Kinetic fracture

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 

Where to, A17, t) are constants for a given fibre, r is the applied stress and Tis the absolute temperature. Zhurkov achieved an excellent correlation between AU calculated from such stress-rupture data and Al/ , the activation energy for thermal degradation for a wide range of fibres. It was also possible to identify the free radicals and degradation products formed 

Campbell and Peterlin and Peterlin concluded from e.s.r. measurements on isotropic and highly drawn nylon 6 and 6.6 fibres that no detectable free radicals were formed in the isotropic state, whereas approximately 1 chain in 250 was fractured in a fibre under high axial tension at failure. These fractured chains were later identified with the tie molecules linking adjacent crystallites together in the fibre direction. Quantitative theories have since been developed by Kausch et and more recently by DeVries et alP which attempt to correlate creep, creep-rupture, and stress-relaxation in fibres in terms of the measured main chain scission. 

DeVries, Peterlin and Zhurkov all specifically identify the dominating mechanism in the tensile deformation of fibres as the scission of tie molecules which are assumed to have a distribution of lengths. 

It should be pointed out that no strong evidence exists for chain scission in unoriented polymers or in oriented polymers in, say, compression or transverse tension or compression. Ito proposed an explanation of the yield drop observed in many polymers as due to the stress required to break secondary bonds superposed on a rubber-like stress-strain curve. However, Steg and Ishai in a study of a plasticised epoxy system concluded that there was no significant change in crosslink density on passing through )rield. 

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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]... 

Evidence suggests that there is a threshold tensile stress at which void nucleation occurs and spall fracture initiates. Materials subject to transient internal tensions can support tensile stresses significantly in excess of this threshold level, however. Such behavior is a consequence of kinetics and inertia associated with the nucleation and growth of voids during spall. A fairly large body of experimental and theoretical literature on spall phenomena exists and many aspects of the effect are reasonably well understood. Review articles on spall (Curran et al., 1977 Davision and Graham, 1979 Curran, 1982 Meyer and Aimone, 1983 Novikov, 1981) provide access to most of the literature on the subject. 

Assuming a complete transfer of the kinetic energy in (8.24) into energy dissipated during the spall fracture process in (8.25) provides an expression for the characteristic spall fragment size... 

If contact with a rough surface is poor, whether as a result of thermodynamic or kinetic factors, voids at the interface are likely to mean that practical adhesion is low. Voids can act as stress concentrators which, especially with a brittle adhesive, lead to low energy dissipation, i/f, and low fracture energy, F. However, it must be recognised that there are circumstances where the stress concentrations resulting from interfacial voids can lead to enhanced plastic deformation of a ductile adhesive and increase fracture energy by an increase in [44]. 

It seems that indeed the answers to many fundamental questions are obtained, at least in qualitative form. Perhaps, the most important exception are thixotropic phenomena. There are many of them and the necessary systematization and mathematical generalization are absent here. Thus, it is not clear how to describe the effect of an amplitude on nonlinear dynamic properties. It is not clear what is the depth and kinetics of the processes of fracture-reduction of structure, formed by a filler during deformation. Further, there is no strict description of wall effects and a friction law for a wall slip is unknown in particular. 

It should be stressed that the observed critical strain-rate for bond fracture (sf) in the case of a polydisperse fraction refers to the longest chain present in the sample. This quantity is significantly different from the critical strain-rate (r ) defined with respect to an average molecular mass whose value could be determined only after careful consideration of the degradation kinetics. 

The parameter R in Eq. (92) can be derived empirically from the experimental SEC traces with a minimum of computational effort and without regard to the details of the degradation kinetics by using the following arguments. Once a macromolecule is fractured, the moieties are immediately driven into another region of space. Due to the decrease in MW, a considerably higher strain rate... 

The dissimilarity in the kinetic laws for chain fracture observed under different flow conditions reflect the deficiencies of the present theories which should be able to incorporate both dependences (M 1 and M 2) into its structure, with either... 

