A Non-Linear Viscoelastic Model for the Relaxation of Tendon

Frances M. Davis · Raffaella De Vita
Virginia Tech

Abstract. Tendons exhibit viscoelastic properties that can be quantified by performing stress relaxation experiments. The quasi-linear viscoelastic (QLV) model proposed by Fung [1] has been successfully used to describe the stress relaxation experimental data for tendons. In the QLV model, the rate of stress relaxation is assumed to be independent of strain. However, recent experimental evidence demonstrated that the rate of relaxation is a non-linear function of strain [2] at physiological strains less that 5%. Therefore a more robust model is necessary to describe the viscoelastic behavior of tendons.

A transversely isotropic viscoelastic model for the stress relaxation behavior of tendons was developed within the nonlinear integral representation framework proposed by Pipkin-Rogers [3,4]. This framework allows for the formulation of a stress relaxation response function, which is a non-separable function of strain and time. The model includes six parameters, two of them describe the elastic behavior of tendon and four of them describe the strain dependant viscous behavior of tendon. After assuming an isochoric axisymmetric deformation and traction-free boundary conditions, the resulting equation was curve-fit to experimental data collected by performing uniaxial relaxation tests on rat-tail tendon fascicles. The specimens were preconditioned for five cycles, allowed to recover for 10 minutes, ramped up to a displacement between 0.25 mm and 1.75 mm and held for 10 minutes while measuring load at 20 Hz. The values for the elastic parameters were determined using isochronal stress-strain data. Stress-relaxation data collected at different strain levels were used to find the remaining model parameters. This model represents a departure from the QLV model because it allows the rate of stress relaxation to depend on strain. Future studies will be conducted to extend the proposed model to describe other viscoelastic phenomena such as hysteresis and creep.

  1. Fung, Y.C. (1993) Springer-Verlag, New York.
  2. Hingorani, R.V. et al. (2004) Ann Biomed Eng, 32(2), pp. 306-312.
  3. Pipkin, A.C. and Rogers, T.G. (1968) J Mech Phys Solids, 16(1), pp. 59-72.
  4. Rajagopal, K.R. and Wineman, A. S. (2009) Math Mech Solids, 29(10), pp. 490-501.

Keywords: tendon, relaxation, quasilinear viscoelastic, Pipkin-Rogers integral series.