Contents -- Preface -- Chapter 1 Long and close range interaction within elastic structures -- 1.1 Dilute composite structures. Scalar problems -- 1.1.1 An elementary example. Motivation -- 1.1.2 Asymptotic algorithm involving a boundary layer -- 1.1.2.1 Formulation of the problem -- 1.1.2.2 The leading-order approximation -- 1.1.2.3 Asymptotic formula for the energy -- 1.1.3 The dipole matrix -- 1.1.3.1 Definition of the dipole matrix -- 1.1.3.2 Symmetry of the dipole matrix -- 1.1.3.3 The energy asymptotics for a body with a small void -- 1.1.4 Dipole matrix for a 2D void in an infinite plane -- 1.1.5 Dipole matrices for inclusions -- 1.1.6 A note on homogenization of dilute periodic structures -- 1.2 Dipole fields in vector problems of linear elasticity -- 1.2.1 Definitions and governing equations -- 1.2.2 Physical interpretation -- 1.2.3 Evaluation of the elements of the dipole matrix -- 1.2.4 Examples -- 1.2.5 The energy equivalent voids -- 1.3 Circular elastic inclusions -- 1.3.1 Inclusions with perfect bonding at the interface -- 1.3.2 Dipole tensors for imperfectly bonded inclusions -- 1.3.2.1 Derivation of transmission conditions at the zero-thickness interface -- 1.3.2.2 Neutral coated inclusions -- 1.4 Close-range contact between elastic inclusions -- 1.4.1 Governing equations -- 1.4.2 Complex potentials -- 1.4.3 Analysis for two circular elastic inclusions -- 1.4.4 Square array of circular inclusions -- 1.4.5 Integral approximation for the multipole coefficients. Inclusions close to touching -- 1.4.5.1 Scalar problem -- 1.4.5.2 Vector problem -- 1.5 Discrete lattice approximations -- 1.5.1 Illustrative one-dimensional example -- 1.5.2 Two-dimensional array of obstacles -- Chapter 2 Dipole tensors in spectral problems of elasticity.

2.1 Asymptotic behaviour of fields near the vertex of a thin conical inclusion -- 2.1.1 Spectral problem on a unit sphere -- 2.1.2 Boundary layer solution -- 2.1.2.1 The leading term -- 2.1.2.2 Problem for w(2) -- 2.1.3 Stress singularity exponent A2 -- 2.2 Imperfect interface. "Coated" conical inclusion -- 2.2.1 Formulation of the problem -- 2.2.2 Boundary layer solution -- 2.2.2.1 Change of coordinates for the "coating" layer -- 2.2.2.2 Problem for w(1) -- 2.2.2.3 Problem for w(2) -- 2.2.2.4 Asymptotic behaviour of w(2) at infinity -- 2.2.3 Stress singularity exponent A2 -- 2.2.4 Some examples. Discussion and conclusions -- Chapter 3 Multipole methods and homogenisation in two-dimensions -- 3.1 The method of Rayleigh for static problems -- 3.1.1 The multipole expansion and effective properties -- 3.1.2 Solution to the static problem -- 3.2 The spectral problem -- 3.2.1 Formulation and Bloch waves -- 3.2.2 The dynamic multipole method -- 3.2.3 The dynamic lattice sums -- 3.2.4 The integral equation and the Rayleigh identity -- 3.2.5 The dipole approximation -- 3.3 The singularly perturbed problem and non-commuting limits -- 3.3.1 The Neumann problem and non-commuting limits -- 3.3.2 The Dirichlet problem and source neutrality -- 3.4 Non-commuting limits for the effective properties -- 3.5 Elastic waves in doubly-periodic media -- 3.5.1 Governing equations -- 3.5.2 Convergence of the Rayleigh matrix -- 3.5.3 Numerical results and comments -- 3.6 Concluding remarks -- Bibliography -- Index.