Griffith crack model

Let us note in conclusion, that the adduced above results have shown that the fractal Griffith crack model application can not only improve quantitative conformity of theoretical and experimental data, but also obtain qualitatively new picture of fracture processes. 

Fig. 18. Adhesive contact of elastic spheres. pH(r) and pa(r) are the Hertz pressure and adhesive tension distributions, (a) JKR model uses a Griffith crack with a stress singularity at the edge of contact (r = a) (b) Maugis model uses a Dugdale crack with a constant tension aa in a < r < c [1111.

Next, in 1963, a means of reconciling the differences between theoretical and measured metal strength was published, which modelled ductile failure as arrays of atomic-level dislocations. This model, known as the BCS model after its authors Bilby, Cottrell and Swinden [8], would later be combined with the Griffith crack equation to enable a mathematical model for both ductile failure and brittle fracture, the R6 method, which we will come back to shortly. 

In this section we find the derivative of the energy functional in the three-dimensional linear elasticity model. The derivative characterizes the behaviour of the energy functional provided that the crack length is changed. The crack is modelled by a part of the two-dimensional plane removed from a three-dimensional domain. In particular, we derive the Griffith formula. 

Figure 5.38 The Griffith model for micro-crack induced brittle failure. Cracks at the surface have a length of a, whereas internal cracks have a length of 2a.

Their analysis of experimental data shows that tensile strength was the only parameter that varied as a function of particle size. Model simulation indicate that larger lumps were stronger than smaller lumps which is contradictory to Waters et al. [8], Teo and Waters [9], and Griffith [10] theory of fracture, which implies that larger particle are more likely to contain larger cracks and hence be more susceptible to breakage. 

In a disordered solid, as modelled by the percolation model (discussed in Section 1.2) with intact bond concentration p on a lattice, there occur various vacancy clusters (voids), or pre-existing cracks, of different sizes and shapes. The typical crack size in such a percolating solid being of the size of the correlation length (see Section 1.2.1) the application of the Griffith fracture criterion gives (Chakrabarti 1988, Ray and Chakrabarti 1985a,b) from equation (3.4)... 

The use of the generalized Griffith criterion not only allows cracks to extend under tensile loading, but also allows cracks to extend in shear, even under moderate normal compression. This means that closed shear cracks can extend under intense loading (as near an explosive charge), causing the material to be much weaker under subsequent tensile loading. The reduction in the elastic moduli is also important in that it leads to the correct directional response as the rock is fractured. Thus, phenomena such as spall are modeled in a realistic fashion. 

There are two principal theories, or models, that attempt to describe what happens during brittle fracture, the Griffith fracture theory and the Irwin model. Both assume that fracture takes place through the presence of preexisting cracks or flaws in the polymer and are concerned with what happens near such a crack when a load is applied. Each leads to the definition of a fracture-toughness parameter and the two parameters are closely related to each other. The Griffith theory is concerned with the elastically stored energy near the crack, whereas the Irwin model is concerned with the distribution of stresses near the crack. Both theories apply strictly only for materials that are perfectly elastic for small strains and are therefore said to describe linear fracture mechanics. 

The second term expresses the released energy in the area around the flaw during cracking, and the third one denotes the additional surface energy of the flaw due to crack. According to the Griffith model ", the flaw will increase its size spontaneously only when a decrease of energy in the sample takes place, that is. 

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