Modeling and Analysis of Time-Dependent Creep and Relaxation Behavior of Polymeric Materials Using Fractional Derivative Three-Component Standard Viscoelastic Models and Nanoindentation Experimental Data

In the modeling of viscoelastic materials, two-component elements such as the Maxwell or Kelvin models, which consist of a spring and a dashpot arranged in series or parallel configurations, fail to accurately capture the complex behavior of polymer materials. To address this limitation, this study employs fractional derivative equations within the frameworks of three-component Zener and Boltzmann models to simulate the viscoelastic response of polymeric substances. Two distinct numerical methods are utilized to identify and estimate the parameters of these fractional derivative models. In the first method, model parameters are derived by fitting experimental data to hysteresis loops and their corresponding equations. The second method leverages time-series data, applying the least squares technique to determine the models' parameters and coefficients. Additionally, a data-fitting approach is employed to align the proposed mathematical models with experimental results from nanoindentation tests, ensuring their validation and accuracy. Key outcomes include the extraction of storage and loss moduli: the storage modulus consistently increased with rising dimensionless frequency across all fractional derivative orders. In contrast, the loss modulus initially increased to a dimensionless frequency of one before exhibiting a decreasing trend. Hysteresis loops, representing the energy dissipated per unit volume of material, revealed a reduction in damping with lower fractional derivative orders. Moreover, both methods demonstrated a small relative error when subjected to noise, indicating their robustness and high accuracy in estimating viscoelastic parameters from laboratory data within a narrow range of excitation frequencies.