Solution (Problem 7)
This question tests your understanding of the Griffith theory, and is representative of typical examination questions in this area. It should take about 15 minutes to complete.

Griffith postulated that brittle fracture occurred when the energy release rate during crack growth exceeded the rate that energy was required. His original analysis was performed for a brittle material (glass).

a)    Sketch curves showing how the changes in surface energy required, and potential energy released, as a crack extends.

b)    Sketch the changes in their differentials (rates R and G) with respect to crack length extension. Label the sketches and define the parameters R and G.

c)    State what the critical condition is for fracture, in terms of R and G.

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a)    Griffith was concerned with the energetics of fracture, and considered the energy changes associated with incremental crack extension. For a loaded brittle body undergoing incremental crack extension, the only contributors to energy changes are the energy of the new fracture surfaces (two surfaces per crack tip) and the change in potential energy in the body. The surface energy term (S) represents energy absorbed in crack growth, while the some stored strain energy (U) is released as the crack extends (due to unloading of regions adjacent to the new fracture surfaces). Surface energy has a constant value per unit area (or unit length for a unit thickness of body) and is therefore a linear function of (crack length), while the stored strain energy released in crack growth is a function of (crack length)2, and is hence parabolic. These changes are indicated in the figure below:

Griff1.jpg (97770 bytes)

b)    The next step in the development of Griffith's argument was consideration of the rates of energy change with crack extension, because the critical condition corresponds to the maximum point in the total energy curve, i.e. dW/da = 0, where a = a*. For crack lengths greater than this value (under a given applied stress), the body is going to a lower energy state, which is favourable, and hence fast fracture occurs. dW/da = 0 occurs when dS/da = dU/da and the sketch below shows the energy rates, or differentials with respect to a.

Griff2.jpg (21736 bytes)

R is the resistance to crack growth (= dS/da) and G is the strain energy release rate (= dU/da).

c)    When fracture occurs, R = G and we can define Gcrit as the critical value of strain energy release, and equate this to R. Hence Gcrit represents the fracture toughness of the material.

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