Rivlin-Sawyers models

In applied viscoelasticity not all the constitutive equations are formulated by an a-priori defined internal energy y/, but the constitutive model is expressed directly by the functional relation between the stress and the strain through an hereditary integral. In rheology this class of constitutive models is called Rivlin-Sawyers models Fxmg s [164], Fosdick and Yu s [165] and many other models currently used belong to this constitutive class. 

Because Rivlin-Sawyers models are not obtained with the Coleman and Noll s procedure, their thermodynamic consistency must be verified a posteriori. 

If we assume that the initial stress S(—oo) is a pressure, then, to within isotropic tenns, we can generalize Eq. 102 to include incompressible fluids. These obey the so-called the Rivlin-Sawyers model given by... 

Another special case of the Rivlin-Sawyer model that is a generalization of Eq. 10.10 was proposed by Wagner and Demarmels [12] who added a dependency on the Cauchy tensor to Eq. 10.10 to provide for a non-zero second normal stress difference and a better fit to data for planar extension, which is defined in Section 10.9. Their model is shown as Eq. 10.11. 

The K BKZ model differs from the Rivim Sawyer model in that in the former the coefficients of the strain tensors are derivatives of a strain energy function like that describing free energy in a deformed rubber. In the Rivlin-Sawyer and tteigner models, there is no general strain energy function. 

Wagner s Eq. 10.10 is special case of a more general integral model of Rivlin and Sawyer, which is, in turn, a generalization called the K-BKZ model [7]. In the K-BKZ model, the strain is described by a linear combination of the Finger and Cauchy tensors, and as a result it is possible to accommodate a non-zero second normal stress difference. ... 

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