Many problems of practical interest involve non-linear behavior of solids and structures. In the present context a solid means a body with a firm shape, as opposed to a fluid, while a structure refers to a solid composed of slender elements such as beams, plates and shells. Typical problems are the motion of robots, collapse scenarios of structures, metal forming processes in industrial production, and material deformation and failure in geotechnical engineering. These problems typically involve a considerable change of shape, often accompanied by non-linear material behavior.
The finite element method is an important tool for the analysis of nonlinear problems, such as geometrical and material non-linear behavior of solids and structures. The solution of non-linear problems by the finite element method involves modeling, leading to the formulation of an appropriate set of non-linear equations describing the problem, followed by an appropriate strategy for the numerical solution of these equations. In contrast to linear problems, where the solution strategy reduces to the solution of a system of linear equations, the solution phase in a non-linear problem typically involves an iterative procedure.
Non-linear modeling and analysis is a very active research area with many engineering applications. The many different aspects involved are not covered in any single text. However, some central references to general texts should be given here. A brief introduction to some of the basic problems of non-linear finite element analysis of solids and structures is included in the book by Cook et al. (1989).