Normal Stress Relaxation

Turning to the normal stresses, we note that in an incompressible material, normal stresses are themselves of no rheological significance, because if they are equal in all directions they cause no deformation. However, differences between normal stress components are significant, because they do cause deformation. For simple shear, the two theologically significant differences are the irst and second normal stress differences Nj and N2, which are defined by Eqs. 10.33 and 10.34. 

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

This suggests that at sufficiently small strains, the stress ratio (N la) should become equal to the strain. This relationship is known as the Lodge-Meissner rule [36]. 

This is similar to the result for an isotropic, perfectly elastic solid [11, p. 78], which has a constant modulus of elasticity  

the stress ratio (Nj/o) is equal to the strain for the perfecdy elastic solid. 

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Hint Note that the normalized stress relaxation modulus for reptation is the probability (5, t) that a segment s is still a part of the tube after time t averaged over all segments s ... 

Fig. 6. Shear and normal stress relaxation for rigid dumbbell suspensions...

In stress relaxation after cessation of steady shear flow, the elastic dumbbells give no dependence of the relaxation process on the steady-state shear rate, but the rigid dumbbells do. In addition the elastic dumbbells show the shear and normal stresses relaxing with exactly the same... 

Fig. 19. Normal stress relaxation following steady shear flow for 60 mole % PHB/PET. 

Stammen et al. measured the stress relaxation curves for 20 and 25 % PVA-C by applying a 20 % constant strain and monitoring the stress relaxation for 24 h. They did not report the normalized stress-relaxation curves, instead they plotted the stress (in MPa) variation over time [53]. 

MiUon et al. studied the stress relaxation properties of 10 % PVA-C after 6 FTCs by applying a 45 % constant strain and measuring the normalized stress relaxation for 1 h. They observed that the stress remaining after 1 h did not completely level off [45]. 

Wang and Campbell studied the stress relaxation curves for their samples by applying a 25 % constant strain and measuring the normalized stress relaxation for 30 s. They also performed creep measurement. The initial load was applied by compressing the samples at 4 mm/s ( 33 %/s) up to 25 % strain or when it reached 223 N and holding the force for 30 s [54]. 

D. C. Venerus and H. Kahvand, Normal Stress Relaxation in Reversing Double Step Strain Flows J. Rheol. 38, 1297-1315 (1994). 

The increase of stress during approach to steady-state flow can also be studied it often passes through a maximum. With the device of Pierson and Kovacs, a small correction is made for the compliance of the apparatus. To obtain the normalized stress relaxation function, no form factor b is... 

Venerus, D. C. and Kahvand, H. Normal stress relaxation in reversing double-step flows./. Rheol. (1994) 38, pp. 1297-1315... 

In an ideal fluid, the stresses are isotropic. There is no strength, so there are no shear stresses the normal stress and lateral stresses are equal and are identical to the pressure. On the other hand, a solid with strength can support shear stresses. However, when the applied stress greatly exceeds the yield stress of a solid, its behavior can be approximated by that of a fluid because the fractional deviations from stress isotropy are small. Under these conditions, the solid is considered to be hydrodynamic. In the absence of rate-dependent behavior such as viscous relaxation or heat conduction, the equation of state of an isotropic fluid or hydrodynamic solid can be expressed in terms of specific internal energy as a function of pressure and specific volume E(P, V). A familiar equation of state is that for an ideal gas... 

A representative measure of rubbery elasticity of a material may be two quantities dimensionless ratio (ct/t) and characteristic relaxation time 9 = ct/2ty. According to the data of works [37, 38] when fibers are introduced into a melt, ct/t increases (i.e. normal stresses grow faster than stresses) and 0 also increases on a large scale, by 102-103 times. However, discussing in this relation the papers published earlier, we noted in the paper cited that the data were published according to which if fibers were used as a filler (as in work [37]), 9 indeed increased [39], but if a filler represented disperse particles of the type Ti02 or CaC03, the value of 0 decreased [40],... 

The stress-relaxation behavior of a material is normally determined in either the tensile or the flexural mode. In these experiments, a material specimen is rapidly elongated or compressed to produce a specified strain level and the load exerted by the specimen on the test apparatus is measured as a function of time. Specimens of certain plastics may fail during tensile or flexural stress-relaxation experiments. 

Analogous results have been found for stress relaxation. In fibers, orientation increases the stress relaxation modulus compared to the unoriented polymer (69,247,248,250). Orientation also appears in some cases to decrease the rate, as well as the absolute value, at which the stress relaxes, especially at long times. However, in other cases, the stress relaxes more rapidly in the direction parallel to the chain orientation despite the increase in modulus (247.248,250). It appears that orientation can in some cases increase the ease with which one chain can slip by another. This could result from elimination of some chain entanglements or from more than normal free volume due to the quench-cooling of oriented polymers. 

Several attempts have been made to superimpose creep and stress-relaxation data obtained at different temperatures on styrcne-butadiene-styrene block polymers. Shen and Kaelble (258) found that Williams-Landel-Ferry (WLF) (27) shift factors held around each of the glass transition temperatures of the polystyrene and the poly butadiene, but at intermediate temperatures a different type of shift factor had to be used to make a master curve. However, on very similar block polymers, Lim et ai. (25 )) found that a WLF shift factor held only below 15°C in the region between the glass transitions, and at higher temperatures an Arrhenius type of shift factor held. The reason for this difference in the shift factors is not known. Master curves have been made from creep and stress-relaxation data on partially miscible graft polymers of poly(ethyl acrylate) and poly(mcthyl methacrylate) (260). WLF shift factors held approximately, but the master curves covered 20 to 25 decades of time rather than the 10 to 15 decades for normal one-phase polymers. 

A stress relaxation experiment can be performed on a wide range of materials. If we perform such a test on a real material a number of deviations are normally observed from the behaviour of a single Maxwell model. Some of these deviations are associated with the application of the strain itself. For example it is very difficult to apply an instantaneous strain to a sample. This influences the measured response at short experimental times. It is often difficult to apply a strain small enough to provide a linear response. A Maxwell model is only applicable to linear responses. Even if you were to imagine an experiment where a strain is... 

Fig. 3.14. The data is for a very broad range of times and temperatures. The superposition principle is based on the observation that time (rate of change of strain, or strain rate) is inversely proportional to the temperature effect in most polymers. That is, an equivalent viscoelastic response occurs at a high temperature and normal measurement times and at a lower temperature and longer times. The individual responses can be shifted using the WLF equation to produce a modulus-time master curve at a specified temperature, as shown in Fig. 3.15. The WLF equation is as shown by Eq. 3.31 for shifting the viscosity. The method works for semicrystalline polymers. It works for amorphous polymers at temperatures (T) greater than Tg + 100 °C. Shifting the stress relaxation modulus using the shift factor a, works in a similar manner.

Stress decay (relaxation) measurements of propellant binders are a way to obtain insight into the network structure of binder systems (29). In addition, high hysteretical losses appear to be associated with good tensile properties. Figure 5 shows a normalized stress-decay vs. time plot of a polyurethane elastomer. If the reference stress, [Pg.105]

Constant strain for stress relaxation tests and constant load creep tests may be conducted in simple devices. Temperature control is critical since the results are usually applied as a spectral representation for structural analysis or research purposes. Figure 8 illustrates a multistation creep tester with automated data recorders. Strain and load endurance tests are conducted in similar devices, but the conditions existing at failure and time to failure are normally the only data required. The endurance tests are used frequently to supplement the constant displacement rate tests for routine evaluation. 

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