A general elongational flow experiment: Inflation and extension of a viscoelastic tube

Abstract

The simultaneous inflation and extension of a tube is investigated as a method for conducting elongational flow experiments. This generally unsteady inhomogeneous deformation is restricted so that the material particles at the outer radius of the tube (r 0 ) experience steady elongational flow. The solution to the stress equations of motion, including inertia and interfacial tension, is given for a tube made of an incompressible, isotropic simple material. This solution is used to relate the contributions of interfacial tension, inertia and the normal stress differences to two measurable surface tractions: the pressure difference across the tube wall and the axial force on the ends of the tube. Explicit predictions for dimensionless force and pressure functions (related to the axial force and pressure difference) are obtained using a simplified form ofLodge's rubberlike liquid constitutive equation to evaluate the integrals containing the normal stress differences. For typical elongational flow experiments with high-viscosity materials, the contribution of inertia and interfacial tension to the predicted force and pressure functions is negligible, but the effect of the cylindrical geometry is significant, except for uniaxial elongation which is a homogeneous flow. For certain elongational flows the cylindrical geometry causes an “overshoot” in the predicted pressure function which is not observed in the related normal stress difference for the corresponding steady elongational flow atr 0 . As the time since the inception of any inhomogeneous elongational flow increases, the ratio of the tube wall thickness to the outer radius of the tube decreases and the force and pressure functions approach values predicted for the corresponding steady elongational flow atr 0 , provided that a critical value of one strain rate is not exceeded. Under these conditions the normal stress differences for the steady elongational flow atr 0 can be determined approximately from experimental measurements of the pressure difference and axial force. The restriction on one strain rate noted above indicates that it is not always valid to assume that the effect of the cylindrical geometry on the measured surface tractions decreases as the ratio of the tube wall thickness to the outer radius decreases.