Phenomenology of Yielding

It is always very useful to be able to predict at what level of external stress and in which directions the macroscopic yielding will occur under different loading geometry. Mathematically, the aim is to find functions of all stress components which reach their critical values equal to some material properties for all different test geometries. This is mathematically equivalent to derivation of some plastic instability conditions commonly termed as the yield criterion. Historically, the yield criteria derived for metals were appHed to polymers and, later, these criteria have been modified as the knowledge of the differences in deformation behavior of polymers compared to metals has been acquired [20,25,114,115]. 

The simplest yield criterion is that of Tresca. This criterion states that yielding will occur when the maximum shear stress on any plane in the tested sohd reaches its critical value [20]  

Lower values of the yield stress measured in tension compared to those measured in compression suggest that the effect of pressure, which is important for polymers, is not accounted for in this criterion. Hence, appropriate correction has to be made in order to account for the effect from external pressure. The most frequent version of pressure-dependent yield criterion is the modified von Mises criterion [20]  

In the case of compressive loading, the Coulomb yield criterion is often utilized in the form [20]  

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The resulting, isolated, flow stress contribution is shown in Figure 31 (left). As was discussed in the first part of this chapter, dealing with the phenomenology of yielding, upon plastic deformation of the material all influence of its prior thermomechanical history is erased and the material shows a response independent of prior history. From the resulting minimum in stress 0, 0, we can, therefore, calculate with the help of eqn [5] a first estimate of the rejuvenated reference viscosity 7r,o, since fi and tq are known, Sa= 0, and, finally, the strain rate fi is prescribed. Note that we obtain an estimate of this parameter since its calculation is based on a number of assumptions as listed earlier. At the same time we can also obtain an estimate of the value of the state parameter Sa since the softening drop A[Pg.739]

The results are shown in Fig. 5. We notice that the EOS calculated with the microscopic TBF produces the largest gravitational masses, with the maximum mass of the order of 2.3 M , whereas the phenomenological TBF yields a maximum mass of about 1.8 M . In the latter case, neutron stars are characterized by smaller radii and larger central densities, i.e., the Urbana TBF produce more compact stellar objects. For completeness, we also show a sequence of stellar configurations obtained using only two-body forces. In this case the maximum mass is slightly above 1.6 M , with a radius of 9 km and a central density equal to 9 times the saturation value. 

Eyring s equation may be regarded as a good phenomenological description of yield stress as a function of test parameters (T, e), but it cannot be related to physical processes at the molecular scale. The equation can be used at high e for impact properties and for the prediction of the ductile brittle transition temperature. Eyring s equation can be modified with two sets of parameters if two relaxations are involved in the range of temperatures and strain rates (Bauwens-Crowet et al., 1972). 

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. 

Kennedy and Squires studied the effect of hydrogen chloride on the course of isobutene polymerisation catalysed by aluminium chloride. They showed that at —78 °C HCl increased the polymer yield if introduced before the catalyst, but had no effect if added to a quiescent mixture obtained by direct initiation giving a limited conversion. These observations are entirely consistent with our interpretation of the phenomenology of direct initiation HCl is a cocatalyst in the presence of free aluminium chloride, i.e. when added at the beginning of the experiment, but is ineffective if the Lewis acid is tied up in conjugation products which, as we have seen, bring the polymerisation to a halt before all the monomer is consumed (cf. Sect. IV-B). 

Previously introduced, the thermodynamic surface tension 7 represents the elastic resistance to surface dilation. Furthermore, two types of viscosities are defined within the interface, a dilational viscosity and a shear viscosity. For a surfactant monolayer, the surface shear viscosity rjS is analogous to the three-dimensional shear viscosity the rate of yielding of a layer of fluid due to an applied shear stress. The phenomenological coefficient s represents the surface dilational viscosity, and expresses the magnitude of the viscous forces during a rate expansion of a surface element. Figures 10a and 10b illustrate the difference between the two surface viscosities. 

Here, the reciprocal rules hold, and we have L12 = T2i- Introduction of the explicit form of chemical potential for a single component Afi = —SAT -I- VAP into the phenomenological equations yields... 

The preceding is a general overview of the main features of yield in pol5uners. From a practical point of view, structures are rarely subjected to simple imiaxial or shear loads and it is instructive to be able to determine when a material might yield under more complicated stress states. The following yield criteria attempt to do this from a phenomenological point of view, that is they do not address the fundamental mechanisms of yield but rather provide yield criteria for multiaxial loading conditions. 

FRA techniques were used to evaluate the risk Impact of the containment flooding strategy at the Peach Bottom plant. The FRA framework accounts for the complex system and phenomenological responses and interactions, such that the overall merit of the strategy can be assessed. The uncertainty in the phenomenology, which yields a wide range of potential outcomes, is also considered. 

There is a large volume of contemporary literature dealing with the structure and chemical properties of species adsorbed at the solid-solution interface, making use of various spectroscopic and laser excitation techniques. Much of it is phenomenologically oriented and does not contribute in any clear way to the surface chemistry of the system included are many studies aimed at the eventual achievement of solar energy conversion. What follows here is a summary of a small fraction of this literature, consisting of references which are representative and which also yield some specific information about the adsorbed state. 

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