Affine deformations

Johnson, M. W. and Segalman, D., 1977. A model for viscoelastic fluid behaviour which allows non-affine deformation. J. Non-Newtonian Fluid Mech. 2, 255-270. 

Even better agreement between theory and experiment has been obtained in other theories by abandoning the notion of affine deformation and recognizing that shorter subchains experience a greater strain than do longer subchains for a given stress. We shall not pursue this development any further, however, and shall turn next to a consideration of other types of deformation. 

Equation (3.47) is used with each of the force components given by (3.44). The limits of integration are different for affine deformation under shear at constant volume. In terms of the coordinates shown in Fig. 3.6, there is no change in x, z increases by a, and y decreases by 1/a. 

A plausible assumption would be to suppose that the molecular coil starts to deform only if the fluid strain rate (s) is higher than the critical strain rate for the coil-to-stretch transition (ecs). From the strain rate distribution function (Fig. 59), it is possible to calculate the maximum strain (kmax) accumulated by the polymer coil in case of an affine deformation with the fluid element (efl = vsc/vcs v0/vcs). The values obtained at the onset of degradation at v0 35 m - s-1, actually go in a direction opposite to expectation. With the abrupt contraction configuration, kmax decreases from 19 with r0 = 0.0175 cm to 8.7 with r0 = 0.050 cm. Values of kmax are even lower with the conical nozzles (r0 = 0.025 cm), varying from 3.3 with the 14° inlet to a mere 1.6 with the 5° inlet. In any case, the values obtained are lower than the maximum stretch ratio for the 106 PS which is 40. It is then physically impossible for the chains to become fully stretched in this type of flow. 

Equation (32a) has been very successful in modelling the development of birefringence with extension ratio (or equivalently draw ratio) in a rubber, and this is of a different shape from the predictions of the pseudo-affine deformation scheme (Eq. (30a)). There are also very significant differences between the predictions of the two schemes for P400- In particular, the development of P400 with extension ratio is much slower for the network model than for the pseudo-affine scheme. 

Fig. 3a. P200 and P400 as a function of draw ration for the pseudo-affine deformation scheme (uniaxially oriented sample) b P20o and P400 as a function of draw ratio X for the rubber network affine deformation scheme (N = 6, uniaxially oriented sample). Reproduced from Journal of Polymer Science by permission of the publishers, John Wiley Sons Incs (C)...

Such considerations appear to be very relevant to the deformation of polymethylmethacrylate (PMMA) in the glassy state. At first sight, the development of P200 with draw ratio appears to follow the pseudo-affine deformation scheme rather than the rubber network model. It is, however, not possible to reconcile this conclusion with the temperature dependence of the behaviour where the development of orientation reduces in absolute magnitude with increasing temperature of deformation. It was proposed by Raha and Bowden 25) that an alternative deformation scheme, which fits the data well, is to assume that the deformation is akin to a rubber network, where the number of cross-links systematically reduces as the draw ratio is increased. It is assumed that the reduction in the number of cross-links per unit volume N i.e. molecular entanglements is proportional to the degree of deformation. 

Pseudo-affine deformation scheme 96, 97 Pseudohexagonal (rotator) phase 67 Pseudorotator phase 67 Pulsed NMR techniques 30... 

The large deformability as shown in Figure 21.2, one of the main features of rubber, can be discussed in the category of continuum mechanics, which itself is complete theoretical framework. However, in the textbooks on rubber, we have to explain this feature with molecular theory. This would be the statistical mechanics of network structure where we encounter another serious pitfall and this is what we are concerned with in this chapter the assumption of affine deformation. The assumption is the core idea that appeared both in Gaussian network that treats infinitesimal deformation and in Mooney-Rivlin equation that treats large deformation. The microscopic deformation of a single polymer chain must be proportional to the macroscopic rubber deformation. However, the assumption is merely hypothesis and there is no experimental support. In summary, the theory of rubbery materials is built like a two-storied house of cards, without any experimental evidence on a single polymer chain entropic elasticity and affine deformation. 

In this section, we show the morphological changes of stretched NR without filler by AFM. Two-dimensional mappings of topography and elasticity for elongated NR will be given to confirm the breakdown of the long-beheved assumption of affine deformation. 

