Neo-Hookeian solid

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. 

A particxilar t e of rubber behaves like a neo-Hookeian solid with a value of ( = 4 X 10 Pa. Calculate (a) the force F required to extend a long piece of this rubber of unstretched cross-section 1 cm to twice its original length and (b) the force required to compress a thin sheet of cross-section 1 cm to half its original thickness if this could be done in such a way that it could expand freely in the lateral directions. What would be the true stresses for (a) and (b) ... 

It is shown below that this equation describes a neo-Hookeian sohd as defined earlier if C = E/6, where E is the modulus for vanishingly low uniaxial stress. A neo-Hookeian solid can thus more generally be defined as one that obeys equation (6.26). 

The conclusions to be drawn from the results above are that, although the predictions of equation (6.26) for the simple neo-Hookeian solid do not describe the behaviour of rubbers well at high extensional strains, they describe it well for low extensional strains, for compressional strains and for quite large simple shear strains. Discussion of modifications to the neo-Hookeian equation is deferred until section 6.5, after consideration of a more physical theory of rubber elasticity in the next section. 

A cylindrical piece of rubber 10 cm long and 2 mm in diameter is extended by a simple tensile force F to a length of 20 cm. If the rubber behaves as a neo-Hookeian solid with a Young s modulus of 1.2 N mm, calculate (i) the diameter of the stretched cylinder, (ii) the value of the nominal stress, (iii) the value of the true stress and (iv) the value of F. 

---