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E-grāmata: Applications of Homogenization Theory to the Study of Mineralized Tissue

, , , (University of Delaware, Newark, USA), (Washington State University, Pullman, USA)
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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
Preface ix
Chapter 1 Introductory Remarks 1(20)
1.1 Some functional spaces
1(8)
1.1.1 Periodic functions
4(1)
1.1.2 Lax-Milgram Theorem
5(4)
1.2 Variational formulation
9(3)
1.3 Geometry of the two-phase composite
12(2)
1.4 Two-scale convergence method
14(1)
1.5 The concept of a homogenized equation
15(4)
1.6 Two-scale convergence with time dependence
19(1)
1.7 Potential and solenoidal fields
19(2)
Chapter 2 The Homogenization Technique Applied to Soft Tissue 21(12)
2.1 Homogenization of soft tissue
21(3)
2.2 Galerkin approximations
24(5)
2.3 Derivation of the effective equation of u0
29(4)
Chapter 3 Acoustics in Porous Media 33(16)
3.1 Introduction
33(2)
3.2 Diphasic macroscopic behavior
35(7)
3.2.1 Derivation of the effective equations for u0
39(3)
3.3 Well-posedness for problems (3.2.48) and (3.2.55)
42(2)
3.4 The slightly compressible diphasic behavior
44(5)
Chapter 4 Wet Ionic, Piezoelectric Bone 49(16)
4.1 Introduction
49(1)
4.2 Wet bone with ionic interaction
50(7)
4.2.1 Nondimentionalized equations
54(2)
4.2.2 Fluid equations with slight compressibility
56(1)
4.2.3 Nernst-Plank equations
57(1)
4.3 Homogenization using formal power series
57(3)
4.4 Wet bone without ionic interaction
60(3)
4.4.1 Reuss bound on the energy
60(1)
4.4.2 Fluid displacement
61(1)
4.4.3 Kinetic energy
62(1)
4.4.4 Constitutive equations
62(1)
4.5 Electrodynamics
63(2)
4.5.1 Electrically isotropic solid
63(1)
4.5.2 Electromagnetism in the fluid
63(1)
4.5.3 Effective electro-magnetic equations
64(1)
Chapter 5 Viscoelasticity, and Contact Friction between the Phases 65(24)
5.1 Kelvin-Voigt Material
65(8)
5.1.1 Two-scale convergence approach
69(4)
5.2 Rigid particles in a visco-elastic medium
73(1)
5.3 Equations of motion and contact conditions
74(3)
5.3.1 Boundary conditions
74(1)
5.3.2 Approximation of the contact conditions
75(1)
5.3.3 Microscale equations
76(1)
5.4 Two-scale expansions and formal homogenization
77(2)
5.5 Model case I: Linear contact conditions
79(3)
5.5.1 Cell problems
80(1)
5.5.2 Averaged equations for Model I
81(1)
5.6 Model II: Quadratic contact conditions
82(2)
5.6.1 Averaged equation for Model II
83(1)
5.7 Model III: Power type contact condition
84(5)
5.7.1 Contact conditions, ansatz and cell problems
84(1)
5.7.2 The relation between ξ1 and ξ°
85(1)
5.7.3 Effective stress
86(1)
5.7.4 Effective drag force
87(2)
Chapter 6 Acoustics in a Random Microstructure 89(8)
6.1 Introduction
89(2)
6.2 Stochastic two-scale limits
91(2)
6.3 Periodic approximation
93(4)
Chapter 7 Non-Newtonian Interstitial Fluid 97(20)
7.1 The slightly compressible polymer: Microscale problem
97(2)
7.2 A priori estimates
99(8)
7.3 Two-scale system
107(1)
7.4 Description of the effective stress
108(6)
7.5 Effective equations
114(3)
Chapter 8 Multi scale FEM for the Modeling of Cancellous Bone 117(46)
8.1 Concept of the multiscale FEM
117(2)
8.2 Microscale: The RVE proposal and effective properties
119(11)
8.2.1 Modeling of the RVE for cancellous bone
119(2)
8.2.2 Modeling of the solid phase
121(3)
8.2.3 Modeling of the fluid phase
