Continuum description of the Poisson׳s ratio of ligament and tendon under finite deformation

Ligaments and tendons undergo volume loss when stretched along the primary fiber axis, which is evident by the large, strain-dependent Poisson׳s ratios measured during quasi-static tensile tests. Continuum constitutive models that have been used to describe ligament material behavior generally assume incompressibility, which does not reflect the volumetric material behavior seen experimentally. We developed a strain energy equation that describes large, strain dependent Poisson׳s ratios and nonlinear, transversely isotropic behavior using a novel method to numerically enforce the desired volumetric behavior. The Cauchy stress and spatial elasticity tensors for this strain energy equation were derived and implemented in the FEBio finite element software (www.febio.org). As part of this objective, we derived the Cauchy stress and spatial elasticity tensors for a compressible transversely isotropic material, which to our knowledge have not appeared previously in the literature. Elastic simulations demonstrated that the model predicted the nonlinear, upwardly concave uniaxial stress-strain behavior while also predicting a strain-dependent Poisson׳s ratio. Biphasic simulations of stress relaxation predicted a large outward fluid flux and substantial relaxation of the peak stress. Thus, the results of this study demonstrate that the viscoelastic behavior of ligaments and tendons can be predicted by modeling fluid movement when combined with a large Poisson׳s ratio. Further, the constitutive framework provides the means for accurate simulations of ligament volumetric material behavior without the need to resort to micromechanical or homogenization methods, thus facilitating its use in large scale, whole joint models.