Mechanics and Physics of Structured Media

Mechanics and Physics of Structured Media

Asymptotic and Integral Equations Methods of Leonid Filshtinsky.

1st Edition - January 20, 2022

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  • Editors: Igor Andrianov, Simon Gluzman, Vladimir Mityushev
  • Paperback ISBN: 9780323905435
  • eBook ISBN: 9780323906531

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Description

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.

Key Features

  • Covers key issues around the mechanics of structured media, including modeling techniques, fracture mechanics in various composite materials, the fundamentals of integral equations, wave mechanics, and more
  • Discusses boundary value problems of materials, techniques for predicting elasticity of composites, and heterogeneous nanomaterials and their statistical description
  • Includes insights on asymptotic methods, wave mechanics, the mechanics of piezo-materials, and more
  • Applies homogenization concepts to various physical systems

Readership

Researchers, graduate students, and professional engineers/R&D professionals in mechanical engineering, material science, physics

Table of Contents

  • 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

Product details

  • No. of pages: 524
  • Language: English
  • Copyright: © Academic Press 2022
  • Published: January 20, 2022
  • Imprint: Academic Press
  • Paperback ISBN: 9780323905435
  • eBook ISBN: 9780323906531

About the Editors

Igor Andrianov

Igor V. Andrianov is Professor Emeritus, RWTH Aachen University. He is the author or co-author of 14 books and more than 300 papers in peer-reviewed journals. He has presented papers at more than 150 international conferences and seminars and also supervised 21 PhD. Students. His research interests include mechanics of solids, mechanics of composite materials, nonlinear dynamics, and asymptotic methods.

Affiliations and Expertise

Professor Emeritus, RWTH Aachen University, Germany

Simon Gluzman

Simon Gluzman is presently an Independent Researcher (Toronto, Canada) and formerly a Research Associate at PSU in Applied Mathematics. He is interested in Re-summation methods in theory of random and regular composites and the method of self-similar and rational approximants.

Affiliations and Expertise

Independent Researcher, Toronto, Canada

Vladimir Mityushev

Vladimir Mityushev is the Professor of Cracow University of Technology, a leader of the research group www.materialica.plus. He is interested in mathematical modeling and computer simulations, Industrial mathematics and boundary value problems and their applications. .

Affiliations and Expertise

Institute of Mathematics, Faculty of Computer Science and Telecommunications, Cracow University of Technology, Kraków, Poland

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