Introduction to the Variational Formulation in Mechanics

EDGARDO OMAR TAROCO, PHD, was a Full Researcher at the National Laboratory for Scientific Computing (LNCC/MCTIC) and at the National Institute of Science and Technology in Medicine Assisted by Scientific Computing (INCT-MACC), Petrópolis, Brazil. PABLO JAVIER BLANCO, PHD, is a Full Researcher at the National Laboratory for Scientific Computing (LNCC/MCTIC) and at the National Institute of Science and Technology in Medicine Assisted by Scientific Computing (INCT-MACC), Petrópolis, Brazil, and Associate Professor at the Catholic University of Petrópolis, Brazil. RAÚL ANTONINO FEIJÓO, is a Full Researcher at the National Laboratory for Scientific Computing (LNCC/MCTIC) and at the National Institute of Science and Technology in Medicine Assisted by Scientific Computing (INCT-MACC), Petrópolis, Brazil.

• Preface xvPart I Vector and Tensor Algebra and Analysis 11 Vector and Tensor Algebra 31.1 Points and Vectors 31.2 Second-Order Tensors 61.3 Third-Order Tensors 171.4 Complementary Reading 222 Vector and Tensor Analysis 232.1 Differentiation 232.2 Gradient 282.3 Divergence 302.4 Curl 322.5 Laplacian 342.6 Integration 352.7 Coordinates 382.8 Complementary Reading 45Part II Variational Formulations in Mechanics 473 Method of Virtual Power 493.1 Introduction 493.2 Kinematics 503.2.1 Body and Deformations 503.2.2 Motion: Deformation Rate 553.2.3 Motion Actions: Kinematical Constraints 613.3 Duality and Virtual Power 663.3.1 Motion Actions and Forces 673.3.2 Deformation Actions and Internal Stresses 693.3.3 Mechanical Models and the Equilibrium Operator 713.4 Bodies without Constraints 743.4.1 Principle of Virtual Power 753.4.2 Principle of Complementary Virtual Power 803.5 Bodies with Bilateral Constraints 813.5.1 Principle of Virtual Power 813.5.2 Principle of Complementary Virtual Power 863.6 Bodies with Unilateral Constraints 873.6.1 Principle of Virtual Power 893.6.2 Principle of Complementary Virtual Power 923.7 Lagrangian Description of the Principle of Virtual Power 943.8 Configurations with Preload and Residual Stresses 973.9 Linearization of the Principle of Virtual Power 1003.9.1 Preliminary Results 1013.9.2 Known Spatial Configuration 1023.9.3 Known Material Configuration 1023.10 Infinitesimal Deformations and Small Displacements 1033.10.1 Bilateral Constraints 1043.10.2 Unilateral Constraints 1053.11 Final Remarks 1063.12 Complementary Reading 1074 Hyperelastic Materials at Infinitesimal Strains 1094.1 Introduction 1094.2 Uniaxial Hyperelastic Behavior 1094.3 Three-Dimensional Hyperelastic Constitutive Laws 1134.4 Equilibrium in Bodies without Constraints 1164.4.1 Principle of Virtual Work 1174.4.2 Principle of Minimum Total Potential Energy 1174.4.3 Local Equations and Boundary Conditions 1184.4.4 Principle of Complementary Virtual Work 1204.4.5 Principle of Minimum Complementary Energy 1214.4.6 Additional Remarks 1224.5 Equilibrium in Bodies with Bilateral Constraints 1234.5.1 Principle of Virtual Work 1254.5.2 Principle of Minimum Total Potential Energy 1254.5.3 Principle of Complementary Virtual Work 1264.5.4 Principle of Minimum Complementary Energy 1274.6 Equilibrium in Bodies with Unilateral Constraints 1284.6.1 Principle of Virtual Work 1284.6.2 Principle of Minimum Total Potential Energy 1284.6.3 Principle of Complementary Virtual Work 1294.6.4 Principle of Minimum Complementary Energy 1304.7 Min–Max Principle 1314.7.1 Hellinger–Reissner Functional 1314.7.2 Hellinger–Reissner Principle 1334.8 Three-Field Functional 1344.9 Castigliano Theorems 1364.9.1 First and Second Theorems 1364.9.2 Bounds for