3-D contact problem of a transversely isotropic, transversely homogeneous biphasic cartilage layer

The thin cartilage tissue which covers the diathrodial joints plays a central role in articular mechanics. The highly heterogeneous material by which it is constituted allows the bones to exchange forces with a substantial absence of friction and an advantageous distribution of stresses. The rapid solicitations to which the articulation undergoes are carried, in the short term, mainly by the self-pressurized fluid phase. Such constraint of the interstitial fluid and the shear stresses are instead ascribed to a structurally complicated solid matrix. The latter appears inhomogeneous both in stiffness and permeability. Given the thinness of the cartilage layer with respect to the contact surface dimensions, solutions for the biphasic deformation problem have been traditionally obtained by means of asymptotic analysis. The closed-form analytical solutions for a homogeneous isotropic linear elastic solid matrix was studied by [1, 4], transverse isotropy was introduced later by [2]. Finally the effects of the complex microstructure composed of collagen fibrils and distributed porosity were taken into account through in-depth exponential inhomogeneity in [3]. Following the three-dimensional generalization of the axisymmetric contact problem by [5] which was applied to a homogeneous cartilage layer in [6], we study the integral characteristics for a transversely isotropic, transversely homogenous biphasic layer based on the constitutive equations provided in [3]. The cartilage is firmly attached to a rigid substrate shaped as elliptic paraboloid, the fluid phase is not allowed to flow through the confining surfaces.