Problems of Nonlinear Deformation

B.1. Two Forms of the Method of Continuation of the Solution with Respect to a Parameter.- B.2. The Problem of Choosing the Continuation Parameter and Its Relation to the Behaviour of the Solution in the Neighbourhood of Singular Points.- 1. Generalized Forms of the Continuation Method.- 1.1. Generalized Forms of Continuous Continuation of the Solution.- 1.2. Generalized Forms of Discrete Continuation of the Solution.- 1.3. Examples of Applying Different Forms of the Continuation Method.- 1.4. Optimum and Near-Optimum Continuation Parameters.- 1.5. Forms of the Continuation Method with Partial Optimization of the Continuation Parameter.- 2. Continuation of the Solution Near Singular Points.- 2.1. Classification of Singular Points.- 2.2. The Simplest Form of Bifurcation Equations.- 2.3. The Simplest Case of Branching (rank
$$
(/bar J^ /circ ) = m - 1
$$.- 2.4. The Case of Branching When rank
$$
(/bar J^ /circ ) = m - 2
$$.- 3. The Continuation Method for Nonlinear Boundary Value Problems.- 3.1. Continuous Continuation of the Solution in Nonlinear One-Dimensional Boundary Value Problems.- 3.2. Discrete Continuation of the Solution in Nonlinear One-Dimensional Boundary Value Problems.- 3.3. The Discrete Orthogonal Shooting Method.- 3.4. Algorithms for Continuous and Discrete Continuation of the Solution with Respect to a Parameter for Nonlinear One-Dimensional Boundary Value Problems.- 4. Large Deflections of Arches and Shells of Revolution.- 4.1. Large Elastic Deflections of Plane Arches in Their Plane.- 4.2. Stability of an Inextensible Circular Arch under Uniform Pressure.- 4.3. Algorithms for the Method of Continuation of the Solution with Respect to a Parameter for Large Deflections of a Circular Arch.- 4.4. Large Deflections of a Circular Arch Interacting with a Rigid Half-Plane.- 4.5. Equations for Large Axisymmetric Deflections of Shells of Revolution.- 4.6. Toroidal Shell of Circular Section under Uniform External Pressure.- 5. Eigenvalue Problems for Plates and Shells.- 5.1. General Formulation of the Continuation Method in Eigenvalue Problems.- 5.2. Natural Vibrations of a Parallelogram Membrane.- 5.3. Natural Vibrations of a Trapezoidal Membrane.- 5.4. Eigenvalue Problems for Homogeneous and Sandwich Plates and Spherical Panels of Parallelogram and Trapezoidal Form in Plan. Membrane Analogy.- 5.5. Solution for a Parallelogram Membrane by the Perturbation Method.- Appendix I. A Survey of Literature on the Use of the Continuation Method for Nonlinear Problems in the Mechanics of Deformable Solids.- 1.1. General Formulation of the Continuation Method.- 1.2. Continuation of the Solution in the Neighbourhood of Singular Points and the Problem of Choosing the Continuation Parameter.- 1.3. Different Forms of the Continuation Method.- 1.4. Application to Geometrically Nonlinear Systems.- 1.5. The Use of the Continuation Method in Conjunction with the Finite Element Method.- 1.6. The Continuation Method in Physically Nonlinear Problems.- 1.7. A Comparison of the Different Forms of the Continuation Method.- Appendix II. A Brief Summary of the Notation and Basic Definitions in the Algebra of Vector Spaces.- Author’s index.