Fractional-order viscoelastic model for tendons with multilevel self-similar structures

Tendons share multilevel self-similar structures from the molecular to macroscopic levels, and their long-range viscoelastic responses have significant biomechanical implications in clinical settings. Existing research has not adequately studied the correlation between the tendon's multilevel structure and its viscoelastic behaviour. This study proposes a fractional-order viscoelastic model for tendons that describes their multilevel self-similar structures. A self-similar spring-dashpot network is abstracted from the multilevel structures of tendons, and the spring-dashpot parameters corresponding to adjoint structural levels are assumed to be described by a power-law relationship. The constitutive equations of the viscoelastic model can be derived using Heaviside's operational calculus, which shows that the long-range viscoelastic response of the tendon can be described by a tempered fractional-order operator and that the fractional order nonlinearly combines the effect of the power-law scaling factor and structural self-similarity. The proposed model is validated to fit the tendon relaxation response described at the macroscopic and micro-nano levels in the literature. The fitting performance is also found to be better than that without consideration of power-law characteristics. In addition, the geometrical definition of the fractal dimension is extended to the proposed fractional-order viscoelastic model, which provides a theoretical foundation for understanding the connections between the multilevel structure of soft biomaterials and their time-dependent biomechanical functions.