Finger deformation tensor invariants

Here we describe the strain history with the Finger strain tensor C 1(t t ) as proposed by Lodge [55] in his rubber-like liquid theory. This equation was found to describe the stress in deforming polymer melts as long as the strains are small (second strain invariant below about 3 [56] ). The permanent contribution GcC 1 (r t0) has to be added for a linear viscoelastic solid only. C 1(t t0) is the strain between the stress free state t0 and the instantaneous state t. Other strain measures or a combination of strain tensors, as discussed in detail by Larson [57], might also be appropriate and will be considered in future studies. A combination of Finger C 1(t t ) and Cauchy C(t /. ) strain tensors is known to express the finite second normal stress difference in shear, for instance. 

Map of invariants of the Finger tensor for deformation of an incompressible material, IIIb = 1. All types of deformation must occur in the shaded regions and thus may be considered to be a combination of the three simple ones indicated by the lines. 

The second and third invariants of 2D are the only ones that vary during incompressible flows. As the invariants of the Finger tensor bound the possible deformations in a material (Figure 1.4.3), the invariants of the rate of deformation bind the possibleflows. The domain of all possible flows is shown in Figure 2.2.S. 

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