A review of three-dimensional viscoelastic models with an application to viscoelasticity characterization using nanoindentation

This work reviewed different three-dimensional viscoelastic models, including Hooke, Newton, Maxwell, Voigt, Boltzmann, Zener, Tsay, Burgers, Weichert, and Kelvin models. The relaxation moduli and creep compliances are derived and related via the viscoelastic parameters. Physical meanings of viscoelastic parameters are also explained for each model. The formulae of relaxation test, creep test, and dynamic loading test for each viscoelastic models are formulated. Relaxation moduli and creep compliances are drawn for visualizing and comparing. The less discussed time-dependent Poisson’s ratios are also emphasized and compared in this work. All viscoelastic functions can be represented as the relaxation-creep duality representation. The instantaneous and permanent moduli and compliances as well as the fractions of exponential and complementary exponential pairs with different characteristic times can be immediately understood via the relaxation-creep duality representation. The three-dimensional Burgers model is selected to describe the viscoelastic behavior of PMMA with nanoindentation test. The two-step curve fitting method is introduced to fit the P–t curve and h–t curve separately. The fitting results are better than the direct fitting of P–h curve in the literature. The relaxation moduli and creep compliances can then be used to understand the viscoelastic behavior of PMMA.