# Linear and Nonlinear Waves in Microstructured Solids

## Homogenization and Asymptotic Approaches

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. Les mer
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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.

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

Igor Andrianov is Professor with 25 years of experience in Mathematics, Applied Mechanics and Mechanics of Solids. Researcher with 41 years of experience in Applied Mathematics and Mechanics of Solids. Supervisor of 18 Ph.D. students. Research interests: Asymptotic Approaches, Nonlinear Dynamics, Composite Materials, Theory of Plates and Shells.

Jan Awrejcewicz is a Head of the Department of Automation, Biomechanics and Mechatronics at Lodz University of Technology, Head of Ph.D. School on 'Mechanics' (since 1996) and of graduate/postgraduate programs on Mechatronics (since 2006). He is also recipient of Doctor Honoris Causa (Honorary Professor) of Academy of Arts and Technology (Poland, Bielsko-Biala, 2014) and of Czestochowa University of Technology (Poland, Czestochowa, 2014), Kielce University of Technology (2019), National Technical University "Kharkiv Polytechnic Institute" (2019), and Gdansk University of Technology (2019). His papers and research cover various disciplines of mechanics, material science, biomechanics, applied mathematics, automation, physics and computer oriented sciences, with main focus on nonlinear processes.

Vladyslav V Danishevskyy is Professor at Prydniprovska State Academy of Civil Engineering and Architecture, Ukraine. His research area includes mechanical and physical properties of composite materials, metamaterials and heterogeneous structures; non-linear dynamics; waves in heterogeneous media; asymptotic methods.