Finger deformation tensor

The Finger deformation tensor for this flow is obtained from Eq. (1-19) with Aj = A.3 = A Hence, taking A = A, we obtain... 

The dot product of two tensors is called a tensor product because it generates a new tensor just as matrix multiplicaton of one 3x3 matrix by another generates a new 3x3 matrix. In this case the new tensor is called the Finger deformation tensor after J. Finger (1894), who was the first to use it. 

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. 

On the other hand, rubber can deform elastically up to extensions of as much as 7. As rubber came into use as an engineering material during World War n, a need arose to express Hooke s law for large deformations. Using the Finger deformation tensor, we can come up with the result quite easily. If the stress at any point is linearly proportional to deformation and if the material is isotropic (i.e., has the same proportionality in all directions), then the extra stress due to tteformation should be determined by a constant times the deformation. 

In which C is the Finger deformation tensor (see Chapter 2) and m t) is the memory function defined by... 

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... 

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... 

For the deformations illustrated in Figure 1.4.2 and Example 1.4.1, evaluate the components of the Finger and the Cauchy deformation tensors. 

Inverse Deformation Tensors (a) Show that the inverse of the Finger tensor operates on unit vectors in the deformed state to give inverse square of length change ... 

This little exercise is significant. It tells us that one of the four key rheological phenomena laid out in the introduction to these chapters on constitutive relations-normal stresses in steady shear flows-cannot be explained by any function of the rate of the deformation tensor. On the other hand, almost any function of B, the Finger tensor, does generate proper shear normal stresses. We will wait until Chapter 4 to pursue this reasoning further. 

The deviatoric part of the left Cauchy-Green deformation tensor, or Finger tensor, is given as... 

For uniaxial deformation the Finger and Cauchy tensors read... 

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... 

In codeformational equations, the basic kinematic quantities are the displacement functions. This generally means using the respective Cauchy and Finger tensors deformation gradient tensor X /9 Xfi ] by the following equations... 

By this definition, B is a symmetric tensor, whereas E can be asymmetric. The Finger tensor naturally arises when one considers the length of a deformed material line. From Eq. (1-13), the square of the length of an embedded vector <5x following a deformation 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. 

A direct way for obtaining the Finger tensor, which also describes the deformation, is to consider the change in a local area. In Fig. 5.4, we can follow the change of either the area dA or the length dX to achieve the same purpose. dA and dX are related by the volume dV ... 

Physically, the Finger tensor B describes the change in area at a certain point in the material body and operates on the unit vector n in the deformed state or at the present time. This is what we hope to have for relating to the stress tensor, which is measured with respect to the present form of the material body. 

Both the Cauchy and Finger tensors are unit tensors. This means that, as expected, the solid-body rotation does not deform the material. 

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... 

In Chapter 5, we defined the deformation gradient tensor E, Cauchy tensor C, and Finger tensor B, respectively, as... 

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