Finger strain

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... 

The inverse of the Cauchy-Green tensor, Cf, is called the Finger strain tensor. Physically the single-integral constitutive models define the viscoelastic extra stress Tv for a fluid particle as a time integral of the defonnation history, i.e. 

Therefore the Eulerian description of the Finger strain tensor, given in terms of the present and past position vectors x and x of the fluid particle as > x ), can now be 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. 

With the gel equation, we can conveniently compute the consequences of the self-similar spectrum and later compare to experimental observations. The material behaves somehow in between a liquid and a solid. It does not qualify as solid since it cannot sustain a constant stress in the absence of motion. However, it is not acceptable as a liquid either, since it cannot reach a constant stress in shear flow at constant rate. We will examine the properties of the gel equation by modeling two selected shear flow examples. In shear flow, the Finger strain tensor reduces to a simple matrix with a shear component... 

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

Another combination of the displacement gradient tensors which are often used are the Cauchy strain tensor and the Finger strain tensor defined by B —1 = Afc A and B = EEt, respectively. 

This can also be rewritten in terms of the Cauchy and Finger strain tensors as ... 

The Wagner equation finds its theoretical basis in the derivation of the more general K-BKZ equation. Unfortunately, it loses part of its original thermod3mamic consistency since, for simplification purposes, only the Finger strain measure is taken into account. Doing so, it is no more derivable from any potential function and additionally it does not predict second normal stress differences any more. 

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. 

L /polymer extensibility smectic-layer compressive modulus E E, Finger strain tensor B , Cauchy strain tensor yriso/r, capillary number characteristic ratio, defined by R )q — Ccotib translational diffusivity-------------------------... 

In Sect. 12 five assumptions are introduced in order to carry the development further. In Sect 13 the equation for the singlet distribution is solved for simple models, both for isothermal and nonisothermal conditions the solutions are given in terms of the Finger strain tensor, which describes the kinematics of the fluid motion. 

Here B is the standard Finger strain tensor used in continuum mechamcs, and Yfo] = 8 — B IS a relative finite strain tensor, defined in DPL, Eq. D.3-4. We note in passing that it follows from Eq. (13.5) that the quantity HfkT) QQ P, dQ is equal to oi and thus satisfies Eq. (13.6). 

This is the same as DPL, Eq. (15.3-17). Thus the stress tensor is given in terms of the Finger strain tensor via Eq. (13.10). The polymer contribution to the stress tensor for the Hookean dumbbell model is obtained by replacing in Eq. (14.12) by the a tensor defined m Eq. (13.7)... 

For finite strains, however, several measures of strain are available, and each of these reduces to the same quantity in the limit of infinitesimal strains. The situation is therefore similar to the one encountered previously in connection with the multiplicity of time derivatives for the stress. The simplest molecular network theories o) suggest the use of the so-called Finger measure of strain, and the resulting equation is called the Lodge rubberlike liquid. Not surprisingly, one finds(9,8i) that, with the use of the Finger strain measure, Eq. (31) is mathematically the same as Eq. (26). 

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)/... 

There is an alternative form of Finger s equation which gives us a choice and is, indeed, to be preferred when dealing with rubbers. We introduce the Finger strain tensor B, being defined as the reciprocal of the Cauchy strain tensor... 

As the directions of the principal axes of B and C must coincide, the diagonal form of the Finger strain tensor is simply... 

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... 

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... 

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... 

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