1.8. Nonlinear Elasticity

1.8.1. Stress strain relations

For a linear elastic material, there is always a constant relationship between the applied stress and the resulting strain, regardless the magnitude of the stress and the strain. The stress strain relation (Eq. (1.91)) for such a material is therefore a straight line, as shown in Fig. 1.19a. The elastic modulus corresponding to this stress strain pair is the slope of the curve.

Fig. 1.19: Stress strain relations for: a) linear elastic material. b) perfectly elastic material. c) elastic material, with hysteresis. d) material suffering permanent deformation.

Any material not obeying a linear stress strain relation is said to behave nonlinearly. For a nonlinear elastic material, the stress strain relation may be written as

Remembering that ? and ? generally are tensors, it is clear that nonlinear elasticity may be very complicated mathematically.

Further, we can not ignore the higher order terms in the strain tensor, which were neglected in the derivation on page 15. The full expression for the strain tensor, replacing Eq. (1.74), is (see e.g. Landau and Lifshitz, 1986):

Nonlinear behaviour may have various causes, and appear in many different ways. Fig. 1.19b shows one example. This material has a nonlinear stress strain relation, since the ratio of stress to strain is not the same for all stresses. The relation is, however, identical for the loading and unloading process. Such materials are said to be perfectly elastic.

For nonlinear stress strain relations the elastic modulus is no longer uniquely defined, not even for a specific...