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:
Preface xv Part I Vector and Tensor Algebra and Analysis 1 1 Vector and Tensor Algebra 3 1.1 Points and Vectors 3 1.2 Second-Order Tensors 6 1.3 Third-Order Tensors 17 1.4 Complementary Reading 22 2 Vector and Tensor Analysis 23 2.1 Differentiation 23 2.2 Gradient 28 2.3 Divergence 30 2.4 Curl 32 2.5 Laplacian 34 2.6 Integration 35 2.7 Coordinates 38 2.8 Complementary Reading 45 Part II Variational Formulations in Mechanics 47 3 Method of Virtual Power 49 3.1 Introduction 49 3.2 Kinematics 50 3.2.1 Body and Deformations 50 3.2.2 Motion: Deformation Rate 55 3.2.3 Motion Actions: Kinematical Constraints 61 3.3 Duality and Virtual Power 66 3.3.1 Motion Actions and Forces 67 3.3.2 Deformation Actions and Internal Stresses 69 3.3.3 Mechanical Models and the Equilibrium Operator 71 3.4 Bodies without Constraints 74 3.4.1 Principle of Virtual Power 75 3.4.2 Principle of Complementary Virtual Power 80 3.5 Bodies with Bilateral Constraints 81 3.5.1 Principle of Virtual Power 81 3.5.2 Principle of Complementary Virtual Power 86 3.6 Bodies with Unilateral Constraints 87 3.6.1 Principle of Virtual Power 89 3.6.2 Principle of Complementary Virtual Power 92 3.7 Lagrangian Description of the Principle of Virtual Power 94 3.8 Configurations with Preload and Residual Stresses 97 3.9 Linearization of the Principle of Virtual Power 100 3.9.1 Preliminary Results 101 3.9.2 Known Spatial Configuration 102 3.9.3 Known Material Configuration 102 3.10 Infinitesimal Deformations and Small Displacements 103 3.10.1 Bilateral Constraints 104 3.10.2 Unilateral Constraints 105 3.11 Final Remarks 106 3.12 Complementary Reading 107 4 Hyperelastic Materials at Infinitesimal Strains 109 4.1 Introduction 109 4.2 Uniaxial Hyperelastic Behavior 109 4.3 Three-Dimensional Hyperelastic Constitutive Laws 113 4.4 Equilibrium in Bodies without Constraints 116 4.4.1 Principle of Virtual Work 117 4.4.2 Principle of Minimum Total Potential Energy 117 4.4.3 Local Equations and Boundary Conditions 118 4.4.4 Principle of Complementary Virtual Work 120 4.4.5 Principle of Minimum Complementary Energy 121 4.4.6 Additional Remarks 122 4.5 Equilibrium in Bodies with Bilateral Constraints 123 4.5.1 Principle of Virtual Work 125 4.5.2 Principle of Minimum Total Potential Energy 125 4.5.3 Principle of Complementary Virtual Work 126 4.5.4 Principle of Minimum Complementary Energy 127 4.6 Equilibrium in Bodies with Unilateral Constraints 128 4.6.1 Principle of Virtual Work 128 4.6.2 Principle of Minimum Total Potential Energy 128 4.6.3 Principle of Complementary Virtual Work 129 4.6.4 Principle of Minimum Complementary Energy 130 4.7 Min–Max Principle 131 4.7.1 Hellinger–Reissner Functional 131 4.7.2 Hellinger–Reissner Principle 133 4.8 Three-Field Functional 134 4.9 Castigliano Theorems 136 4.9.1 First and Second Theorems 136 4.9.2 Bounds for Displacements and Generalized Loads 139 4.10 Elastodynamics Problem 144 4.11 Approximate Solution to Variational Problems 148 4.11.1 Elastostatics Problem 148 4.11.2 Hellinger–Reissner Principle 154 4.11.3 Generalized Variational Principle 156 4.11.4 Contact Problems in Elastostatics 158 4.12 Complementary Reading 162 5 Materials Exhibiting Creep 165 5.1 Introduction 165 5.2 Phenomenological Aspects of Creep in Metals 165 5.3 Influence of Temperature 168 5.4 Recovery, Relaxation, Cyclic Loading, and Fatigue 170 5.5 Uniaxial Constitutive Equations 173 5.6 Three-Dimensional