Micromechanics of Composites: Multipole Expansion Approach

Contents
Preface to the first edition
Preface to the second edition
1 Multipole expansion approach
	1.1 Introduction
	1.2 Structure models
		1.2.1 Single inhomogeneity
		1.2.2 Finite arrays of inhomogeneities
		1.2.3 Composite band and layer
		1.2.4 Representative unit cell
	1.3 Bulk and interface field models
		1.3.1 Conductivity
		1.3.2 Elasticity
	1.4 Method of solution
		1.4.1 Multipole expansion: why and how?
		1.4.2 Local expansion
		1.4.3 Superposition principle
		1.4.4 Summary of the method
	1.5 Induced dipole moment of inhomogeneity
		1.5.1 Definition
		1.5.2 Conservation law
		1.5.3 Relationship to the property contribution tensor
	1.6 Macroscopic field parameters
		1.6.1 Conductivity
			Definition of the macroscopic quantities
			Formula for the macroscopic flux
		1.6.2 Elasticity
	1.7 Homogenization problem
		1.7.1 Maxwell scheme
		1.7.2 Rayleigh scheme
2 Potential fields of interacting spherical inhomogeneities
	2.1 Single inhomogeneity
		2.1.1 Series expansion
		2.1.2 Resolving equations
	2.2 Particle coating vs imperfect interface
	2.3 Finite cluster model
		2.3.1 FCM boundary-value problem
			Direct (superposition) sum
			Local series expansion
			Infinite linear system
		2.3.2 Convergence proof
		2.3.3 Modified Maxwell scheme for effective conductivity
			Formal representation
			Convergence check
	2.4 Composite sphere
		2.4.1 Outer boundary condition
		2.4.2 Interface conditions
		2.4.3 RSV and effective conductivity of composite
	2.5 Half-space FCM
		2.5.1 Formal solution
		2.5.2 Half-space boundary condition
		2.5.3 Interface conditions
3 Periodic multipoles and RUC model of a composite
	3.1 Composite layer
		3.1.1 2P fundamental solution of Laplace equation
		3.1.2 2P solid harmonics
		3.1.3 Heat flux through the composite layer
	3.2 Periodic composite as a sandwich of composite layers
	3.3 Representative unit cell model
	3.4 3P scalar solid harmonics
		3.4.1 Direct summation
		3.4.2 Hasimoto\'s approach
		3.4.3 2P harmonics based approach
	3.5 Local temperature field
	3.6 Effective conductivity
		3.6.1 Rayleigh homogenization scheme
		3.6.2 Numerical results
4 Elastic solid with spherical inhomogeneities
	4.1 Single inhomogeneity in an unbounded solid
		4.1.1 Multipole series expansion
		4.1.2 Induced elastic dipole moment
		4.1.3 Far field expansion
		4.1.4 Resolving set of linear equations
	4.2 Coated spherical inhomogeneity
	4.3 Application to nanocomposite: Gurtin & Murdoch theory
		4.3.1 Imperfect interface conditions
		4.3.2 Formal solution
		4.3.3 Numerical examples
			Single cavity under hydrostatic far-field load
			Single cavity under uniaxial tension
	4.4 Isotropic solid with anisotropic inhomogeneity
		4.4.1 Formal solution
		4.4.2 Resolving set of equations
	4.5 Finite cluster of inhomogeneities
		4.5.1 Direct (superposition) sum
		4.5.2 Local expansion
		4.5.3 Infinite system of linear equations
		4.5.4 Numerical examples
			Two cavities under uniaxial tension
			Interface-induced stress in a nanostructured solid
			Stress concentration factor
	4.6 Effective stiffness of composite
		4.6.1 Modified Maxwell scheme
		4.6.2 Cubic symmetry
			Bulk modulus k*
			Shear modulus μ1*
			Shear modulus μ2*
	4.7 Elastic composite sphere
		4.7.1 Elastic fields
		4.7.2 Effective elastic moduli
			Macroscopic strain and stress tensors
			Effective bulk modulus
			Effective shear modulus
5 Elasticity of composite half-space, layer, and bulk
	5.1 Finite cluster of spherical inhomogeneities in a half-space
		5.1.1 Problem statement
		5.1.2 Homogeneous half-space
		5.1.3 Heterogeneous half-space
	5.2 Doubly periodic structures
		5.2.1 2P solutions of Lamé equation
		5.2.2 Composite layer
		5.2.3 Periodic composite as a sandwich of composite layers
