The Finger Tensor

As mentioned above, the tensor E contains information about both the stretching and the rotation of a material element. Yet, if a material element is rotated only and not stretched, no 

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. 

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

From the definition of B given in Eq. (1-16), along with the relationship of Eq. (1-15), we find that 

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Note that for t < 0, A(r, t) = exp(et) is independent of t, since the sample is not being stretched at times less than 0. From Eq. (A3-2), we obtain the components of the Finger tensor... 

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. 

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 this case inversion of the Cauchy tensor, which is necessary when deriving the Finger tensor, is trivial. For non-orthogonal deformations this is more complicated, and here one can make use of a direct expression for the components of B which reads... 

Here, while retaining the relaxation function G t — t ), the Finger tensor is replaced by a functional depending on B(t, t ) and its invariants /B(t, t ) and Ilsit t ). The modification enables us to account for the viscosity reduction effects. As an example, one can factorize and write... 

Thus, physically the Finger tensor describes the area change around a point on a plane whose normal is n. B can give the deformation at any point in terms of area change by operating on the normal to the area defined in the present or deformed state. Because area is a scalar, we need to operate on the vector twice. 

Note that we have merely switched the order of the tensor product from that given in the Finger tensor, but as we will see, in general, this switch gives us different results. By a similar derivation, (see Exercise 1.10.4) as given for p, the relative area change, we can... 

Length, area, and volume change can also be expressed in terms of the invariants of B or C (see eqs. 1.4.45-1.4.47). Note that the Cauchy tensor operates on unit vectors that are defined in the past state. In the next section we will see that the Cauchy tensor is not as useful as the Finger tensor for describing the stress response at large strain for an elastic solid. But first we illustrate each tensor in Example 1.4.2. This example is particularly important because we will use the results direcfiy in the next section with our neo-Hookean constitutive equation. 

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. 

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 ... 

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. 

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. 

In fact we have lost something over Chapter 1. The Finger tensor B is able to give us normal stresses and extensional thickening. We will have to wait until Chapter 4 to get these factors back into our models. But in the next chapter we will see how to bring in the phenomenon of time dependence, which is so important for polymeric systems. We should note that for concentrated suspensions, especially flocculated systems, there is little elastic recovery, and time dependence is often either very short or extremely long. Thus the viscous models of this chapter are often quite adequate (recall Figures 2.5.3 and 2.5.4 and look ahead to Chapter 10). 

The upper-convected time derivative is a time derivative in a special coordinate system whose base coordinate vectors stretch and rotate with material lines. With this definition of the upper-convected time derivative, stresses are produced only when material elements are deformed mere rotation produces no stress (see Section 1.4). Because of the way it is defined, the upper-convected time (teriva-tive of the Finger tensor is identically zero (see eqs. 2.2.3S and 1.4.13) ... 

Before closing this section it should be pointed out that Eiq. (4.158) can be written in the form of the K-BKZ model (see Chapter 3), by expressing the orientation tensor Q in terms of the Finger tensor (see Chapter 2) as (Doi and Edwards 1978c)... 

A primitive model of nonlinear behavior can be obtained by simply replacing the infinitesimal strain tensor in Eq. 10.3 by a tensor that can describe finite strain. However, there is no unique way to do this, because there are a number of tensors that can describe the configuration of a material element at one time relative to that at another time. In this book we will make use of the Finger and Cauchy tensors, B and C, respectively, which have been found to be most useful in describing nonlinear viscoelasticity. We note that the Finger tensor is the inverse of the Cauchy tensor, i.e., B = C. A strain tensor that appears in constitutive equations derived from tube models is the Doi-Edwards tensor Q, which is defined below and used in Chapter 11. The definitions of these tensors and their components for shear and uniaxial extension are given in Appendix B. 

Lodge [10] constructed a very simple model of nonlinear behavior that can be represented as Eq. 10.3 with the Finger tensor B t,t ) replacing He called the material described by... 

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... 

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