Mechanics and Physics of Structured Media: Asymptotic and Integral Equations Methods of Leonid Filshtinsky.

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Mechanics and Physics of Structured Media
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Contents
List of contributors
Acknowledgements
1 L.A. Filshtinsky\'s contribution to Applied Mathematics and Mechanics of Solids
	1.1 Introduction
		1.1.1 Personality and career
		1.1.2 Lessons of collaboration (V. Mityushev)
		1.1.3 Filshtinsky\'s contribution to the theory of integral equations
	1.2 Double periodic array of circular inclusions. Founders
		1.2.1 Preliminaries
		1.2.2 Contribution by Eisenstein
		1.2.3 Contribution by Rayleigh
		1.2.4 Contribution by Natanzon
		1.2.5 Contribution by Filshtinsky
	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
	Acknowledgment
	References
2 Cracks in two-dimensional magneto-electro-elastic medium
	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
		A crack in MME plane
	2.6 Numerical example
	2.7 Conclusion
	References
3 Two-dimensional equations of magneto-electro-elasticity
	3.1 Introduction
	3.2 2D equations of magneto-electro-elasticity
		3.2.1 Linear equations of magneto-electro-elasticity and potentials
		3.2.2 Complex representation of field values
	3.3 Boundary value problem
	3.4 Dielectrics
	3.5 Circular hole
		Numerical example
	3.6 MEE equations and homogenization
	3.7 Homogenization of 2D composites by decomposition of coupled fields
		3.7.1 Straley-Milgrom decomposition
		3.7.2 Rylko decomposition
		3.7.3 Example
	3.8 Conclusion
	References
4 Hashin-Shtrikman assemblage of inhomogeneous spheres
	4.1 Introduction
	4.2 The classic Hashin-Shtrikman assemblage
	4.3 HSA-type structure
	4.4 Conclusion
	Acknowledgments
	References
5 Inverse conductivity problem for spherical particles
	5.1 Introduction
	5.2 Modified Dirichlet problem
		5.2.1 Reduction to functional equations
		5.2.2 Explicit asymptotic formulas
	5.3 Inverse boundary value problem
	5.4 Discussion and conclusion
	Acknowledgments
	References
6 Compatibility conditions: number of independent equations and boundary conditions
	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
	Acknowledgments
	References
7 Critical index for conductivity, elasticity, superconductivity. Results and methods
	7.1 Introduction
	7.2 Critical index in 2D percolation. Root approximants
		7.2.1 Minimal difference condition according to original
		7.2.2 Iterated roots. Conditions imposed on thresholds
		7.2.3 Conditions imposed on the critical index
		7.2.4 Conditions imposed on amplitudes
		7.2.5 Minimal derivative (sensitivity) condition
	7.3 3D Conductivity and elasticity
		7.3.1 3D elasticity, or high-frequency viscosity
	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
		7.7.1 Comment on unbiased estimates. Iterated roots
	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
	7.A Failure of the DLog Padé method
	7.B Polynomials for the effective conductivity of graphene-type composites with vacancies
	References
8 Double periodic bianalytic functions
	8.1 Introduction
	8.2 Weierstrass and Natanzon-Filshtinsky functions
	8.3 Properties of the generalized Natanzon-Filshtinsky functions
	8.4 The function p1,2
	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
9 The slowdown of group velocity in periodic waveguides
	9.1 Introduction
	9.2 Acoustic waves
		9.2.1 Equal impedances
		9.2.2 Small scatterers
		9.2.3 Highly mismatched impedances
	9.3 Electromagnetic waves
	9.4 Elastic waves
	9.5 Discussion
	Acknowledgments
	References
10 Some aspects of wave propagation in a fluid-loaded membrane
	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
	Acknowledgment
	References
11 Parametric vibrations of axially compressed functionally graded sandwich plates with a complex plan form
	11.1 Introduction
	11.2 Mathematical problem
	11.3 Method of solution
	11.4 Numerical results
	11.5 Conclusions
	Conflict of interest
	References
12 Application of volume integral equations for numerical calculation of local fields and effective properties of elastic composites
	12.1 Introduction
	12.2 Integral equations for elastic fields in heterogeneous media
		12.2.1 Heterogeneous inclusions in a homogeneous host medium
		12.2.2 Cracks in homogeneous elastic media
		12.2.3 Medium with cracks and inclusions
	12.3 The effective field method
		12.3.1 The effective external field acting on a representative volume element
		12.3.2 The effective compliance tensor of heterogeneous media
	12.4 Numerical solution of the integral equations for the RVE
