Robust Recovery of Optimally Smoothed Polymer Relaxation Spectrum from Stress Relaxation Test Measurements

Area of Science:

• Polymer Science and Engineering
• Rheology
• Materials Science

Background:

• The relaxation spectrum is crucial for understanding polymer viscoelasticity and rheological properties.
• Existing methods for relaxation spectrum identification are often ill-posed, requiring built-in smoothing and noise-robustness mechanisms.
• Regularization and solution space limitations are common but imperfect remedies for ill-posed problems.

Purpose of the Study:

• To directly state and solve the problem of determining an optimally smoothed continuous relaxation time spectrum.
• To develop a method for identifying the spectrum from discrete-time, noise-corrupted relaxation modulus measurements.
• To find the best-smoothed relaxation time spectrum model within the class of continuous square-integrable functions.

Main Methods:

• Utilized the Hilbert projection theorem to derive the structure of the best-smoothed relaxation spectrum.
• Introduced a quadratic term with Lagrange multipliers to enhance robustness for noise-corrupted data.
• Derived necessary and sufficient optimality conditions leading to a direct analytical formula for the spectrum.
• Developed a computational algorithm based on singular value decomposition.

Main Results:

• The optimally smoothed relaxation spectrum is represented by a finite sum of specific exponential-hyperbolic basis functions.
• The proposed method yields a small model error for the relaxation modulus, increasing robustness.
• Numerical studies confirm successful application for one- and two-mode Gaussian-like relaxation spectra.
• The approach shows good approximation for the Baumgaertel, Schausberger, and Winter (BSW) spectrum at higher frequencies and near local maxima.

Conclusions:

• A direct, analytical method for obtaining an optimally smoothed continuous relaxation time spectrum from experimental data has been established.
• The developed algorithm is computationally efficient and robust against noise.
• While effective for various spectra, the model's approximation is limited in the close neighborhood of zero-relaxation time compared to the BSW model's asymptotic properties.