TABLE OF CONTENTS Many introductory texts on the finite element method discuss the solution for linear problems of elasticity and field equations. 1-3 In practical applications the limitation of linear elasticity, or more generally of linear behaviour, often precludes obtaining an accurate assessment of the solution because of the presence of 'non-linear' effects and/or because the geometry has a 'thin' dimension in one or more directions. In this book we describe extensions to the formulations introduced to solve linear problems to permit solutions to both classes of problems.
Non-linear behaviour of solids takes two forms: material non-linearity and geometric non-linearity. The simplest form of non-linear material behaviour is that of elasticity for which the stress is not linearly proportional to the strain. More general situations are those in which the loading and unloading response of the material is different. Typical here is the case of classical elastic-plastic behaviour.
When the deformation of a solid reaches a state for which the undeformed and deformed shapes are substantially different a state of finite deformation occurs. In this case it is no longer possible to write linear strain-displacement or equilibrium equations on the undeformed geometry. Even before finite deformation exists it is possible to observe buckling or load bifurcations in some solids and non-linear equilibrium effects need to be considered. The classical Euler column, where the equilibrium equation for buckling includes the effect of axial loading, is an example of this class of problem. When deformation is large the boundary...
