1 Models and Methods to Study Elastic Waves 1.1 Brief literature overview 1.2 Small "tutorial" 1.3 Analytical and numerical solutions in the theory of composite materials 1.4 Some general results of the homogenization theory 2 Waves in Layered Composites: Linear Problems 2.1 One-dimensional (1D) dynamic problem 2.2 Higher order homogenization method 2.3 The Bloch-Floquet method and exact dispersion equation 2.4 Numerical results 3 Waves in Fibre Composites: Linear Problems 3.1 Two-dimensional (2D) dynamic problem 3.2 Method of higher order homogenization 3.3 The Bloch-Floquet method and solution based on Fourier series 3.4 Numerical results 3.5 Shear waves dispersion in cylindrically structured cancellous viscoelastic bones 4 Longitudinal Waves in Layered Composites 4.1 Fundamental relations of nonlinear theory of elasticity 4.2 Input boundary value problems 4.3 Macroscopic wave equation 4.4 Analytical solution for stationary waves 4.5 Analysis of solution and numerical results 5 Antiplane ShearWaves in Fibre Composites withStructural Nonlinearity 5.1 Boundary value problem for imperfect bonding conditions. 5.2 Macroscopic wave equation 5.3 Analytical solution for stationary waves 5.4 Analysis of solution and numerical result 6 Formation of Localized Nonlinear Waves in Layered Composites 6.1 Initial model and pseudo-spectral method 6.2 The Fourier-Pade approximation 6.3 Numerical modeling of non-stationary nonlinear waves 7 Vibration Localization in 1D Linear and Nonlinear Lattices 7.1 Introduction 7.2 Monatomic lattice with a perturbed mass 7.3 Monatomic lattice with a perturbed mass
the continuous approximation 7.4 Diatomic lattice 7.5 Diatomic lattice with a perturbed mass 7.6 Diatomic lattice with a perturbed mass
the continuous approximation 7.7 Vibrations of a lattice on the support with a defect 7.8 Nonlinear vibrations of a lattice 7.9 Effect of nonlinearity on pass bands and stop bands 8 Spatial Localization of Linear Elastic Waves in Composite Materials With Defects 8.1 Introduction 8.2 Wave localization in a layered composite material:transfer-matrix method 8.3 Wave localization in a layered composite material: lattice approach 8.4 Antiplane shear waves in a fibre composite 9 Non-Linear Vibrations of Viscoelastic Heterogeneous Solids of Finite Size 9.1 Introduction 9.2 Input problem and homogenised dynamical equation 9.3 Discretization procedure 9.4 Method of multiple time scales 9.5 Numerical simulation of the modes coupling 9.6 Concluding remarks 10 Nonlocal, Gradient and Local Models of Elastic Media: 1D Case 10.1 Introduction 10.2 A chain of elastically coupled masses 10.3 Classical continuous approximations 10.4 "Splashes" 10.5 Envelope continualization 10.6 Intermediate continuous models 10.7 Using of Pade approximations 10.8 Normal modes expansion 10.9 Theories of elasticity with couple-stresses 10.10 Correspondence between functions of discrete argumentsnand approximating analytical functions 10.11 The kernels of integro-differential equations of the discrete and continuous systems 10.12 Dispersive wave propagation 10.13 Green's function 10.14 Double- and triple- dispersive equations 10.15Toda lattice 10.16Discrete kinks 10.17Continualization of b-FPU lattice 10.18Acoustic branch of a-FPU lattice 10.19Anti-continuum limit 10.202D lattice 10.21 Molecular dynamics simulations and continualization: handshake 10.22 Continualization and discretization 10.23Possible generalization and applications and open problems 11 Regular and Chaotic Dynamics Based on Continualization and Discretization 11.1 Introduction 11.2 Integrable ODE 11.3 Continualization with Pade approximants 11.4 Numerical results References Index