1st Edition

Applications of Homogenization Theory to the Study of Mineralized Tissue

    297 Pages 40 B/W Illustrations
    by Chapman & Hall

    297 Pages 40 B/W Illustrations
    by Chapman & Hall

    297 Pages 40 B/W Illustrations
    by Chapman & Hall

    Homogenization is a fairly new, yet deep field of mathematics which is used as a powerful tool for analysis of applied problems which involve multiple scales. Generally, homogenization is utilized as a modeling procedure to describe processes in complex structures.

    Applications of Homogenization Theory to the Study of Mineralized Tissue functions as an introduction to the theory of homogenization. At the same time, the book explains how to apply the theory to various application problems in biology, physics and engineering.

    The authors are experts in the field and collaborated to create this book which is a useful research monograph for applied mathematicians, engineers and geophysicists. As for students and instructors, this book is a well-rounded and comprehensive text on the topic of homogenization for graduate level courses or special mathematics classes.

    Features:

    • Covers applications in both geophysics and biology.
    • Includes recent results not found in classical books on the topic
    • Focuses on evolutionary kinds of problems; there is little overlap with books dealing with variational methods and T-convergence
    • Includes new results where the G-limits have different structures from the initial operators

    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

    Biography

    Robert P. Gilbert is a Unidel Professor of Mathematics at the University of Delaware and has authored numerous papers and articles for journals and conferences. He is also a founding editor of Complex Variables and Applicable Analysis and serves on many editorial boards. His current research involves the area of inverse problems, homogenization and the flow of viscous fluids.

    Ana Vasilic is an associate professor of mathematics at Northern New Mexico College. She has contributed to mathematical publications such as Mathematical and Computer Modeling and Applicable Analysis. Her research interests include applied analysis, partial differential equations, homogenization and multiscale problems in porous media.

    Sandra Klinge is an assistant professor in computational mechanics at Technische Universitat, Dortmund, Germany. She has written several articles for science journals and contributed to many books. Homogenization, modeling of polymers and multiscale modeling are among her research interests.

    Alex Panchenko is a professor of mathematics at Washington State University and has written for several publications, many of which are collaborations with Robert Gilbert. These include journals such as SIAM Journal Math. Analysis and Mathematical and Computer Modeling.

    Klaus Hackl is a professor of mechanics at Ruhr-Universitat Bochum, Germany and has made many scholarly contributions to various journals. His research interests include continuum mechanics, numerical mechanics, modeling of materials and multiscale problems.