Rubber-like liquids

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. 

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... 

From Eq. (3-24) for a rubber-like liquid, assuming a single relaxation time r and modulus G, calculate formulas for the extensional viscosity as a function of time after start-up of steady uniaxial extension at extension rate e. 

We now wish to solve Eq. (3-24), the equation for a rubber-like liquid ... 

In order to improve the internal stress performance even further or to provide toughness for potting and adhesive applications it is advantageous to introduce reactive additives to the basic epoxide compound formulation. One of the most popular additives that has found considerable use in this type of application is a rubber-like liquid copolymer known as carboxyl terminated butadiene/acrylonitrile (CTBN). The generalised molecular structure of this material is ... 

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... 

As we can see, both are independent of the strain rate 7. Hence, as a first conclusion. Lodge s equation of state cannot describe the shear thinning phenomenon. Equation (7.147) is in fact identical with Eq. (5.107) derived in the framework of linear response theory. The new result contributed by Lodge s formula is the expression Eq. (7.148) for the primary normal stress difference. It is interesting to note that the right-hand side of this equation has already appeared in Eq. (5.108) of the linear theory, formulating the relationship between G t) and the recoverable shear compliance. If we take the latter equation, we realize that the three basic parameters of the Lodge s rubber-like liquid, rjo, and 1,0 are indeed related, by... 

Insertion of B(t,t ) in the equation of state for rubber-like liquids yields the normal stress diflFerence... 

This is a second form for the equation of state of rubber-like liquids. m t—t ) is called a memory function , since it characterizes in a general view the fading of the memory on the past. As Eqs. (7.190) and (7.194) have equal forms we find... 

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

This is a second form for the equation of state of rubber-like liquids. is... 

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. 

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 ... 

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