Tensile stress relaxation modulus

Here m is the usual small-strain tensile stress-relaxation modulus as described and observed in linear viscoelastic response [i.e., the same E(l) as that discussed up to this point in the chapter). The nonlinearity function describes the shape of the isochronal stress-strain curve. It is a simple function of A, which, however, depends on the type of deformation. Thus for uniaxial extension,... 

Finally, tensile deformations provide the same information as shear deformation as long as the incompressibility assumption is not violated. In this case, the tensile stress relaxation modulus E(t) is directly related to the shear modulus E(t) = 3G(f), and all other relationships follow accordingly. 

Stress-strain curves are often measured by monitoring the tensile stress as a sample, originally at rest, is subjected to a constant tensile strain rate starting at t = 0. Show that, at any subsequent time during the constant-strain-rate period, the slope of the stress-strain curve is the tensile stress relaxation modulus ... 

Suppose that a material with a tensile stress relaxation modulus given as... 

Here again, tensile stress relaxation modulus. 

Figure 3-11. Continuous distribution of relaxation times expressed as H r) and the corresponding tensile stress relaxation modulus E(t).

A straight section of polypropylene pipe is fixed rigidfy at its ends. Its tensile stress relaxation modulus at time t and coefficient of linear thermal expansion at 20°C are, respective. ... 

The tensile-stress relaxation modulus of a certain pofymer can be... 

Tensile relaxation modulus Tensile stress relaxation modulus Threshold energy of coagulation Activation energy Exchange energy of i-j contact Helmholtz free energy (F = E—TS)... 

This analysis turns out to be a simple extension of that for the deflection of a linear elastic cantilever. Young s modulus is replaced by the appropriate viscoelastic counterpart—in this case the tensile stress relaxation modulus. That this is the appropriate viscoelastic property to be employed can be thought of as arising from the fact that, when the cantilever is subjected to constant deflection, every element of it is subjected to a constant tensile strain (i.e. to tensile stress relaxation conditions). 

A nylon bolt of diameter 8 mm is used to join two rigid plates (Fig. 4.34). The nylon can be assumed to be linear viscoelastic with a tensile stress relaxation modulus... 

The tensile-stress relaxation modulus of a certain polymer can be approximated by an expression of the form given in eqn (4.S3). Relaxed and unrelaxed tensile moduli are r = O.SGPa and u = 1.5GPa, respectively, with relaxation time t 5s. The polymer is subjected to a constant rate of tensile strain e = 10 s" . Derive the stress-strain relation Boltzmann superposition principle. 

Figure 18 Tensile stress relaxation modulus for PP homopolymer at 23 C 4.3.5 Dynamic Fatigue...

Fig. 6.14 The master time-humidity stress relaxation curve for ASl/3501-6. Tensile stress relaxation modulus vs. Log time. (Crossman and Warren 1985)...

Solution. The tensile stress-relaxation modulus for a Maxwell element is... 

Returning to the Maxwell element, suppose we rapidly deform the system to some state of strain and secure it in such a way that it retains the initial deformation. Because the material possesses the capability to flow, some internal relaxation will occur such that less force will be required with the passage of time to sustain the deformation. Our goal with the Maxwell model is to calculate how the stress varies with time, or, expressing the stress relative to the constant strain, to describe the time-dependent modulus. Such an experiment can readily be performed on a polymer sample, the results yielding a time-dependent stress relaxation modulus. In principle, the experiment could be conducted in either a tensile or shear mode measuring E(t) or G(t), respectively. We shall discuss the Maxwell model in terms of shear. 

Using this factorizability of response into a time-dependent and a strain-dependent function. Landel et ai. (61,62) have proposed a theory that would express tensile stress relaxation in the nonlinear regime as the product of a time-dependent modulus and a function of the strain ... 

The buffering action of a coating in this situation is determined by the relaxation modulus of the coating material. The relaxation modulus may be measured on a film cast from the material by carrying out tensile-stress relaxation measurements with a suitable apparatus such as a Rheovibron dynamic viscoelastometer operated in a static mode. Figure 13 (inset) displays such measurements for the four coating materials used on the fibers measured in Figure 12. The measurements were carried out at 23 °C at small tensile strains, where the materials exhibit linear viscoelastic behavior. 

In the same studies, Moehlenpah et al (1970, 1971) obtained master curves for the stress relaxation of their epoxy systems, at least into the glass-to-rubber transition region (Figure 12.4), and demonstrated similar behavior of both the stress relaxation modulus and the tensile modulus as a function of strain rate. As with the strain rate studies mentioned, no effect of filler type on the WLF shift factor was observed. All solid fillers increased the modulus of the system, the fibers being more effective than the spheres. The bubbles, as expected (Nielsen, 1967a), decreased the modulus. 

Because we are dealing with relatively small strains, shear normal stresses and the type of deformation— for example, shear versus uniaxial extension—will not be important. To impose a small strain on a material in one direction only requires exerting a stress in that direction (Lockett, 1972). Thus, we could just as well define a tensile relaxation modulus from tensile stress relaxation (recall eq. 1.5.10)... 

Although time-temperature superposition is applicable to any viscoelastic response test (creep, dynamic, etc.), here, we will focus on its application to stress relaxation. Figure 16.12 shows tensile stress relaxation data at various temperatures for polyisobutylene, plotted in the form of a time-dependent tensile (Young s) modulus E t) versus, time on a log-log scale ... 

Several functions are used to characterize tire response of a material to an applied strain or stress [4T]. The tensile relaxation modulus E(t) describes tire response to tire application of a constant tensile strain l/e -. 

Here is tire tensile stress and = lL-/L-, where is tire initial lengtli of tire sample and AL is tire sample elongation. In shear experiments, tire shear relaxation modulus G(t) is defined as where... 

Not only are the creep compliance and the stress relaxation shear modulus related but in turn the shear modulus is related to the tensile modulus which itself is related to the stress relaxation time 0. It is therefore in theory possible to predict creep-temperature relationships from WLF data although in practice these are still best determined by experiment. 

For elastomers, factorizability holds out to large strains (57,58). For glassy and crystalline polymers the data confirm what would be expected from stress relaxation—beyond the linear range the creep depends on the stress level. In some cases, factorizability holds over only limited ranges of stress or time scale. One way of describing this nonlinear behavior in uniaxial tensile creep, especially for high modulus/low creep polymers, is by a power... 

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