In this chapter, AFM palpation was introduced to verify the entropic elasticity of a single polymer chain and affine deformation hypothesis, both of which are the fundamental subject of mbber physics. The method was also applied to CB-reinforced NR which is one of the most important product from the industrial viewpoint. The current status of arts for the method is still unsophisticated. It would be rather said that we are now in the same stage as the ancients who acquired fire. However, we believe that here is the clue for the conversion of rubber science from theory-guided science into experiment-guided science. AFM is not merely high-resolution microscopy, but a doctor in the twenty-first century who can palpate materials at nanometer scale. 

Affine deformation method, description, 109 Agitated glass ampule method apparatus, 510,511/312 limitations, 515... 

For Ca > Cacri, a drop continually stretches until it breaks. If Ca > KCacr , where k is about 2 for simple shear flow and 5 for elongational flow (Janssen, 1993), the drop undergoes affine deformation, i.e., the drop acts as a material element, and it is stretched into an extended cylindrical thread with length L and radius R according to... 

Fig. 15. Affine stretching of a filament in the journal bearing flow. Experiments (top) agree well with computations (bottom) carried out assuming that the filament deforms as the suspending fluid (i.e., affine deformation) (Tjahjadi and Ottino, 1991).

Illustration Drop size distributions produced by chaotic flows. Affinely deformed drops generate long filaments with a stretching distribution based on the log-normal distribution. The amount of stretching (A) determines the radius of the filament locally as... 

When we begin to stretch a semicrystalline polymer it deforms affinely, that is, each element of the sample within the gauge region experiences identical stress and strain. As we continue to stretch the sample, we reach a point at which affine deformation ceases and the sample yields. At this point, it typically develops a local region of reduced cross-sectional area, known... 

According to the importance of the cross-links, various models have been used to develop a microscopic theory of rubber elasticity [78-83], These models mainly differ with respect to the space accessible for the junctions to fluctuate around their average positions. Maximum spatial freedom is warranted in the so-called phantom network model [78,79,83], Here, freely intersecting chains and forces acting only on pairs of junctions are assumed. Under stress the average positions of the junctions are affinely deformed without changing the extent of the spatial fluctuations. The width of their Gaussian distribution is predicted to be... 

The elastic free energy of the constrained-junction model, similar to that of the slip-link model, is the sum of the phantom network free energy and that due to the constraints. Both the slip-link and the constrained-junction model free energies reduce to that of the phantom network model when the effect of entanglements diminishes to zero. One important difference between the two models, however, is that the constrained-junction model free energy equates to that of the affine network model in the limit of infinitely strong constraints, whereas the slip-link model free energy may exceed that for an affine deformation, as may be observed from Equation (41). 

Shape Change of Structural Entities. In many cases the growing anisotropy is not only a phenomenon of rotating structural entities, but also goes along with a deformation of the structural entities themselves. This case will be studied here. Only affine deformations shall be discussed. In practice, such processes are observed while thermoplastic elastomers are subjected to mechanical load, but also while fibers are spun. 

According to his deduction the common finding of ellipsoidal deformation of the reflections is indicative for affine deformation. Moreover, he arrives at an equation that permits to determine with high accuracy the microscopical draw ratio, Xd, of the structural entities from the ellipticity of the deformed Debye sphere. This value can be compared to the macroscopical draw ratio. Even the intensity distribution along the ellipsoidal ridge is predicted for a bcc-lattice of spheres, and deviations of experimental data are discussed. 

The concept of affine deformation is central to the theory of rubber elasticity. The foundations of the statistical theory of rubber elasticity were laid down by Kuhn (JJ, by Guth and James (2) and by Flory and Rehner (3), who introduced the notion of affine deformation namely, that the values of the cartesian components of the end-to-end chain vectors in a network vary according to the same strain tensor which characterizes the macroscopic bulk deformation. To account for apparent deviations from affine deformation, refinements have been proposed by Flory (4) and by Ronca and Allegra (5) which take into account effects such as chain-junction entanglements. 

Because of the grandiose scale of the apparatus involved, SANS facilities are few in number worldwide access to them is limited and expensive. We have attempted to devise an experiment which employs solid state nuclear magnetic resonance to examine some aspects of affine deformation. 

Recent small angle neutron scattering experiments shed uncertainty on the assumption of affine deformation (14-18). There... 

Birefringence. The birefringence of a crosslinked Gaussian rubber subjected to an affine deformation is described by the theories of Kuhn and Grun (1 ) and Treloar (2). These predict a stress-optical coefficient given by... 

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