A theoretical and experimental study of bone´s microstructural effect on the dispersion of ultrasonic guided waves

Ultrasonic evaluation of bone has been based on the classical linear elastic theory. However, this theory cannot adequately describe bone´s mechanical behavior since the microstructure is neglected. In this study, we use the simplest form of gradient theory (Mindlin FormII) to theoretically derive the velocity dispersion curves for free isotropic bone-mimicking plates and we investigate whether that theory can characterize the modes propagating in real bones better than the Lamb wave theory. Two additional terms are included in the constitutive equations representing the characteristic length in bone: (a) the gradient coefficient g and (b) the micro-inertia term h whose values were at the order of the osteons size. The velocity dispersion curves of guided waves were numerically obtained for four combinations between g and h and were superimposed on the time-frequency representations of the signals obtained from ex-vivo measurements. For the first time it was made feasible to detect fast waves with velocity higher than the S0 mode. Overall the gradient theory seems to be more efficient in mode identification than the classical theory, providing thus better understanding of ex-vivo measurements.