Stress relaxation modulus curve

Figure 6.9 Stress relaxation modulus curve illustrating polymer stress-response behavior.2... 

Equation (3) is plotted with two different time scales in Figures 1 and 2 for values somewhat typical of an elastomer. All the initial deformation takes place in the spring at a later time the dashpot starts to relax and allows the spring to contract. Most of the relaxation takes place within one decade of time on both sides of the relaxation time, but this is shown clearly only in Figure 2. On the logarithmic time scale, the stress-relaxation curve has a maximum slope at the time / = T and the stress ratio cr/cr is 0.3679 ore. The stress relaxation may also be given in terms of a stress-relaxation modulus Er(t) ... 

The temperature-time superposition principle is illustrated in Figure 8 by a hypothetical polymer with a TK value of 0°C for the case of stress relaxation. First, experimental stress relaxation curves are obtained at a series of temperatures over as great a time period as is convenient, say from 1 min to 10 min (1 week) in (he example in Figure 8. In making the master curve from the experimental data, the stress relaxation modulus ,(0 must first be multiplied by a small temperature correction factor/(r). Above Tg this correction factor is where Ttrt is the chosen reference... 

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,... 

Figure 22 Stress-relaxation modulus as a function of Crystallinity at temperatures above I f Numbers on the curves are rough values of the degree of Crystallinity.

Because of equipment limitations in measuring stress and strain in polymers, the time-temperature superposition principle is used to develop the viscoelastic response curve for real polymers. For example, the time-dependent stress relaxation modulus as a function of time and temperature for a PMMA resin is shown in... 

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 relaxation modulus predicted by the Rouse model for a melt of unentangled chains with jV= 10 . The solid curve is the exact Rouse result [Eq. (8.55)] and the dotted curve is the approximate Rouse result [Eq. (8.48)]. 

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 ... 

Computer) Determine the 10-s modulus-versus-temperature curve for a material whose tensile stress relaxation modulus is given as... 

FIGURE 6.5 Stress relaxation modulus of natural mbber at different types and magnitudes of strain. The curves are approximately parallel, indicating that strain and time effects are decoupled (Fuller, 1988). 

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. 

Viscoelastic characteristics of polymers may be measured by either static or dynamic mechanical tests. The most common static methods are by measurement of creep, the time-dependent deformation of a polymer sample under constant load, or stress relaxation, the time-dependent load required to maintain a polymer sample at a constant extent of deformation. The results of such tests are expressed as the time-dependent parameters, creep compliance J t) (instantaneous strain/stress) and stress relaxation modulus Git) (instantaneous stress/strain) respectively. The more important of these, from the point of view of adhesive joints, is creep compliance (see also Pressure-sensitive adhesives - adhesion properties). Typical curves of creep and creep recovery for an uncross-Unked rubber (approximated by a three-parameter model) and a cross-linked rubber (approximated by a Voigt element) are shown in Fig. 2. 

The data can he obtained from dynamic mechanical analysis (DMA) tests used to construct the stress relaxation master cnrves at different levels of conversion, within the frame of time-temperature superposition principle. Curves of stress relaxation modulus can he modeled using the stretched Kohlrausch-William-Watts (KWW) exponential function at each level of conversion, as follows ... 

As relaxation curves extend over many decades of time it is convenient to use a logarithmic time-scale, so that the stress relaxation modulus becomes... 

Wu [5] got the MWD from storage modulus G and stress relaxation modulus G(f) using approximations derived from the Doi-Edwards description of chain dynamics. Wu s method accurately predicted the MWD of polymers with narrow distribution. However, often, it led to a distorted shape of the MWD for the sample with bimodal distributions. Therefore, Tuminello [6-8] developed a theory based on a diluted assumption in 1986. His method rigorously applies only to linear polymers. Especially, it works better for linear polymers with PI < 3.5. According to his theory, the relative differential AIFVD of polymer can be determined well from dynamic modulus master curve. 

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)...

Stress-relaxation modulus-time curves taken at different temperatures can be related to a master curve at a specific temperature. More detailed discussion of the WLF equation and time-temperature superposition can be found in [15]. 

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