Constitutive Equations 182 5.7 Generalization of the Constitutive Law 188 5.8 Constitutive Equations for Structural Components 191 5.8.1 Bending of Beams 192 5.8.2 Bending, Extension, and Compression of Beams 195 5.9 Equilibrium Problem for Steady-State Creep 199 5.9.1 Mechanical Equilibrium 199 5.9.2 Variational Formulation 201 5.9.3 Variational Principles of Minimum 205 5.10 Castigliano Theorems 209 5.10.1 First and Second Theorems 209 5.10.2 Bounds for Velocities and Generalized Loads 211 5.11 Examples of Application 214 5.11.1 Disk Rotating with Constant Angular Velocity 214 5.11.2 Cantilevered Beam with Uniform Load 217 5.12 Approximate Solution to Steady-State Creep Problems 219 5.13 Unsteady Creep Problem 225 5.14 Approximate Solutions to Unsteady Creep Formulations 227 5.15 Complementary Reading 228 6 Materials Exhibiting Plasticity 229 6.1 Introduction 229 6.2 Elasto-Plastic Materials 229 6.3 Uniaxial Elasto-Plastic Model 235 6.3.1 Elastic Relation 235 6.3.2 Yield Criterion 236 6.3.3 Hardening Law 238 6.3.4 Plastic Flow Rule 240 6.4 Three-Dimensional Elasto-Plastic Model 243 6.4.1 Elastic Relation 244 6.4.2 Yield Criterion and Hardening Law 246 6.4.3 Potential Plastic Flow 249 6.5 Drucker and Hill Postulates 253 6.6 Convexity, Normality, and Plastic Potential 255 6.6.1 Normality Law and a Rationale for the Potential Law 255 6.6.2 Convexity of the Admissible Region 257 6.7 Plastic Flow Rule 258 6.8 Internal Dissipation 260 6.9 Common Yield Functions 262 6.9.1 The von Mises Criterion 263 6.9.2 The Tresca Criterion 264 6.10 Common Hardening Laws 266 6.11 Incremental Variational Principles 267 6.11.1 Principle of Minimum for the Velocity 268 6.11.2 Principle of Minimum for the Stress Rate 269 6.11.3 Uniqueness of the Stress Field 270 6.11.4 Variational Inequality for the Stress 270 6.11.5 Principle of Minimum with Two Fields 271 6.12 Incremental Constitutive Equations 272 6.12.1 Constitutive Equations for Rates 273 6.12.2 Constitutive Equations for Increments 275 6.12.3 Variational Principle in Finite Increments 278 6.13 Complementary Reading 279 Part III Modeling of Structural Components 281 7 Bending of Beams 285 7.1 Introduction 285 7.2 Kinematics 285 7.3 Generalized Forces 289 7.4 Mechanical Equilibrium 290 7.5 Timoshenko Beam Model 294 7.6 Final Remarks 298 8 Torsion of Bars 301 8.1 Introduction 301 8.2 Kinematics 301 8.3 Generalized Forces 304 8.4 Mechanical Equilibrium 305 8.5 Dual Formulation 309 9 Plates and Shells 315 9.1 Introduction 315 9.2 Geometric Description 316 9.3 Differentiation and Integration 320 9.4 Principle of Virtual Power 323 9.5 Unified Framework for Shell Models 326 9.6 Classical Shell Models 332 9.6.1 Naghdi Model 332 9.6.2 Kirchhoff–Love Model 335 9.6.3 Love Model 340 9.6.4 Koiter Model 342 9.6.5 Sanders Model 344 9.6.6 Donnell–Mushtari–Vlasov Model 346 9.7 Constitutive Equations and Internal Constraints 347 9.7.1 Preliminary Concepts 348 9.7.2 Model with Naghdi Hypothesis 350 9.7.3 Model with Kirchhoff–Love Hypothesis 357 9.8 Characteristics of Shell Models 360 9.8.1 Relation Between Generalized Stresses 360 9.8.2 Equilibrium Around the Normal 361 9.8.2.1 Kirchhoff–Love Model 361 9.8.2.2 Love Model 362 9.8.2.3 Koiter Model 363 9.8.2.4 Sanders Model 363 9.8.3 Reactive Generalized Stresses 364 9.8.3.1 Reactions in the Naghdi Model 364 9.8.3.2 Reactions in the Kirchhoff–Love Model 366 9.9 Basics Notions of Surfaces 369 9.9.1 Preliminaries 369 9.9.2 First Fundamental