Materials Science

Foundations of the Theory of Elasticity, Plasticity, and Viscoelasticity
Eduard Starovoitov, DSc
Faig Bakhman Ogli Naghiyev, DSc

Foundations of the Theory of Elasticity, Plasticity, and Viscoelasticity

Published. Now Available
Pub Date: July 2012
Hardback Price: see ordering info
Hard ISBN: 9781926895116
Paperback ISBN: 978-1-77463-199-7
E-Book ISBN: 9781466558793
Pages: 364 pp with index
Binding Type: hardbound / ebook / paperback


Reviews
“The book is devoted to the mechanics of deformable bodies. The authors present the results of many years of basic research. The book features a deep, comprehensive, and rigorous study of the issues covered. The authors have carefully and thoroughly studied the mechanics of deformable bodies and provide solutions. The methods will undoubtedly be useful to many researchers in this field. The methods illustrated in Chapter 8, on the theory of plasticity, and Chapter 11, which deals with the thermal effects on a number of specific materials, can be used to advance the construction of the theory of thermoviscoelastoplasticity, which is extremely important not only in theoretical terms, but also for the development of industrial production and processing of conductive material. The book will be of great benefit in the work of researchers and students in the field of solid mechanics.”
--Professor Mark Liberzon, Head, Department of Analytical Mechanics, and Vice President, Tsyolkovsky Moscow Aviation Technological University; Vice President, Russian Engineering Academy, Moscow, Russia

“In this new book, the authors present contemporary approaches to the formulation and solution of problems in the theory of elasticity, plasticity, and viscoelasticity. The traditional sections of the course of the theory of elasticity and plasticity are presented in the modern interpretation, clearly and in an easy-to-understand manner. The tensor form of writing is used along with the coordinate form, which facilitates the student to understand the material. The analytical solutions and numerical results for elastic, elastic-plastic, linear viscoelastic and viscous elastoplastic three-layer circular plates under quasi-static and dynamic stresses are the results of the authors’ own work.”
--D. V. Tarlakovsky, DSc, Head, Department of Resistance of Materials, Dynamics and Strength of Machines, Moscow State Aviation Institute, Russia



Now Available in Paperback


Readers of this book will be informed about fundamental and practical skills and approaches for carrying out research in the field of modern problems in the mechanics of deformed solids, which involves the theories of elasticity, plasticity, and viscoelasticity. The book includes all modern methods of research, as well as the results of the authors? recent work and is presented with sufficient mathematical strictness and proof.

The first six chapters are devoted to the foundations of the theory of elasticity. Theory of stress-strain state, physical relations and problem statements, variation principles, contact and 2D problems, and the theory of plates are presented, and the theories are accompanied by examples of solving typical problems.

The last six chapters will be useful to postgraduates and scientists engaged in nonlinear mechanics of deformed inhomogeneous bodies. The foundations of the modern theory of plasticity (general, small elastoplastic deformations and the theory of flow), linear, and nonlinear viscoelasticity are set forth. Corresponding research of three-layered circular plates of various materials is included to illustrate methods of problem solving. Analytical solutions and numerical results for elastic, elastoplastic, lineaer viscoelastic and viscoelastoplastic plates are also given. Thermoviscoelastoplastic characteristics of certain materials need for numerical account are presented in the eleventh Chapter.

The informative book is intended for scientists, postgraduates and higher-level students of engineering spheres and will provide important practical skills and approaches.

CONTENTS:
Preface
Introduction

Chapter 1. The theory of stress-strain state
- 1.1. The concept of elastic continuum
- 1.2. Stress tensor
- 1.3. Stress tensor properties
- 1.4. Equilibrium equations for elastic bodies
- 1.5. Boundary equilibrium conditions
- 1.6. Principal axes and principal values of stress tensor
- 1.7. Maximal tangential stresses
- 1.8. Deviator and spherical part of stress tensor
- 1.9. Displacements and deformations. Strain tensor
- 1.10. Principal axes and principal values of strain tensor
- 1.11. Deviator and spherical part of strain tensor
- 1.12. Compatibility equation of deformations
- References
Chapter 2. Physical relations in the elasticity theory
- 2.1. Strain energy and elastic potential
- 2.2. Hooke’s law
- 2.3. Hooke’s law for anisotropic materials
- 2.4. Clapeyron formula
- 2.5. Temperature effects
- 2.6. SSS case study
- References
Chapter 3. Statement and problem solving procedures of elasticity theory
- 3.1. Boundary problems
- 3.2. Statement of the elasticity theory problem in displacements (Lame’s equation
- 3.3. Statement of the elasticity theory problem in stresses (Beltrami–Michell equation
- 3.4. Clapeyron theorem
- 3.5. Existence and uniqueness of the elasticity theory problem solution
- 3.6. Semi-inverse Saint-Venant’s method
- 3.7. Statement of the elasticity theory problem in cylindrical and spherical coordinates
- References
Chapter 4. Variational methods
- 4.1. Lagrangian principle of virtual displacements
- 4.2. Castigliano’s principle of virtual forces
- 4.3. Castigliano’s theorems
- 4.4. Betti’s reciprocal theorem
- 4.5. The variational Rayleigh–Ritz method
- 4.6. Bubnov-Galerkin’s method
- 4.7. Ritz–Lagrange’s method
- References
Chapter 5. The plane elastic problems
- 5.1. Plane strain state
- 5.2. Plane stress state
- 5.3. Compatibility equations for stresses
- 5.4. Airy’s stress function
- 5.5. Examples of solutions of the plane elastic problem
- References
Chapter 6. Plate Bending
- 6.1. Basic concepts and hypotheses
- 6.2. Displacements and deformations in a plates
- 6.3. Stresses and internal forces in a plates
- 6.4. Differential equation of plate bending
- 6.5. Boundary conditions
- 6.6. Cylindrical bending of a rectangular plate
- 6.7. A rectangular plate at sinusoidal load
- 6.8. Solution in double trigonometric series
- 6.9. Application of a single trigonometric series
- 6.10. A rectangular plate on elastic foundation
- 6.11. Circular plate bending
- 6.12. Symmetric bending of a circular plate
- 6.13. Elliptic plate
- 6.14. A circular elastic three-layer plate
- 6.15. A circular three-layer plate on an elastic foundation
- References

