Comprehensive treatment of metal plasticity requires an understanding of the fundamental nature of stresses and strains. A stress can be understood at a basic level as a force per unit area on which it acts, while a strain is an extension divided by an original length. However, the limitations of these definitions rapidly become clear when considering anything other than very simple loading situations. Analysis of various practical situations can in fact be rigorously implemented without becoming embroiled in mathematical complexity, most commonly via usage of commercial (finite element) numerical modeling packages. However, there are various issues involved in such treatments, which need to be appreciated by practitioners if outcomes are to be understood in detail. This chapter covers the necessary fundamentals, relating to stresses and strains, and to their relationship during elastic (reversible) deformation. How this relationship becomes modified when the material undergoes plastic (permanent) deformation is covered in the following chapter.

In the previous chapter, it was shown that an aligned composite is usually stiff along the fibre axis, but much more compliant in the transverse directions. Sometimes, this is all that is required. For example, in a slender beam, such as a fishing rod, the loading is often predominantly axial and transverse or shear stiffness are not important. However, there are many applications in which loading is distributed within a plane: these range from panels of various types to cylindrical pressure vessels. Equal stiffness in all directions within a plane can be produced using a planar random assembly of fibres. This is the basis of chopped-strand mat. However, demanding applications require material with higher fibre volume fractions than can readily be achieved in a planar random (or woven) array. The approach adopted is to stack and bond together a sequence of thin ‘plies’ or ‘laminae’, each composed of long fibres aligned in a single direction, into a laminate. It is important to be able to predict how such a construction responds to an applied load. In this chapter, attention is concentrated on the stress distributions that are created and the elastic deformations that result. This involves consideration of how a single lamina deforms on loading at an arbitrary angle to the fibre direction. A summary is given first of some matrix algebra and analysis tools used in elasticity theory.