Strain state dependent anisotropic viscoelasticity of tendon-to-bone insertion

Tendon-to-bone insertion tissues may be considered as functionally-graded connective tissues, providing a gradual transition from soft tendon to hard bone tissue, and functioning to alleviate stress concentrations at the junction of these tissues. The tendon-to-bone insertion tissues demonstrate pronounced viscoelastic behavior, like many other biological tissues, and are designed by the nature to alleviate stress at physiological load rates and strains states. In this paper we present experimental data showing that under biaxial tension tendon-to-bone insertion demonstrates rate-dependent behavior and that stress-strain curves for the in-plane components of stress and strain become less steep when strain rate is increased, contrary to a typical viscoelastic behavior, where the opposite trend is observed. Such behavior may indicate the existence of a protective viscoelastic mechanism reducing stress and strain during a sudden increase in mechanical loading, known to exist in some biological tissues. The main purpose of the paper is to show that such viscoelastic stress reduction indeed possible and is thermodynamically consistent. We, therefore, propose an anisotropic viscoelasticity model for finite strain. We identify the range of parameters for this model which yield negative viscoelastic contribution into in-plane stress under biaxial state of strain and simultaneously satisfy requirements of thermodynamics. We also find optimal parameters maximizing the observed protective viscoelastic effect for this particular state of strain. This model will be useful for testing and describing viscoelastic materials and for developing interfaces for dissimilar materials, considering rate effect and multiaxial loadings.