Although significant insight into the process of bond rupture has been gathered from the studies on the scission kinetics, it remains desirable at this stage to carry out further experimentation at a molecular level to get information on the chain conformation at the moment of fracture. As a first step in this direction, birefringence measurements have been attempted recently in the single jet cell... 

As shown in Sect. 2, the fracture envelope of polymer fibres can be explained not only by assuming a critical shear stress as a failure criterion, but also by a critical shear strain. In this section, a simple model for the creep failure is presented that is based on the logarithmic creep curve and on a critical shear strain as the failure criterion. In order to investigate the temperature dependence of the strength, a kinetic model for the formation and rupture of secondary bonds during the extension of the fibre is proposed. This so-called Eyring reduced time (ERT) model yields a relationship between the strength and the load rate as well as an improved lifetime equation. 

Linear combination of atomic orbitals (LCAO) method, 16 736 Linear condensation, in silanol polycondensation, 22 557-558 Linear congruential generator (LCG), 26 1002-1003 Linear copolymers, 7 610t Linear density, 19 742 of fibers, 11 166, 182 Linear dielectrics, 11 91 Linear elastic fracture mechanics (LEFM), 1 509-510 16 184 20 350 Linear ethoxylates, 23 537 Linear ethylene copolymers, 20 179-180 Linear-flow reactor (LFR) polymerization process, 23 394, 395, 396 Linear free energy relationship (LFER) methods, 16 753, 754 Linear higher a-olefins, 20 429 Linear internal olefins (LIOs), 17 724 Linear ion traps, 15 662 Linear kinetics, 9 612 Linear low density polyethylene (LLDPE), 10 596 17 724-725 20 179-211 24 267, 268. See also LLDPE entries a-olefin content in, 20 185-186 analytical and test methods for,... 

The possible fatigue failure mechanisms of SWCNT in the composite were also reported (Ren et al., 2004). Possible failure modes mainly include three stages, that is, splitting of SWCNT bundles, kink formation, and subsequent failure in SWCNTs, and the fracture of SWCNT bundles. As shown in Fig. 9.12, for zigzag SWCNT, failure of defect-free tube and tubes with Stone-Wales defect of either A or B mode all resulted in brittle-like, flat fracture surface. A kinetic model for time-dependent fracture of CNTs is also reported (Satapathy et al., 2005). These simulation results are almost consistent with the observed fracture surfaces, which can be reproduced reasonably well, suggesting the possible mechanism should exist in CNT-polymer composites. 

Xiao T, Fang N, Chan V, Liao K (2004). A kinetic model for time-dependent fracture of carbon nanotubes. Nano Lett. 4 1139-1142. 

Most researchers attribute slow kinetics to some sort of diffusion limitation (e.g., diffusion is random movement under the influence of a concentration gradient [193]), because sorbing molecules are subject to diffusive constraints throughout almost the entire sorption/desorption time course due to the porous nature of particles. Particles are porous by virtue of their aggregated nature and because the lattice of individual grains in the aggregate may be fractured. 

Natarajan, K. A., Riemer, S. C., Iwasaki, I., 1984. Influence of pyrrhotite on the corrosive wear of grinding balls in magnetite ore grinding. Inter. J. Miner. Process, 13(1) 73-81 Nesbitt, H. W., Bancroft, G. M., Pratt, A. R., Scaini, M. J., 1998. Sulfur and iron surface states on fractured pyrite surfaces. American Mineralogist, 83 1067 - 1076 Neeraj, K. M., 2000. Kinetic studies of sulphide mineral oxidition and xanthate adsorption. Doctor thesis of Virginia Polytechnic Institute and State University. A Bell Howell Company UMI dissertation Services... 

Fig. 50. Contribution of energy for crack propagation to the total fracture energy for sol-vent-modified epoxies prepared via CIPS with 1 wt % catalyst porous, and macroporous epoxies prepared via kinetically controlled CIPS with 1 wt % catalyst...

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