Mathematical modeling of linear viscoelastic impact: Application to drop impact testing of articular cartilage

In recent years, a number of experimental studies have been conducted to investigate the mechanical behavior and damage mechanisms of articular cartilage under impact loading. Some experimentally observed results have been explained using a non-linear viscoelastic impact model. At the same time, there is the need of simple mathematical models, which allow comparing experimental results obtained in drop impact testing with impact loads of different weights and incident velocities. The objective of this study was to investigate theoretically whether the main features of articular impact could be qualitatively predicted using a linear viscoelastic theory or the linear biphasic theory. Based on the short-time asymptotic solution of the contact problem for a thin biphasic layer [Ateshian GA, Lai WM, Zhu WB, Mow VC. An asymptotic solution for the contact of two biphasic cartilage layers. Journal of Biomechanics 1994;27(11):1347–1360.], an asymptotic model for blunt impact of articular cartilage is developed, which is mathematically equivalent to the Maxwell impact model. In the present paper, exact analytical solutions are obtained for the main parameters of the Kelvin–Voigt and Maxwell impact models. Perturbation analysis of the impact process according to the standard viscoelastic solid model is performed. Asymptotic solutions are obtained for the drop weight impact test. The dependence of the coefficient of restitution on the impactor parameters has been studied in detail.