The flexure and torsion of bones viewed as anisotropic poroelastic bodies

Two-phase poroelastic material is taken as a model of the bone, in the sense that the osseous tissue is considered as a perfectly elastic solid and the fluid substances filling the cavities as a viscous compressible fluid. Biot's theory of consolidation is adopted, assuming interconnections between the cavities. For a transversely isotropic body the theory leads to seven linear constitutive equations, connecting seven stress components with seven deformation components, by means of eight material coefficients. The theory is applied to beam-like structural elements by using the procedure of Michell, consisting of representation of stress and deformation components as rational integral functions of the axial coordinate z.

General equations are derived for two particular cases; (a) when the stress and strain components do not depend on z, (b) when the latter are linear functions of z. The first case corresponds to the action of terminal couples (both flexural and torsional); the second case is associated with the action of terminal forces.

Illustrative examples are solved, involving (a) cylindrical bending of poroelastic plates by terminal couples, (b) pure bending of poroelastic beams of arbitrary cross section having one axis of symmetry, (c) pure torsion of poroelastic beams of arbitrary cross section.

The explicit solution of the case (b) shows that the behavior of the poroelastic material is analogous to that of a three-element visco-elastic model. This agrees qualitatively with the experimental findings of Sedlin for bones.