Homogenizing the acoustics of cancellous bone with an interstitial non-Newtonian fluid

We study the problem of derivation of an effective model of acoustic wave propagation in a two-phase medium composed of a linear KelvinVoight viscoelastic solid and a shear-thinning non-Newtonian fluid. Bone tissue is an important example of such composite materials. The microstructure is modeled as a periodic arrangement of fluid-saturated pores inside the solid matrix. The ratio ε of the macroscopic length scale and the size of the microstructural periodicity cell is a small parameter of the problem. We employ two-scale convergence and some other weak convergence techniques to pass to the limit ε→0 in the nonlinear governing equations. The effective model is a two-velocity system for the effective velocity v̄ and a corrector velocity w. The latter describes the influence of the high-frequency oscillations on the effective wave propagation. The effective constitutive equation provides an explicit dependence of the effective stress on e(v̄)+ey(w).