Applying the fractional derivative Zener model to fitting the time‐dependent material viscoelasticity tested by nanoindentation

Abstract

Material viscoelasticity is of particular importance for polymers, biotissues and many others, due to the significant influence not only on the elastic response of structures, but also on their failure mechanism in actual applications. In this study, the experimental tests by nanoindentations are categorized according to the objective parameter combined with the indenter shape into four distinct cases, and the Fractional Derivative Zener model (FDZM) is employed to characterize this time-dependent property. The explicit data-fitting formulas are derived to determine the relaxation modulus and creep compliance tested with the conical and spherical indenters. Besides, a fitting scheme is provided to facilitate the identification of these viscoelastic parameters. In comparison with the general methods adopting the Prony series or empirical models, two silent merits present: (1) fewer model coefficients necessitate being determined based on the experimental data; (2) the relaxation modulus G(t) and the creep compliance J(t) can be obtained through one indentation test, and their exact interconversion is guaranteed as well. Several sets of experimental data are applied to examine the validity of the developed approach, and three more examples are adopted to demonstrate the exact interconversion between G(t) and J(t). A few concluding remarks are drawn eventually.