Homogenisation: Averaging Processes in Periodic Media: Mathematical Problems in the Mechanics of Composite Materials by N.S. BakhvalovHomogenisation: Averaging Processes in Periodic Media: Mathematical Problems in the Mechanics of Composite Materials by N.S. Bakhvalov

Homogenisation: Averaging Processes in Periodic Media: Mathematical Problems in the Mechanics of…

byN.S. Bakhvalov, G. Panasenko

Paperback | September 27, 2011

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'Et moi, .... si j'avait su comment en revenir, One service mathematics has rendered the je n'y semis point all,,: human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non­ The series is divergent: therefore we may be sense'. able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non­ !inearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com­ puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.
Title:Homogenisation: Averaging Processes in Periodic Media: Mathematical Problems in the Mechanics of…Format:PaperbackPublished:September 27, 2011Publisher:Springer NetherlandsLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:9401075069

ISBN - 13:9789401075060

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Table of Contents

1. Formulation of Elementary Boundary Value Problems.- §1. The Concept of the Classical Formulation of a Boundary Value Problem for Equations with Discontinuous Coefficients.- §2. The Concept of Generalized Solution.- §3. Generalized Formulations of Problems for the Basic Equations of Mathematical Physics.- 2. The Concept of Asymptotic Expansion. A Model Example to Illustrate the Averaging Method.- §1. Asymptotic expansion. A Formal Asymptotic Solution.- §2. Asymptotic Expansion of a Solution of the Equation u = 1 + ?u3.- §3. Asymptotic Expansion of a Solution of the Equation (K(x/?)u?)?= f(x) by the Averaging Method.- §4. Generalization of the Averaging Method in the Case of a Piecewise Smooth Coefficient.- §5. Averaging the System of Differential Equations.- 3. Averaging Processes in Layered Media.- §1. Problem of Small Longitudinal Vibrations of a Rod.- §2. Nonstationary Problem of Heat Conduction.- §3. Averaging Maxwell Equations.- §4. Averaging Equations of a Viscoelastic Medium.- §5. Media with Slowly Changing Geometric Characteristics.- §6. Heat Transfer Through a System of Screens.- §7. Averaging a Nonlinear Problem of the Elasticity Theory in an Inhomogeneous Rod.- §8. The System of Equations of Elasticity Theory in a Layered Medium.- §9. Considerations Permitting Reduction of Calculations in Constructing Averaged Equations.- §10. Nonstationary Nonlinear Problems.- §11. Averaging Equations with Rapidly Oscillating Nonperiodic Coefficients.- §12. Problems of Plasticity and Dynamics of Viscous Fluid as Described by Functions Depending on Fast Variables.- 4. Averaging Basic Equations of Mathematical Physics.- §1. Averaging Stationary Thermal Fields in a Composite.- §2. Asymptotic Expansion of Solution of the Stationary Heat Conduction Problem.- §3. Stationary Thermal Field in a Porous Medium.- §4. Averaging a Stationary System of Equations of Elasticity Theory in Composite and Porous Materials.- §5. Nonstationary Systems of Equations of Elasticity and Diffusion Theory.- §6. Averaging Nonstationary Nonlinear System of Equations of Elasticity Theory.- §7. Averaging Stokes and Navier-Stokes Equations. The Derivation of the Percolation Law for a Porous Medium (Darcy's Law).- §8. Averaging in case of Short-Wave Propagation.- §9. Averaging the Transition Equation for a Periodic Medium.- §10. Eigenvalue Problems.- 5. General Formal Averaging Procedure.- §1. Averaging Nonlinear Equations.- §2. Averaged Equations of Infinite Order for a Linear Periodic Medium and for the Equation of Moment Theory.- §3. A Method of Describing Multi-Dimensional Periodic Media that does not Involve Separating Fast and Slow Variables.- 6. Properties of Effective Coefficients. Relationship Among Local and Averaged Characteristics of a Solution.- §1. Maintaining the Properties of Convexity and Symmetry of the Minimized Functional in Averaging.- §2. On the Principle of Equivalent Homogeneity.- §3. The Symmetry Properties of Effective Coefficients and Reduction of Periodic Problems to Boundary Value Problems.- §4. Agreement Between Theoretically Predicted Values of Effective Coefficients and Those Determined by an Ideal Experiment.- 7. Composite Materials Containing High-Modulus Reinforcement.- §1. The Stationary Field in a Layered Material.- §2. Composite Materials with Grains for Reinforcement.- §3. Dissipation of Waves in Layered Media.- §4. High-Modulus 3D Composite Materials.- §5. The Splitting Principle for the Averaged Operator for 3D High-Modulus Composites.- 8. Averaging of Processes in Skeletal Structures.- §1. An Example of Averaging a Problem on the Simplest Framework.- §2. A Geometric Model of a Framework.- §3. The Splitting Principle for the Averaged Operator for a Periodic Framework.- §4. The Splitting Principle for the Averaged Operator for Trusses and Thin-walled Structures.- §5. On Refining the Splitting Principle for the Averaged Operator.- §6 Asymptotic Expansion of a Solution of a Linear Equation in Partial Derivatives for a Rectangular Framework.- §7 Skeletal Structures with Random Properties.- 9. Mathematics of Boundary-Layer Theory in Composite Materials.- §1. Problem on the Contact of Two Layered Media.- §2. The Boundary Layer for an Elliptic Equation Defined on a Half-Plane.- §3. The Boundary Layer Near the Interface of Two Periodic Structures.- §4. Problem on the Contact of Two Media Divided by a Thin Interlayer.- §5. The Boundary Layer for the Nonstationary System of Equations of Elasticity Theory.- §6. On the Ultimate Strength of a Composite.- §7. Boundary Conditions of Other Types.- §8. On the Averaging of Fields in Layer Media with Layers of Composite Materials.- §9. The Time Boundary Layer for the Cauchy Parabolic Problem.- Supplement: Existence and Uniqueness Theorems for the Problem on a Cell.