Analytical characterization of viscoelasticity in Zener and generalized Zener models under typical loading conditions

This study presents a comprehensive analytical investigation of linear viscoelastic behavior based on the classical Zener model and its generalized extension. Closed-form expressions are derived for key mechanical quantities, such as stress, tangent stiffness, hysteresis area, complex modulus, relaxation modulus, and creep modulus, under typical loading conditions, including constant strain rate, sinusoidal cyclic loading, stress relaxation, and creep. Particular attention is given to energy-dissipation phenomena, with formulations that elucidate the relationships between measurable mechanical responses and the underlying constitutive parameters. Notably, under cyclic loading, the analysis reveals linear correlations between the relaxation time and the time period corresponding to maximum energy dissipation. A linear extrapolation technique is also proposed to identify viscous parameters from stress-relaxation or creep data. These findings provide a practical reference for interpreting experimental results and support parameter estimation through analytical or computational approaches. Overall, this work offers a structured and interpretable framework for viscoelastic modeling, complementing numerical methods and enhancing the physical understanding of time-dependent behavior in polymers and time-dependent materials.