Variational, Incremental and Energy Methods in Solid Mechanics and Shell Theory

PrefaceIntroduction Chapter 1. Mathematical Preliminaries 1.1 Tensor Formulas in Three-Dimensional Euclidean Spaces 1.2 Tensor Fomulas For Surfaces 1.3 Basic Concepts and Definitions in the Theory of Abstract Spaces and Operators 1.4 Fundamentals of the Classical Calculus of Variations 1.5 Differentials and Derivatives of Operators — Variations 1.6 Variational Boundary Value Problems 1.7 Variational Methods of ApproximationPart I — Three-Dimensional Theory Chapter 2. Kinematics and Stress in Three-Dimensional Solid Continua 2.1 Description of Motion, Coordinates, Reference and Deformed States 2.2 Kinematics of Deformation 2.3 Kinematics of Incremental Deformation 2.4 Definitions of Stress — Stress Rates 2.5 Incremental Stresses 2.6 External Loadings — Boundary Conditions 2.7 Illustrative Example of Physical Components of Displacements and Stress Chapter 3. General Principles - The First and the Second Laws of Thermodynamics 3.1 Kinematics of Rigid Body Motions — Invariance Requirements 3.2 the First Law of Thermodynamics 3.3 the Second Law of Thermodynamics 3.4 the Principles of Virtual Displacements and Virtual Forces 3.5 Equations of Equilibrium in Physical Components For Cylindrical and Spherical Coordinates Chapter 4. Constitutive Material Laws 4.1 Elastic, Hypoelastic and Hyperelastic Materials 4.2 Elastic-Plastic Materials 4.3 Visco-Elastic MaterialsPart II - Shell Theory Chapter 5. Deformation and Stress in Shells 5.1 Shell Middle Surface: Coordinates, Base Vectors 5.2 Description of Deformation and Motion in Shells 5.3 Kinematics of Incremental Deformation in Shells 5.4 Stress Resultants and Stress Couples for Shells - Three-Dimensional Approach 5.5 Load Vectors and Incremental Load Vectors Chapter 6. the First Law of Thermodynamics Applied to Shells — General Principles 6.1 Formulation of the First Law - Invariance Requirements 6.2 Reduced and Local Forms of the First Law 6.3 The First Law in the Reference State 6.4 Some Consequences of the First Law — Discussion — Boundary Condition 6.5 Incremental Forms of the First Law of Thermodynamics 6.6 Incremental Theory and Stability 6.7 the Principles of Virtual Displacements and Virtual Forces For Shells Chapter 7. Constitutive Relations and Material Laws For Shells 7.1 Material Laws For Thin Elastic Shells 7.2 Hyperelastic Incremental Material Relations 7.3 A Note on the Constitutive Relations for Inelastic ShellsPart III. Variational Principles, Finite Element Models and Applications, Elastic Stability Chapter 8. Variational Principles of Elasticity and Shell Theory — Classical Approach 8.1 Variational Principles of Three-Dimensional Elasticity 8.2 Variational Principles of Elastic Shell Theory 8.3 Special Cases — The Principles of Reciprocity, Clapeyron, Castigliano and Menabrea 8.4 General Remarks — The Role of Legendre Transformation 8.5 Illustrative Examples 8.6 Variational Principles — Weak Solutions Chapter 9. Variational and Incremental Principles — General Abstract Approach 9.1 Basic Equations and Operators 9.2 Inner Products-Formal Adjoints — Special Notations and Admissible Variations 9.3 the Principles of Virtual Displacements and Virtual Forces 9.4 General Conditions for Potentialness of the Operators of Dynamics and Kinematics 9.5 Strain and Complementary Energy — Loading Potentials 9.6 General Variational Principles Derived from the Principles of Virtual Displacements and Virtual Forces 9.7 the Variational Principles of Non-Linear Elasticity and Elastic Shell Theory 9.8 Incremental Variational Principles 9.9 Incremental Principles of Non-Linear Elasticity and Shallow Shell Theory Chapter 10. Finite Element Models — Selected Applications 10.1 Matrix Representation of Important Results 10.2 Matrix Formulas for Physical Components 10.3 Finite Element Models — Stiffness, Flexibility and Tangent Matrices 10.4 the Numerical Solution of Finite Element Equations in Statics and Dynamics 10.5 Illustrative Examples 10.6 the Problem of Initial Stresses and Temperature Effects 10.7 Variational Problems with Discontinuous Fields 10.8 Concluding Remark Chapter 11. Elastic Stability 11.1 Incremental Expansions for the Potential Energy 11.2 Neutral Equilibrium — Critical State 11.3 Illustrative Examples 11.4 General Theory of Elastic Stability — Postbuckling Behaviour 11.5 Stability of an Equilibrium State 11.6 Critical Behaviour 11.7 Limit Points and Branching Points 11.8 Imperfection Sensitivity, Coincident Branching Points and Multiparameter Loadings 11.9 Critical and Limit Points — Illustrative ExamplesSelected ReferencesIndex