Nonlinear Waves In Elastic Crystals

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.

Nonlinear Waves in Elastic Media explores the theoretical results of one-dimensional nonlinear waves, including shock waves, in elastic media. It is the first book to provide an in-depth and comprehensive presentation of the nonlinear wave theory while taking anisotropy effects into account. The theory is completely worked out and draws on 15 years of research by the authors, one of whom also wrote the 1965 classic Magnetohydrodynamics. Nonlinear Waves in Elastic Media emphasizes the behavior of quasitransverse waves and analyzes arbitrary discontinuity disintegration problems, illustrating that the solution can be non-unique - a surprising result. The solution is shown to be especially interesting when anisotropy and nonlinearity effects interact, even in small-amplitude waves. In addition, the text contains an independent mathematical chapter describing general methods to study hyperbolic systems expressing the conservation laws. The theoretical results described in Nonlinear Waves in Elastic Media allow, for the first time, discovery and interpretation of many new peculiarities inherent to the general problem of discontinuous solutions and so provide a valuable resource for advanced students and researchers involved with continuum mechanics and partial differential equations.

The translation into English of Academician Fedorov's ex cellent treatise on elastic wave propagation in solids has come at an opportune time. His systematic exposition of all aspects of this field is most lucid and straightforward. The author has gone to considerable pains to develop in his mathematical background a consistent tensor framework which acts as a unifying motif through out the various aspects of the subject. In many respects his approach will appear quite novel as his treatment introduces several concepts and parameters previously unfamiliar to the literature of the West. Extensive tables in the final chapters illustrate the application of these ideas to the exist ing body of experimental data. The book is both extensive and comprehensive in al1 phases of the subject. Workers in the fields of ultrasonic propagation and elastic properties will find this treatise of great interest and direct concern. H. B. Huntington Rensselaer Polytechnic Institute Troy, New York November 1967 v Preface to the American Edition In preparing this edition I have corrected various misprints and errors appearing in the Russian edition, but I have also incorpo rated some substantial changes and additions, the latter representing some results I and my colleagues have recently obtained and pub_ lished in Russian journals. For example, in section 32 I have added a general derivation of the equation for the seetion of the wave surface by a symmetry plane for cubic, hexagonal, tetragonal, and orthorhombic crystals.

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 book describes behavior of crystalline solids primarily via methods of modern continuum mechanics. Emphasis is given to geometrically nonlinear descriptions, i.e., finite deformations. Primary topics include anisotropic crystal elasticity, plasticity, and methods for representing effects of defects in the solid on the material's mechanical response. Defects include crystal dislocations, point defects, twins, voids or pores, and micro-cracks. Thermoelastic, dielectric, and piezoelectric behaviors are addressed. Traditional and higher-order gradient theories of mechanical behavior of crystalline solids are discussed. Differential-geometric representations of kinematics of finite deformations and lattice defect distributions are presented. Multi-scale modeling concepts are described in the context of elastic and plastic material behavior. Representative substances towards which modeling techniques may be applied are single- and poly- crystalline metals and alloys, ceramics, and minerals. This book is intended for use by scientists and engineers involved in advanced constitutive modeling of nonlinear mechanical behavior of solid crystalline materials. Knowledge of fundamentals of continuum mechanics and tensor calculus is a prerequisite for accessing much of the text. This book could be used as supplemental material for graduate courses on continuum mechanics, elasticity, plasticity, micromechanics, or dislocation mechanics, for students in various disciplines of engineering, materials science, applied mathematics, and condensed matter physics.

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 marks the 60th birthday of Prof. Vladimir Erofeev – a well-known specialist in the field of wave processes in solids, fluids, and structures. Featuring a collection of papers related to Prof. Erofeev’s contributions in the field, it presents articles on the current problems concerning the theory of nonlinear wave processes in generalized continua and structures. It also discusses a number of applications as well as various discrete and continuous dynamic models of structures and media and problems of nonlinear acoustic diagnostics.

Wave Processes in Solids with Microstructure is useful for undergraduates, graduate students, researchers and practitioners in the mechanics of solids as well as in physical and technical acoustics.