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

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 Rectangu