Nonlinear Elastic Waves In Materials

The main goal of the book is a coherent treatment of the theory of propagation in materials of nonlinearly elastic waves of displacements, which corresponds to one modern line of development of the nonlinear theory of elastic waves. The book is divided on five basic parts: the necessary information on waves and materials; the necessary information on nonlinear theory of elasticity and elastic materials; analysis of one-dimensional nonlinear elastic waves of displacement – longitudinal, vertically and horizontally polarized transverse plane nonlinear elastic waves of displacement; analysis of one-dimensional nonlinear elastic waves of displacement – cylindrical and torsional nonlinear elastic waves of displacement; analysis of two-dimensional nonlinear elastic waves of displacement – Rayleigh and Love nonlinear elastic surface waves. The book is addressed first of all to people working in solid mechanics – from the students at an advanced undergraduate and graduate level to the scientists, professionally interesting in waves. But mechanics is understood in the broad sense, when it includes mechanical and other engineering, material science, applied mathematics and physics and so forth. The genesis of this book can be found in author’s years of research and teaching while a head of department at SP Timoshenko Institute of Mechanics (National Academy of Sciences of Ukraine), a member of Center for Micro and Nanomechanics at Engineering School of University of Aberdeen (Scotland) and a professor at Physical-Mathematical Faculty of National Technical University of Ukraine “KPI”. The book comprises 11 chapters. Each chapter is complemented by exercises, which can be used for the next development of the theory of nonlinear waves.

This revised and updated edition expands on its explanations of methods used to analyze waves in solid materials, such as the waves created by earthquakes and the ultrasonic waves used to detect flaws in materials and for medical diagnoses. In addition to the traditional methods used to analyze steady-state and transient waves in elastic materials, the book contains introductions to advanced areas that no other single text covers. These topics include the use of finite elements to solve wave problems, the Cagniard-de Hoop method, the four-pole technique for analyzing waves in layered media, and the growth and decay of shock and acceleration waves. The authors explain the theory of linear elasticity through the displacement equations of motion, methods used to analyze steady-state and transient waves in layered media, and include an appendix on functions of a complex variable. Originally developed for a graduate course for which no suitable text existed, the new edition retains its classroom-tested treatment of the theories of linear elasticity and complex variables for students needing background in those subjects.

This volume outlines the basic concepts and methods of the theory of wave propagation in elastic materials. The linear theory of elasticity is covered, culminating in the displacement equations of motion. One-dimensional waves are analyzed through the D'Alembert solution.

The mathematical modelling of changing structures in materials is of increasing importance to industry where applications of the theory are found in subjects as diverse as aerospace and medicine. This book deals with aspects of the nonlinear dynamics of deformable ordered solids (known as elastic crystals) where the nonlinear effects combine or compete with each other. Physical and mathematical models are discused and computational aspects are also included. Different models are considered - on discrete as well as continuum scales - applying heat, electricity, or magnetism to the crystal structure and these are analysed using the equations of rational mechanics. Students are introduced to the important equations of nonlinear science that describe shock waves, solitons and chaos and also the non-exactly integrable systems or partial differential equations. A large number of problems and examples are included, many taken from recent research and involving both one-dimensional and two-dimensional problems as well as some coupled degress of freedom.

Comprehensive introduction to nonlinear elasticity for graduates and researchers, covering new developments in the field.

Papers in this book provide a state-of-the-art examination of waves in pre-stressed materials. You’ll gain new perspectives via a multi-disciplinary approach that interweaves key topics. These topics include the mathematical modeling of incremental material response (elastic and inelastic), an analysis of the governing differential equations, and boundary-value problems. Detailed illustrations help you visualize key concepts and processes.

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.

Summary: This book presents necessary background knowledge on mechanics to understand and analyze elastic wave propagation in solids and fluids. This knowledge is necessary for elastic wave propagation modeling and for interpreting experimental data generated during ultrasonic nondestructive testing and evaluation (NDT&E). The book covers both linear and nonlinear analyses of ultrasonic NDT&E techniques. The materials presented here also include some exercise problems and solution manual. Therefore, this book can serve as a textbook or reference book for a graduate level course on elastic waves and/or ultrasonic nondestructive evaluation. It will be also useful for instructors who are interested in designing short courses on elastic wave propagation in solids or NDT&E. The materials covered in the first two chapters provide the fundamental knowledge on linear mechanics of deformable solids while Chapter 4 covers nonlinear mechanics. Thus, both linear and nonlinear ultrasonic techniques are covered here. Nonlinear ultrasonic techniques are becoming more popular in recent years for detecting very small defects and damages. However, this topic is hardly covered in currently available textbooks. Researchers mostly rely on published research papers and research monographs to learn about nonlinear ultrasonic techniques. Chapter 3 describes elastic wave propagation modeling techniques using DPSM. Chapter 5 is dedicated to an important and very active research field – acoustic source localization – that is essential for structural health monitoring and for localizing crack and other type of damage initiation regions. Features • Introduces Linear and Nonlinear ultrasonic techniques in a single book. • Commences with basic definitions of displacement, displacement gradient, traction and stress. • Provides step by step derivations of fundamental equations of mechanics as well as linear and nonlinear wave propagation analysis. • Discusses basic theory in addition to providing detailed NDE applications. • Provides extensive example and exercise problems along with an extensive solutions manual.