Griffith theory discussion

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. 

Stress Intensity Factors. A seemingly different approach to fracture on the basis of stress intensity factors, (, 1 6, 17) is frequently encountered in the fracture mechanics literature. It is, therefore, appropriate to briefly discuss this approach and its relation to the Griffith theory. 

Thus, if the actual stress imposed a < Ucr. the material will sustain that stress without the crack growing. The equation is identical for both constant load and constant displacement conditions, and hence it should work also for any intermediate conditions. Equation 21 has been the inspiration for much further work. Pascoe (44) provides a fairly detailed discussion of the Griffith theory. 

A brief review is given here of the spectroscopic vector model of an atom or ion. In crystal-field theory, the wave function of the isolated ion is taken as the unperturbed state, and the perturbing effect of the electric and magnetic fields is computed. Thus crystal field theory uses the language, nomenclature, and methods employed in the theory of atomic spectra. A complete discussion of these methods can be found in books by Condon and Shortley (5) and by Griffith (/). 

If the values of E found by Wulf and Tolman or the other recent value, 27,800, found by Belton, Griffith, and McKeown are substituted in the equation number of molecules reacting = number entering into collision x e ElRT, it appears that the rate of activation is not great enough to account for thd rate of reaction. The reason for this departure from the simple theory is discussed on page 102. 

Restricting our discussion to the subspace spanned by the terms 6Aig and 4 Tig, the matrix element of the spin-orbit operator have been evaluated by Weissbluth [59] using the formalism pioneered by Griffith [56] and ending at the eigenvalue problem of the 18 x 18 dimension (which is partly factored— Table 34). Then the second-order perturbation theory yields the energies of the lowest multiplets as... 

The idea that the strength of bulk solids is controlled by flaws was advanced by Griffith in 1921 and has led to the development of a mudi more sophisticated continuum approach to fracture, known as fracture mechanics. Fracture mechanics is concerned always with the conditions for the propagation of an existing crack, and it is important to bear this in mind when comparing different theories of fracture. Griffith s ideas are well known and do not need to be elaborated here. There are some aspects of his theory which are relevant to the present discussion, however. Griffith s equation for the fracture stress of an elastic material is (for plane stress). 

Pauling used his valence bond approach to explain differences in magnetic behavior among coordination compounds by use of either metal ion 3d or 4d orbitals. Griffith and OrgeF developed ligand field theory, derived from the crystal field theory of Bethe and Van Vleck on the behavior of metal ions in crystals and from the molecular orbital treatment of Van Vleck. Chapter 10 discusses these theories. 

The fracture of most semicrystalline polymers, notably polyethylene, polypropylene and nylon, cannot be described by LEFM based on the theory of Griffith and Irwin because large-scale yielding occurs at the crack tip prior to failure. For these materials and for toughened polymers and polymer blends other approaches have been developed, three of which will be discussed in detail ... 

Cracking occurs when the stress in the network exceeds its strength. Since the classic work of Griffith [70], it has been understood that fracture of brittle materials depends on the presence of flaws that amplify the stress applied to the body. That is, if a uniform stress is applied to a body containing a crack with a length of c, the stress at the tip of the crack is proportional to ct,Vc, and failure occurs when that stress exceeds the strength of the material. The theory of linear elastic fracture mechanics (LEFM), which is discussed in several excellent textbooks [71-73], indicates that catastrophic crack propagation occurs when... 

Methods for mechanical testing of materials are briefly introduced along with various strengthening mechanisms. The number and siu-face area of the slip systems in metals and in ceramics are shown to be responsible for the ductility (or the lack of it) and for ductile-to-brittle transitions. Griffith s theory of brittle fracture is used to introduce fracture mechanics and to develop the concept of fracture toughness. The viscoelastic behavior of polymers is briefly discussed. 

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