Applications of Homogenization Theory to the Study of Mineralized Tissue
Applications to the Biological and Physical Sciences
Chapman & Hall/CRC Monographs and Research Notes in Mathematics Series

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

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.