Shear Waves Dispersion in Cylindrically Structured Cancellous Viscoelastic Bones

In this chapter we study anti-plane shear waves propagating through a cylindrically structured cancellous bone represented by a two-dimensional mesh of elastic trabeculae filled by a viscous marrow. In the long-wave limit, the original heterogeneous medium can be approximately substituted by a homogeneous one characterized by an effective complex shear modulus. The effect of dispersion is caused by the transmission of mechanical energy to heat due to the viscosity of the marrow (viscoelastic damping). We derive an approximate analytical solution using the asymptotic homogenization method; the cell problem is solved by means of a boundary shape perturbation and a lubrication theory approaches. For short waves, when the wavelength is comparable to the trabeculae size, the effect of dispersion is caused by successive reflections and refractions of local waves at the trabecula-marrow interfaces (Bloch dispersion). Decrease in the wavelength reveals a sequence of pass and stop frequency bands, so the heterogeneous bone can act like a discrete wave filter.