Mechanics and Physics of Structured Media

Asymptotic and Integral Equations Methods of Leonid Filshtinsky.

1st Edition - January 20, 2022

Editors: Igor Andrianov, Simon Gluzman, Vladimir Mityushev

Paperback ISBN:

9 7 8 - 0 - 3 2 3 - 9 0 5 4 3 - 5

eBook ISBN:

9 7 8 - 0 - 3 2 3 - 9 0 6 5 3 - 1

Mechanics and Physics of Structured Media: Asymptotic and Integral Methods of Leonid Filshtinsky provides unique information on the macroscopic properties of various composite… Read more

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Mechanics and Physics of Structured Media: Asymptotic and Integral Methods of Leonid Filshtinsky provides unique information on the macroscopic properties of various composite materials and the mathematical techniques key to understanding their physical behaviors. The book is centered around the arguably monumental work of Leonid Filshtinsky. His last works provide insight on fracture in electromagnetic-elastic systems alongside approaches for solving problems in mechanics of solid materials. Asymptotic methods, the method of complex potentials, wave mechanics, viscosity of suspensions, conductivity, vibration and buckling of functionally graded plates, and critical phenomena in various random systems are all covered at length.

Other sections cover boundary value problems in fracture mechanics, two-phase model methods for heterogeneous nanomaterials, and the propagation of acoustic, electromagnetic, and elastic waves in a one-dimensional periodic two-component material.

