Mathematical Theory of Elastic and Elasto-Plastic Bodies: Volume 3 An Introduction(Volume 3 Studies in Applied Mechanics)

The book acquaints the reader with the basic concepts and relations of elasticity and plasticity, and also with the contemporary state of the theory, covering such aspects as the nonlinear models of elasto-plastic bodies and of large deflections of plates, unilateral boundary value problems, variational principles, the finite element method, and so on.

Table of Contents:
PrefaceSummary of NotationChapter 1. Stress Tensor 1.1. Tensors. Green's Theorem 1.2. Stress Vector 1.3. Components of the Stress Tensor 1.4. Equations of Equilibrium 1.5. Tensor Character of Stress 1.6. Principal Stresses and the Quadric of StressChapter 2. Strain Tensor 2.1. Finite Strain Tensor 2.2. Small Strain Tensor 2.3. Equations of the Compatibility of StrainChapter 3. Generalized Hooke's Law 3.1. Tension Test 3.2. Generalized Hooke's Law 3.3. Elasto-Plastic Materials. Deformation Theory. (A Special Case of the Nonlinear Hooke's Law) 3.4. Elasto-Inelastic Bodies. A Model with Internal State Variables 3.5. Hooke's Law with a Perfectly Plastic Domain 3.6. Flow Theory of PlasticityChapter 4. Formulation of Boundary Value Problems of the Theory of Elasticity 4.1. Lamé Equations. Beltrami-Michell Equations 4.2. The Classical Formulation of Basic Boundary Value Problems of ElasticityChapter 5. Variational Principles in Small Displacement Theory 5.1. Principles of Virtual Wo;k, Virtual Displacements and Virtual Stresses 5.2. Principle of Minimum Potential Energy in the Theory of Elasticity 5.3. Principle of Minimum Complementary Energy in the Theory of Elasticity 5.4. Hybrid Principles in the Theory of Elasticity. The Hellinger-Reissner PrincipleChapter 6. Functions with Finite Energy 6.1. The Space of Functions with Finite Energy 6.2. The Trace Theorem. Equivalent Norms, Rellich's Theorem 6.3. Coerciveness of Strains. Korn's InequalityChapter 7. Variational Formulation and Solution of Basic Boundary Value Problems of Elasticity 7.1. Weak (Generalized) Solution 7.2. Solution of Basic Boundary Value Problems by the Variational Method 7.2.1. Solution of the Abstract Variational Problem 7.2.2. Application to Basic Problems of the Theory of Elasticity 7.3. Solution of the First Basic Boundaiy Value Problem of Elasticity 7.4. Contact and Other Boundary Value Problems 7.5. Variational Formulation in Terms of Stresses. Method of Orthogonal Projections and Castigliano's Principle 7.6. Basic Boundary Value Problems of Elasticity in Orthogonal Curvilinear Coordinates 7.6.1. Tensors in Curvilinear Coordinates 7.6.2. Physical Components of Strain and Stress Tensors 7.6.3. Formulation of Variational Principles in Curvilinear Coordinates 7.6.4. Weak Solution of Basic Boundary Value Problems Formulated in Terms of Displacements or StressesChapter 8. Solution of Boundary Value Problems for the Elasto-Plastic Body. Deformation Theory 8.1. Formulation of the Weak Solution 8.2. Application of the Variational Method to the Solution of Basic Boundary Value ProblemsChapter 9. Solution of Boundary Value Problems for the Elasto-Inelastic Body 9.1. Elasto-Inelastic Material 9.2. Solution of the First Boundary Value Problem for the Elasto-Inelastic Body 9.3. Solution of the Second Boundary Value ProblemChapter 10. Two- and One-Dimensional Problems 10.1. Saint-Venant's Principle 10.2. Plane Elasticity 10.2.1. Basic Cases of Plane Elasticity 10.2.2. Solution of Problems of Plane Elasticity in Terms of Displacements 10.2.3. Solution of Problems of Plane Elasticity in Terms of Stresses 10.3. Axisymmetric Boundary Value Problems 10.4. Reduction of Dimension in the Theory of Elasticity 10.4.1. Kantorovics Method 10.4.2. Bending of a Beam 10.4.3. Bending of a Plate 10.4.4. Shells 10.4.5. Solution of a Boundary Value Problem for a Cylindrical Shell 10.5. Torsion of a BarChapter 11. Ritz-Galerkin and Other Approximate Methods 11.1. Minimizing Sequence 11.2. The Ritz-Galerkin Method 11.3. Finite Element Method 11.3.1. Compatible Models 11.3.2. Equilibrium Models 11.3.3. Mixed Models 11.4. A Posteriori Error Bounds. Two-Sided Energy Bounds. The Hypercircle Method 11.5. The Kacanov Method 11.6. Method of Steepest Descent 11.7. Method of ContractionChapter 12. Large Deflections of Plates. The Equations of von Kármán 12.1. Finite Elasticity 12.2. Large Deflections of Plates 12.3. Theory of Von Kármán's EquationsChapter 13. Variational Inequalities with Applications to Problems of Signorini's Type and to the Theory of Plasticity 13.1. Signorini's Problem 13.2. Elasto-Plastic Body with a Perfectly Plastic Domain 13.3. Approximate Solution of Variational Inequalities 13.4. Flow Theory of Plasticity. Elasto-Inelastic Body with a Perfectly Plastic Domain 13.5. Flow Theory. Elasto-Inelastic Body with Strain HardeningBibliographySubject Index