Computational study of wave propagation in materials with microstructure effect: bone application

Abstract

Fractures are common at human bones. So, a callus is formed and the procedureof osteogenesis is initiated. Medical doctors need to have a tool that allowsthem to evaluate the healing procedure without taking X-ray photos every week.Such a variety of tools can be provided by non-destructive inspection techniques.But rst, one has to create a model for predicting phenomena such as size-eectsand in particular dispersive acoustic waves propagation.Before this thesis, there has been made an attempt by (Vavva, 2009), topredict modal wave propagation with Mindlin's Form-II. Herein, for the rsttime there are presented dynamic solutions of this theory.To begin with, the bone is considered to be a dampless homogeneous (ortho)isotropic composite material, with interstitial tissue being the matrix andthe osteons being the bres. So, Mindlin's theory can be applied in this case.Next, a fundamental solution is obtained for Mindlin's Form-II of his gradientelasticity theory. In conjunction to an existing integral representation, there canbe obtained solutions using the Boundary Element Method. With the help of aconsidered Representative Volume Element, simulations have been conducted andresults are presented for the cases of P, S and Rayleigh waves, as well as guidedwaves in plates. The dispersion diagrams as given by Wigner-Ville representationsare compared to the theoretical ones. What is more, the validity and accuracy ofthe BEM code have been checked using analytical solutions of one-dimensionalproblems.Furthermore, relaxation functions from viscoelastic theories are consideredand are taken into account using the correspondence principle. So, both viscoelasticand gradient-visco-elastic models have been considered and the resultsof various cases (P, S, Rayleigh and Lamb waves) have been compared to theabove.Finally, since the present thesis has to do with information extracted fromdispersive wave propagation, some studies have been made and measures havebeen proposed for velocities and dispersion.All in all, this has been a work dealing with the fact that micro-structureaects the macro-behavior of a material concerning waves propagation and, inthe framework of Mindlin's Form-II, there have been extracted several conclusionsconcerning bone-like materials.