Kinematic non-linearity is a recognized part of continuum mechanics often termed ‘large’ or ‘finite’ displacement theory. The non-linearity arises because equilibrium is considered in the current, and initially unknown, state of the body. In order to describe finite deformation of a continuous body it is necessary to have a non-linear measure of strain and a stress definition that can be used in the deformed state. It turns out that the Green strain, introduced for axial strain in Chapter 2, can be generalized to multi-component form describing the deformation of a continuous body. This is the subject of Section 6.1.
For any continuum mechanics theory it is very desirable to use stresses and strains that satisfy some form of virtual work principle. It was demonstrated in Chapter 2 that the use of the Green strain, which was a convenient quadratic strain measure with exact invariance with respect to arbitrary rigid body motion, led to a slightly modified interpretation of the normal force N appearing in the principle of virtual work. In a similar way the use of the Green strain for a continuous body leads to a special stress definition, the second Piola–Kirchhoff stress. For small strains this stress definition has a simple physical interpretation, precisely as N in the case of a bar element. This stress is introduced in Section 6.2, and it is demonstrated how it serves as a convenient reference for other stress measures of practical importance.