Problems of Nonlinear Deformation: The Continuation Method Applied to Nonlinear Problems in Solid Mechanics

Interest in nonlinear problems in mechanics has been revived and intensified by the capacity of digital computers. Consequently, a question offundamental importance is the development of solution procedures which can be applied to a large class of problems. Nonlinear problems with a parameter constitute one such class. An important aspect of these problems is, as a rule, a question of the variation of the solution when the parameter is varied. Hence, the method of continuing the solution with respect to a parameter is a natural and, to a certain degree, universal tool for analysis. This book includes details of practical problems and the results of applying this method to a certain class of nonlinear problems in the field of deformable solid mechanics. In the Introduction, two forms of the method are presented, namely continu­ ous continuation, based on the integration of a Cauchy problem with respect to a parameter using explicit schemes, and discrete continuation, implementing step­ wise processes with respect to a parameter with the iterative improvement of the solution at each step. Difficulties which arise in continuing the solution in the neighbourhood of singular points are discussed and the problem of choosing the continuation parameter is formulated.

Table of Contents:
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.

Review :
` Problems in Nonlinear Deformation should be of value to those interested in nonlinear problems of small strain elastic deformation of solids. It may also be a useful reference for graduate students working in certain areas of applied mechanics. ' Applied Mechanics Review