Griffith proposed that a brittle material contains number of micro-cracks which causes stress rise in localized regions at a nominal stress which is well below the theoretical value. When one of the cracks spreads into a brittle fracture, it produces an increase in the surface energy of the sides of the crack. Source of the increased surface energy is the elastic strain energy, released as crack spreads. Griffith’s criteria for propagation of crack include these terms as: a crack will propagate when the decrease in elastic energy is at least equal to the energy required to create the new crack surface. This criterion is useful in determining the tensile stress which will just cause a critical sized crack to propagate as a brittle fracture.
Elastic energy stored under tensile stress will be released as crack propagates. Part of this energy is expended in forming the surface of the crack, while the remaining is transformed into kinetic energy. According to Griffith, such as crack will propagate and produce brittle fracture when an incremental increase in its length does not change the net energy of the system. Strain energy released in a thin plate of unit thickness is given by Inglis as follows:
$$U_e = \frac{\pi \sigma^2 c^2}{E}$$
Where E is Young’s modulus, and is the applied stress. At the same time, surface energy gained by the system due to new surfaces formed as a crack, of length 2c, propagates can be given as
$$U_z = 4 \gamma c$$
Griffith’s criterion can be expressed as follows for incremental change in systems energy with crack length:
$$\frac{\delta U_e}{\delta C} = \frac{\delta U_s}{\delta c} \Rightarrow \frac{2\pi \sigma^2c}{E} = 4\gamma \Rightarrow \sigma = \bigg(\frac{2E\gamma}{c \pi} \bigg)^{1/2}$$
The equation gives the stress required to propagate a crack in a thin plate under plane stress. The stress necessary to cause fracture varies inversely with length of existing cracks, thus largest crack determines the strength of material. The sensitivity of the fracture of solids to surface conditions has been termed Joffe effect. For a plate which is thick compared with crack size (i.e. plane strain condition), the stress is given as
$$\sigma = \bigg( \frac{2E \gamma}{(1 - \nu^2)c \pi} \bigg) ^{1/2}$$
where ν is Poisson’s ratio. The Griffith theory applies only to completely brittle materials. In crystalline materials which fracture in an apparently brittle mode, plastic deformation usually occurs next to fracture surface. Orowan, thus, modified the Griffith equation to make it more compatible by including plastic energy term as follows
$$\sigma = \bigg( \frac{2E(\gamma + p)}{c \pi} \bigg)^{1/2} /approx \bigg( \frac{Ep}{c} \bigg)^{1/2}$$
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