Displacements and Generalized Loads 1394.10 Elastodynamics Problem 1444.11 Approximate Solution to Variational Problems 1484.11.1 Elastostatics Problem 1484.11.2 Hellinger–Reissner Principle 1544.11.3 Generalized Variational Principle 1564.11.4 Contact Problems in Elastostatics 1584.12 Complementary Reading 1625 Materials Exhibiting Creep 1655.1 Introduction 1655.2 Phenomenological Aspects of Creep in Metals 1655.3 Influence of Temperature 1685.4 Recovery, Relaxation, Cyclic Loading, and Fatigue 1705.5 Uniaxial Constitutive Equations 1735.6 Three-Dimensional Constitutive Equations 1825.7 Generalization of the Constitutive Law 1885.8 Constitutive Equations for Structural Components 1915.8.1 Bending of Beams 1925.8.2 Bending, Extension, and Compression of Beams 1955.9 Equilibrium Problem for Steady-State Creep 1995.9.1 Mechanical Equilibrium 1995.9.2 Variational Formulation 2015.9.3 Variational Principles of Minimum 2055.10 Castigliano Theorems 2095.10.1 First and Second Theorems 2095.10.2 Bounds for Velocities and Generalized Loads 2115.11 Examples of Application 2145.11.1 Disk Rotating with Constant Angular Velocity 2145.11.2 Cantilevered Beam with Uniform Load 2175.12 Approximate Solution to Steady-State Creep Problems 2195.13 Unsteady Creep Problem 2255.14 Approximate Solutions to Unsteady Creep Formulations 2275.15 Complementary Reading 2286 Materials Exhibiting Plasticity 2296.1 Introduction 2296.2 Elasto-Plastic Materials 2296.3 Uniaxial Elasto-Plastic Model 2356.3.1 Elastic Relation 2356.3.2 Yield Criterion 2366.3.3 Hardening Law 2386.3.4 Plastic Flow Rule 2406.4 Three-Dimensional Elasto-Plastic Model 2436.4.1 Elastic Relation 2446.4.2 Yield Criterion and Hardening Law 2466.4.3 Potential Plastic Flow 2496.5 Drucker and Hill Postulates 2536.6 Convexity, Normality, and Plastic Potential 2556.6.1 Normality Law and a Rationale for the Potential Law 2556.6.2 Convexity of the Admissible Region 2576.7 Plastic Flow Rule 2586.8 Internal Dissipation 2606.9 Common Yield Functions 2626.9.1 The von Mises Criterion 2636.9.2 The Tresca Criterion 2646.10 Common Hardening Laws 2666.11 Incremental Variational Principles 2676.11.1 Principle of Minimum for the Velocity 2686.11.2 Principle of Minimum for the Stress Rate 2696.11.3 Uniqueness of the Stress Field 2706.11.4 Variational Inequality for the Stress 2706.11.5 Principle of Minimum with Two Fields 2716.12 Incremental Constitutive Equations 2726.12.1 Constitutive Equations for Rates 2736.12.2 Constitutive Equations for Increments 2756.12.3 Variational Principle in Finite Increments 2786.13 Complementary Reading 279Part III Modeling of Structural Components 2817 Bending of Beams 2857.1 Introduction 2857.2 Kinematics 2857.3 Generalized Forces 2897.4 Mechanical Equilibrium 2907.5 Timoshenko Beam Model 2947.6 Final Remarks 2988 Torsion of Bars 3018.1 Introduction 3018.2 Kinematics 3018.3 Generalized Forces 3048.4 Mechanical Equilibrium 3058.5 Dual Formulation 3099 Plates and Shells 3159.1 Introduction 3159.2 Geometric Description 3169.3 Differentiation and Integration 3209.4 Principle of Virtual Power 3239.5 Unified Framework for Shell Models 3269.6 Classical Shell Models 3329.6.1 Naghdi Model 3329.6.2 Kirchhoff–Love Model 3359.6.3 Love Model 3409.6.4 Koiter Model 3429.6.5 Sanders Model 3449.6.6 Donnell–Mushtari–Vlasov Model 3469.7 Constitutive Equations and Internal Constraints 3479.7.1 Preliminary Concepts 3489.7.2 Model with Naghdi Hypothesis 3509.7.3 Model with Kirchhoff–Love Hypothesis 3579.8 Characteristics of Shell Models 3609.8.1 Relation Between Generalized Stresses 3609.8.2 Equilibrium Around the Normal 3619.8.2.1 Kirchhoff–Love Model 