Introduction to the Variational Formulation in Mechanics

Introduction to the Variational Formulation in Mechanics

by Edgardo O. Taroco, Pablo J. Blanco, Raúl A. Feijóo
Released February 2020
Publisher(s): Wiley
ISBN: 9781119600909

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Book description

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

Product information

  • Title: Introduction to the Variational Formulation in Mechanics
  • Author(s): Edgardo O. Taroco, Pablo J. Blanco, Raúl A. Feijóo
  • Release date: February 2020
  • Publisher(s): Wiley
  • ISBN: 9781119600909