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

Description

This book uses asymptotic methods to obtain simple approximate analytic solutions to various problems within mechanics, notably wave processes in heterogeneous materials. Presenting original solutions to common issues within mechanics, this book builds upon years of research to demonstrate the benefits of implementing asymptotic techniques within mechanical engineering and material science. Focusing on linear and nonlinear wave phenomena in complex micro-structured solids, the book determines their global characteristics through analysis of their internal structure, using homogenization and asymptotic procedures, in line with the latest thinking within the field. The book’s cutting-edge methodology can be applied to optimal design, non-destructive control and in deep seismic sounding, providing a valuable alternative to widely used numerical methods. Using case studies, the book covers topics such as elastic waves in nonhomogeneous materials, regular and chaotic dynamics based on continualisation and discretization and vibration localization in 1D Linear and Nonlinear lattices. The book will be of interest to students, research engineers, and professionals specialising in mathematics and physics as well as mechanical and civil engineering. 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