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Fundamentals of Continuum Mechanics

Book Description

Fundamentals of Continuum Mechanics provides a clear and rigorous presentation of continuum mechanics for engineers, physicists, applied mathematicians, and materials scientists. This book emphasizes the role of thermodynamics in constitutive modeling, with detailed application to nonlinear elastic solids, viscous fluids, and modern smart materials. While emphasizing advanced material modeling, special attention is also devoted to developing novel theories for incompressible and thermally expanding materials. A wealth of carefully chosen examples and exercises illuminate the subject matter and facilitate self-study.
  • Uses direct notation for a clear and straightforward presentation of the mathematics, leading to a better understanding of the underlying physics
  • Covers high-interest research areas such as small- and large-deformation continuum electrodynamics, with application to smart materials used in intelligent systems and structures
  • Offers a unique approach to modeling incompressibility and thermal expansion, based on the authors’ own research

Table of Contents

  1. Cover image
  2. Title page
  3. Table of Contents
  4. Copyright
  5. Dedication
  6. Preface
    1. Continuum Mechanics: The New Pedagogy
    2. Acknowledgments
  7. PART I: THE BEGINNING
    1. Chapter 1: What Is a Continuum?
      1. Abstract
    2. Chapter 2: Our Mathematical Playground
      1. Abstract
      2. 2.1 Real numbers and euclidean space
      3. 2.2 Tensor algebra
      4. 2.3 Eigenvalues, eigenvectors, polar decomposition, invariants
      5. 2.4 Tensors of order three and four
      6. 2.5 Tensor calculus
      7. 2.6 Curvilinear coordinates
  8. PART II: KINEMATICS, KINETICS, AND THE FUNDAMENTAL LAWS OF MECHANICS AND THERMODYNAMICS
    1. Chapter 3: Kinematics: Motion and Deformation
      1. Abstract
      2. 3.1 Body, configuration, motion, displacement
      3. 3.2 Material derivative, velocity, acceleration
      4. Exercises
      5. 3.3 Deformation and strain
      6. Exercises
      7. Exercises
      8. Exercises
      9. 3.4 Velocity gradient, rate of deformation tensor, vorticity tensor
      10. Exercises
      11. 3.5 Material point, material line, material surface, material volume
      12. 3.6 Volume elements and surface elements in volume and surface integrations
    2. Chapter 4: The Fundamental Laws of Thermomechanics
      1. Abstract
      2. 4.1 Mass
      3. 4.2 Forces and moments, linear and angular momentum
      4. 4.3 Equations of motion (mechanical conservation laws)
      5. 4.4 The first law of thermodynamics (conservation of energy)
      6. 4.5 The transport and localization theorems
      7. Exercises
      8. 4.6 Cauchy stress tensor, heat flux vector
      9. 4.7 The energy theorem and stress power
      10. 4.8 Local forms of the conservation laws
      11. Exercises
      12. 4.9 Lagrangian forms of the integral conservation laws
      13. 4.10 Piola-kirchhoff stress tensors, referential heat flux vector
      14. Exercises
      15. 4.11 The lagrangian form of the energy theorem
      16. 4.12 Local conservation laws in lagrangian form
      17. Exercises
      18. 4.13 The second law of thermodynamics
      19. Exercises
  9. PART III: CONSTITUTIVE MODELING
    1. Chapter 5: Constitutive Modeling in Mechanics and Thermomechanics
      1. Abstract
      2. Part I: Mechanics
      3. Part II: Thermomechanics
      4. 5.3 Fundamental laws, constitutive equations, thermomechanical processes
      5. Exercises
      6. 5.4 Restrictions on the constitutive equations
      7. Exercises
    2. Chapter 6: Nonlinear Elasticity
      1. Abstract
      2. 6.1 Mechanical theory
      3. 6.2 Thermomechanical theory
      4. Exercises
      5. Exercises
      6. 6.3 Strain energy models
    3. Chapter 7: Fluid Mechanics
      1. Abstract
      2. 7.1 Mechanical theory
      3. Exercises
      4. Exercises
      5. 7.2 Thermomechanical theory
      6. Exercises
      7. Exercises
    4. Chapter 8: Incompressibility and Thermal Expansion
      1. Abstract
      2. 8.1 Introduction
      3. 8.2 Newtonian fluids
      4. Exercise
      5. Exercises
      6. Exercises
      7. Exercises
      8. Exercises
      9. Exercises
      10. Exercise
      11. 8.3 Nonlinear elastic solids
      12. Exercises
  10. PART IV: BEYOND MECHANICS AND THERMOMECHANICS
    1. Chapter 9: Modeling of Thermo-Electro-Magneto-Mechanical Behavior, with Application to Smart Materials
      1. Abstract
      2. 9.1 The fundamental laws of continuum electrodynamics: integral forms
      3. Exercises
      4. 9.2 The fundamental laws of continuum electrodynamics: pointwise forms
      5. Exercises
      6. Exercises
      7. 9.3 Modeling of the effective electromagnetic fields
      8. Exercise
      9. 9.4 Modeling of the electromagnetically induced coupling terms
      10. Exercises
      11. 9.5 Thermo-electro-magneto-mechanical process
      12. 9.6 Constitutive model development for thermo-electro-magneto-elastic materials: large-deformation theory
      13. Exercise
      14. Exercise
      15. Exercises
      16. 9.7 Constitutive model development for thermo-electro-magneto-elastic materials: small-deformation theory
      17. 9.8 Linear, reversible, thermo-electro-magneto-mechanical processes
      18. 9.9 Specialization of the small-deformation thermo-electro-magneto-elastic framework to piezoelectric materials
      19. Exercise
  11. Appendix A: Different Notions of Invariance
  12. Appendix B: The Physical Basis of Constitutive Assumptions
  13. Appendix C: Isotropic Tensors
  14. Appendix D: A Family of Thermomechanical Processes
  15. Appendix E: Energy Formulations and Stability Conditions for Newtonian Fluids
    1. E.1 Governing equations
    2. E.2 Stability conditions
  16. Appendix F: Additional Energy Formulations for Thermo-Electro-Magneto-Mechanical Materials
    1. F.1 Deformation-temperature-electric displacement-magnetic induction formulation
  17. Bibliography
  18. Index