Rubberlike liquid

Lodge rubberlike liquid (3.3-15) Constant = constant l/2 = 0 No predicts elongational stress growth rj1 (l, k) Yes... 

Equations 3.4-3 and 3.4-4 form the molecular theory origins of the Lodge rubberlike liquid constitutive Eq. 3.3-15 (23). For large strains, characteristic of processing flows, the nonlinear relaxation spectrum is used in the memory function, which is the product of the linear spectrum and the damping function h(y), obtained from the stress relaxation melt behavior after a series of strains applied in stepwise fashion (53)... 

The DSC is widely used to measure the glass-rubber transition temperature (Tg-value), which is an important parameter for polymer characterisation. The Tg-value represents the temperature region at which the (amorphous phase) of a polymer is transformed from a brittle, glassy material into a tough rubberlike liquid. This effect is accompanied by a step-wise increase of the DSC heat flow/temperature or specific heat/ temperature curve. Enthalpy relaxation effects can hamper the... 

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). 

Generally, it is found that while the upper convected Maxwell fluid, Eq. (26), and the Lodge rubberlike liquid, Eq. (31), predict the correct qualitative features of polymeric fluid behavior, the representation is not quantitative. In particular, in a stress-relaxation experiment, the relaxation takes place over too broad a range of time to be described by a single exponential. One therefore uses a spectrum of relaxation times, and modifies Eq. (30) to... 

The relaxation of the primary normal stress difference after cessation of steady-state flow at strain rate 7 can also be expressed in terms of linear viscoelastic properties by these models. For example, in terms of the relaxation spectrum, the rubberlike liquid theory of Lodge ° provides ... 

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. 

The rubberlike liquid model is able to predict, qualitatively, certain nonlinear viscoelastic phenomena. In particular, some effects arising from the finite orientation of chain segments are predicted, for example a nonzero first normal stress difference. However, it fails to describe many other nonlinear effects. For example, it predicts that the viscosity is constant with shear rate and the second normal stress difference is zero. In fact, all its predictions for the shear stress in simple shear are the same as those of the Boltzmann superposition principle. We can gain some insight into the origins of nonlinearity by examining the features of the rubberlike liquid model that limit its predictive ability. 

For the relaxation of the first normal stress difference following a step strain, the rubberlike liquid model (Eq. 10.6) predicts that [35] ... 

For start-up of steady-simple shear the rubberlike liquid model predicts that the shear stress is given Eq. 4.8. While h/2 is predicted to be zero, the first normal stress difference is ... 

Another symbol sometimes used in place of Pj is 6. The rubberlike liquid model (Eq. 10.5), predicts that Pj is independent of shear rate and related to the linear relaxation modulus ... 

Figure 10.14 Recoverable shear for steady simple shear for an LDPE. Also shown is the shear-rate dependent recoverable shear 2 aiy), the predictions of the rubberlike liquid model (straight line.fromEq. 10.5) and Wagner s equation (solid curve,from Eq. 10.10).FromLaun...

Turning to the behavior of typical melts, it is found that the damping function is not nearly as sensitive to molecular structure as are the linear viscoelastic properties, e.g. the storage and loss moduli. The rubberlike liquid, as well as the tube model, predict that the ratio of the first normal stress difference to the shear stress in step shear should be equal to the strain at all strains, and this is in fact observed. The other quantity measured in simple shear experiments is the second normal stress difference, but this is difficult to measure and few data are available. Of the shear histories other than step strain than have been used to study nonlinear viscoelasticity, start-up of steady simple shear has been the most used. If the shear rate is sufficiently large, some degree of chain stretch can be generated in the early stages. 

Here y[o] is shorthand for YiojitXl- This can be regarded as an expansion about the Lodge rubberlike liquid, which in turn includes the general linear viscoelastic model. By expanding the strain tensors in equation (49) about time t, the retarded-motion expansion of equation (38) is obtained, with the... 

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