On Sampling-Times-Independent Identification of Relaxation Time and Frequency Spectra Models of Viscoelastic...

Area of Science:

• Materials Science
• Rheology
• Computational Mechanics

Background:

• Viscoelastic relaxation time and frequency spectra are crucial for characterizing material mechanical properties.
• Traditional methods for obtaining these spectra rely on stress or oscillatory shear measurements.
• Existing mathematical models and algorithms for spectrum identification have limitations regarding material specificity and dependence on measurement conditions.

Purpose of the Study:

• To develop a robust method for identifying relaxation spectra models that are asymptotically independent of experimental sampling times.
• To ensure the stability and accuracy of identified models despite noise in relaxation modulus measurements.
• To present a complete algorithm for relaxation spectra identification applicable to various rheological materials.

Main Methods:

• Approximation of relaxation spectra using finite series of orthogonal (Legendre, Laguerre, Chebyshev) and non-orthogonal (power exponential, modified Bessel) basis functions.
• Random selection of sampling times in stress relaxation experiments to achieve sampling-time independence.
• Application of Tikhonov regularization to stabilize the ill-posed identification problems.
• Solving a sequence of weighted least-squares relaxation modulus approximation problems.

Main Results:

• Demonstrated that sampling time randomization enables the determination of optimal spectra models independent of sampling times, even for unknown true spectra.
• Recovered spectra models are strongly consistent estimates of the true models.
• Achieved stable solutions for ill-posed identification problems through regularization.
• Stochastic convergence analysis showed convergence at an exponential rate.
• Simulation studies confirmed the effectiveness for various known spectra (Kohlrausch-Williams-Watts, Gauss-like, Baumgaertel-Schausberger-Winter).

Conclusions:

• The proposed method effectively identifies stable and accurate relaxation spectra models, independent of experimental sampling times.
• Randomization of sampling times is a key strategy for achieving robust and generalizable rheological models.
• The developed algorithm provides a reliable approach for analyzing material viscoelastic properties, enhancing model predictability and applicability.