The transition to large-strain elasticity

In section 6.2 the formalism of the elastic behaviour of an ideal linear elastic solid for small strains was considered. Rubbers may, however, be reversibly extended by hundreds of per cent, implying that a different approach is required. The previous ideas suggest a possible plausible generalisation, as follows. 

For finite strain in isotropic media, only states of homogeneous pure strain will be considered, i.e. states of uniform strain in the medium, with all shear components zero. This is not as restrictive as it might first appear to be, because for small strains a shear strain is exactly equivalent to equal compressive and extensional strains applied at 90° to each other and at 45° to the original axes along which the shear was applied (see problem 6.1). Thus a shear is transformed into a state of homogeneous pure strain simply by a rotation of axes by 45°. A similar transformation can be made for finite strains, but the rotation is then not 45°. All states of homogeneous strain can thus be regarded as pure if suitable axes are chosen. 

Extension ratios X in the directions of the three axes can then be used, where X is defined as (new length)/(original length) = l/l (see fig. 6.5). 

be the change in length of unit length in the Ox, direction, so that d, corresponds to the strain e, in the small-strain theory. Thus (see fig. 6.5) 

Equation (6.9) for the small-strain elasticity of an incompressible solid. 

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