Adaptations of linear theory - integral models

Leaderman s approach was to modify the basic Boltzmann superposition principle of linear viscoelasticity, so that the strain was given by 

Another simple adaptation of the Boltzmann superposition principle is that of Findlay and Lai [14], who worked with step stress histories applied to specimens of poly(vinylchloride). Their theory was reformulated by Pipkin and Rogers [15] for general stress and strain histories. Pipkin and Rogers took a non-linear stress relaxation modulus R t, e), defined somewhat differently from G in Equation (10.4)  

Note that Equations (10.4) and (10.14) reduce to the same form in the linear case but may differ significantly in the non-linear case. The Pipkin and Rogers integral for the stress in terms of the strain history is 

Similarly, for the strain in terms of the stress history a creep function C was defined as 

The most significant difference between the Leaderman and the Pipkin and Rogers approach is that in the former the material response is separable into time and stress dependence. Thus, in Equation (10.13) there is a fimction / of stress multiplied by a function J of time. By contrast, in Equation (10.17) C is explicitly a function of two variables that may or may not be separable in this sense. The Pipkin and Rogers approach is thus more general and we would expect it to be capable of modelling a greater range of material behaviour. 

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