Lodge rubber-liquid

Shear flow for a Lodge rubber-liquid. If we consider the flow field,... 

Comment how the viscometric functions for the shear flow of a Lodge rubber-liquid develop in Example 2.4 compare with experimental observations. 

Develop expressions for the elongational viscosities for the Lodge rubber-liquid in steady shearfree flow. 

The constitutive equations benefiting from the specific representations of reptation theory have the general form of the Lodge rubber-like liquid equation, since they are all... 

M(t — / ) is called the memory function. Eq. 101 is the constitutive relation for the Lodge rubber-like liquid (LRL). 

As we can see, both are independent of the strain rate 7. Hence, as a first conclusion, Lodge s equation of state caimot describe the shear thinning phenomenon. Equation (9.182) is in fact identical with Eq. (6.107) derived in the framework of linear response theory. The new result contributed by Lodge s formula is the expression Eq. (9.183) for the primary normal stress difference. It is interesting to note that the right-hand side of this equation already appeared in Eq. (6.108) of the linear theory, formulating the relationship between G t) and the recoverable shear compliance J. If we take the latter equation, we realize that the three basic parameters of the Lodge rubber-hke liquid, and are indeed related, by... 

Other possibilities exist to solve the frame invariant problem Cauchy-Maxwell equation uses the Cauchy tensor, C, which is also independent of the system of reference, the Lodge rubber-like liquid model uses the Finger tensor but contrarily to the Lodge model, it uses a generalized memory function ... 

Rubber elasticity theory can also be used to derive a general class of singleintegral equations, as opposed to a particular equation such as that for the Lodge rabberlike liquid. To proceed, consider a cube of rabber, initially of unit edge, in an extensional deformation as shown in Figure 14.23. In the deformed state, the block of rabber has dimensions Aj, X2, and /I3, which also happen to be the extension ratios. 

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. 

Lodge was the first to formulate such an equation by a combination of Eq. (7.74) with the Boltzmann superposition principle as expressed by Eq. (5.111). Explicitly, the Lodge equation of state of rubber-like liquids , when written for homogeneous deformations, has the following form... 

Representing a combination of the equation of state of ideal rubbers and Boltzmann s superposition principle, Lodge s equation provides an interpolation between the properties of rubbers and viscous liquids. The limiting cases of an elastic rubber and the Newtonian liquid are represented by... 

Validity of the linear stress-optical rule points at the dominant role of the network forces in pol3mier melts. The Lodge equation of state can be interpreted on this basis. We introduced the equation empirically, as an ap>-propriate combination of properties of rubbers with those of viscous liquids. It is possible to associate the equation with a microscopic model. Since the entanglement network, although temporary in its microscopic structure, leads under steady state conditions to stationary viscoelastic properties, we have to assume a continuous destruction and creation of stress-bearing chain sequences. This implies that at any time the network will consist of sequences of different ages. As long as a sequence exists, it can follow all imposed deformations. 

An equation like 10.5, obtained from the Boltzmann principle by replacing the infinitesimal strain tensor by one that can describe a large deformation, is sometimes called a model of finite linear viscoelasticity . If the memory function in the rubberlike liquid is taken to be the relaxation modulus of a single Maxwell element [G(f) = Gq exp(f/T)], we obtain the special case of the rubber like liquid that we will call Lodge s equation this is shown as Eq. 10.6. 

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