	5.3 Triply periodic solutions of Lamé equation
		5.3.1 Scalar 3P biharmonics
		5.3.2 Periodic solutions of Lamé equation
	5.4 RUC model
		5.4.1 Formal solution
			Superposition sum
			Local expansion
			Algebraic set of equations
		5.4.2 Effective stiffness tensor
			Approximate formula for the bulk modulus of nanocomposite
			Average stress as a governing parameter
	5.5 Numerical study
		5.5.1 Local stress field
		5.5.2 Effective stiffness tensor
			Periodic composite
			Random structure composite
			Nanoporous solid
6 Conductivity of a solid with spheroidal inhomogeneities
	6.1 Single inhomogeneity
		6.1.1 Series expansion
		6.1.2 Induced dipole moment
		6.1.3 Resolving equations for perfect interface
		6.1.4 Limiting cases: spherical, penny-shaped, and needle-like inhomogeneities
			Spherical inhomogeneity
			Penny-shaped crack and superconducting flake
			Flux intensity factor
		6.1.5 LC imperfect interface
		6.1.6 HC imperfect interface
	6.2 Finite cluster model
	6.3 Modified Maxwell scheme
		6.3.1 Analytical results
		6.3.2 Numerical examples
			Single inhomogeneity with imperfect interface
			FCM
	6.4 Heat conduction in a periodic composite
		6.4.1 Doubly periodic harmonics
		6.4.2 Triply periodic harmonics
		6.4.3 Temperature field in a periodic composite
			3P approach
			2P approach
		6.4.4 Multiple inhomogeneity RUC model
	6.5 Rayleigh homogenization scheme
		6.5.1 Composite with perfect interface
			Spheroidal cavities and inhomogeneities
			Penny-shaped cracks
			Superconducting flakes
		6.5.2 Composite with imperfect interface
			LC interface
			HC interface
7 Elastic solid with spheroidal inhomogeneities
	7.1 Single inhomogeneity with perfect interface
		7.1.1 Displacement field
		7.1.2 Elastic dipole moment
		7.1.3 Stress intensity factors for a penny-shaped crack
	7.2 Single inhomogeneity with imperfect interface
		7.2.1 Spring type interface
			Resolving system
			Numerical examples
		7.2.2 Membrane type interface
			Resolving system
			Numerical examples
	7.3 Finite cluster of spheroidal inhomogeneities
		7.3.1 Formal solution
		7.3.2 Local expansion
		7.3.3 Numerical examples
			Penny-shaped crack interacting with another crack or inhomogeneity
			Interacting nanocavities
	7.4 Modified Maxwell scheme
		7.4.1 Single inhomogeneity model
			Weak interface
			Stiff interface
		7.4.2 Multiple cavity model
	7.5 Elastic half-space with spheroidal inhomogeneities
	7.6 RUC model of an elastic spheroidal particle composite
		7.6.1 Periodic solutions of Lamé equation
		7.6.2 Displacement solution
		7.6.3 Numerical study
			Stress concentration
			Stress intensity factor
			Nanoporous solid
			Nanocomposite
		7.6.4 Rayleigh homogenization scheme
			Composite with a perfect interface
			Effective stiffness of a nanocomposite
8 Composites with transversely isotropic constituents
	8.1 Transversely isotropic conductivity
		8.1.1 Partial solutions
		8.1.2 Problem statement
		8.1.3 Temperature field
		8.1.4 Effective conductivity tensor
	8.2 Transversely isotropic elasticity
		8.2.1 Partial vector solutions
		8.2.2 Single inhomogeneity
		8.2.3 Finite array of inhomogeneities
	8.3 RUC model
		8.3.1 Displacement field
		8.3.2 Effective stiffness tensor
	8.4 Numerical examples
		8.4.1 Stress concentration
		8.4.2 Effective stiffness
9 Conductivity of an ellipsoidal particle composite
	9.1 Composite with isotropic constituents
		9.1.1 Problem statement
		9.1.2 Single ellipsoidal inhomogeneity
			Series expansion
			Resolving system
			Induced dipole moment
		9.1.3 Finite array of ellipsoids
		9.1.4 Periodic array of ellipsoids
	9.2 Composite with anisotropic constituents
		9.2.1 Problem statement
		9.2.2 Reduction to isotropic case
			Affine mapping
			Global and local variables