Cover image
Title page
Table of Contents
Copyright
Leonid Anshelovich Filshtinsky
List of contributors
Acknowledgements
Chapter 1: L.A. Filshtinsky's contribution to Applied Mathematics and Mechanics of Solids
Abstract
Acknowledgement
1.1. Introduction
1.2. Double periodic array of circular inclusions. Founders
1.3. Synthesis. Retrospective view from the year 2021
1.4. Filshtinsky's contribution to the theory of magneto-electro-elasticity
1.5. Filshtinsky's contribution to the homogenization theory
1.6. Filshtinsky's contribution to the theory of shells
1.7. Decent and creative endeavor
References
Chapter 2: Cracks in two-dimensional magneto-electro-elastic medium
Abstract
2.1. Introduction
2.2. Boundary-value problems for an unbounded domain
2.3. Integral equations for an unbounded domain
2.4. Asymptotic solution at the ends of cracks
2.5. Stress intensity factors
2.6. Numerical example
2.7. Conclusion
References
Chapter 3: Two-dimensional equations of magneto-electro-elasticity
Abstract
3.1. Introduction
3.2. 2D equations of magneto-electro-elasticity
3.3. Boundary value problem
3.4. Dielectrics
3.5. Circular hole
3.6. MEE equations and homogenization
3.7. Homogenization of 2D composites by decomposition of coupled fields
3.8. Conclusion
References
Chapter 4: Hashin-Shtrikman assemblage of inhomogeneous spheres
Abstract
Acknowledgements
4.1. Introduction
4.2. The classic Hashin-Shtrikman assemblage
4.3. HSA-type structure
4.4. Conclusion
References
Chapter 5: Inverse conductivity problem for spherical particles
Abstract
Acknowledgements
5.1. Introduction
5.2. Modified Dirichlet problem
5.3. Inverse boundary value problem
5.4. Discussion and conclusion
References
Chapter 6: Compatibility conditions: number of independent equations and boundary conditions
Abstract
Acknowledgements
6.1. Introduction
6.2. Governing relations and Southwell's paradox
6.3. System of ninth order
6.4. Counterexamples proposed by Pobedrya and Georgievskii
6.5. Various formulations of the linear theory of elasticity problems in stresses
6.6. Other approximations
6.7. Generalization
6.8. Concluding remarks
Conflict of interest
References
Chapter 7: Critical index for conductivity, elasticity, superconductivity. Results and methods
Abstract
7.1. Introduction
7.2. Critical index in 2D percolation. Root approximants
7.3. 3D Conductivity and elasticity
7.4. Compressibility factor of hard-disks fluids
7.5. Sedimentation coefficient of rigid spheres
7.6. Susceptibility of 2D Ising model
7.7. Susceptibility of three-dimensional Ising model. Root approximants of higher orders
7.8. 3D Superconductivity critical index of random composite
7.9. Effective conductivity of graphene-type composites
7.10. Expansion factor of three-dimensional polymer chain
7.11. Concluding remarks
Appendix 7.A. Failure of the DLog Padé method
Appendix 7.B. Polynomials for the effective conductivity of graphene-type composites with vacancies
References
Chapter 8: Double periodic bianalytic functions
Abstract
8.1. Introduction
8.2. Weierstrass and Natanzon-Filshtinsky functions
8.3. Properties of the generalized Natanzon-Filshtinsky functions
8.4. The function ℘1,2
8.5. Relation between the generalized Natanzon-Filshtinsky and Eisenstein functions
8.6. Double periodic bianalytic functions via the Eisenstein series
8.7. Conclusion
References
Chapter 9: The slowdown of group velocity in periodic waveguides
Abstract
Acknowledgements
9.1. Introduction
9.2. Acoustic waves
9.3. Electromagnetic waves
9.4. Elastic waves
9.5. Discussion
References
Chapter 10: Some aspects of wave propagation in a fluid-loaded membrane
Abstract
Acknowledgement
10.1. Introduction
10.2. Statement of the problem
10.3. Dispersion relation
10.4. Moving load problem
10.5. Subsonic regime
10.6. Supersonic regime
10.7. Concluding remarks
References
Chapter 11: Parametric vibrations of axially compressed functionally graded sandwich plates with a complex plan form
Abstract
11.1. Introduction
11.2. Mathematical problem
11.3. Method of solution
11.4. Numerical results
11.5. Conclusions
Conflict of interest
References
Chapter 12: Application of volume integral equations for numerical calculation of local fields and effective properties of elastic composites
Abstract
12.1. Introduction
12.2. Integral equations for elastic fields in heterogeneous media
12.3. The effective field method
12.4. Numerical solution of the integral equations for the RVE
12.5. Numerical examples and optimal choice of the RVE
12.6. Conclusions
References
Chapter 13: A slipping zone model for a conducting interface crack in a piezoelectric bimaterial
Abstract
13.1. Introduction
13.2. Formulation of the problem
13.3. An interface crack with slipping zones at the crack tips
13.4. Slipping zone length
13.5. The crack faces free from electrodes
13.6. Numerical results and discussion
13.7. Conclusion
References
Chapter 14: Dependence of effective properties upon regular perturbations
Abstract
Acknowledgements
14.1. Introduction
14.2. The geometric setting
14.3. The average longitudinal flow along a periodic array of cylinders
14.4. The effective conductivity of a two-phase periodic composite with ideal contact condition
14.5. The effective conductivity of a two-phase periodic composite with nonideal contact condition
14.6. Proof of Theorem 14.5.2
14.7. Conclusions
References
Chapter 15: Riemann-Hilbert problems with coefficients in compact Lie groups
Abstract
15.1. Introduction
15.2. Recollections on classical Riemann-Hilbert problems
15.3. Generalized Riemann-Hilbert transmission problem
15.4. Lie groups and principal bundles
15.5. Riemann-Hilbert monodromy problem for a compact Lie group
References
Chapter 16: When risks and uncertainties collide: quantum mechanical formulation of mathematical finance for arbitrage markets
Abstract
16.1. Introduction
16.2. Geometric arbitrage theory background
16.3. Asset and market portfolio dynamics as a constrained Lagrangian system
16.4. Asset and market portfolio dynamics as solution of the Schrödinger equation: the quantization of the deterministic constrained Hamiltonian system
16.5. The (numerical) solution of the Schrödinger equation via Feynman integrals
16.6. Conclusion
Appendix 16.A. Generalized derivatives of stochastic processes
References
Chapter 17: Thermodynamics and stability of metallic nano-ensembles
Abstract
17.1. Introduction
17.2. Vacancy-related reduction of the metallic nano-ensemble's TPs
17.3. Increase of the metallic nano-ensemble's TPs due to surface tension
17.4. Balance of the vacancy-related and surface-tension effects
17.5. Conclusions
References
Chapter 18: Comparative analysis of local stresses in unidirectional and cross-reinforced composites
Abstract
18.1. Introduction
18.2. Homogenization method as applied to composite reinforced with systems of fibers
18.3. Numerical analysis of the microscopic stress-strain state of the composite material
18.4. The “anisotropic layers” approach
18.5. The “multicomponent” approach by Panasenko
18.6. Solution to the periodicity cell problem for laminated composite
18.7. The homogenized strength criterion of composite laminae
18.8. Conclusions
References
Chapter 19: Statistical theory of structures with extended defects
Abstract
19.1. Introduction
19.2. Spatial separation of phases
19.3. Statistical operator of mixture
19.4. Quasiequilibrium snapshot picture
19.5. Averaging over phase configurations
19.6. Geometric phase probabilities
19.7. Classical heterophase systems
19.8. Quasiaverages in classical statistics
19.9. Surface free energy
19.10. Crystal with regions of disorder
19.11. System existence and stability
19.12. Conclusion
References
Chapter 20: Effective conductivity of 2D composites and circle packing approximations
Abstract
20.1. Introduction
20.2. General polydispersed structure of disks
20.3. Approximation of hexagonal array of disks
20.4. Checkerboard
20.5. Regular array of triangles
20.6. Discussion and conclusions
References
Chapter 21: Asymptotic homogenization approach applied to Cosserat heterogeneous media
Abstract
Acknowledgements
21.1. Introduction
21.2. Basic equations for micropolar media. Statement of the problem
21.3. Example. Effective properties of heterogeneous periodic Cosserat laminate media
21.4. Numerical results
21.5. Conclusions
References
Appendix A: Finite clusters in composites
Index