Applications of Homogenization Theory to the Study of Mineralized Tissue

Introductory Remarks Some Functional Spaces Variational Formulation Geometry of Two Phase Composite Two-scale Convergence Method The Concept of a Homogenized Equation Two-Scale convergence with time dependence Potential and Solenoidal Fields The Homogenization Technique Applied to Soft Tissue Homogenization of Soft Tissue Galerkin approximations Derivation of the effective equation of U0 Acoustics in Porous Media Introduction Diphasic Macroscopic Behavior Well-posedness for problem (3.2.49 and 3.2.55) The slightly compressible di-phasic behavior Wet Ionic, Piezo-electric Bone Introduction Wet bone with ionic interaction Homogenization using Formal Power Series Wet bone without ionic interaction Electrodynamics Visco-elasticity and Contact Friction Between the Phases Kelvin-Voigt Material Rigid Particles in a Visco-elastic Medium Equations of motion and contact conditions Two-scale expansions and formal homogenization Model case I: Linear contract conditions Model case II: Quadratic contract conditions Model case III: Power type contact condition Acoustics in a Random Microstructure Introduction Stochastic Two-scale limits Periodic Approximation Non-Newtonian Interstitial Fluid The Slightly Compressible Polymer. Microscale Problem A Priori Estimates Two-Scale System Description of the effective stress Effective equations Multiscale FEM for the modeling of cancellous bone Concept of the multiscale FEM Microscale: Modeling of the RVE and calculation of the effective material properties Macroscale: Simulation of the ultrasonic test Simplified version of the RVE and comparison with the experimental results Anisotropy of cancellous bone Investigation of the influence of reflection on the attenuation of cancellous bone Determination of the geometry of the RVE for cancellous bone by using the effective complex shear modulus G-convergence and Homogenization of Viscoelastic Flows Introduction Main definitions. Corrector operators for G-convergence A scalar elliptic equation in divergence form Homogenization of two-phase visco-elastic flows with time-varying interface Main theorem and outline of the proof Corrector operators and oscillating test functions Inertial terms in the momentum balance equation Effective deviatoric stress. Proof of the main theorem Fluid-structure interaction Biot Type Models for Bone Mechanics Bone Rigidity Anisotropic Biot Systems The Case of a non-Newtonian Interstitial Fluid Some Time-Dependent Solutions to the Biot System Creation of RVE for Bone Microstructure The RVE Model Reformulation as a Graves-like scheme Absorbring boundary condition-perfectly matched layer Discretized systems Bone Growth and Adaptive Elasticity The Model Scalings of Unknowns Asymptotic Solutions Further Reading