Sheared Networks

Shearing can be treated in the same way as drawing. When sheared, the sample is elongated in the.v direction and correspondingly contracted in thej direction. The coordinates in the z direction remain constant. Thus, a = otx, a = lay, and az = 1. Equation (11-33) therefore becomes 

The shear strain 7 produced by a shearing load is given by 7 = a — a From 

The relationship between shear stress 021 and shear strain 7 is given, in analogy to Equation (11-35), by 

After differentiating Equation (11-44), we obtain from Equation (11-45) 

According to Equation (11-46), the shear stress 021 is directly proportional to the shear strain 7. The rubber, therefore, deforms on shearing according to Hooke s law with the shear modulus G [see Equation (11-3)], but is non-Hookean during elongation [see Equation (11-39)]. 

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Under compression or shear most polymers show qualitatively similar behaviour. However, under the application of tensile stress, two different defonnation processes after the yield point are known. Ductile polymers elongate in an irreversible process similar to flow, while brittle systems whiten due the fonnation of microvoids. These voids rapidly grow and lead to sample failure [50, 51]- The reason for these conspicuously different defonnation mechanisms are thought to be related to the local dynamics of the polymer chains and to the entanglement network density. 

Colloidal dispersions often display non-Newtonian behaviour, where the proportionality in equation (02.6.2) does not hold. This is particularly important for concentrated dispersions, which tend to be used in practice. Equation (02.6.2) can be used to define an apparent viscosity, happ, at a given shear rate. If q pp decreases witli increasing shear rate, tire dispersion is called shear tliinning (pseudoplastic) if it increases, tliis is known as shear tliickening (dilatant). The latter behaviour is typical of concentrated suspensions. If a finite shear stress has to be applied before tire suspension begins to flow, tliis is known as tire yield stress. The apparent viscosity may also change as a function of time, upon application of a fixed shear rate, related to tire fonnation or breakup of particle networks. Thixotropic dispersions show a decrease in q, pp with time, whereas an increase witli time is called rheopexy. 

The shear modulus for an ideal elastomer in a perfect network is not difficult to derive ... 

Dispersion of a soHd or Hquid in a Hquid affects the viscosity. In many cases Newtonian flow behavior is transformed into non-Newtonian flow behavior. Shear thinning results from the abiHty of the soHd particles or Hquid droplets to come together to form network stmctures when at rest or under low shear. With increasing shear the interlinked stmcture gradually breaks down, and the resistance to flow decreases. The viscosity of a dispersed system depends on hydrodynamic interactions between particles or droplets and the Hquid, particle—particle interactions (bumping), and interparticle attractions that promote the formation of aggregates, floes, and networks. 

If the dispersion particles are attracted to each other, they tend to flocculate and form a stmcture. At low concentrations the particles form open aggregates, which give a fractal stmcture (93,94). At higher concentrations a network stmcture results, which can be so pronounced that the mixture has a yield point and behaves like a soHd when at rest. Shearing breaks up this stmcture, and viscosity decreases. 

As one example, in thin films of Na or K salts of PS-based ionomers cast from a nonpolar solvent, THF, shear deformation is only present when the ion content is near to or above the critical ion content of about 6 mol% and the TEM scan of Fig. 3, for a sample of 8.2 mol% demonstrates this but, for a THF-cast sample of a divalent Ca-salt of an SPS ionomer, having only an ion content of 4.1 mol%, both shear deformation zones and crazes are developed upon tensile straining in contrast to only crazing for the monovalent K-salt. This is evident from the TEM scans of Fig. 5. For the Ca-salt, one sees both an unfibrillated shear deformation zone, and, within this zone, a typical fibrillated craze. The Ca-salt also develops a much more extended rubbery plateau region than Na or K salts in storage modulus versus temperature curves and this is another indication that a stronger and more stable ionic network is present when divalent ions replace monovalent ones. Still another indication that the presence of divalent counterions can enhance mechanical properties comes from... 

From the results obtained in [344] it follows that the composites with PMF are more likely to develop a secondary network and a considerable deformation is needed to break it. As the authors of [344] note, at low frequencies the Gr(to) relationship for Specimens Nos. 4 and 5 (Table 16) has the form typical of a viscoelastic body. This kind of behavior has been attributed to the formation of the spatial skeleton of filler owing to the overlap of the thin boundary layers of polymer. The authors also note that only plastic deformations occurred in shear flow. 

Above a critical hller concentration, the percolation threshold, the properties of the reinforced rubber material change drastically, because a hller-hUer network is estabhshed. This results, for example, in an overproportional increase of electrical conductivity of a carbon black-hUed compound. The continuous disruption and restorahon of this hller network upon deformation is well visible in the so-called Payne effect [20,21], as represented in Figure 29.5. It illustrates the strain-dependence of the modulus and the strain-independent contributions to the complex shear or tensUe moduli for carbon black-hlled compounds and sUica-hUed compounds. 

In pseudoplastic substances shear thinning depends mainly on the particle or molecular orientation or alignement in the direction of flow, this orientation is lost or regained at the same speed. Additionally many dispersions show this potential for particle or molecule interactions, this leads to bonds creating a three-dimensional network structure. They are often build-up from relatively weak hydrogen or ionic bonds. When the network is disturbed. 

Finally, in instances in which a bulky solute molecule with several functional groups can be added to the system, a fragile sort of structure can be built up by simultaneous attachment of these molecules to create a network with the characteristics of a gel. This system is then permanently metastable toward settling and caking, but may not withstand the ravages of shear or high temperature. 

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