of terms grows with time are solved by a novel analytical approach in conjunction with the second-order Runge–Kutta time-marching, bound-varying trapezoidal technique, and an iterative/updating scheme. In comparison to the commercial finite element analysis codes, the present analytical formulation and solution algorithm lead to higher accuracies from the mathematical point of view. The visco-hyperelastic properties are extracted from previously reported experimental results, using an error minimization fitting technique. Results reveal that in contrast to the hyperelastic vessel, the oscillations of the displacements and stresses of the visco-hyperelastic cylinder occur around levels that grow with time. Moreover, smaller relaxation parameters lead to smaller displacements and stresses and the responses intend earlier to the steady-state condition for higher values of the stiffness elements of the Prony series. In long term, the mean stresses and mean radial displacements of the inner and outer layers of the cylinder grow asymptotically with time but the largest overshoots and amplitudes of the responses occur at the early times. However, smaller values of the relaxation parameter expedite the convergence to the steady-state condition.