Form 370 9.9.3 Second Fundamental Form 372 9.9.4 Third Fundamental Form 375 9.9.5 Complementary Properties 375 Part IV Other Problems in Physics 377 10 Heat Transfer 379 10.1 Introduction 379 10.2 Kinematics 379 10.3 Principle of Thermal Virtual Power 381 10.4 Principle of Complementary Thermal Virtual Power 386 10.5 Constitutive Equations 388 10.6 Principle of Minimum Total Thermal Energy 390 10.7 Poisson and Laplace Equations 390 11 Incompressible Fluid Flow 393 11.1 Introduction 393 11.2 Kinematics 394 11.3 Principle of Virtual Power 396 11.4 Navier–Stokes Equations 403 11.5 Stokes Flow 405 11.6 Irrotational Flow 407 12 High-Order Continua 411 12.1 Introduction 411 12.2 Kinematics 412 12.3 Principle of Virtual Power 418 12.4 Dynamics 425 12.5 Micropolar Media 427 12.6 Second Gradient Theory 429 Part V Multiscale Modeling 435 13 Method of Multiscale Virtual Power 439 13.1 Introduction 439 13.2 Method of Virtual Power 439 13.2.1 Kinematics 439 13.2.2 Duality 442 13.2.3 Principle of Virtual Power 445 13.2.4 Equilibrium Problem 446 13.3 Fundamentals of the Multiscale Theory 447 13.4 Kinematical Admissibility between Scales 449 13.4.1 Macroscale Kinematics 449 13.4.2 Microscale Kinematics 451 13.4.3 Insertion Operators 453 13.4.4 Homogenization Operators 456 13.4.5 Kinematical Admissibility 458 13.5 Duality in Multiscale Modeling 462 13.5.1 Macroscale Virtual Power 462 13.5.2 Microscale Virtual Power 464 13.6 Principle of Multiscale Virtual Power 467 13.7 Dual Operators 468 13.7.1 Microscale Equilibrium 468 13.7.2 Homogenization of Generalized Stresses 470 13.7.3 Homogenization of Generalized Forces 472 13.8 Final Remarks 473 14 Applications of Multiscale Modeling 475 14.1 Introduction 475 14.2 Solid Mechanics with External Forces 475 14.2.1 Multiscale Kinematics 476 14.2.2 Characterization of Virtual Power 479 14.2.3 Principle of Multiscale Virtual Power 480 14.2.4 Equilibrium Problem and Homogenization 482 14.2.5 Tangent Operators 487 14.3 Mechanics of Incompressible Solid Media 490 14.3.1 Principle of Virtual Power 491 14.3.2 Multiscale Kinematics 493 14.3.3 Principle of Multiscale Virtual Power 495 14.3.4 Incompressibility and Material Configuration 497 14.4 Final Remarks 500 Part VI Appendices 501 A Definitions and Notations 503 A.1 Introduction 503 A.2 Sets 503 A.3 Functions and Transformations 504 A.4 Groups 507 A.5 Morphisms 509 A.6 Vector Spaces 509 A.7 Sets and Dependence in Vector Spaces 512 A.8 Bases and Dimension 513 A.9 Components 514 A.10 Sum of Sets and Subspaces 516 A.11 Linear Manifolds 516 A.12 Convex Sets and Cones 516 A.13 Direct Sum of Subspaces 517 A.14 Linear Transformations 517 A.15 Canonical Isomorphism 522 A.16 Algebraic Dual Space 523 A.16.1 Orthogonal Complement 524 A.16.2 Positive and Negative Conjugate Cones 525 A.17 Algebra in V 526 A.18 Adjoint Operators 528 A.19 Transposition and Bilinear Functions 529 A.20 Inner Product Spaces 532 B Elements of Real and Functional Analysis 539 B.1 Introduction 539 B.2 Sequences 541 B.3 Limit and Continuity of Functions 542 B.4 Metric Spaces 544 B.5 Normed Spaces 546 B.6 Quotient Space 549 B.7 Linear Transformations in Normed Spaces 550 B.8 Topological Dual Space 552 B.9 Weak and Strong Convergence 553 C Functionals and the Gâteaux Derivative 555 C.1 Introduction 555 C.2 Properties of Operator 𝒦 555 C.3 Convexity and Semi-Continuity 556 C.4 Gâteaux Differential 557 C.5 Minimization of Convex Functionals 557 References 559 Index 575