Finger strain tensor invariants

In Equation (1.28) function M(t - r ) is the time-dependent memory function of linear viscoelasticity, non-dimensional scalars 4>i and 4>2 and are the functions of the first invariant of Q(t - f ) and F, t t ), which are, respectively, the right Cauchy Green tensor and its inverse (called the Finger strain tensor) (Mitsoulis, 1990). The memory function is usually expressed as... 

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. 

In simple extension the difference between the stress in the elongation direction and that in the direction perpendicular to the elongation direction is measured. For conciseness this stress difference will be denoted by Cg. According to Eq. (2) Og is determined by the difference between the 11-and 22-components of the strain tensor, which will be denoted by Sg. represents a tensorial strain measure determined by the first and second invariants I Ct ) and Il Ct ) of the relative Finger strain tensor. In the case of uniaxial extension these invariants can always be expressed in the ratio of the stretch ratios at times t and t , so that [ (t)/... 

Whether to use the first or the second form of Finger s constitutive equa tion is just a matter of convenience, depending on the expression obtained for the free energy density in terms of the one or the other set of invariants. For the system under discussion, a body of rubbery material, the choice is clear The free energy density of an ideal rubber is most simply expressed when using the invariants of the Finger strain tensor. Equation (7.22), giving the result of the statistical mechanical treatment of the fixed junction model, exactly corresponds to... 

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... 

Three scalar invariants of the Finger strain tensor may defined in the following way... 

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. 

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. 

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. 

The Doi-Edwards equation is a K-BKZ model, since the scalar functions and are derivatives of a strain energy function and depend on the first and second invariants of the Finger tensor, which are defined by Eqs. 10.8 and 10.9. While these two functions cannot be written in a closed form, Currie [13] has shown that they can be approximated by the following analytical expressions. 

---