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

✍ Scribed by Igor V. Andrianov, Jan Awrejcewicz, Vladyslav Danishevskyy

• Publisher: CRC Press
• Year: 2021
• Tongue: English
• Leaves: 251
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

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