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Introduction to the Variational Formulation in Mechanics

Author Edgardo O. Taroco , Pablo J. Blanco , Raúl A. Feijóo
Release Date: 2020/02/01
ISBN: 9781119600909
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Introduces readers to the fundamentals and applications of variational formulations in mechanics

Nearly 40 years in the making, this book provides students with the foundation material of mechanics using a variational tapestry. It is centered around the variational structure underlying the Method of Virtual Power (MVP). The variational approach to the modeling of physical systems is the preferred approach to address complex mathematical modeling of both continuum and discrete media. This book provides a unified theoretical framework for the construction of a wide range of multiscale models.

Introduction to the Variational Formulation in Mechanics: Fundamentals and Applications enables readers to develop, on top of solid mathematical (variational) bases, and following clear and precise systematic steps, several models of physical systems, including problems involving multiple scales. It covers: Vector and Tensor Algebra; Vector and Tensor Analysis; Mechanics of Continua; Hyperelastic Materials; Materials Exhibiting Creep; Materials Exhibiting Plasticity; Bending of Beams; Torsion of Bars; Plates and Shells; Heat Transfer; Incompressible Fluid Flow; Multiscale Modeling; and more.

  • A self-contained reader-friendly approach to the variational formulation in the mechanics
  • Examines development of advanced variational formulations in different areas within the field of mechanics using rather simple arguments and explanations
  • Illustrates application of the variational modeling to address hot topics such as the multiscale modeling of complex material behavior
  • Presentation of the Method of Virtual Power as a systematic tool to construct mathematical models of physical systems gives readers a fundamental asset towards the architecture of even more complex (or open) problems

Introduction to the Variational Formulation in Mechanics: Fundamentals and Applications is a ideal book for advanced courses in engineering and mathematics, and an excellent resource for researchers in engineering, computational modeling, and scientific computing.

