The Kinetic Theory of Fracture

Developed chiefly in Russia, the kinetic theory of fracture at first appears to represent an entirely different account of fracture phenomena to that discussed in Section 1.2. In fact some Russian authors have claimed that the kinetic theory contradicts the Griffith theory of fracture. As we shall see, however, this is not the case. 

The surface energy represented by AG is of the nature of referred to earlier. 

In contrast the fracture surface energy S is related to the activation energy G%, ance this is the energy that must be supplied to break the bond in the first instance. 

The point to notice is that a threshold level of stored energy in the bond is necessary before AG - 0 and thus before the forward reaction can proceed. This stored energy is, of course, equal to AG and thus (when normalised) to the surface energy of the material. Thus we see that the kinetic model of fracture requires a minimum energy supply equal to the surface energy, which is the fracture criterion employed by Griffith. No contradiction therefore exists between the kinetic and Griffith theories. 

In contrast, as Fig. 2c shov , the stress on the bond may be so high that -AG G and only very small themK -fluetuations are required to drive the forward reaction i e. crack propagation. In this situation growth rates will be high and governed by the ability of the stress field around the crack to adjust itself to translation of the crack tip. This rate of adjustment is controlled in turn by the inelastic properties of the solid. 

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We shall see that the situation is far from simple and that the data on molecular processes cannot be used to make quantitative predictions about macroscopic deformation and failure properties. In particular it will beconw evident that the kinetic theory of fracture initiation is an oveRimpliflcation. 

Our discussion of the kinetic theory of fracture in Section 1 has already indicated the manner in which applied stress can bring about a net accumulation of nwlecular breakages in a jxrlymeric solid. Since the stress is continuous throughout a specimen loaded in tension, these breakages are distributed throughout the material and can be detected by ESR in terms of a volume concentration of free radicals. 

The major thrust of the stress MS studies to date has focused on gathering evidence to support the kinetic theory of fracture proposed by Zhurkov. For the Zhurkov kinetic theory of fracture to be applicable to the material in question, not only must the mechanical and thermal degradation products be identical, but both processes must follow the same degradation mechanism. 

Figure 11. Important questions in the use of stress MS data as evidence for the kinetic theory of fracture...

Based on the kinetic theory of fracture, He (1986) calculated the theoretical strength of polymers, cr ax, as ... 

Following the earlier works of Eyring, Zhurkov, and Bueche, these and other authors have continued to develop different aspects of the kinetic theory of fracture. Particularly in the USSR fracture data have been interpreted in terms of the failure of regular, nonmorphological model lattices (e.g., 54—58). Gubanov et al. 

Within the range of validity of this approximation the logarithm of lifetime, log tb, is almost linearly related to w, i.e., to the applied stress. Exactly this behavior is exhibited by a large number of stressed metals, ceramics, or polymers. Together with Eyring s original interpretation this has laid the foundations of the kinetic theory of fracture, to which — as indicated above — subsequently a large number of researchers have contributed. 

A final consequence of this reconciliation between the kinetic theory and fracture mechanics concepts, is that the effective surface energy S should vary somewhat as Regime I gives way to Regime II. In the former, the stress field is required to provide energy (in the absence of losses) of the order of AG upwards, since the balance of the activation energy G is provided by thermal fluctuations. In Regime II virtually the whole of G must be provided by the stress field so that we have. 

Kinetic Theory of Fracture. Catastrophic failure of a polymeric material is a complex process in which a sequence of partially understood events occurs at both the molecular and macroscopic levels. The stress-induced cleavage of the main-chain polymer bond is one event occurring on the molecular level which has been studied by both stress MS and electron spin resonance spectroscopy (ESR). 

Only if the answers to all these questions are aflBrmative can we use the Zhurkov kinetic theory of fracture to explain the degradation processes. The evidence gathered to date which is germane to this subject is summarized below. 

The network of Fig. 2 is deformed at a constant temperature T and a rate of elongation e. This leads to straining of the van der Waals (vdW) bonds which are broken according to the Eyring kinetic theory of fracture (Kausch, 1987), at a rate... 

In the last fifteen years modern spectroscopical methods (ESR, IR) and conventional methods of structure research have permitted considerable progress in the investigation of deformation and fracture of polymeric materials. For the first time in western languages a unified view of the kinetic theory of polymer fracture is presented by one of the scientists contributing to its development. 

In terms of the kinetic theory of the resistance and fracture of the adhesive joint, the relation between the stress, y, and durability, Y, is given ... 

A kinetic theory of fracture intends to interrelate the motion and response of molecules to the ultimate properties of a stressed sample. A kinetic theory, therefore, entails a molecular description of the deformation of the microscopically heterogeneous and anisotropic aggregates of chains to such an extent that critical deformation processes can be identified. The macroscopic deformation of any aggregate potentially involves the deformation, displacement, and/or reorientation of so different substructural elements as bond vectors, chain segments or crystal lamellae. 

As stated in the introduction to this section the subcritical crack growth in a polymer is due to the thermomechanical activation of different molecular deformation processes such as chain slip and orientation or void opening. The energy dissipated depends on the frequency, nature, kinetics, and interaction of these processes. But there are many and notable attempts to treat the subcritical crack propagation as one thermally activated multi-step process characterized by one enthalpy or energy of activation and one activation volume. Several of these kinetic theories of fracture have been treated in Chapters 3 and 8. 

As mentioned in the introduction, a kinetic theory of fracture deals with and particularly considers the existence and discrete size, the motion, and the response of molecules or their parts in a stressed sample. It thus requires a molecular description of polymer structure and deformation. Such a description is available in the cited books [1—12], [20] or papers [15, 38]. In Chapter 2 only a very brief summary is offered which is not meant to be self-supporting or complete. 

In the kinetic theory, attrition is focused on the event of bond fracture and the latter is described in terms of chemical rate theory. The mechanical stress on the bond (normally the result of an applied load) modifies the free energy barrier that must be crossed if the bond is to change from an unbroken to a broken state. Figure 2 illustrates the situaticm. 

The basic equations of the kinetic theory can be preserved in more sophisticated treatments of fracture which allow for some of the effects neglected by the simple theory. 

The kinetic theory can also be applied to crack propagation, using Eq. (10) to describe the fracture of molecules at the tip of the crack As already indicated. 

In dynamics with the inclusion of the force of inertia, the above stated principle can be extended to the principle of virtual work, expressed in the form of an evolution inequality. But in contrast to the classical theory, in the evolution inequality of fracture mechanics the virtual flux of kinetic energy entering into a tip of a slit (or a propagating crack) must be taken into account. It is resulted from analysis of an energy balance equation. 

This book on Polymer Fracture might as well have been called Kinetic Theory of Polymer Fracture . The term kinetic theory , however, needs some definition or, at least, some explanation. A kinetic theory deals with and particularly considers the effect of the existence and discrete size, of the motion and of the physical properties of molecules on the macroscopic behavior of an ensemble, gaseous or other. A kinetic theory of strength does have to consider additional aspects such as elastic and anelastic deformations, chemical and physical reactions, and the sequence and distribution of different disintegration steps. 

In the fracture theory of Hsiao-Kausch [60, 61] the state of orientation of a polymeric solid is explicitly recognized. The theory combines the kinetic concept of Zhurkov or Bueche and the theory of deformation of anisotropic solids developed by Hsiao [59]... 

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... 

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