The Reduced Stress

The reduced stress, or, was introduced by Krieger4 from a dimensional analysis and has the form  

The similarity to the Peclet number is obvious but we should also bear in mind the relationship to the Deborah number. This becomes clear when we consider the fact that the mechanism of stress relaxation is due to the 

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The reduced stress is defined as the force per cross-sectional area of the undeformed sample, divided by the term X-X- with X being the relative elongation L/L0. With undiluted rubber, this is not found experimentally. In most cases, however, the elastic behaviour in a moderate elongation range is satisfactorily described fcy the empirical Mooney-Rivlin equation, which predicts a linear dependence of on reciprocal elongation X- (32-34)... 

Plots of G at 0.5 Hz and the reduced stress ore(j obtained from stress-strain measurements at small strains against temperature, give almost identical straight lines (Figure 5). This similarity was expected because no frequency dependence of G had been observed. Hence G equals the equilibrium modulus G G moreover equals the reduced stress ore(j, if the latter is measured in the vicinity of X= 1. The measurements were always performed at X = 1.02 - 1.04, so that this requirement is fulfilled. 

Figure 5. From left to right temperature dependence of the storage modulus at 0.5 Hz, and of the reduced stress, ( 1.03). Key , PDMS-B1 O, PDMS-B2 X, PDMS-B6 A, PDMS-B7.

Comparison with Statistical Theory at Moderate Strains. So far we have shown, that a transition between the two limiting classical theories, i.e. affine theory and phantom theory, is possible by a suitable choice of the network microstructure. This argument goes beyond the revised theory by Ronca and Allegra and by Flory, which predicts such a transition as a result of increasing strain, thus explaining the experimentally observed strain dependence of the reduced stress. 

The results of uniaxial stress-strain experiments are often analyzed in terms of the reduced stress defined by... 

Classical molecular theories of rubber elasticity (7, 8) lead to an elastic equation of state which predicts the reduced stress to be constant over the entire range of uniaxial deformation. To explain this deviation between the classical theories and reality. Flory (9) and Ronca and Allegra (10) have separately proposed a new model based on the hypothesis that in a real network, the fluctuations of a junction about its mean position may may be significantly impeded by interactions with chains emanating from spatially, but not topologically, neighboring junctions. Thus, the junctions in a real network are more constrained than those in a phantom network. The elastic force is taken to be the sum of two contributions (9) ... 

The results of stress-strain measurements can be summarized as follows (1) the reduced stress S (A- A ) (Ais the extension ratio) is practically independent of strain so that the Mooney-Rivlin constant C2 is practically zero for dry as well as swollen samples (C2/C1=0 0.05) (2) the values of G are practically the same whether obtained on dry or swollen samples (3) assuming that Gee=0, the data are compatible with the chemical contribution and A 1 (4) the difference between the phantom network dependence with the value of A given by Eq.(4) and the experimental moduli fits well the theoretical dependence of G e in Eq.(2) or (3). The proportionality constant in G for series of networks with s equal to 0, 0.2, 0.33, and 0. Ewas practically the same -(8.2, 6.3, 8.8, and 8.5)x10-4 mol/cm with the average value 7.95x10 mol/cm. Results (1) and (2) suggest that phantom network behavior has been reached, but the result(3) is contrary to that. Either the constraints do survive also in the swollen and stressed states, or we have to consider an extra contribution due to the incrossability of "phantom" chains. The latter explanation is somewhat supported by the constancy of in Eq.(2) for a series of samples of different composition. 

The shear rate range over which the shear thinning of hard sphere suspensions occurs can be determined from the equations due to Krieger26 or Cross27 in conjunction with the reduced stress or Peclet number respectively ... 

Note that err = y (crr)a3/k Tand recall that in a concentrated dispersion the Peclet number is Pe = 67ry (crr)a3/k T. The use of the suspension viscosity implies that the particle diffusion can be estimated from an effective medium approach. Both Krieger and Cross gave the power law indices (n and m) as 1 for monodisperse spherical particles. In this formulation, the subscript c indicates the characteristic value of the reduced stress or Peclet number at the mid-point of the viscosity curve. The expected value of Pec is 1, as this is the point at which diffusional and convective timescales are equal. This will give a value of ac 5 x 10 2. Figure 3.15 shows a plot of Equation (3.57a) with this value and n = 1... 

