Transient internal pressure loading of a viscoelastic cylinder

The extensional relaxation function of the viscoelastic material is defined using a six-term Prony series:

The instantaneous modulus, $E_0$, is 1651.59 MPa; and the six pairs of relative moduli ${\overline{g}}_i^P$ and time constants ${\tau }_i$ are

This model results in a very low long-term elastic modulus (0.4955 MPa), so the material almost behaves as a viscoelastic fluid. Because the viscoelastic material is incompressible throughout the problem, the relative moduli ${\overline{g}}_i^P$ and time constants ${\tau }_i$ that constitute the extensional relaxation function can be used directly in the definition of the shear relaxation function. Contrast this with Viscoelastic rod subjected to constant axial load, in which the material is slightly compressible, so that the shear modulus and time constant were related to the extensional values through the bulk modulus.

A solution to the present problem is also obtained by modeling the behavior of the viscoelastic cylinder with large-strain linear viscoelastic theory. The relaxation behavior is defined in the same way, but the short-term elastic properties are specified as part of the hyperelastic material definition. The polynomial formulation with $N=$1 is used, and the constants are $C_{10}$ = 275.247 MPa , $C_{01}$ = 0 (neo-Hookean material) and $D_1$ = 7. × 10⁻⁷ MPa⁻¹. These constants are such that the initial Young's modulus and initial Poisson's ratio are equal to $E_0$ and ${\nu }_0$, respectively. The steel case is assumed to be linear elastic, with a Young's modulus of 200 GPa and a Poisson's ratio of 0.3.