Yield strain-rate dependence

From the weak dependence of ef on the surrounding medium viscosity, it was proposed that the activation energy for bond scission proceeds from the intramolecular friction between polymer segments rather than from the polymer-solvent interactions. Instead of the bulk viscosity, the rate of chain scission is now related to the internal viscosity of the molecular coil which is strain rate dependent and could reach a much higher value than r s during a fast transient deformation (Eqs. 17 and 18). This representation is similar to the large loops internal viscosity model proposed by de Gennes [38]. It fails, however, to predict the independence of the scission yield on solvent quality (if this proves to be correct). 

The strain rate dependence of the yield stress is shown at various temperatures in Fig. 20. To go further in the analysis, it is interesting to use the Eyring approach presented in Sect. 2.2.1.1. For this purpose, the ratio oy/T, K is plotted versus log( , s-1) at various temperatures in Fig. 21. A linear dependence is observed at each temperature, in agreement with the Eyring expression. However, the slopes show two different temperature regimes at low and high temperatures. Of course, the activation volume, Vo, directly related to the slope, reflects the change in behaviour, as shown in Fig. 22. At low temperature, the activation volume is small (around 0.1 nm3) and independent of temperature, whereas it increases rapidly above room temperature... 

Fig. 20 Strain rate dependence of yield stress, ay, and plastic flow stress, apf, of PMMA at the indicated temperatures (From [33])...

Fig. 23. Comparison of the predicted strain rate dependence of plastic yield of PS under uniaxial compression, tension, and shear...

Fig. 26. Comparison of the predicted (solid lines [46]) and measured (points [45]) strain rate dependence of the compressive yield stress of silica filled epoxy at different filler concentrations...

It is usual to assume that the shear yield stress has the same temperature dependence and strain rate dependence as the flow stress of the polymer in the active zone of the craze, i.e., n, = n, and in fact usually one go even further and sets How-... 

Equation (14.20) describes the temperature and strain rate dependence of the yield stress, Gy. 

Figure 11.4 shows the variation of yield stress with strain rate for the a-PP and 3-PP specimens. Compared to a-PP, the incorporation of (3-nucleator brings about distinct softening as evidenced by lower values of yield stress at various strain rates. The strain rate dependence of yield stress can be described by Frying equation given by... 

Figure 11.4 Strain rate dependence of yield stress for a-PP and (3-PP specimens. (From Reference 15 with permission from The Society of Plastics Engineers.)...

Polymers are very sensitive to the rate of testing. As the strain rate increases, polymers in general show a decrease in ductility while the modulus and the yield or tensile strength increase. Figure 13.32 illustrates this schematically. The sensitivity of polymers to strain rate depends on the type of polymer for brittle polymers the effect is relatively small, whereas for rigid, ductile polymers and elastomers, the effects can be quite substantial if the strain rate covers several decades. 

Correspondingly, eq. (13.2) representing the strain-rate dependence of the plastic resistance is taken to be given by a standard uniaxial reference experiment at a reference strain rate e gf, typically of magnitude 10 s that evokes a reference tensile uniaxial plastic resistance o-j-gf, which in this case would be the tensile yield stress o-q. The form of the idealized power law relating eg to Ug is given by the exponent m of the equivalent stress, which must be temperature-dependent in a form given in Chapter 8 as... 

Noting that the initial uniaxial yield strength uo will be strain-rate-dependent, making sq (= ffg jE) also strain-rate-dependent, we note that the tensile toughness fkp will also be strain-rate-dependent. Thus,... 

This is also true for PBT, as is shown by the study of as-extruded and heat-treated films of Feldman et al. [210]. The maximum values obtained for modulus and strength, 238 G Nm and 1.51 G Nm " respectively, are considerably lower than the largest fiber values, see Table 3. A study of the temperature and strain rate dependence of the deformation behaviour of these films revealed the onset of a structural reorganization near 300 °C, while the stress activation volume characterizing the activated rate process of the yield stress increased considerably above 200 °C [172]. 

Additional experiments on stress and strain, providing information on ductility, may be seen in Fig. 2.75. This figure illustrates the brittle and ductile failure modes and the strain-rate dependence of yield stress. These different failure... 

Other estimates of the ultimate shear strength of amorphous polymers have been made by a number of authors and generally all fall within a factor of 2 of each other (38,77,78). Stachurski (79) has expressed doubt as to the validity of the concept of an intrinsic shear strength based on the value of the shear modulus, G, for an amorphous solid. He questions which modulus is the correct value to use— the initial small strain value or the value at higher strain (the yield point or the ultimate extension). Further, the temperature and strain-rate dependence of both the yield strength and modulus (however defined) suggests that perhaps the ratio of yield strength to modulus is not a true intrinsic material property. We remark however that the temperature and strain-rate dependence of both the yield stress and the shear modulus are often similar. 

A related issue is that the modulus is a viscoelastic property, as evidenced by the temperature/strain-rate dependence, and that for most poljnners (at least those without a large beta transition near the alpha transition) time-temperature superposition of, for example, the shear relaxation modulus is valid (80). Further, G Sell and McKenna (81) have shown that the 5neld stress vs strain rate also seems to obey time-temperature superposition. Hence there is a correlation between the viscoelastic properties and the yield response of pol5uners, though one that is not generally stated explicitly. We note that some of the models mentioned previously, such as those of Caruthers group (41,42), Tervoort and co-workers (40), and Knauss and Emri (35), are (nonlinear) viscoelastic models that have yield arising due to the nonlinear response induced by the material clock (see Viscoelasticity). 

N. W. J. Brooks, R. A. Duckett, and I. M. Ward, Modeling of Double Yield Points in Polyethylene Temperature and Strain Rate Dependence J. Rheol. 39, 425-436 (1995). 

Bao S P and S. C. Tjong (2009) Temperature and strain rate dependences of yield stress of polypropylene composites reinforced with carbon nanofibers, Polym Compos 30 1749-1760. 

Mindel and Brown [27] performed a Sherby-Dom type of analysis on data for the compressive creep of polycarbonate. Superposition was achieved using an equation of the form of Equation (10.30) with an activation volume of 5.7 nm, which was very close to the values of the activation volume obtained from measurements of the strain rate dependence of the yield stress (Section 11.5.1). 

Our first task in this chapter is to discuss the relevance of classical ideas of plasticity to the yielding of polymers. Although the yield behaviour is temperature and strain rate dependent it will be shown that, provided that the test conditions are chosen suitably, 3deld stresses can be measured that satisfy conventional yield... 

We have already seen in Section 10.3.4 that yield can be modelled using the Eyring process. It provides a convincing representation of both the temperature-and strain rate dependence of the yield stress. However, the discussions of Chapter 10 were confined to one-dimensional states of stress, whereas we now appreciate that yield criteria are essentially functions of the three-dimensional stress state. Also, in view of the discussion in the previous section, it is of interest to explore its applicability to pressure dependence. Both pressure dependence and the extension of the Eyring process to general stress states are considered here. 

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