More complicated single-integral representations

Schapery [43,44] has used the theory of the thermodynamics of irreversible processes to produce a model that may be viewed as a further extension of Leaderman s. Schapery continued Leaderman s technique of replacing the stress by a function of stress f(o) in the superposition integral and also replaced time by a function of time, the reduced time i/r. The material is assumed to be linear viscoelastic at small strains, with a creep compliance function of the form [44] 

The constant term Do corresponds to the instantaneous elastic response (the unrelaxed compliance of Equation (5.22)). In general, the strain in terms of the stress history is given by 

In the theory of Equation (11.21), four functions of stress go, gi,gi and a characterise the non-linearity and must be evaluated over the required stress range. Experimental regimes that involve periods of constant stress, during which the functions are constants, have proved useful for this purpose. For a single-step creep test at stress a applied at time t = 0, the Equation (11.21) can be evaluated, noting that it contains a Duhamel integral like Equation (5.3), to give the result 

He showed how the use of double logarithmic plots of the recovery strain against time, obtained for different stress levels, could be related to one another by shift factors the shift factors could then be simply related to gi and Oa. The technique of step loading combined with Equation (11.23) has also been used by Crook [45] and Lai and Bakker [46]. Schapery s model has been applied to nitrocellulose, fibre-reinforced phenolic resin and polyisobutylene [44] polycarbonate [45] high-density polyethylene [46] and graphite-epoxy composites [10]. 

The theory of Bernstein, Kearsley and Zapas [47] and developments of it (e.g. Zapas and Craft [48]) - so-called BKZ theories - are aimed in particular at large-deformation 

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