A Class of Algorithms for Recovery of Continuous Relaxation Spectrum from Stress Relaxation Test Data Using Orthonormal Functions

Abstract

The viscoelastic relaxation spectrum provides deep insights into the complex behavior of polymers. The spectrum is not directly measurable and must be recovered from oscillatory shear or relaxation stress data. The paper deals with the problem of recovery of the relaxation spectrum of linear viscoelastic materials from discrete-time noise-corrupted measurements of relaxation modulus obtained in the stress relaxation test. A class of robust algorithms of approximation of the continuous spectrum of relaxation frequencies by finite series of orthonormal functions is proposed. A quadratic identification index, which refers to the measured relaxation modulus, is adopted. Since the problem of relaxation spectrum identification is an ill-posed inverse problem, Tikhonov regularization combined with generalized cross-validation is used to guarantee the stability of the scheme. It is proved that the accuracy of the spectrum approximation depends both on measurement noises and the regularization parameter and on the proper selection of the basis functions. The series expansions using the Laguerre, Legendre, Hermite and Chebyshev functions were studied in this paper as examples. The numerical realization of the scheme by the singular value decomposition technique is discussed and the resulting computer algorithm is outlined. Numerical calculations on model data and relaxation spectrum of polydisperse polymer are presented. Analytical analysis and numerical studies proved that by choosing an appropriate model through selection of orthonormal basis functions from the proposed class of models and using a developed algorithm of least-square regularized identification, it is possible to determine the relaxation spectrum model for a wide class of viscoelastic materials. The model is smoothed and robust on measurement noises; small model approximation errors are obtained. The identification scheme can be easily implemented in available computing environments.

Two-Level Scheme for Identification of the Relaxation Time Spectrum Using Stress Relaxation Test Data with the Optimal Choice of the Time-Scale Factor

“…Following [40], on the basis of the relaxation modulus course, the range of applicability is specified as the time t, for which the first K basis functions φ k (t, α) no longer permanently exceed; i.e., for any θ > t, ε = 0.5% of its maximum value. Specifically,…”

“…Similarly, in [40], the range of applicability specified directly for the relaxation times τ was defined on the basis of the variability in the basis functions h k (τ); i.e.,…”

“…Therefore, computationally efficient algorithms to determine the relaxation spectrum applied to time measurements of the relaxation modulus are still desirable. Recently, for the dominant definition of the relaxation spectrum in the literature, a class of algorithms for relaxation spectrum recovery, which combines the technique of an expansion of a function into a series in an orthonormal basis with the least-squares regularized identification, has been derived by Stankiewicz [40]. Legendre, Laguerre, Chebyshev and Hermite functions were used as the basis functions for the spectrum model.…”

“…Legendre, Laguerre, Chebyshev and Hermite functions were used as the basis functions for the spectrum model. In [40], the problem of determining the spectrum of relaxation frequencies was considered. The complementary problem of determining the spectrum of relaxation times based on the stress relaxation data is both analytically and numerically more difficult than the determination of the spectrum of relaxation frequencies, because the relaxation time occurs in the denominator of the exponential function in the spectrum definitional formula.…”

“…This very task is the subject of this article. Previous studies [18,39,40] have suggested that by selecting an appropriate time-scale factor, a better fit of the model to the measurement data can be obtained.…”