An anisotropic visco-hyperelastic model for ligaments at finite strains. Formulation and computational aspects

In this paper we present a fully three-dimensional finite strain anisotropic visco-hyperelastic model for ligaments and tendons. The structural model is formulated within the framework of non-linear continuum mechanics and is well-suited for its finite element implementation. This model 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 linear viscous models. Also, we use a local multiplicative decomposition of the deformation gradient into volume-preserving and dilatational parts that permits to model the incompressible properties of soft biological tissues. To simulate the viscoelastic properties of this kind of tissues, we consider different viscoelastic behaviours for the matrix and the different families of fibers. A second-order accurate numerical integration procedure is used, established entirely in the reference configuration. Expressions for the stress and elasticity tensors in the spatial description are also presented. Of all soft tissues, we have focused in ligaments due to the importance of their viscoelastic properties in the clinical practise. In order to show clearly the performance of the constitutive model, we present 3D simulations of the behaviour of the anterior cruciate ligament and patellar tendon graft. The model was also tested for various multi-axial loading situations. The relaxation and creep responses and the strain rate dependent behaviour of anterior cruciate ligament and patellar tendon graft were accurately predicted.