Applications of Homogenization Theory to the Study of Mineralized Tissue

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.

• Introductory RemarksSome Functional SpacesVariational FormulationGeometry of Two Phase CompositeTwo-scale Convergence MethodThe Concept of a Homogenized EquationTwo-Scale convergence with time dependencePotential and Solenoidal FieldsThe Homogenization Technique Applied to Soft TissueHomogenization of Soft TissueGalerkin approximationsDerivation of the effective equation of U0Acoustics in Porous MediaIntroductionDiphasic Macroscopic BehaviorWell-posedness for problem (3.2.49 and 3.2.55)The slightly compressible di-phasic behaviorWet Ionic, Piezo-electric BoneIntroductionWet bone with ionic interactionHomogenization using Formal Power SeriesWet bone without ionic interactionElectrodynamicsVisco-elasticity and Contact Friction Between the PhasesKelvin-Voigt MaterialRigid Particles in a Visco-elastic MediumEquations of motion and contact conditionsTwo-scale expansions and formal homogenizationModel case I: Linear contract conditionsModel case II: Quadratic contract conditionsModel case III: Power type contact conditionAcoustics in a Random MicrostructureIntroductionStochastic Two-scale limitsPeriodic ApproximationNon-Newtonian Interstitial FluidThe Slightly Compressible Polymer. Microscale ProblemA Priori EstimatesTwo-Scale SystemDescription of the effective stressEffective equationsMultiscale FEM for the modeling of cancellous boneConcept of the multiscale FEMMicroscale: Modeling of the RVE and calculation of the effective material propertiesMacroscale: Simulation of the ultrasonic testSimplified version of the RVE and comparison with the experimental resultsAnisotropy of cancellous boneInvestigation of the influence of reflection on the attenuation of cancellous boneDetermination of the geometry of the RVE for cancellous bone by using the effective complex shear modulusG-convergence and Homogenization of Viscoelastic FlowsIntroductionMain definitions. Corrector operators for G-convergenceA scalar elliptic equation in divergence formHomogenization of two-phase visco-elastic flows with time-varying interfaceMain theorem and outline of the proofCorrector operators and oscillating test functionsInertial terms in the momentum balance equationEffective deviatoric stress. Proof of the main theoremFluid-structure interactionBiot Type Models for Bone MechanicsBone RigidityAnisotropic Biot SystemsThe Case of a non-Newtonian Interstitial FluidSome Time-Dependent Solutions to the Biot SystemCreation of RVE for Bone MicrostructureThe RVE ModelReformulation as a Graves-like schemeAbsorbring boundary condition-perfectly matched layerDiscretized systemsBone Growth and Adaptive ElasticityThe ModelScalings of UnknownsAsymptotic SolutionsFurther Reading