Finger tensors

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... 

The Cauchy and Finger tensors may be expressed in terms of the mapping fimction / and its derivatives [26,60], using the vector positions Xj. and Xx and... 

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... 

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 the three kinds of material body motions shown in Fig. 5.5, we calculate the Cauchy and Finger tensors below ... 

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... 

Internal energy per unit volume Internal energy per unit mass Internal energy associated with bead Contribution to equilibrium internal energy Averaged bead velocity Mass-average fluid velocity Species velocity Set of phase-space coordinates Contributions to relative velocity vector Tensor in dumbbell distribution function Tensors m Rouse distribution function Contributions to the a tensor Finger tensor... 

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... 

This general expression first accounts for the principal of causality by stating that the state of stress at a time t is dependent on the strains in the past only. Secondly, by using the time dependent Finger tensor B, one extracts from the fiow fields only those properties which produce stress and eliminates motions like translations or rotations of the whole body which leave the stress invariant. Equation (7.128) thus provides us with a suitable and sound basis for further considerations. 

The Boltzmann superposition principle represents the stress as a result of changes in the state of strain at previous times. In the linear theory valid for small strains, these can be represented by the linear strain tensor. In Lodge s equation the changes in the latter are substituted by changes in the time dependent Finger tensor... 

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... 

Now, we consider the second class of experiments and check for the predictions of Lodge s model with regard to extensional flows. Using again an equation from the ideal rubbers we can directly write down the time dependent Finger tensor B(f,t ). It has the form... 

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... 

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