Generalised Hooke’s law

When using the generalised Hooke s law strain energy function there are a number of possible strain definitions that can be used depending on the situation. When material deformation is very small the infinitesimal strain approach is a valid approximation with the strain defined as... 

The mechanical properties of an anisotropic elastic solid for small strains are defined by the generalised Hooke s law... 

The classical elastic theory of anisotropic materials at infinitesimal strains has been discussed in an earlier chapter. The starting point is the generalised Hooke s law which can be written as... 

For high strength composites, the theoretical models are based on the generalised Hooke s law, with all necessary complements for local effects, time and temperature influences, non-homogeneity, etc. 

It is customary to adopt an abbreviated nomenclature in which the generalised Hooke s law relates the six independent components of stress to the six independent components of the engineering strains. 

Using the above definitions of finite strain and stress, we wish to construct stress-strain relationships for finite strains that are analogous to the generalised Hooke s law for small strain elasticity. Each component of stress can be a function of every component of strain, and vice versa. For a linear relation, we would expect equations such as... 

Thus the relationships (6.21) and (6.21a) are compatible with the isotropy and incompressibility of a rubber and reduce to Hooke s law at small strains. Materials that obey these relationships are sometimes called neo-Hookeian solids. Equation (6.21a) is compared with experimental data in fig. 6.6, which shows that, although equation (6.21a) is only a simple generalisation of small-strain elastic behaviour, it describes the behaviour of a real rubber to a first approximation. In particular, it describes qualitatively the initial fall in the ratio of to k that occurs once k rises above a rather low level. It fails, however, to describe either the extent of this fall or the subsequent increase in this ratio for high values of k. 

Finally, in many practical applications (such as films and synthetic fibres) polymers are used in an oriented or anisotropic form, which requires a considerable generalisation of Hooke s law. 

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