Introduction to the Variational Formulation in Mechanics : Fundamentals and Applications.

Intro -- Title Page -- Copyright Page -- Contents -- Preface -- Part I Vector and Tensor Algebra and Analysis -- Chapter 1 Vector and Tensor Algebra -- 1.1 Points and Vectors -- 1.2 Second-Order Tensors -- 1.3 Third-Order Tensors -- 1.4 Complementary Reading -- Chapter 2 Vector and Tensor Analysis -- 2.1 Differentiation -- 2.2 Gradient -- 2.3 Divergence -- 2.4 Curl -- 2.5 Laplacian -- 2.6 Integration -- 2.7 Coordinates -- 2.8 Complementary Reading -- Part II Variational Formulations in Mechanics -- Chapter 3 Method of Virtual Power -- 3.1 Introduction -- 3.2 Kinematics -- 3.2.1 Body and Deformations -- 3.2.2 Motion: Deformation Rate -- 3.2.3 Motion Actions: Kinematical Constraints -- 3.3 Duality and Virtual Power -- 3.3.1 Motion Actions and Forces -- 3.3.2 Deformation Actions and Internal Stresses -- 3.3.3 Mechanical Models and the Equilibrium Operator -- 3.4 Bodies without Constraints -- 3.4.1 Principle of Virtual Power -- 3.4.2 Principle of Complementary Virtual Power -- 3.5 Bodies with Bilateral Constraints -- 3.5.1 Principle of Virtual Power -- 3.5.2 Principle of Complementary Virtual Power -- 3.6 Bodies with Unilateral Constraints -- 3.6.1 Principle of Virtual Power -- 3.6.2 Principle of Complementary Virtual Power -- 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 -- 3.9.1 Preliminary Results -- 3.9.2 Known Spatial Configuration -- 3.9.3 Known Material Configuration -- 3.10 Infinitesimal Deformations and Small Displacements -- 3.10.1 Bilateral Constraints -- 3.10.2 Unilateral Constraints -- 3.11 Final Remarks -- 3.12 Complementary Reading -- Chapter 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 -- 4.4.1 Principle of Virtual Work -- 4.4.2 Principle of Minimum Total Potential Energy -- 4.4.3 Local Equations and Boundary Conditions -- 4.4.4 Principle of Complementary Virtual Work -- 4.4.5 Principle of Minimum Complementary Energy -- 4.4.6 Additional Remarks -- 4.5 Equilibrium in Bodies with Bilateral Constraints -- 4.5.1 Principle of Virtual Work -- 4.5.2 Principle of Minimum Total Potential Energy -- 4.5.3 Principle of Complementary Virtual Work -- 4.5.4 Principle of Minimum Complementary Energy -- 4.6 Equilibrium in Bodies with Unilateral Constraints -- 4.6.1 Principle of Virtual Work -- 4.6.2 Principle of Minimum Total Potential Energy -- 4.6.3 Principle of Complementary Virtual Work -- 4.6.4 Principle of Minimum Complementary Energy -- 4.7 Min-Max Principle -- 4.7.1 Hellinger-Reissner Functional -- 4.7.2 Hellinger-Reissner Principle -- 4.8 Three-Field Functional -- 4.9 Castigliano Theorems -- 4.9.1 First and Second Theorems -- 4.9.2 Bounds for Displacements and Generalized Loads -- 4.10 Elastodynamics Problem -- 4.11 Approximate Solution to Variational Problems -- 4.11.1 Elastostatics Problem -- 4.11.2 Hellinger-Reissner Principle -- 4.11.3 Generalized Variational Principle -- 4.11.4 Contact Problems in Elastostatics -- 4.12 Complementary Reading -- Chapter 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 -- 5.8.1 Bending of Beams -- 5.8.2 Bending, Extension, and Compression of Beams -- 5.9 Equilibrium Problem for Steady-State Creep -- 5.9.1 Mechanical Equilibrium.

5.9.2 Variational Formulation -- 5.9.3 Variational Principles of Minimum -- 5.10 Castigliano Theorems -- 5.10.1 First and Second Theorems -- 5.10.2 Bounds for Velocities and Generalized Loads -- 5.11 Examples of Application -- 5.11.1 Disk Rotatingwith Constant Angular Velocity -- 5.11.2 Cantilevered Beam with Uniform Load -- 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 -- Chapter 6 Materials Exhibiting Plasticity -- 6.1 Introduction -- 6.2 Elasto-Plastic Materials -- 6.3 Uniaxial Elasto-Plastic Model -- 6.3.1 Elastic Relation -- 6.3.2 Yield Criterion -- 6.3.3 Hardening Law -- 6.3.4 Plastic Flow Rule -- 6.4 Three-Dimensional Elasto-Plastic Model -- 6.4.1 Elastic Relation -- 6.4.2 Yield Criterion and Hardening Law -- 6.4.3 Potential Plastic Flow -- 6.5 Drucker and Hill Postulates -- 6.6 Convexity, Normality, and Plastic Potential -- 6.6.1 Normality Law and a Rationale for the Potential Law -- 6.6.2 Convexity of the Admissible Region -- 6.7 Plastic Flow Rule -- 6.8 Internal Dissipation -- 6.9 Common Yield Functions -- 6.9.1 The von Mises Criterion -- 6.9.2 The Tresca Criterion -- 6.10 Common Hardening Laws -- 6.11 Incremental Variational Principles -- 6.11.1 Principle of Minimumfor the Velocity -- 6.11.2 Principle of Minimumfor the Stress Rate -- 6.11.3 Uniqueness of the Stress Field -- 6.11.4 Variational Inequality for the Stress -- 6.11.5 Principle of Minimum with Two Fields -- 6.12 Incremental Constitutive Equations -- 6.12.1 Constitutive Equations for Rates -- 6.12.2 Constitutive Equations for Increments -- 6.12.3 Variational Principle in Finite Increments -- 6.13 Complementary Reading -- Part III Modeling of Structural Components -- Chapter 7 Bending of Beams -- 7.1 Introduction -- 7.2 Kinematics.

