Simple shear Finger tensor

Note 5 The Finger strain tensor for simple shear flow is... 

The tensor E E, called the Piola tensor (Astarita and Marrucci 1974), is closely related to B. In an extensional deformation, E E is exactly equal to B. B, a symmetric tensor, contains information about the orientation of the three principal axes of stretch and about the magnitudes of the three principal stretch ratios, but no information about rotations of material lines that occurred during that deformation. Thus, for example, from the Finger tensor alone, one could not determine whether a deformation was a simple shear (which has rotation of material lines) or a planar extensional deformation (which does not). The Finger tensor B(r, f) describes the change in shape of a small material element between times t and t, not whether it was rotated during this time interval. 

For simple shear, E is given in Fig. 1-16 hence from Eq. (1-16), the Finger tensor for simple shear is... 

To see the consequences implied by this equation of state, it is instructive to consider first simple shear flow conditions. We may write down the time dependent Finger tensor immediately, just by replacing in Eq. (7.98), derived for a deformed rubber, 7 by the increment t) — 7(t ) This results in... 

The strain-memory function is derived from the first and second invariants of the Finger strain tensor. For simple shear flow, the strain-memory function is given as... 

With the gel equation, we can conveniently compute the consequences of the self-similar spectrum and later compare to experimental observations. The material behaves somehow in between a liquid and a solid. It does not qualify as solid since it cannot sustain a constant stress in the absence of motion. However, it is not acceptable as a liquid either, since it cannot reach a constant stress in shear flow at constant rate. We will examine the properties of the gel equation by modeling two selected shear flow examples. In shear flow, the Finger strain tensor reduces to a simple matrix with a shear component... 

Here h(/i, 12), the damping function , is a function of the invariants of the Finger strain tensor given in equations (31) and (32) the damping function is determined by requiring the constitutive equation to describe shear and elongational flow data. Extensive comparisons with experimental data show that this rather simple empiricism is extremely useful. Equation (47) gives a value of zero for the second normal stress coefficient. 

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