Griffith crack equation

The Griffith crack equation has been shown to apply, albeit with some scatter of results, to the brittle polymeric materials poly(methyl methacrylate) and poly(styrene) when cracks of controlled size have been introduced deliberately into the specimens. Such experiments give values of surface energy that are very large, typically 10 - 10 J m , which is about 100 times greater than the theoretical value calculated from the energy of the chemical bonds involved. This value of y thus seems to be made up of two terms, Le. 

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. 

If Vt 1240 meters/sec in the matrix and branching will occur in the rubber at 29 meters/sec, we calculate A/Co = 0.047. Thus, branching can occur after a matrix crack acceleration distance of only 2 to 5/x (assuming a Griffith crack length of 50-100fi) hence, ample room for the development of fast cracks or fast crazes exists in the ABS structure. Note that the expressions for craze instability, acceleration, and speed (Equations 1, 6, 7) show that the macro strain rate of the specimen is irrelevant— fast cracks and crazes propagate in specimens strained even at slow creep rates. 

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). 

The second approach, due to lrwin is to characterise the stress field surrounding a crack in a stressed body by a stress-field parameter K (the stress intensity factor ). Fracture is then supposed to occur when K achieves a critical value K - Although, like Griffith s equation, this formulation of fracture mechanics is based on the assumptions of linear elasticity, it is found to work quite effectively provided that inelastic deformations are limited to a small zone around the crack tip. Like, however, the critical parameter remains an empirical quantity it cannot be predicted or related explicitly to the hysical properties of the solid. Like,, K. is time and temperature de ndent. 

To obviate this inadequacy, Griffith s equation had to be modified to include the energy expended in plastic deformation in the fracture process. Accordingly, Irwin defined a parameter, G, the strain-energy-release rate or crack extension force, which he showed to be related to the applied stress and crack length by the equation ... 

The tensile strength of classic cement pastes conforms to the curve plotted on the basis of the Griffith s equation, at the assumption that the width of crack is about 1 mm. The further part of this curve corresponds to the data obtained by Birchall [97], for the specially prepared macro defects free pastes (Fig. 5.40). These pastes are discussed in Chap. 9. They exhibit significantly higher strength because the macropores do not exceed 90 pm. 

Griffith calculated the change in elastically stored energy using a solution obtained by Inglis [2] for the problem of a plate, pierced by a small elliptical crack, that is stressed at right angles to the major axis of the crack. Equation (12.1) then allows the fracture stress of the material to be defined in terms of... 

The colloidal pores do not start cracks as their dimensions are extremely small and this fact is correctly represented by the Griffith s equation ... 

This section is concerned with the two-dimensional elasticity equations. Our aim is to find the derivative of the energy functional with respect to the crack length. The nonpenetration condition is assumed to hold at the crack faces. We derive the Griffith formula and prove the path independence of the Rice-Cherepanov integral. This section follows the publication (Khludnev, Sokolowski, 1998c). 

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. 

Griffith derived a similar equation using an energy balance approach, equating stored energy with the energy required for crack propagation ... 

As a result of this additional work to create a surface over and above the thermodynamic surface energy assumed by Griffith, we need to rewrite his equation for the relationship between breaking stress and crack length. We modify equation (7.2) to ... 

The value of o obtained from this equation is the critical value of the applied tensile stress normal to the crack (also known as the Griffith stress) necessary to cause the crack to propagate and is given by... 

Generally, for any dimension therefore, if a crack of length I already exists in an infinite elastic continuum, subject to uniform tensile stress a perpendicular to the length of the crack, then for the onset of brittle fracture, Griffith equates (the differentials of) the elastic energy E with the surface energy E ... 

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