Pipkin-Rogers model

In particular, the LS-Dyna finite viscoelastic relationship [175] takes into accotmt rate effects through linear viscoelasticity by a convolution integral. The model corresponds to a Maxwell fluid consisting of dampers and springs in series. The Abaqus FEA model is reminiscent of, and similar to, a well-established model of finite viscoelasticity, namely the Pipkin-Rogers model [161]. This model, with an appropriate choice of the constitutive parameters, reduces to the Fung (QLV) model [173, 177]. 

The Pipkin-Rogers model was and remains a significant theoretical development since, as noted by Drapaca, Sivaloganathan and Tenti [31], many of the alternative approaches can be shown to be special cases of it. 

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. 

There have, however, been criticisms of the practicality of this approach. Stafford [30] has pointed out that the second term requires the evaluation of a function of four variables. He argued that this would make it less experimentally tractable than the two-variable kernel in Green and Rivlin s second term. Nevertheless Pipkin and Rogers demonstrated that the first term of their expansion was more effective than a three-term Green-Rivlin model. 

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