Chapter 7.Deformation of a half-space and contact problems
- 7.1. An elastic half-space effected by surface forces
- 7.2. Fundamental solutions for an elastic half-plane
- 7.3. The problem of a punch on an elastic half-plane
- 7.4. The plane contact problem for two elastic bodies
- 7.5. Interaction of a punch on an elastic half-space
- 7.6. Hertzian problem
- References
Chapter 8. Foundations of the theory of plasticity
- 8.1. Plasticity of materials at tension and compression
- 8.2. Plasticity conditions
- 8.3. Simple and complex loading
- 8.4. Hypotheses of the theory of small elastoplastic deformations
- 8.5. Formulation of the problem of small elastoplastic deformations
- 8.6. A method of elastic solutions
- 8.7. Geometric interpretation of loading process
- 8.8. The theory of plastic flow
- 8.9. An example of limiting surface
- 8.10. Foundations of the general mathematical plasticity theory by A. A. Ilyushin
- 8.11. Elastoplastic bending of a circular three-layer plate
- 8.12. Variable loading of elastoplastic bodies
- 8.13. Cyclic loading in temperature field
- 8.14. Cyclic deformation of elastoplastic bodies in neutron field
- References
Chapter 9. Linear viscoelastic continua
- 9.1. Creep and relaxation
- 9.2. Statement of the linear viscoelasticity problems
- 9.3. Types of creep and relaxation kernels
- 9.4. Volterra’s principle
- 9.5. Ilyushin’s experimental and theoretical method
- 9.6. Time-temperature analogy
- 9.7. The theory of ageing
- 9.8. A circular linearly viscoelastic three-layer plate
- References
Chapter 10. Thermoviscoelastoplasticity
- 10.1. Nonlinear viscoelastic continua
- 10.2. Nonlinear viscoelasticity equations accounting for the type of SSS effect
- 10.3. Viscoelastoplastic continua
- 10.4. A method of successive approximations in viscoelastoplasticity problems
- 10.5. A circular viscoelastoplastic three-layer plate
- References
Chapter 11. Thermoviscoelastoplastic characteristics of materials
- 11.1. Aluminum alloy D-16?
- 11.2. Ceramic materials
- 11.3. Polytetrafluoroethylene
- 11.4. Radiation effect on mechanical properties of materials
- 11.5. Calculation of temperature fields in a three-layer plate
- References
Chapter 12. Dynamic problems of the elasticity theory
- 12.1. Statement of dynamic problem of the elasticity theory
- 12.2. Variational principle in dynamics
- 12.3. Free vibrations of elastic bodies
- 12.4. Forced vibrations of elastic bodies
- 12.5. Rayleigh’s inequality and Ritz method
- 12.6. Vibrations of an elastic circular three-layer plate
- 12.7. Resonance vibrations of a circular three-layer plate
- References
- Appendix
Chapter 13
- 13.1. Tensors in Cartesian coordinates
- 13.2. Boundary conditions and displacements in Airy’s functions
- 13.3. Generalized functions
- 13.4. Coefficients of linearly viscoelastic three-layer plate
- 13.5. Special functions
- 13.5.1. Bessel functions
- 13.5.2. Kelvin’s functions
- References

References
Subject index
Author’s index


About the Authors / Editors:
Eduard Starovoitov, DSc
Head of the Ministry of Education, Republic of Belarus, Byelorussian State University of Transport

Dr. Eduard Starovoitov is Head of the Ministry of Education at the Republic of Belarus, Byelorussian State University of Transport. He is the author of over 100 articles and thirteen books and more than 250 analytical reports on fundamental and applied studies in mechanics of solid, mathematics, computer programming, education, and sciences.

Faig Bakhman Ogli Naghiyev, DSc
Leading Research Officer, Ministry of Communications and Information Technology of Azerbaijan Republic, Baku, Azerbaijan.
Formerly Professor of Development and Production of Oil Fields, Azerbaijan State Oil Academy, Baku, Azerbaijan.


Faig Bakhman Ogli Naghiyev, Dsc, is currently Leading Research Officer at the Ministry of Communications and Information Technology of Azerbaijan Republic, Baku, Azerbaijan. He was formerly Professor of Department of Development and Production of Oil Fields at Azerbaijan State Oil Academy, Baku, Azerbaijan. He is the author of over 100 articles and 2 books and more than 200 analytical reports, projects on fundamental and applied studies in hydro-mechanics and thermal physics of multiphase media, mathematics, computer programming, research studies in oil and gas sphere,power-engineering and technological complexes, education and sciences.




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