An analytical study on guided wave propagation in bone-mimicking plates using Mindlin's Form II gradient elasticity

The dipolar gradient theory of elasticity has been recently very attractive for the ultrasonic characterization of bone since it includes intrinsic parameters that combine microstructural effects with the macrostructural behavior. Nevertheless this is the special case of Mindlin's Form-II strain gradient elastic theory, which includes two intrinsic parameters i.e., one for micro-stiffness and one for micro-inertia. However the general Form II strain gradient theory is characterized by four internal length scale parameters indicating different microstructural effects on longitudinal and transverse waves, which is necessary for realistic modeling of media with microstructural effects. In this work this theory is employed to analytically determine the velocity dispersion curves of guided waves propagating in isotropic bone-like plates. Calculations are performed for various combinations between the internal constants which corresponded to values close to the osteon's size. Comparisons are made with the classical elastic case as well as with corresponding analytical results from the dipolar theory. The results indicate that when microstructural effects are described by different stiffness and inertia intrinsic parameters the microstructural effects are more pronounced. This work could provide supplementary information to elucidate wave guidance mechanisms in bones.