Temperature and strain dependence of elastic response

The elastic relations introduced above are always defined at zero strain (or stress). As elastic strains increase in magnitude, the relations become progressively nonlinear and ultimately lead to tensile or volumetric de-cohesion or ideal shear, not considering plastic yielding, which can intervene much earlier. 

The symmetry-preserving bulk de-cohesion obeys a remarkably simple scaling relationship arising from a universal binding-energy relation demonstrated by Rose et al. (1983) to be given, for uniaxial tension, simply by 

From this it is clear that the strain-dependent decrease of the uniaxial modulus with increasing tensile strain can be obtained readily by differentiation  

The corresponding ideal shear response at large shear strains is less well understood because of shear-induced breakdown of symmetry. 

It is useful to note that eqs. (4.19) and (4.20) for uniaxial tension (or compression), in addition to being applicable to uniaxial strain deformation, as stated above, are also applicable to the case of dilatation responding to negative pressure where the basic symmetry of the deformation is maintained. In that case, however, ffii is replaced with r, the triaxial tensile stress (negative pressure), n is replaced with e, the dilatation, and Eq, Young s modulus, is replaced with the bulk modulus K(). Moreover, a must be replaced by fi, which represents the reciprocal of the critical athermal cavitation dilatation. 

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