Vol.1: Static Problems

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9781402066641

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384

S.K. Kanaun

148, Solid Mechanics and Its Applications

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Englisch

"The book is dedicated to the application of self-consistent methods to the solution of static and dynamic problems of the mechanics and physics of composite materials. The effective elastic, electric, dielectric, thermo-conductive and other properties of composite materials reinforced by ellipsoidal, spherical multi-layered inclusions, thin hard and soft inclusions, short fibers and unidirected multi-layered fibers are considered. Explicit formulas and efficient computational algorithms for the calculation of the effective properties of the composites are presented and analyzed. The method of the effective medium and the method of the effective field are developed for the calculation of the phase velocities and attenuation of the mean (coherent) wave fields propagating in the composites. The predictions of the methods are compared with experimental data and exact solutions for the composites with periodical microstructures. The book may be useful for material engineers creating new composite materials and scholars who work on the theory of composite and non-homogeneous media. TOC:From the contents

1. Introduction2. An elastic medium with sources of external and internal stresses

2.1 Medium with sources of external stresses

2.2 Medium with sources of internal stresses

2.3 Discontinuities of elastic fields in a medium with sources of external and internal stresses

2.4 Elastic fields far from the sources

2.5 Notes

3. Equilibrium of a homogeneous elastic medium with an isolated inclusion

3.1 Integral equations for a medium with an isolated inhomogeneity

3.2 Conditions on the interface between two media

3.3 Ellipsoidal inhomogeneity

3.4 Ellipsoidal inhomogeneity in a constant external field

3.5 Inclusion in the form of a plane layer

3.6 Spheroidal inclusion in a transversely isotropic medium

3.7 Crack in an elastic medium

3.8 Elliptical crack

3.9 Radially heterogeneous inclusion

3.9.1 Elastic fields in a medium with a radially heterogeneous inclusion

3.9.2 Thermoelastic problem for a medium with a radially heterogeneous inclusion

3.10 Multi-layered spherical inclusion

3.11 Axially symmetric inhomogeneity in an elastic medium

3.12 Multi-layered cylindrical inclusion

3.13 Notes

4. Thin inclusion in a homogeneous elastic medium

4.1 External expansions of elastic fields

4.2 Properties of potentials (4.4) and (4.5)

4.3 External limit problems for a thin inclusion

4.3.1 Thin soft inclusion

4.3.2 Thin hard inclusion

4.4 Internal limiting problems and the matching procedure

4.5 Singular models of thin inclusions

4.6 Thin ellipsoidal inclusions

4.7 Notes

5. Hard fiber in a homogeneous elastic medium

5.1 External and internal limiting solutions

5.2 Principal terms of the stress field inside a hard fiber

5.3 Stress fields inside fibers of various forms

5.3.1 Cylindrical fiber

5.3.2 Prolate ellipsoidal fiber

5.3.3 Fiber in the form of a double cone

5.4 Curvilinear fiber

5.5 Notes

6. Thermal and electric fields in a medium with an isolated inclusion

6.1 Fields with scalar potentials in a homogeneous medium with an isolated inclusion

6.2 Ellipsoidal inhomogeneity

6.2.1 Constant external field

6.2.2 Linear external field

6.2.3 Spheroidal inhomogeneity in a transversely isotropic medium

6.3 Multi-layered spherical inclusion in a homogeneous medium

6.4 Thin inclusion in a homogeneous medium

6.5 Axisymmetric fiber in a homogeneous media

7. Homogeneous elastic medium with a set of isolated inclusion

7.1 The homogenization problem

7.2 Integral equations for the elastic fields in a medium with isolated inclusions

7.3 Tensor of the effective elastic moduli

7.4 The effective medium method and its versions

7.4.1 Differential effective medium method

7.5 The effective field method

7.5.1 Homogeneous elastic medium with a set of ellipsoidal inclusions

7.5.2 Elastic medium with a set of spherically layered inclusion

7.6 The Mon-Tanaka method

7.7 Regular lattices

7.8 Thin inclusions in a homogeneous elastic medium

7.9 Elastic medium reinforced with hard thin flakes or bands

7.9.1 Elastic medium with thin hard spheroids (flakes) of the same orientation

7.9.2 Elastic medium with thin hard spheroids homoge neousl distributed over the orientations

7.9.3 Elastic medium with thin hard unidirected bands of the same orientation

7.10 Elastic media with thin soft inclusions and cracks

7.10.1 Thin soft inclusions of the same orientation

7.10.2 Homogeneous distribution of thin soft inclusions over the orientations

7.10.3 Elastic medium with regular lattices of thin inclusions

7.11 Plane problem for a medium with a set of thin inclusions

7.11.1 A set of thin soft elliptical inclusions of the same orientation

7.11.2 Homogeneous distribution of thin inclusions over the orientations

7.11.3 Regular lattices of thin inclusions in plane

7.11.4 A triangular lattice of cracks

7.11.5 Col

2.1 Medium with sources of external stresses

2.2 Medium with sources of internal stresses

2.3 Discontinuities of elastic fields in a medium with sources of external and internal stresses

