Strain Finger tensor Deformation

For a fluid that does not possess any elastic property, what needs to be measured is the deformation rate, instead of the total strain. Such a fluid has no memory of past deformations. Thus, we need to express the present changing rate of the Finger tensor B, while moving the past point position X in B infinitely close to its present position X. Mathematically, these two ideas can be expressed as... 

The use of the Finger tensor to build a model for the response of a polymer to large deformations implies that the deformation is affine. This means that the strain at the microscopic level, i.e., of the molecules, is the same as the strain in a macroscopic sample. This will require... 

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. 

Note 3 The Finger strain tensor for a homogeneous orthogonal deformation or flow of incompressible, viscoelastic liquid or solid is... 

One major discrepancy of the previous model can be attributed to the use of the infinitesimal strain tensor and to derivatives restricted to time changes. Indeed, in the case of large deformations, one has to refer to finite strain tensors, such as the Finger (, t (t ) or Cauchy C t(t ) strain tensors (t being the... 

The flow field in Eq. (Al-7) is really just a solid-body rotation which rotates, but does not deform, the fluid element. As a result, the rate-of-strain tensor D is the zero tensor, and the Finger strain tensor is the unit tensor. 

Due to the unknown hydrostatic pressure, p, the individual normal stresses an are indeterminate. However, the normal stress differences are well-defined. Let us consider the difference between Gzz and Gxx- Insertion of the Finger strain tensor associated with uniaxial deformations, Eq. (7.80), in the constitutive equation (7.74) yields... 

Since it is often simpler to write the Finger deformation tensor, and since it only differs fiom the strain tensor by unity, we usually write constitutive equations in terms of B. 

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