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 selfcontained readerfriendly 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
 Cover
 Preface
 Part I: Vector and Tensor Algebra and Analysis

Part II: Variational Formulations in Mechanics

3 Method of Virtual Power
 3.1 Introduction
 3.2 Kinematics
 3.3 Duality and Virtual Power
 3.4 Bodies without Constraints
 3.5 Bodies with Bilateral Constraints
 3.6 Bodies with Unilateral Constraints
 3.7 Lagrangian Description of the Principle of Virtual Power
 3.8 Configurations with Preload and Residual Stresses
 3.9 Linearization of the Principle of Virtual Power
 3.10 Infinitesimal Deformations and Small Displacements
 3.11 Final Remarks
 3.12 Complementary Reading

4 Hyperelastic Materials at Infinitesimal Strains
 4.1 Introduction
 4.2 Uniaxial Hyperelastic Behavior
 4.3 Three‐Dimensional Hyperelastic Constitutive Laws
 4.4 Equilibrium in Bodies without Constraints
 4.5 Equilibrium in Bodies with Bilateral Constraints
 4.6 Equilibrium in Bodies with Unilateral Constraints
 4.7 Min–Max Principle
 4.8 Three‐Field Functional
 4.9 Castigliano Theorems
 4.10 Elastodynamics Problem
 4.11 Approximate Solution to Variational Problems
 4.12 Complementary Reading

5 Materials Exhibiting Creep
 5.1 Introduction
 5.2 Phenomenological Aspects of Creep in Metals
 5.3 Influence of Temperature
 5.4 Recovery, Relaxation, Cyclic Loading, and Fatigue
 5.5 Uniaxial Constitutive Equations
 5.6 Three‐Dimensional Constitutive Equations
 5.7 Generalization of the Constitutive Law
 5.8 Constitutive Equations for Structural Components
 5.9 Equilibrium Problem for Steady‐State Creep
 5.10 Castigliano Theorems
 5.11 Examples of Application
 5.12 Approximate Solution to Steady‐State Creep Problems
 5.13 Unsteady Creep Problem
 5.14 Approximate Solutions to Unsteady Creep Formulations
 5.15 Complementary Reading

6 Materials Exhibiting Plasticity
 6.1 Introduction
 6.2 Elasto‐Plastic Materials
 6.3 Uniaxial Elasto‐Plastic Model
 6.4 Three‐Dimensional Elasto‐Plastic Model
 6.5 Drucker and Hill Postulates
 6.6 Convexity, Normality, and Plastic Potential
 6.7 Plastic Flow Rule
 6.8 Internal Dissipation
 6.9 Common Yield Functions
 6.10 Common Hardening Laws
 6.11 Incremental Variational Principles
 6.12 Incremental Constitutive Equations
 6.13 Complementary Reading

3 Method of Virtual Power
 Part III: Modeling of Structural Components
 Part IV: Other Problems in Physics
 Part V: Multiscale Modeling

Part V: Appendices

A Definitions and Notations
 A.1 Introduction
 A.2 Sets
 A.3 Functions and Transformations
 A.4 Groups
 A.5 Morphisms
 A.6 Vector Spaces
 A.7 Sets and Dependence in Vector Spaces
 A.8 Bases and Dimension
 A.9 Components
 A.10 Sum of Sets and Subspaces
 A.11 Linear Manifolds
 A.12 Convex Sets and Cones
 A.13 Direct Sum of Subspaces
 A.14 Linear Transformations
 A.15 Canonical Isomorphism
 A.16 Algebraic Dual Space
 A.17 Algebra in
 A.18 Adjoint Operators
 A.19 Transposition and Bilinear Functions
 A.20 Inner Product Spaces
 B Elements of Real and Functional Analysis
 C Functionals and the Gâteaux DerivativeFunctionals and the Gâteaux Derivative

A Definitions and Notations
 References
 Index
 End User License Agreement
Product information
 Title: Introduction to the Variational Formulation in Mechanics
 Author(s):
 Release date: February 2020
 Publisher(s): Wiley
 ISBN: 9781119600909
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