2.4 Elastic fields far from the sources

2.5 Notes

3. Equilibrium of a homogeneous elastic medium with an isolated inclusion

3.1 Integral equations for a medium with an isolated inhomogeneity

3.2 Conditions on the interface between two media

3.3 Ellipsoidal inhomogeneity

3.4 Ellipsoidal inhomogeneity in a constant external field

3.5 Inclusion in the form of a plane layer

3.6 Spheroidal inclusion in a transversely isotropic medium

3.7 Crack in an elastic medium

3.8 Elliptical crack

3.9 Radially heterogeneous inclusion

3.9.1 Elastic fields in a medium with a radially heterogeneous inclusion

3.9.2 Thermoelastic problem for a medium with a radially heterogeneous inclusion

3.10 Multi-layered spherical inclusion

3.11 Axially symmetric inhomogeneity in an elastic medium

3.12 Multi-layered cylindrical inclusion

3.13 Notes

4. Thin inclusion in a homogeneous elastic medium

4.1 External expansions of elastic fields

4.2 Properties of potentials (4.4) and (4.5)

4.3 External limit problems for a thin inclusion

4.3.1 Thin soft inclusion

4.3.2 Thin hard inclusion

4.4 Internal limiting problems and the matching procedure

4.5 Singular models of thin inclusions

4.6 Thin ellipsoidal inclusions

4.7 Notes

5. Hard fiber in a homogeneous elastic medium

5.1 External and internal limiting solutions

5.2 Principal terms of the stress field inside a hard fiber

5.3 Stress fields inside fibers of various forms

5.3.1 Cylindrical fiber

5.3.2 Prolate ellipsoidal fiber

5.3.3 Fiber in the form of a double cone

5.4 Curvilinear fiber

5.5 Notes

6. Thermal and electric fields in a medium with an isolated inclusion

6.1 Fields with scalar potentials in a homogeneous medium with an isolated inclusion

6.2 Ellipsoidal inhomogeneity

6.2.1 Constant external field

6.2.2 Linear external field

6.2.3 Spheroidal inhomogeneity in a transversely isotropic medium

6.3 Multi-layered spherical inclusion in a homogeneous medium

6.4 Thin inclusion in a homogeneous medium

6.5 Axisymmetric fiber in a homogeneous media

7. Homogeneous elastic medium with a set of isolated inclusion

7.1 The homogenization problem

7.2 Integral equations for the elastic fields in a medium with isolated inclusions

7.3 Tensor of the effective elastic moduli

7.4 The effective medium method and its versions

7.4.1 Differential effective medium method

7.5 The effective field method

7.5.1 Homogeneous elastic medium with a set of ellipsoidal inclusions

7.5.2 Elastic medium with a set of spherically layered inclusion

7.6 The Mon-Tanaka method

7.7 Regular lattices

7.8 Thin inclusions in a homogeneous elastic medium

7.9 Elastic medium reinforced with hard thin flakes or bands

7.9.1 Elastic medium with thin hard spheroids (flakes) of the same orientation

7.9.2 Elastic medium with thin hard spheroids homoge neousl distributed over the orientations

7.9.3 Elastic medium with thin hard unidirected bands of the same orientation

7.10 Elastic media with thin soft inclusions and cracks

7.10.1 Thin soft inclusions of the same orientation

7.10.2 Homogeneous distribution of thin soft inclusions over the orientations

7.10.3 Elastic medium with regular lattices of thin inclusions

7.11 Plane problem for a medium with a set of thin inclusions

7.11.1 A set of thin soft elliptical inclusions of the same orientation

7.11.2 Homogeneous distribution of thin inclusions over the orientations

7.11.3 Regular lattices of thin inclusions in plane

7.11.4 A triangular lattice of cracks

7.11.5 Col

The theory of heterogeneous materials has been intensively developed during the past few decades. The main reason for the interest of many researchers in this part of the mechanics of solids is the wide area of application of hete- geneous materials in modern material engineering. Self-consistent methods form a well-known branch of the theory of heterogeneous materials. In most books devoted to the mechanics and physics of heterogeneous media, the reader can ?nd self-consistent solutions. But there are no books covering the entire spectrum of self-consistent methods in application to the calculation of static and dynamic properties of heterogeneous materials. This book has been written to cover this gap. It is written for engineers because here they can ?nd the equations for the e?ective properties of composites reinforced with various types of inclusions. The main advantage of self-consistent methods is that they give relatively simple equations for the e?ective parameters of composites. Such equations for static and dynamic properties of matrix composites reinforced with va- oustypesofinclusions,forporousmedia,mediawithcracksandotherdefects, for polycrystals, etc., are widely used in engineering practice, and many new self-consistent solutions are presented in the book. This book is written also for scholars who wish to develop the theory of heterogenousmedia.Inthebooktheywill ?ndthebasicideasandalgorithms for the construction of self-consistent solutions. The book shows how these methods may be applied to composites with inclusions of complex structures, toproblemsofwavepropagation,forcalculationofhigherstatisticalmoments of physical ?elds in composites. Various ways for improving self-consistent solutions are proposed and discussed.