Distributed Relaxation Spectrum Delay Differential Model for Viscoelastic Materials: Stability and Bifurcation Analysis

In our research, we developed a Distributed Relaxation Spectrum Delay Differential Equation (DRSDDE) model to simulate viscoelastic responses exhibited by materials with multiple-scale relaxation mechanisms and finite delay times. Our model expanded upon traditional integer-order viscoelastic models to include a continuum relaxation process using a log-time-space Gaussian distribution representing a continuum of relaxation processes, including a direct representation of the effect of delayed feedback via an explicit time delay term. Consequently, the resultant model can be viewed as a generalized Maxwell-type formulation where the viscoelastic behavior exhibits distributed relaxation dynamics and has finite signal propagation characteristics. We then used experimental data obtained from three representative materials: PDMS Sylgard 184, bovine brain white matter, and polyurethane foam to calibrate the model. Calibration was achieved by estimating model parameters through the use of Gauss-Legendre quadrature combined with non-linear optimization of the relaxation spectrum. The results indicate that the coefficients of determination for each of the materials exceeded R² > 0.83. Therefore, the proposed DRSDDE model outperformed the classical Zener model when simulating materials that exhibit a wide relaxation spectrum. The parameter values estimated for each of the examined materials provided additional insight into their physical behaviors. Specifically, the characteristic relaxation times for the studied materials were determined based upon \(\tau\)_c = 10^µ ranging from about 63 s to 158 s. These results illustrate different dominant relaxation regimes for the investigated materials. Additionally, both characteristic equations and frequency domain analyses were utilized to study the stability and bifurcation properties of the DRSDDE model. A significant finding resulted from identifying a delay-insensitive stability regime for materials with \(\tilde{K} < 1\) (as illustrated by bovine brain white matter). For materials with \(\tilde{K} > 1\), the analysis revealed Hopf bifurcation results illustrating critical delay thresholds and frequencies for the onset of oscillations. Further, it was established that all calibrated delay values were significantly less than these threshold values. This indicates that all identified models functioned well below the oscillation thresholds at realistic delay times. Ultimately, the proposed DRSDDE model represents a physically intuitive, robust, and flexible method for modeling complex viscoelastic systems. Future research will involve investigating temperature-dependent effects, nonlinear bifurcations, and experimental validations of predicted oscillatory dynamics