Factorability time-strain

Since these assumptions are not always justifiable when applied to plastics, the classic equations cannot be used indiscriminately. Each case must be considered on its merits, with account being taken of such factors as the time under load, the mode of deformation, the service conditions, the fabrication method, the environment, and others. In particular, it should be noted that the traditional equations are derived using the relationship that stress equals modulus times strain, where the modulus is a constant. From the review in Chapter 2 it should be clear that the modulus of a plastic is generally not a constant. Several approaches have been used to allow for this condition. The drawback is that these methods can be quite complex, involving numerical techniques that are not attractive to designers. However, one method has been widely accepted, the so-called pseudo-elastic design method. 

Firstly, it has been shown that there may be many experimental problems in a direct determination of the experimental fimction. In shear, damping functions obtained from step strain and from step strain rate experiments do not match each other. This poses an important question on the validity of the separability assumption in the short time rai e. Significant departures from this factorization have already been observed in the case of narrow polystyrene fractions by Takahashi et al. [54]. These authors found that time-strain superposition of the linear and nonlinear relaxation moduli was only possible above a cert2un characteristic time. It is interesting to note that this is predicted by the Doi-Edwards theory [10] and according to this theory, this phenomena is attributed to an additional decrease of the modulus connected to a tube contraction process and time-strain separability may hold after this equilibration process has been completed. Other examples of non-separability were also reported by Einaga et al. [55] and more recently by Venerus et al. [56] for solutions. 

For small strains S(E) is only negh bly different fi om unity, so inclusion of this term does not change the e3q)r on for plateau modulus (Eq. 37). It also does not affect the factorization of strain and time depmidence for t > r, but it does change the strain dependent part rather substantially, principally by shifting the onset of strain dependence in h(y) to smaller strains (Fig. 14). 

Figure 3. Tonperature shift ftu tors, log as a function of temperature. Table II. Vertical shift factors for time - temperature and time - strain superposition...

Such a behavior of stress relaxation is called time-strain separability (Larson 1988). The factor h(y) is called the damping function. The empirical damping functions that are often used in polymer melts include the exponential function... 

In a series of articles by Salem [20-22], based on the results of the study of the effect of strain rate and draw temperature on the crystallization of PET, it was shown that the crystallization curves at different strain rates and temperatures could be superimposed by using a shift factor. This strain rate/draw-time superposition can be used to predict the degree of crystallinity at any strain rate and temperature. 

Equation (9) has an empirical origin but a theoretical foundation can be proposed as follows. Indeed, quite a common assumption in many approaches of nonlinear viscoelasticity consists in considering time-strain separability (or factorability). Such an assumption readily means that the nonlinear relaxation modulus function G(t, y) can be separated into a time-dependent and a strain-dependent contributions, so that ... 

For deformations other than shear—such as step biaxial extension (Soskey and Winter, 1985) and step planar extension (Khan and Larson, 1991)—time-strain factorability has also been found to hold, at least approximately, for commercial polymer melts. If... 

Factorization of the function concentrated polystyrene solution, time-strain factorability is not valid at short times after the imposition of the step shear. An accurate K-BKZ constitutive equation for shearing flows of this material will be much more complex than that for melt I. Furthermore, in strain histories in which a strain reversal takes place, such as constrained recoil (Wagner and Laun, 1978) or double-step strains with the second strain of sign opposite the first (Doi, 1980 Larson and Valesano, 1986), good agreement... 

The study that produced the data shown in Fig. 10.2 [2] included several other solutions, and it was found that molecular weight and concentration had little effect on the damping function for cM around 5 lO. Todemonstrate the degree of time-strain separability, the data ofFig. 10.2 are replotted in Fig. 10.5 as relaxation modulus divided by the vertical shift factor required to superpose them, i.e., The superposibility is excellent for times longer than a... 

Step 2. After a contact time t, the material is fractured or fatigued and the mechanical properties determined. The measured properties will be a function of the test configuration, rate of testing, temperature, etc., and include the critical strain energy release rate Gic, the critical stress intensity factor K[c, the critical... 

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