Figure 6.7 Plot of the stress-dependent packing fraction

Stress is thought to be an important factor in the pathophysiology of depression and anxiety disorders. It seems possible that the reduced stress reactivity of NKl receptor- and tael-deficient mice has contributed to the behavioral phenotypes observed in the animal models of anxiety and depression. 

Equation (5.131) can be inverted to give the reduced stress response to the inplane stress ... 

Other interesting features of elastomeric networks can be revealed using the plots of the reduced stress, crred = /( — -2) against inverse extension ratio 1. This can be analyzed from the stress-strain behavior described by a phenomenological expression suggested by Mooney [78] and Rivlin and Saunders [79] ... 

As is easily seen, the reduced stress components occurring in this equation can be interpreted as contributions per chain molecule, which... 

Subscript "w means that the weight average molecular weight is used in eqs. (3.44a), 3.44b) or (3.43). The angular brackets are omitted here, since the reduced stress components are no longer simple averages. As is seen from eq. (3.83), a somewhat modified polydispersity factor is obtained in this way. Its free-draining representation is quite simple and is quoted here ... 

Theory also predicts the reduced stress or modulus [/ ], defined as the ratio of the nominal stress to the strain function (a - or2), to be independent of the elongation a. Experimentally, however, the modulus is found to change with a, generally decreasing linearly with decreasing reciprocal elongation. 

For this purpose the "reduced" stress and strain are calculated from the following equations ... 

The stress-strain curve for unfilled NR exhibits a large increase in stress at higher deformations. NR displays, due to its uniform microstructure, a very unique important characteristic, that is, the ability to crystallise under strain, a phenomenon known as strain-induced crystallization. This phenomenon is responsible for the large and abrupt increase in the reduced stress observed at higher deformation corresponding, in fact, to a self-toughening of the elastomer because the crystallites act as additional cross-links in the network. This process can be better visualized by using a Mooney-Rivlin representation, based on the so-called Mooney-Rivlin equation ... 

Mackwell et al. (1985) found that when specimens that had been deformed under anhydrous conditions were subsequently further deformed under wet conditions, there was a significant change in microstructure. TEM observations revealed enhanced formation of dislocation walls, despite the reduced stress levels. This observation was interpreted as due to enhanced dislocation climb under wet conditions. However, the two walls illustrated by Mackwell et al. (1985) could be interpreted as healed or partly healed fractures. One wall consists of a very irregular network of dislocations with many bubbles, particularly at dislocation intersections. 

Note that, with this model, the reduced stress defined by Eq. (3) depends on the deformation X, whereas it did not for the simple affine model. 

Figure 1 presents the results for the illustrative cases n = 20 and 40 skeletal bonds, as a function of the elongation a = L/L , where L and Li are the stretched and unstretched sample lengths, respectively. An alternative representation of the same results is in terms of the reduced stress or modulus defined by (27,28)... 

For colloidal suspensions of spherical particles the concentration changes their separation d uniformly. Thus, one might predict the phase transitions between diluted, caged, or close-packed structures by calculating the attractive interactions from Eq. (16.12) and postulating a specific type of repulsive interactions. The transitions are detectable in plots of -qr versus the reduced stress ... 

Upon increasing the stress, the transition temperature of amorphous-mesophase increases (Fig. 18) [169,175], which variation is linear in the reduced stress (stress divided by absolute temperature). De Gennes predicted... 

At the reduced stress equal to 0.30 the conversion is complete, while the three LC regions in the material (in the middle and on both sides) are easily visible. We have thus also seen the LC reinforcement in action until the reduced stress of 0.20 was exceeded, it was the LC region in the middle which protected the cis conformations on the left side from extension to the trans form. The same results confirm also the basic rule of the mechanical behavior of polymeric materials related to the chain relaxation capability (CRC) [19—22] a polymeric material will relax if it only can. None of the mechanical energy coming from outside has been spent on destructive processes. 

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