	12.5 Numerical examples and optimal choice of the RVE
		12.5.1 Periodic system of penny-shaped cracks of the same orientation
		12.5.2 Periodic system of rigid spherical inclusions
	12.6 Conclusions
	References
13 A slipping zone model for a conducting interface crack in a piezoelectric bimaterial
	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
14 Dependence of effective properties upon regular perturbations
	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
		14.6.1 Preliminaries
		14.6.2 An integral equation formulation of problem (14.7)
		14.6.3 Analyticity of the solution of the integral equation
		14.6.4 Analyticity of the effective conductivity
	14.7 Conclusions
	Acknowledgments
	References
15 Riemann-Hilbert problems with coefficients in compact Lie groups
	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
16 When risks and uncertainties collide: quantum mechanical formulation of mathematical finance for arbitrage markets
	16.1 Introduction
	16.2 Geometric arbitrage theory background
		16.2.1 The classical market model
		16.2.2 Geometric reformulation of the market model: primitives
		16.2.3 Geometric reformulation of the market model: portfolios
		16.2.4 Arbitrage theory in a differential geometric framework
			16.2.4.1 Market model as principal fiber bundle
			16.2.4.2 Stochastic parallel transport
			16.2.4.3 Nelson D weak differentiable market model
			16.2.4.4 Arbitrage as curvature
	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
		16.5.1 From the stochastic Euler-Lagrangian equations to Schrödinger\'s equation: Nelson\'s method
		16.5.2 Solution to Schrödinger\'s equation via Feynman\'s path integral
		16.5.3 Application to geometric arbitrage theory
	16.6 Conclusion
	16.A Generalized derivatives of stochastic processes
	References
17 Thermodynamics and stability of metallic nano-ensembles
	17.1 Introduction
		17.1.1 Nano-substance: inception
		17.1.2 Nano-substance: thermodynamics basics
		17.1.3 Nano-substance: kinetics basics
	17.2 Vacancy-related reduction of the metallic nano-ensemble\'s TPs
		17.2.1 Solution in quadrature of the problem of vacancy-related reduction of TPs
		17.2.2 Particle distributions on their radii
		17.2.3 Derivation of equations for TPs reduction
			17.2.3.1 Even distribution of particles on their radii
			17.2.3.2 Linear distributions
			17.2.3.3 Exponential distribution
			17.2.3.4 The normal (truncated) distribution
		17.2.4 Reduction of TPs: results
	17.3 Increase of the metallic nano-ensemble\'s TPs due to surface tension
		17.3.1 Solution in quadrature of the problem of the TP increase due to surface tension
		17.3.2 Derivation of equations for TPs increase
			17.3.2.1 Even distribution of particles on their radii
			17.3.2.2 Linear distribution
			17.3.2.3 Exponential distribution
			17.3.2.4 The normal distribution
		17.3.3 Increase of TPs: results
	17.4 Balance of the vacancy-related and surface-tension effects
	17.5 Conclusions
	References
18 Comparative analysis of local stresses in unidirectional and cross-reinforced composites
	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
		18.3.1 Macroscopic strain ε11 (tension-compression along the Ox-axis)
		18.3.2 Macroscopic strain ε33 (tension-compression along the Oz-axis)
		18.3.3 Macroscopic deformations ε22 (tension-compression along the Oy-axis)
		18.3.4 Macroscopic deformations ε13 (shift in the Oxz-plane)
		18.3.5 Macroscopic strain ε12 (shift in the Oxy-plane)
		18.3.6 Macroscopic strain ε23 (shift in the Oyz-plane)
	18.4 The ``anisotropic layers\'\' approach
		18.4.1 Axial overall elastic moduli A1111 and A3333
		18.4.2 Axial overall elastic modulus A2222
		18.4.3 Shift elastic moduli A1212 and A2323
		18.4.4 Shift elastic modulus A1313
		18.4.5 The local stresses
	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
19 Statistical theory of structures with extended defects
	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
20 Effective conductivity of 2D composites and circle packing approximations
	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
21 Asymptotic homogenization approach applied to Cosserat heterogeneous media
	21.1 Introduction
	21.2 Basic equations for micropolar media. Statement of the problem
		21.2.1 Two-scale asymptotic expansions
	21.3 Example. Effective properties of heterogeneous periodic Cosserat laminate media
	21.4 Numerical results
		21.4.1 Cosserat laminated composite with cubic constituents
	21.5 Conclusions
	Acknowledgments
	References
A Finite clusters in composites
Index
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