An Integral Constitutive Equation for Mixed Flows: Viscoelastic Characterization

A viscoelastic constitutive equation of the single-integral form has been designed. Its memory function is factored into a time-dependent part and a strain-dependent part’. The time function is the usual series of exponential relaxations. Its relaxation times and weighting coefficients are determined by nonlinear regression on linear viscoelastic data: stress relaxation after small-strain and small-amplitude sinusoidal oscillations. The relative accuracy of linear and nonlinear regression fitting is compared. The strain-dependent function is new. It is of a simple sigmoidal form with only two parameters: one determined from shear and the other from extensional data. Its sigmoidal form provides a finite linear viscoelastic region, a steady viscosity in uniaxial extension, and a well-behaved power-law shear viscosity at high shear rate. An efficient strategy for collecting sufficient data to determine the parameters of the equation is described. Predictions of the equation are tested against shear and extension data collected on the Rheometrics System Four for two polydimethylsiloxanes and against data for other polymer melts from the literature. Both uniaxial and biaxial extension as well as shear data are described. Transient shear normal stresses are somewhat underpredicted. The constitutive equation has the potential for modeling mixed shear and extensional flows as encountered in processing operations and is simple enough to be attractive for efficient computer-aided analysis by modern finite element methods.