			Boundary conditions
		9.2.3 Single ellipsoidal inhomogeneity
			Formal solution
			Imperfect interface
			Induced dipole moment
		9.2.4 Eshelby-type problem
		9.2.5 Finite cluster model
			Reexpansion formulas
			Resolving system
		9.2.6 RUC model
	9.3 Effective conductivity of a composite
		9.3.1 Maxwell homogenization scheme
		9.3.2 Rayleigh homogenization scheme
		9.3.3 Numerical study
			Convergence in terms of harmonics
			Convergence in terms of cluster size
			Comparison study
10 Elasticity of an ellipsoidal particle composite
	10.1 Single ellipsoidal inhomogeneity
		10.1.1 Series expansion
		10.1.2 Dipole moment
	10.2 Uniform far filed
		10.2.1 Ellipsoidal cavity, displacement boundary condition
			Normal mode
			Shear mode
		10.2.2 Ellipsoidal cavity: traction boundary condition
			Normal mode
			Shear mode
		10.2.3 Ellipsoidal inhomogeneity
	10.3 Nonuniform far field
		10.3.1 Displacement boundary condition
		10.3.2 Traction boundary condition
		10.3.3 Perfect interface
		10.3.4 Imperfect interface
	10.4 Finite cluster of ellipsoidal inhomogeneities
		10.4.1 Displacement solution
			Inhomogeneity
			Matrix
		10.4.2 Resolving linear system
		10.4.3 Effective stiffness tensor
	10.5 Orthotropic elastic solid with an arbitrarily oriented inhomogeneity
		10.5.1 Problem formulation
		10.5.2 Reduction to isotropic case
			Affine mapping
			Transformed problem
		10.5.3 Multipole expansion solution
			Inhomogeneity
			Matrix
			Resolving linear system
	10.6 Numerical study
		10.6.1 Single inhomogeneity
			Convergence in terms of integration points
			Convergence in terms of harmonics
			Interface stress
			Effective stiffness
		10.6.2 Multiple inhomogeneities
			Convergence
			Interface stress
			Effective stiffness
A Spherical harmonics and related theory
	A.1 Scalar spherical harmonics
		Laplace equation in spherical coordinates
		Selected properties of solid spherical harmonics
		Spherical harmonics vs multipole potentials
		Fourier integral representation
	A.2 Reexpansion formulas for Yts and yts
		Equally oriented coordinate systems
		Arbitrarily oriented coordinate systems
	A.3 Scalar spherical biharmonics
	A.4 Vector spherical surface harmonics
	A.5 Partial solutions of Lamé equation
		Definition
		Explicit expressions
		Normal traction
		Net force and torque
	A.6 Partial solutions for a half-space
		Cartesian vector surface harmonics
		Vector solutions of Lamé equation for a half-space
		Integral transforms and series expansions
	A.7 Reexpansion formulas for Uts(i) and uts(i)
		Translation
		Rotation
B Spheroidal harmonics and related theory
	B.1 Scalar solid harmonics
		Laplace equation in spheroidal coordinates
		Spheroidal solid harmonics
		Relationships between the spherical and spheroidal harmonics
		Alternative set of spheroidal harmonics
		Extended set of spheroidal harmonics
	B.2 Reexpansion formulas
		Formal series expansion
		Translation: integral form of the expansion coefficients
		Translation: rational form of the expansion coefficients
		Rotation
	B.3 Double Fourier integral transform of spheroidal harmonics
	B.4 Vector solutions of Lamé equation
		Definition
		Selected properties
		Integral transforms
	B.5 Reexpansion formulas for Vts(i) and vts(i)
		General form
		Rotation
C Ellipsoidal harmonics and related theory
	C.1 Ellipsoidal harmonics
		Solid harmonics
		Surface harmonics
	C.2 Differentiation and integration
		Derivatives of solid harmonics
		Numerical integration
	C.3 Reexpansion formulas
D Selected properties of functions Rλ and Xλ
	D.1 Function R(ζ)
	D.2 Functions Rλ(ζ) and Xλ(ζ)
E Elliptic harmonics and related theory
	E.1 Elliptic harmonics
	E.2 Reexpansion formulas
	E.3 Integral transforms
	E.4 Periodic complex potentials
	E.5 Evaluation of βk coefficients
Bibliography
Index