Biot's Consolidation Equations Research Articles

On the fundamental fluid transport mechanisms through normal and pathological articular cartilage during function—II. The analysis, solution and conclusions

The articulating process of synovial joints is represented by a spatially fixed cyclically time varying normal surface traction applied over a layered model of the cartilage-subchondral bone system. A two-phase mechanically interacting mixture is used to represent the solid matrix and the interstitial fluid of the cartilage continuum. The resulting equations of motion may be reduced to the currently employed physical laws used to describe the transport of fluid through the tissue, i.e. Darcy's law and Biot's consolidation equations. These equations were simplified by an order of magnitude analysis, and the resulting equations were solved by a double Fourier-Laplace transform procedure. The analytical solution shows that the fluid transport mechanism is strongly dependent upon a nondimensional parameter defined by ϵ 12 = Nk h 2 ω . This parameter is the ratio of the force required to deform the tissue as a whole to the force of frictional resistance due to the rate of movement of the interstitial fluid relative to the solid matrix. It is found that during normal function of healthy articular cartilage the consolidation effects will dominate the movement of interstitial fluid. In degenerative cartilage, as characterized by increasing the surface porosity and permeability and a decrease in tissue stiffness, the direct pressure effects, i.e. Darcy's law, becomes comparable to those of consolidation.

On the fundamental fluid transport mechanisms through normal and pathological articular cartilage during function—I the formulation

The fundamental fluid transport mechanisms associated with articular cartilage are important toward the understanding of the biomechanical processes involved in the physiology of normal and pathological synovial joints. Phenomenologically, articular cartilage is viewed as a mixture of two mechanically interacting continua, i.e. a solid matrix phase and a liquid phase composed of water. In synovial joints the liquid phase may be transported through the solid matrix by a direct pressure gradient as a result of the squeeze film action of synovial fluid during articulation and as a result of the consolidation of the solid matrix. The mechanically interacting mixture is composed of a solid. defined by an elastic internal energy function, and an incompressible liquid. The accompanying frictional resistance of relative motion is considered by a linear diffusive dissipation term. The equations of motion for each phase and the total mixture were derived from the extended Hamilton's Principle, where the Rayleigh dissipative resistance is considered as a generalized body force field. This procedure yields, as special cases, the classical Darcy's Law for the liquid transport due to direct pressure gradients, as well as Biot's consolidation equations for the liquid transport due to the dilatation of the solid phase.