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

Front Cover
Mechanics and Physics of Structured Media
Copyright
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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