7.3 Generalized Forces -- 7.4 Mechanical Equilibrium -- 7.5 Timoshenko BeamModel -- 7.6 Final Remarks -- Chapter 8 Torsion of Bars -- 8.1 Introduction -- 8.2 Kinematics -- 8.3 Generalized Forces -- 8.4 Mechanical Equilibrium -- 8.5 Dual Formulation -- Chapter 9 Plates and Shells -- 9.1 Introduction -- 9.2 Geometric Description -- 9.3 Differentiation and Integration -- 9.4 Principle of Virtual Power -- 9.5 Unified Framework for Shell Models -- 9.6 Classical Shell Models -- 9.6.1 Naghdi Model -- 9.6.2 Kirchhoff-Love Model -- 9.6.3 Love Model -- 9.6.4 Koiter Model -- 9.6.5 Sanders Model -- 9.6.6 Donnell-Mushtari-Vlasov Model -- 9.7 Constitutive Equations and Internal Constraints -- 9.7.1 Preliminary Concepts -- 9.7.2 Model with Naghdi Hypothesis -- 9.7.3 Model with Kirchhoff-Love Hypothesis -- 9.8 Characteristics of ShellModels -- 9.8.1 Relation Between Generalized Stresses -- 9.8.2 Equilibrium Around the Normal -- 9.8.2.1 Kirchhoff-Love Model -- 9.8.2.2 Love Model -- 9.8.2.3 Koiter Model -- 9.8.2.4 Sanders Model -- 9.8.3 Reactive Generalized Stresses -- 9.8.3.1 Reactions in the Naghdi Model -- 9.8.3.2 Reactions in the Kirchhoff-Love Model -- 9.9 Basics Notions of Surfaces -- 9.9.1 Preliminaries -- 9.9.2 First Fundamental Form -- 9.9.3 Second Fundamental Form -- 9.9.4 Third Fundamental Form -- 9.9.5 Complementary Properties -- Part IV Other Problems in Physics -- Chapter 10 Heat Transfer -- 10.1 Introduction -- 10.2 Kinematics -- 10.3 Principle of Thermal Virtual Power -- 10.4 Principle of Complementary Thermal Virtual Power -- 10.5 Constitutive Equations -- 10.6 Principle of Minimum Total Thermal Energy -- 10.7 Poisson and Laplace Equations -- Chapter 11 Incompressible Fluid Flow -- 11.1 Introduction -- 11.2 Kinematics -- 11.3 Principle of Virtual Power -- 11.4 Navier-Stokes Equations -- 11.5 Stokes Flow -- 11.6 Irrotational Flow.

Chapter 12 High-Order Continua -- 12.1 Introduction -- 12.2 Kinematics -- H( -- H -- H( -- Ht -- HtY) -- Ht -- HtY) -- Gta -- Hta) -- G -- HT -- G -- HT -- GtY)], -- GtY) -- GtY -- GtY) -- 0, -- G -- HtY) -- H. -- HtY) -- 0, -- H1( -- 12.3 Principle of Virtual Power -- D -- D), -- D -- D), -- D) -- D)] -- D) -- 0), -- D), -- D) -- D)] -- D) -- D), -- D -- Dn -- D) -- Dn -- D) -- Dn -- D) -- Dn -- D) -- D)] -- D -- Dn -- D) -- Dn -- D -- Dn -- D]] -- D), -- D -- Dn -- D -- Dn -- Dn) -- R -- Dn -- R -- 12.4 Dynamics -- D) -- D), -- D -- Dn -- 12.5 Micropolar Media -- 12.6 Second Gradient Theory -- M -- Mt -- M. -- M -- Mn -- M) -- Mn -- Mn) -- Mn)]] -- M -- Mn) -- Part V MultiscaleModeling -- Chapter 13 Method of Multiscale Virtual Power -- 13.1 Introduction -- 13.2 Method of Virtual Power -- 13.2.1 Kinematics -- 13.2.2 Duality -- 13.2.3 Principle of Virtual Power -- 13.2.4 Equilibrium Problem -- 13.3 Fundamentals of the Multiscale Theory -- 13.4 Kinematical Admissibility between Scales -- 13.4.1 Macroscale Kinematics -- 13.4.2 Microscale Kinematics -- 13.4.3 Insertion Operators -- 13.4.4 Homogenization Operators -- 13.4.5 Kinematical Admissibility -- 13.5 Duality in Multiscale Modeling -- 13.5.1 Macroscale Virtual Power -- 13.5.2 Microscale Virtual Power -- 13.6 Principle ofMultiscale Virtual Power -- 13.7 Dual Operators -- 13.7.1 Microscale Equilibrium -- 13.7.2 Homogenization of Generalized Stresses -- 13.7.3 Homogenization of Generalized Forces -- 13.8 Final Remarks -- Chapter 14 Applications of Multiscale Modeling -- 14.1 Introduction -- 14.2 Solid Mechanics with External Forces -- 14.2.1 Multiscale Kinematics -- 14.2.2 Characterization of Virtual Power -- 14.2.3 Principle of Multiscale Virtual Power -- 14.2.4 Equilibrium Problem and Homogenization -- 14.2.5 Tangent Operators -- 14.3 Mechanics of Incompressible Solid Media.

14.3.1 Principle of Virtual Power.

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