Temperature and strain-rate dependences of yield

Suppose, however, that, instead of two sites, there is a row of sites all with equal energies and separated by equal energy barriers AG. Application of a stress now causes a flow, because the stress lowers the energy of any one site with respect to the adjacent one in the direction opposite to the flow, so that jumps in the direction of flow are favoured over those in the reverse direction. Because there is a series of energy minima, there is in this case no net change in the number of occupants of any site, so that, on release of the stress, there is no tendency for the flow to continue or reverse jumps take place in either direction equally, at a very low rate. 

In adapting the model of chapter 5 to the description of the flow that takes place on yielding the following changes need to be made (i) the applied stress now produces a large change in the free energies of the sites, so that the approximation made in equations (5.22) cannot be made 

The yield stresses of several polymers at low temperatures and high strain-rates have been found to increase more rapidly with increasing strain-rate and decreasing temperature than predicted by equation (8.17). This behaviour can often be described by the addition of terms representing two activated processes with different activation volumes. 

As indicated in section 8.1, fracture can take place in essentially two ways, either following macroscopic yield, when the fracture is said to be ductile fracture, or without macroscopic yield, when it is called brittle fracture. The present section is concerned with the brittle fracture of polymers. Experimental studies of brittle fracture can be divided broadly into two types, those that are undertaken with a view to understanding the details of what happens during fracture and those that are aimed at providing engineering data about a polymer. Experiments of the former type are often designed to test the predictions of a theoretical model, whereas experiments 

The process of brittle fracture involves two stages, crack initiation and crack propagation. Traditionally it has been assumed that, in practice, minor cracks or flaws always exist in real polymer samples, so that the propagation stage is the one of practical consequence and is usually treated first, forming the subject of fracture mechanics. This order is used here. Another reason why it is useful to consider fracture mechanics first is that it leads to a consideration of what happens very close to the tip of a crack, which is very closely related to the initiation stage. 

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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. 

Figure 12.31 Yield strain for polyethylene at strain rates A 2.08 x (Reproduced with permission from Brooks, N.W., Unwin, A.P., Duckett, R.A. etal. (1997) Temperature and strain rate dependence of yield strain and deformation behavior in polyethylene. ]. Polym. Sci. B Phys. Edn., 35, 545. Copyright (1997) John Wiley Sons, Inc.)...

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

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]. 

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. 

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. 

Two principal approaches have been used to model the yield behaviour of polymers. The first approach addresses the temperature and strain-rate dependence of the yield stress in terms of the Eyring equation for thermally activated processes [39]. This approach has been applied to many amorphous and crystalline polymers (see Section 12.5.1) and links have been established with molecular relaxation processes determined by dynamic mechanical and dielectric measurements and with non-linear viscoelastic behaviour determined by creep and stress relaxation. The Eyring approach assumes that the yield process is velocity controlled, i.e. the yield process relates to existing thermally activated processes that are accelerated by the application of the yield stress to the point where the rate of plastic deformation reaches the applied macroscopic strain rate. This approach has... 

Ductile deformation requires an adequate flexibility of polymer chain segments in order to ensure plastic flow on the molecular level. It has been long known that macromoleculai- chain mobility is a crucial factor decisive for either brittle or ductile behavior of a polymer [93-95]. An increase in the yield stress of a polymer with a decrease of the temperature is caused by the decrease of macromoleculai chain mobility, and vice versa the yield stress can serve as a qualitative measure of macromolecular chain mobility. It was shown that the temperature and strain rate dependencies of the yield stress are described in terms of relaxation processes, similarly as in linear viscoelasticity. Also, the kinetic elements taking pai-t in yielding and in viscoelastic response of a polymer are similar segments of chains, part of crystallites, fragments of amorphous phase. However, in crystalline polymei-s above their glass transition temperature the yield stress is determined by the yield stress required for crystal deformation... 

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

Figure 11.5 Strain rate dependence of yield stress for (a) a-PP and (J-PP homopolymers, and (b) 15% toughened iPP/EPR and (J-nucleated iPP/EPR blends at room temperature. (From Reference 31 with permission from Elsevier.)...

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). 

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 will see that this simple form of pressure-dependent yield criterion is more satisfactory than the Coulomb criterion when a representation is developed which includes the effects of temperature and strain rate on the yield behaviour. In physical terms, the hydrostatic pressure can be seen as changing the state of the polymer by compressing the polymer significantly, unlike the situation in metals where the bulk moduli are much larger ( 100 GPa compared... 

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... 

Eyring s equation is the only relationship describing, with a good agreement, the dependence of yield stress on both temperature and strain rate. Unfortunately, this equation is phenomenological, and the determined constants have no physical meaning. 

In addition to the well known dependence of yield stress on temperature and strain rate, Eq. (51) provides a functional relationship between the plastic yield, physical aging, and type of stresses applied. 

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-... 

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