On modelling nonlinear viscoelastic effects in ligaments

Experiments in human ligaments revealed that the rate of stress relaxation in such materials is strain dependent. This nonlinear behavior requires therefore a modified description of the standard quasilinear viscoelasticity theory commonly used in tissue biomechanics. The goal of this study is to characterize and demonstrate the importance of the nonlinear stress-relaxation behavior of ligaments undergoing finite deformation. The structural model presented herein is based on a local additive decomposition of the stress tensor into initial and non-equilibrium parts as resulted from the assumed structure of the free energy density function that generalizes Kelvin-Voigt nonlinear viscous models. We consider different viscoelastic behavior for the matrix and the fibers and the need of considering the strain dependency of this effect is clearly demonstrated. Model parameters were fit to experimental data obtained in specimens undergoing finite deformation in two directions: longitudinal and transversal with respect to the directions of the collagen fibers. The model was then tested against several multi-axial loading situations. The strain dependent relaxation and the strain rate dependent behavior of the human medial collateral ligament were accurately predicted.