3619.8.2.2 Love Model 3629.8.2.3 Koiter Model 3639.8.2.4 Sanders Model 3639.8.3 Reactive Generalized Stresses 3649.8.3.1 Reactions in the Naghdi Model 3649.8.3.2 Reactions in the Kirchhoff–Love Model 3669.9 Basics Notions of Surfaces 3699.9.1 Preliminaries 3699.9.2 First Fundamental Form 3709.9.3 Second Fundamental Form 3729.9.4 Third Fundamental Form 3759.9.5 Complementary Properties 375Part IV Other Problems in Physics 37710 Heat Transfer 37910.1 Introduction 37910.2 Kinematics 37910.3 Principle of Thermal Virtual Power 38110.4 Principle of Complementary Thermal Virtual Power 38610.5 Constitutive Equations 38810.6 Principle of Minimum Total Thermal Energy 39010.7 Poisson and Laplace Equations 39011 Incompressible Fluid Flow 39311.1 Introduction 39311.2 Kinematics 39411.3 Principle of Virtual Power 39611.4 Navier–Stokes Equations 40311.5 Stokes Flow 40511.6 Irrotational Flow 40712 High-Order Continua 41112.1 Introduction 41112.2 Kinematics 41212.3 Principle of Virtual Power 41812.4 Dynamics 42512.5 Micropolar Media 42712.6 Second Gradient Theory 429Part V Multiscale Modeling 43513 Method of Multiscale Virtual Power 43913.1 Introduction 43913.2 Method of Virtual Power 43913.2.1 Kinematics 43913.2.2 Duality 44213.2.3 Principle of Virtual Power 44513.2.4 Equilibrium Problem 44613.3 Fundamentals of the Multiscale Theory 44713.4 Kinematical Admissibility between Scales 44913.4.1 Macroscale Kinematics 44913.4.2 Microscale Kinematics 45113.4.3 Insertion Operators 45313.4.4 Homogenization Operators 45613.4.5 Kinematical Admissibility 45813.5 Duality in Multiscale Modeling 46213.5.1 Macroscale Virtual Power 46213.5.2 Microscale Virtual Power 46413.6 Principle of Multiscale Virtual Power 46713.7 Dual Operators 46813.7.1 Microscale Equilibrium 46813.7.2 Homogenization of Generalized Stresses 47013.7.3 Homogenization of Generalized Forces 47213.8 Final Remarks 47314 Applications of Multiscale Modeling 47514.1 Introduction 47514.2 Solid Mechanics with External Forces 47514.2.1 Multiscale Kinematics 47614.2.2 Characterization of Virtual Power 47914.2.3 Principle of Multiscale Virtual Power 48014.2.4 Equilibrium Problem and Homogenization 48214.2.5 Tangent Operators 48714.3 Mechanics of Incompressible Solid Media 49014.3.1 Principle of Virtual Power 49114.3.2 Multiscale Kinematics 49314.3.3 Principle of Multiscale Virtual Power 49514.3.4 Incompressibility and Material Configuration 49714.4 Final Remarks 500Part VI Appendices 501A Definitions and Notations 503A.1 Introduction 503A.2 Sets 503A.3 Functions and Transformations 504A.4 Groups 507A.5 Morphisms 509A.6 Vector Spaces 509A.7 Sets and Dependence in Vector Spaces 512A.8 Bases and Dimension 513A.9 Components 514A.10 Sum of Sets and Subspaces 516A.11 Linear Manifolds 516A.12 Convex Sets and Cones 516A.13 Direct Sum of Subspaces 517A.14 Linear Transformations 517A.15 Canonical Isomorphism 522A.16 Algebraic Dual Space 523A.16.1 Orthogonal Complement 524A.16.2 Positive and Negative Conjugate Cones 525A.17 Algebra in V 526A.18 Adjoint Operators 528A.19 Transposition and Bilinear Functions 529A.20 Inner Product Spaces 532B Elements of Real and Functional Analysis 539B.1 Introduction 539B.2 Sequences 541B.3 Limit and Continuity of Functions 542B.4 Metric Spaces 544B.5 Normed Spaces 546B.6 Quotient Space 549B.7 Linear Transformations in Normed Spaces 550B.8 Topological Dual Space 552B.9 Weak and Strong Convergence 553C Functionals and the Gâteaux Derivative 555C.1 Introduction 555C.2 Properties of Operator 𝒦 555C.3 Convexity and Semi-Continuity 556C.4 Gâteaux Differential 557C.5 Minimization of Convex Functionals 557References 559Index 575