124(1)
8.2.4 Summary of the equations defining the BVP on the microlevel
125(1)
8.2.5 Effective elasticity tensor: Output from the microscale
126(1)
8.2.6 Effective material parameters
127(2)
8.2.7 Analysis of the dry skeleton
129(1)
8.3 Macroscale: Simulation of the ultrasonic test
130(6)
8.3.1 Ultrasonic attenuation test
130(1)
8.3.2 FEM model of the ultrasonic test
131(1)
8.3.3 Test example
132(1)
8.3.4 Wave attenuation
133(3)
8.4 Simplified version of the RVE
136(5)
8.4.1 RVE II: Solid phase consisting of thin columns
136(1)
8.4.2 Numerical values of the effective material parameters
136(5)
8.5 Anisotropy of cancellous bone
141(3)
8.6 The influence of reflection on the attenuation
144(10)
8.6.1 Principles of the reflection phenomenon
144(2)
8.6.2 Variational formulation for the wave propagation taking the effects of the reflection into account
146(2)
8.6.3 Numerical implementation
148(3)
8.6.4 Numerical results
151(3)
8.7 Multiscale inverse analysis
154(9)
8.7.1 Definition of the merit function
154(1)
8.7.2 The Levenberg-Marquardt method
155(2)
8.7.3 Numerical examples
157(6)
Chapter 9 G-convergence and Homogenization of Viscoelastic Flows 163(32)
9.1 Introduction
163(1)
9.2 Main definitions. Corrector operators for G-convergence
164(1)
9.3 A scalar elliptic equation in divergence form
165(3)
9.4 Two-phase visco-elastic flows with time-varying interface
168(4)
9.4.1 Introduction
168(1)
9.4.2 Equations of balance and constitutive equations
168(3)
9.4.2.1 Choice of a model
168(2)
9.4.2.2 Weak formulation of the micro-scale problem
170(1)
9.4.3 Finite energy weak solutions and bounds
171(1)
9.5 Main theorem and outline of the proof
172(2)
9.6 Corrector operators and oscillating test functions
174(8)
9.6.1 Auxiliary problem for mpq,ξ
175(5)
9.6.2 Auxiliary problems for npq,ξmTpq,ξ
180(2)
9.7 Inertial terms in the momentum balance equation
182(5)
9.8 Effective deviatoric stress. Proof of the main theorem
187(4)
9.9 Fluid-structure interaction
191(4)
Chapter 10 Biot-Type, Models for Bone Mechanics 195(34)
10.1 Bone rigidity
195(8)
10.1.1 The isotropic, Biot model
196(3)
10.1.2 The direct problem
199(4)
10.2 Anisotropic Biot systems
203(8)
10.2.1 Carcione representation
203(3)
10.2.2 Elimination of the fluid displacements
206(3)
10.2.3 Existence theorem for the anisotropic Biot model
209(2)
10.3 The case of a non-Newtonian, interstitial fluid
211(1)
10.4 Some time-dependent solutions to the Biot system
212(17)
10.4.1 The nonlocal boundary value problem
216(3)
10.4.2 Variational formulation
219(2)
10.4.3 Existence and uniqueness
221(4)
10.4.4 The time domain
225(4)
Chapter 11 Creation of RVE for Bone Microstructure 229(14)
11.1 The RVE model
229(1)
11.2 Reformulation as a Graves-like scheme
230(3)
11.3 Absorbing boundary condition: A perfectly matched layer
233(2)
11.4 Discretized systems
235(8)
11.4.1 Orthotropic random bone
239(4)
Chapter 12 Bone Growth and Adaptive Elasticity 243(12)
12.1 The model
243(2)
12.2 Scalings of unknowns
245(1)
12.3 Asymptotic solutions
246(9)
Appendix 255(4)
A.1 Moving interface in the inertial terms and frozen interface in the constitutive equations
255(1)
A.2 Existence of weak solutions, outline of the proof
256(3)
Bibliography 259(22)
Index 281
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.



 



 









 
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