The Griffith concept

An alternative way of expressing the Griffith approach is to equate the energy differential of the mechanical terms with that associated with the creation of new surface, i.e.. 

This alternative will be developed further in Section 8.5. If the failure criterion df//dc=0 is applied to Eq. (8.8), the critical stress for crack extension (the fracture stress tTj.) is obtained, i.e.. 

This approach indicates that the fracture stress depends on the material parameters, E and yand on the crack size c. Equation (8.10) is commonly termed the Griffith equation. In order to confirm this equation, Griffith introduced cracks of known size into glass bodies and measured their effect on the strength of the glasa The data obtained by Griffith are shown in Fig. 8.6, plotted to show that fracture stress is inversely proportional to Vc, as predicted by Eq. (8.10). 

It is useful at this point to compare the equation developed by Griffith with the one developed earlier based solely on the maximum stress at a crack tip. For ease of comparison, Eq. (8.6) can be re-arranged into the form 

For cases where V(ir/o/8 / ) l, the fracture stress from this equation is less than that obtained from the Griffith equation. Fracture is not, however, predicted to occur under these conditions because the energy approach is a more global approach. Equation (8.11) will, however, be useful for cases when V(7rp/8 /(,) 1. 

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To give the Griffith concept of fracture a concrete structure, which can... 

The importance of inherent flaws as sites of weakness for the nucleation of internal fracture seems almost intuitive. There is no need to dwell on theories of the strength of solids to recognize that material tensile strengths are orders of magnitude below theoretical limits. The Griffith theory of fracture in brittle material (Griflfith, 1920) is now a well-accepted part of linear-elastic fracture mechanics, and these concepts are readily extended to other material response laws. 

Although Griffith put forward the original concept of linear elastic fracture mechanics (LEFM), it was Irwin who developed the technique for engineering materials. He examined the equations that had been developed for the stresses in the vicinity of an elliptical crack in a large plate as illustrated in Fig. 2.66. The equations for the elastic stress distribution at the crack tip are as follows. 

In fact, the first quantitative attempt to incorporate the dynamics of the crack into the Griffith energy balance concept was given by Mott (1948). He suggested that unlike the Griffith case of fracture initiation or nucleation... 

Thus, these two concepts are equivalent. In the classical failure context, fracture depends on some critical combination of stress at the crack tip and the tip radius, neither of which are precisely defined (or definable) or accessible to measurement. For experimental accuracy and practical apphcation, it is more appropriate to use the accessible quantities o and a to determine the fracture toughness of the material. It is to be recognized that the quantities involving a a and a represent the crackdriving force, and 2y, in the Griffith sense, represents the material s resistance to crack growth, or its fracture toughness. 

To see how the fracture energy may be used in the initiation of chemical reactions, the concepts of fracture mechanics are introduced, including the strain rate and temperature dependence of the ductile-brittle behavior. The starting point is the Griffith theory which in its simplest form applies to perfectly brittle materials and states that for a crack to form, the elastic strain energy available must be at least sufficient to provide the energy of the new surfaces formed [74]. 

We may thus conclude that the fracture process is determined by crack formation and crack propagation. Griffith crack theory is essentially a static conception of critical crack formation. Crack growth, however, also depends on dissipative processes. Below the critical load, crack propagation may advance very slowly. In such a case there is a dissipation of energy due to creep processes. Therefore, fracture is a time-dependent process. This aspect is neglected in the Griffith-Irwin theory of fracture. 

Following that year, many work groups studied the theoretical concepts concerning this type of catalytic system. Griffiths et al. [27] presented the first theoretical studies on this catalyst system, having assumed the generally accepted Cossee-Arlman polymerization mechanism [25-27] (Scheme 3.13). 

The break behavior of any desired elastic body is described by the Griffith theory. According to Griffith, a crack in an elastic body only propagates further when the elastically stored energy just exceeds the energy required to break chemical bonds. Combination of this with the Ingles concept leads to... 

