Zhurkov equation

There has been a prevailing theory that oxidative degradation is accelerated by mechanical stress [100]. This theory is based on fracture kinetic work by Tobolsky and Eyring [101], Bueche [102, 103, 104], and Zhurkov and coworkers [105, 106, 107]. Their work resulted in an Arrhenius-type expression [108] sometimes referred to as the Zhurkov equation. This expression caused Zhurkov to claim that the first stage in the microprocess of polymer fracture is the deformation of interatomic bonds reducing the energy needed for atomic bond scission to U=U0-yo, where U0 is the activation energy for scission of an interatomic bond, y is a structure sensitive parameter and o is the stress. 

It follows from Eq. (21) that relation (19) is of limited validity. Zhurkov equation becomes inapplicable in the limiting case a 0 and it is restricted to the region in which the activation energy U(a) is a linear function of the tensile stress 0. This approach, however, permits a physical interpretation of fracture in terms of the activation energy U and of the parametef y. In the case of the fracture of polymers, as has been reported by Zhurkov , Uq is closely related to the activation energy for thermal destruction of macromolecules. 

Fig. 11.20 Temperature dependence of the Zhurkov equation for creep rupture for PMMA. (Data from Zurkov, (1965))...

Figure 2 A plot of quantum yield for degradation vs. stress according to (A) the Plotnikov equation, (B) the DRRE hypothesis, and (C) the Zhurkov equation.

The effect of stress on the rates and efficiencies of reactions that occm subsequent to radical formation is complicated. Examples have been noted in which die rates decrease because of decreased oxygen diffusion in the stressed (more ordered) sample. " In other systems, it has been suggested that the important point is that stress changes the conformations of the C-C bonds in the polymer chains " depending on the system eiffier a rate decrease " or a rate increase can occur. The quantitative relationship of quantum yields to stress in such cases is generally considered to be given by the so-called Zhurkov equation, as discussed in the next section. 

The effect of stress on the fliermal degradation rates of polymers can be fitted to an empirical Arrhenius-like equation that is attributed to Zhmkov rate = Aexp[-(AG Bo)/RT], where AG is an apparent activation energy, o is the stress, and A and B are constants. It has been suggested that an equation similar to the Zhurkov equation might apply in photodegradation ( )obs = Aexp[ (AG Bo)/RT]. The Zhurkov equation is empirical and does not fall strictly into any of the three... 

One of the most successful attempts to include the effects of temperature in a relatively simple expression similar to the one above, has been made by Zhurkov and Bueche using an equation of the form... 

For the material in the previous question, use the Zhurkov-Beuche equation to calculate the time to failure under a steady stress of 44 MN/m if the material temperature is 40°C. The activation energy, Uo, may be taken as 150 kj/mol. 

Molecular tl iy for fracture could be traced ba k to an application of the rate-process theory to fracture teiomena (65) and al the similar line of thou t Beuche (1) developed his theory for fracture in p<%mer. Zhurkov (66,67) derived independently the same equation to the Beuche s one the time to fracture. Based on this equation the activation energies for the fracture were estimated from the experimental results on the time to fracture under the unaxial load (20,68). Change of deformation potential in a stressed chain was discussed by Kausch (J9.20). Fracture developement has been discussed from the a >ects of micromori lr of polymers by Peterlin (J5, 69-71), Kausch (19,20) and DeVries(/7,61, 72). 

One further topic merits discussion in this section in view of its success in dealing with the mechanical properties of oriented fibres, which are after all anisotropic polymers. That is the theory of kinetic fracture, developed mainly by Zhurkov and co-workers. Evidence has been presented from electron spin resonance (e.s.r.X " mass spectrometry, and infra-red spectroscopy that when highly oriented fibres or heavily cross-linked rubbers experience a tensile stress (along the axis for fibres) an appreciable fraction of main-chain bonds are broken by the applied stress. These scission events are observed to occur more or less homogeneously throughout the fibre and are not localised in the fracture plane. Many sets of data show that the lifetime tb of a fibre under stress is described approximately by the following equation... 

Becker was the first to suggest, in 1925, [2] that the deformation rate could be expressed by an Arrhenius-type equation. His concept was developed from the recognition of an identity between deformation and chemical reactions. The rate theory of thermally activated processes and, in general, the principles of deformation and fracture were established by Eyring [3], Orowan [4], and J. Frenkel [S]. In the latter part of the 20th century, these principles were further developed for polymers by Krausz and Eyring [6], Dorn [7], Zhurkov [8] and his collaborators [9], Kausch [10], and many others [11-14]. 

As early as 1957, Zhurkov [55] put forward the kmetic concept of the fiacture of solids. The main feature of the concept of kinetic fracture is the durability , tf, i.e. the time of loading until fracture. The well known empirical equation of this concept is [8, 9, 55]... 

In equation 13, t y is the time to failure, Uq is interpreted as the activation energy for chain scission, to is interpreted as the inverse of the molecular oscillation frequency, and y is a structure-sensitive parameter. Experimentally he (47) observed the formation of platelet-shaped nanoscopic voids, which he referred to as incipient cracks, that form very early in the polymer once a tensile load is applied. Both the incipient cracks and their number density were measured from small-angle x-ray scattering studies on samples that had tensile loads applied to them. The characteristic sizes and number densities of the incipient cracks are included in Table 2. Zhurkov showed that the incipient cracks formed at the very early stages and the remaining life was associated with the coalescence of these incipient cracks to ultimately cause failure. 

Another variant of the activation energy approach (Eq. 11.40) is the Zhurkov method, sometimes referred to as the kinetic rate theory, which is based upon tests on more than 50 different materials including both metals and polymers (see Zhurkov, (1965)) and results in an equation for the time to creep to rupture given by. 

Various modifications of Zhurkov s equation have been suggested and a discussion of these can be found in Griffith, (1980). The Zurkov equation can be written in the form,... 

One of the problems with using test schemes based on a simple Arrhenius relation is that it does not readily account for the influence of different imposed stresses on the reaction rate. This is important, since data measured at one stress level will not necessarily provide a good estimate of reaction rate under another condition. Zhurkov investigated the time to failure of polymeric, metallic, and nonmetaflic crystalline materials in uniaxial tension at a number of elevated temperatures and different stresses [9]. His experimental results fit the empirically derived expression shown in Equation 16B.2 that he called the thermofluctuational theory. ... 

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