For 2D and 3D viscoelastic flow, the standard differential viscoelastic models can be used for both isothermal and non-isothermal flows. For shell models, the simplest rheological description consists of selecting a constant viscosity. In some situations it is more desirable to define a simple viscosity model whose magnitude depends on the local strain.

The fluid constitutive equation is written as follows:

(17–7)

Here the viscosity depends on the local strain or deformation , which is locally evaluated on the basis of the stretching undergone by the individual elements with respect to their initial configuration. The relevance of this model lies in the fact that it enables a differentiated behavior as a function of the accumulated deformation. The advantage is that the property can be easily measured in a simple traction experiment or with a simple elongation rheometer.

In a simple traction experiment, a (usually dog-bone shaped) sample of initial length is stretched at a constant stretching velocity and the tensile stress is recorded as a function of the deformation. Data manipulation allows converting this data into a viscosity versus deformation, by taking the viscosity as the ratio of the stress to the strain rate given by

(17–8)

where is the actual deformation. It is therefore possible to extract the curve . A simple traction experiment is appealing for thermoforming applications, as the sample can be stretched at a rather low temperature. Despite the simplicity, the model embraces possible morphological changes, such as strain induced crystallization.

For extrusion blow molding applications, where the parison is inflated right after extrusion, it is probably more appropriate to consider a viscosity measurement using a simple elongation rheometer. Here, the sample is usually stretched at a constant strain rate , and the viscosity is given as a function of time. Here again, data manipulation allows converting this measurement into a viscosity versus deformation, by converting the time into a deformation. Typically, the Cauchy deformation is given by

(17–9)

Since blow molding occurs under relatively fast deformations, it is appropriate to consider the curve data obtained under high strain rate.

Fitting the viscosity-deformation curve is mainly a matter of algebra and three functions are available to facilitate this task.