In many particular cases, there is a direct correspondence between the Irwin and Griffith criteria, as was noted above an connection with the result in (4.27). However, the latter criterion has the distinct advantage that the energy release rate can often be determined, or at least estimated, without the need for a complete solution of the boundary value problem for the stress field in the body. For this reason, it is selected as the basis for the present discussion. Many of its special features and numerous extensions of the basic concept will become evident in the sections that follow, in the course of discussing various issues concerned with delamination and fracture in thin film configurations. 

The studies carried out earlier have shown that polymer film samples strength to a considerable extent is defined by growth parameters of stable crack in local deformation zone (ZD) at a notch tip [1-3], As it has been shown in Refs. [4, 5], the fiactal concept can be used successfully for the similar processes analysis. This concept is used particularly successfully for the relationships between fracture processes on different levels and subjecting fracture material microstructure derivation [5]. This problem is of the interest in one more respect. As it has been shown earlier, both amorphous polymers structure [7] and Griffith crack [4] are fractals. Therefore, the possibility to establish these objects fractal characteristics intercommunication appears. The authors of Refs. [8, 9] consider stable cracks in polyarylatesul-fone (PASF) film samples treatment as fractals and obtain intercommunication of this polymer structure characteristics with samples with sharp notch fracture parameters. 

It is evident that the theory of failure of adhesion joints should be based on the general principles of solid destruction. However, the transfer of the classical concept of Griffith s theory to two-phase systems is very complex. The difficulties are related to determination of two main parameters in the equation for critical stress of fracture ... 

In the case of viscoelastic material, the concept of tensile strength has lost its traditional meaning. The tear follows a time dependent process. Thus, the Griffith theory is not applicabVe to the tearing process surface energy plays no apparent role. 

The work 7" of fracture was identified hy Dupr and Griffith with the surface energy of solids. In reality work is spent on the deformation leading to rupture rather than on the Break itself. In simple instances it is equal to the work needed to extend a column of material (of unit cross-section) in front of the growing crack until the column snaps. This extension may he purely elastic. The new concept contraiy to the old (a) indicates the analogy Between the Breaks of a liquid and a solid (B) accounts for the absolute value of y/ (c) agrees with the effect of cross-linking on (h) is compatible with the rate dependence of... 

One of the purposes of this paper is to differentiate between surface free energy and fracture surface energy. Thus, several basic principles of fracture mechanics are discussed with reference to the Griffith s energy-b J nce concept and the Irwin-Orowan s plastic-zone concept. We also define several frequently misused terms, e.g., effective fracture surface energy, fracture energy and fracture toughness. Actual values of related parameters are presented to illustrate the applicability of fracture mechanics to the selection of polymers for structural materials. 

Ozmen (2004, especially for chemical bonding) and Coll und Treagust (2003, chemistiy of metals) can help to develop a first overview. Numerous authors (e g. Pfund, 1975 Schldpke, 1991 Griffith and Preston, 1992 Mas et al., 1987) describe parallels between students conceptions and historical scientific ideas. Schldpke (1991), for example, points out similarities between students conceptions concerning properties of matter and ideas in alchemist thinking. Lee, Eichinger, Anderson, Berkheimer, and Blakeslee (1993) mentions semblances between the ideas of Aristotle and students conceptions about general aspects of the particulate nature of matter and the horror vacui . 

As the results of Pfundt (1981), Griffith (1987) and Lee et al. (1993) show, students have problems in dealing with the idea of empty spaces between particles. In her studies, Pfundt presented students with different representations of atoms. The students had to choose the one that represented their ideas at best. Numerous students chose representations of cubic or hexagonal atoms, because they fit without gaps between them (Pfundt, 1981, p. 87). Griffith and Preston (1992) could explicitly show that students suppose that the size of a crystal corresponded to the shape of its atoms. Furthermore, they discuss the parallels between this students conception and the corresponding historical idea of Rene Hairy. 

The principle behind the phenomenon of fracture of materials can be described by having recourse to Griffith s theory. Alternatively, this can be done by introducing the concept of fracture toughness. 

Fisk, George W. A new sense of destiny from ancient symbols renewal of vision through the lost language / George W. Fisk illustrations by Robert C. Griffith. St Joseph (MI) Cosmic Concepts Press, 1988. 124, [2] p. ISBN 0962050709... 

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