Linear and Nonlinear Waves in Microstructured Solids: Homogenization and Asymptotic Approaches

Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Preface
Permissions
Chapter 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
Chapter 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
Chapter 3: Waves in Fiber 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
Chapter 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
Chapter 5: Antiplane ShearWaves in Fiber Composites with Structural 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
Chapter 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
Chapter 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
Chapter 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 FIBER composite
Chapter 9: Nonlinear Vibrations of Viscoelastic Heterogeneous Solids of Finite Size
	9.1. Introduction
	9.2. Input problem and homogenized 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
Chapter 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 arguments and 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
	10.15. Toda lattice
	10.16. Discrete kinks
	10.17. Continualization of b-FPU lattice
	10.18. Acoustic branch of a-FPU lattice
	10.19. Anti-continuum limit
	10.20. 2D lattice
	10.21. Molecular dynamics simulations and continualization: handshake
	10.22. Continualization and discretization
	10.23. Possible generalization and applications and open problems
Chapter 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