Semianalytical methods for the determination of the nonlinear parameter of nonlinear viscoelastic constitutive equations from LAOS data

Abstract

Various viscoelastic constitutive equations have been developed to describe nonlinear viscoelastic flows. Most equations contain two kinds of parameters: Nonlinear and linear ones. The linear parameters correspond to relaxation time spectrum and can be determined from linear viscoelastic data. Meanwhile, the nonlinear parameters cannot be determined by linear viscoelastic data. The determination of the nonlinear parameters requires both reliable nonlinear data and complex procedures for fitting numerical solution of differential equations to the nonlinear data. If an analytical solution of viscoelastic model is available then dramatic reduction of difficulty is expected in the determination of the nonlinear parameters. Previous studies on analytical solution of large amplitude oscillatory shear (LAOS) are based on series expansion which is effective up to only third harmonic. Since it is practically impossible to obtain the analytical solution of higher order than fifth, we suggest a new method which extracts semianalytical solutions (SAS) for some relevant quantities of LAOS from the numerical solutions of nonlinear viscoelastic constitutive equations: The Giesekus model and the Phan-Thien/Tanner model. The SAS includes the effects of higher harmonics which cannot be achieved by low-order series expansion technique. The series expansion is applicable to limited Wi ≪ 1 but any De, while the SAS are applicable to De < 1 but any Wi. The methods developed here are helpful not only for the estimation of nonlinear parameters of viscoelastic models but also for the investigation of the origin of strain-frequency superposition in LAOS.