Mechanics of articular cartilage : derivation and numerical implementation of a viscoelastic model with statistical fibre distribution

Articular cartilage is a multiphasic biological material characterized by complex mechanical properties. The solid phase is composed of a soft matrix reinforced by a three-dimensional network of collagen fibres and is particularly challenging to model as it exhibits non-linear anisotropic inhomogeneous viscoelastic response. In this work, a constitutive relation that accounts for these features was developed, while the fluid phase and related phenomena (i.e. poroelasticity) were disregarded. The model was developed in the framework of finite elasticity, as articular cartilage typically exhibits large deformations under physiological load. The elastic strain energy potential was defined as the superposition of an isotropic term associated with the ground matrix and an anisotropic term related to the collagen reinforcement. The fibre network was modelled in statistical terms using a probability density function from the literature, that provides at each point the probability of finding a fibre oriented in a given direction. The information about a single direction was enclosed into a so-called structure tensor, which was then integrated over the unit sphere to get its directional average, weighted by the probability distribution. Collagen fibres carry the load only when they are stretched. Hence, the contribution of compressed fibres to the average structure tensor was suitably modified to include the information about their distribution without influencing the strain energy potential. The intrinsic viscoelastic response was modelled adding a quasi-linear viscoelastic formulation for the collagen fibres, while the matrix was assumed to be elastic. This approach generates an overall non-linear viscoelastic response for the tissue, which is observed in experimental tests. The constitutive equations were linearised to derive the corresponding elasticity tensor and the model was implemented into a finite element analysis program that allows the definition of custom user subroutines. The averaging integrals where computed using the method of spherical designs, in which the result is approximated by a suitable sum over the unit sphere. The evolution equations were integrated using an efficient numerical scheme from the literature. In the last part of the work three representative numerical examples, based on published experimental data, where analysed with the aim of checking the physical response of the model. The results of these tests indicate that the model is capable of predicting the response of articular cartilage under tensile load both in terms of local stress and deformation and in terms of viscoelastic behaviour. Moreover, they suggest that the model is also able to account for the fibre contribution in compression, whereas at the current state it is not suitable for the analysis of the time-dependent response of the tissue in such configuration due to the important role of fluid-related viscoelasticity, not included in the model. The main outcomes of this work are the definition of a very general constitutive model, that can serve as a framework for further enhancements, and in particular the innovative definition of the contribution of compressed fibres to the structure tensor.