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AuthorBakhvalov, N. author
TitleHomogenisation: Averaging Processes in Periodic Media [electronic resource] : Mathematical Problems in the Mechanics of Composite Materials / by N. Bakhvalov, G. Panasenko
ImprintDordrecht : Springer Netherlands, 1989
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Descript XXXVI, 366 p. online resource


'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 nonยญ 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 nonยญ !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 comยญ 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


1. Formulation of Elementary Boundary Value Problems -- ยง1. The Concept of the Classical Formulation of a Boundary Value Problem for Equations with Discontinuous Coefficients -- ยง2. The Concept of Generalized Solution -- ยง3. 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 -- ยง1. Asymptotic expansion. A Formal Asymptotic Solution -- ยง2. Asymptotic Expansion of a Solution of the Equation u = 1 + ?u3 -- ยง3. Asymptotic Expansion of a Solution of the Equation (K(x/?)u?)?= f(x) by the Averaging Method -- ยง4. Generalization of the Averaging Method in the Case of a Piecewise Smooth Coefficient -- ยง5. Averaging the System of Differential Equations -- 3. Averaging Processes in Layered Media -- ยง1. Problem of Small Longitudinal Vibrations of a Rod -- ยง2. Nonstationary Problem of Heat Conduction -- ยง3. Averaging Maxwell Equations -- ยง4. Averaging Equations of a Viscoelastic Medium -- ยง5. Media with Slowly Changing Geometric Characteristics -- ยง6. Heat Transfer Through a System of Screens -- ยง7. Averaging a Nonlinear Problem of the Elasticity Theory in an Inhomogeneous Rod -- ยง8. The System of Equations of Elasticity Theory in a Layered Medium -- ยง9. Considerations Permitting Reduction of Calculations in Constructing Averaged Equations -- ยง10. Nonstationary Nonlinear Problems -- ยง11. Averaging Equations with Rapidly Oscillating Nonperiodic Coefficients -- ยง12. Problems of Plasticity and Dynamics of Viscous Fluid as Described by Functions Depending on Fast Variables -- 4. Averaging Basic Equations of Mathematical Physics -- ยง1. Averaging Stationary Thermal Fields in a Composite -- ยง2. Asymptotic Expansion of Solution of the Stationary Heat Conduction Problem -- ยง3. Stationary Thermal Field in a Porous Medium -- ยง4. Averaging a Stationary System of Equations of Elasticity Theory in Composite and Porous Materials -- ยง5. Nonstationary Systems of Equations of Elasticity and Diffusion Theory -- ยง6. Averaging Nonstationary Nonlinear System of Equations of Elasticity Theory -- ยง7. Averaging Stokes and Navier-Stokes Equations. The Derivation of the Percolation Law for a Porous Medium (Darcyโ{128}{153}s Law) -- ยง8. Averaging in case of Short-Wave Propagation -- ยง9. Averaging the Transition Equation for a Periodic Medium -- ยง10. Eigenvalue Problems -- 5. General Formal Averaging Procedure -- ยง1. Averaging Nonlinear Equations -- ยง2. Averaged Equations of Infinite Order for a Linear Periodic Medium and for the Equation of Moment Theory -- ยง3. 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 -- ยง1. Maintaining the Properties of Convexity and Symmetry of the Minimized Functional in Averaging -- ยง2. On the Principle of Equivalent Homogeneity -- ยง3. The Symmetry Properties of Effective Coefficients and Reduction of Periodic Problems to Boundary Value Problems -- ยง4. Agreement Between Theoretically Predicted Values of Effective Coefficients and Those Determined by an Ideal Experiment -- 7. Composite Materials Containing High-Modulus Reinforcement -- ยง1. The Stationary Field in a Layered Material -- ยง2. Composite Materials with Grains for Reinforcement -- ยง3. Dissipation of Waves in Layered Media -- ยง4. High-Modulus 3D Composite Materials -- ยง5. The Splitting Principle for the Averaged Operator for 3D High-Modulus Composites -- 8. Averaging of Processes in Skeletal Structures -- ยง1. An Example of Averaging a Problem on the Simplest Framework -- ยง2. A Geometric Model of a Framework -- ยง3. The Splitting Principle for the Averaged Operator for a Periodic Framework -- ยง4. The Splitting Principle for the Averaged Operator for Trusses and Thin-walled Structures -- ยง5. On Refining the Splitting Principle for the Averaged Operator -- ยง6 Asymptotic Expansion of a Solution of a Linear Equation in Partial Derivatives for a Rectangular Framework -- ยง7 Skeletal Structures with Random Properties -- 9. Mathematics of Boundary-Layer Theory in Composite Materials -- ยง1. Problem on the Contact of Two Layered Media -- ยง2. The Boundary Layer for an Elliptic Equation Defined on a Half-Plane -- ยง3. The Boundary Layer Near the Interface of Two Periodic Structures -- ยง4. Problem on the Contact of Two Media Divided by a Thin Interlayer -- ยง5. The Boundary Layer for the Nonstationary System of Equations of Elasticity Theory -- ยง6. On the Ultimate Strength of a Composite -- ยง7. Boundary Conditions of Other Types -- ยง8. On the Averaging of Fields in Layer Media with Layers of Composite Materials -- ยง9. The Time Boundary Layer for the Cauchy Parabolic Problem -- Supplement: Existence and Uniqueness Theorems for the Problem on a Cell

Mathematics Partial differential equations Mathematical models Mechanics Mathematics Partial Differential Equations Mathematical Modeling and Industrial Mathematics Mechanics


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