Table of Contents

  1. Cover
  2. Preface
  3. Part I: Vector and Tensor Algebra and Analysis
    1. 1 Vector and Tensor Algebra
    2. 1.1 Points and Vectors
    3. 1.2 Second‐Order Tensors
    4. 1.3 Third‐Order Tensors
    5. 1.4 Complementary Reading
    6. 2 Vector and Tensor Analysis
    7. 2.1 Differentiation
    8. 2.2 Gradient
    9. 2.3 Divergence
    10. 2.4 Curl
    11. 2.5 Laplacian
    12. 2.6 Integration
    13. 2.7 Coordinates
    14. 2.8 Complementary Reading
  4. Part II: Variational Formulations in Mechanics
    1. 3 Method of Virtual Power
    2. 3.1 Introduction
    3. 3.2 Kinematics
    4. 3.3 Duality and Virtual Power
    5. 3.4 Bodies without Constraints
    6. 3.5 Bodies with Bilateral Constraints
    7. 3.6 Bodies with Unilateral Constraints
    8. 3.7 Lagrangian Description of the Principle of Virtual Power
    9. 3.8 Configurations with Preload and Residual Stresses
    10. 3.9 Linearization of the Principle of Virtual Power
    11. 3.10 Infinitesimal Deformations and Small Displacements
    12. 3.11 Final Remarks
    13. 3.12 Complementary Reading
    14. 4 Hyperelastic Materials at Infinitesimal Strains
    15. 4.1 Introduction
    16. 4.2 Uniaxial Hyperelastic Behavior
    17. 4.3 Three‐Dimensional Hyperelastic Constitutive Laws
    18. 4.4 Equilibrium in Bodies without Constraints
    19. 4.5 Equilibrium in Bodies with Bilateral Constraints
    20. 4.6 Equilibrium in Bodies with Unilateral Constraints
    21. 4.7 Min–Max Principle
    22. 4.8 Three‐Field Functional
    23. 4.9 Castigliano Theorems
    24. 4.10 Elastodynamics Problem
    25. 4.11 Approximate Solution to Variational Problems
    26. 4.12 Complementary Reading
    27. 5 Materials Exhibiting Creep
    28. 5.1 Introduction
    29. 5.2 Phenomenological Aspects of Creep in Metals
    30. 5.3 Influence of Temperature
    31. 5.4 Recovery, Relaxation, Cyclic Loading, and Fatigue
    32. 5.5 Uniaxial Constitutive Equations
    33. 5.6 Three‐Dimensional Constitutive Equations
    34. 5.7 Generalization of the Constitutive Law
    35. 5.8 Constitutive Equations for Structural Components
    36. 5.9 Equilibrium Problem for Steady‐State Creep
    37. 5.10 Castigliano Theorems
    38. 5.11 Examples of Application
    39. 5.12 Approximate Solution to Steady‐State Creep Problems
    40. 5.13 Unsteady Creep Problem
    41. 5.14 Approximate Solutions to Unsteady Creep Formulations
    42. 5.15 Complementary Reading
    43. 6 Materials Exhibiting Plasticity
    44. 6.1 Introduction
    45. 6.2 Elasto‐Plastic Materials
    46. 6.3 Uniaxial Elasto‐Plastic Model
    47. 6.4 Three‐Dimensional Elasto‐Plastic Model
    48. 6.5 Drucker and Hill Postulates
    49. 6.6 Convexity, Normality, and Plastic Potential
    50. 6.7 Plastic Flow Rule
    51. 6.8 Internal Dissipation
    52. 6.9 Common Yield Functions
    53. 6.10 Common Hardening Laws
    54. 6.11 Incremental Variational Principles
    55. 6.12 Incremental Constitutive Equations
    56. 6.13 Complementary Reading
  5. Part III: Modeling of Structural Components
    1. 7 Bending of Beams
    2. 7.1 Introduction
    3. 7.2 Kinematics
    4. 7.3 Generalized Forces
    5. 7.4 Mechanical Equilibrium
    6. 7.5 Timoshenko Beam Model
    7. 7.6 Final Remarks
    8. 8 Torsion of Bars
    9. 8.1 Introduction
    10. 8.2 Kinematics
    11. 8.3 Generalized Forces
    12. 8.4 Mechanical Equilibrium
    13. 8.5 Dual Formulation
    14. 9 Plates and Shells
    15. 9.1 Introduction
    16. 9.2 Geometric Description
    17. 9.3 Differentiation and Integration
    18. 9.4 Principle of Virtual Power
    19. 9.5 Unified Framework for Shell Models
    20. 9.6 Classical Shell Models
    21. 9.7 Constitutive Equations and Internal Constraints
    22. 9.8 Characteristics of Shell Models
    23. 9.9 Basics Notions of Surfaces
  6. Part IV: Other Problems in Physics
    1. 10 Heat Transfer
    2. 10.1 Introduction
    3. 10.2 Kinematics
    4. 10.3 Principle of Thermal Virtual Power
    5. 10.4 Principle of Complementary Thermal Virtual Power
    6. 10.5 Constitutive Equations
    7. 10.6 Principle of Minimum Total Thermal Energy
    8. 10.7 Poisson and Laplace Equations
    9. 11 Incompressible Fluid Flow
    10. 11.1 Introduction
    11. 11.2 Kinematics
    12. 11.3 Principle of Virtual Power
    13. 11.4 Navier–Stokes Equations
    14. 11.5 Stokes Flow
    15. 11.6 Irrotational Flow
    16. 12 High‐Order Continua
    17. 12.1 Introduction
    18. 12.2 Kinematics
    19. 12.3 Principle of Virtual Power
    20. 12.4 Dynamics
    21. 12.5 Micropolar Media
    22. 12.6 Second Gradient Theory
  7. Part V: Multiscale Modeling
    1. 13 Method of Multiscale Virtual Power
    2. 13.1 Introduction
    3. 13.2 Method of Virtual Power
    4. 13.3 Fundamentals of the Multiscale Theory
    5. 13.4 Kinematical Admissibility between Scales
    6. 13.5 Duality in Multiscale Modeling
    7. 13.6 Principle of Multiscale Virtual Power
    8. 13.7 Dual Operators
    9. 13.8 Final Remarks
    10. 14 Applications of Multiscale Modeling
    11. 14.1 Introduction
    12. 14.2 Solid Mechanics with External Forces
    13. 14.3 Mechanics of Incompressible Solid Media
    14. 14.4 Final Remarks
  8. Part V: Appendices
    1. A Definitions and Notations
    2. A.1 Introduction
    3. A.2 Sets
    4. A.3 Functions and Transformations
    5. A.4 Groups
    6. A.5 Morphisms
    7. A.6 Vector Spaces
    8. A.7 Sets and Dependence in Vector Spaces
    9. A.8 Bases and Dimension
    10. A.9 Components
    11. A.10 Sum of Sets and Subspaces
    12. A.11 Linear Manifolds
    13. A.12 Convex Sets and Cones
    14. A.13 Direct Sum of Subspaces
    15. A.14 Linear Transformations
    16. A.15 Canonical Isomorphism
    17. A.16 Algebraic Dual Space
    18. A.17 Algebra in
    19. A.18 Adjoint Operators
    20. A.19 Transposition and Bilinear Functions
    21. A.20 Inner Product Spaces
    22. B Elements of Real and Functional Analysis
    23. B.1 Introduction
    24. B.2 Sequences
    25. B.3 Limit and Continuity of Functions
    26. B.4 Metric Spaces
    27. B.5 Normed Spaces
    28. B.6 Quotient Space
    29. B.7 Linear Transformations in Normed Spaces
    30. B.8 Topological Dual Space
    31. B.9 Weak and Strong Convergence
    32. C Functionals and the Gâteaux DerivativeFunctionals and the Gâteaux Derivative
    33. C.1 Introduction
    34. C.2 Properties of Operator
    35. C.3 Convexity and Semi‐Continuity
    36. C.4 Gâteaux Differential
    37. C.5 Minimization of Convex Functionals
  9. References
  10. Index
  11. End User License Agreement
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