Theoretical study of bone´s microstructural effects on Rayleigh wave propagation

The linear theory of classical elasticity cannot effectively describe bone´s mechanical behavior since only homogeneous media and local stresses are assumed. Additionally, it cannot predict the dispersive nature of Rayleigh wave which has been experimental observed. By adopting Mindlin Form II gradient elastic theory and performing Boundary Element (BEM) simulations we also recently demonstrated Rayleigh dispersion. In this work we use this theory to analytically determine the dispersion of Rayleigh wave. We assume an isotropic semi-infinite space with mechanical properties equal to those of bone and microstructure and microstructural effects. Calculations are performed for various combinations between the internal constants l₁, l₂, h₁, h₂ which corresponded to a) values from closed form relations derived from a realistic model and b) values close to the osteon´s size. Comparisons are made with the corresponding computational results as well as with the classical elastic case. The agreement between the computational and the analytical results was perfect demonstrating the effectiveness of Mindlin´s Form II gradient theory of elasticity to predict the dispersive nature of Rayleigh wave. This study could be regarded as a step towards the ultrasonic characterization of bone.