Griffith's Theroy: Mechanism of Brittle Fracture

      It has been observed that the stress required for a material, at which it fractures, is only a small fraction of cohesive strength. This discrepancy led Griffith to suggest that the low observed strengths were due to presence of micro-cracks, which act as the points of stress concentration. 

       According to the Griffith's criterion. the crack will propagate under the effect of a constant applied stress if an incremental increase in length produces no change in total energy of the systems. Mathematically the above criterion is explained as 

A proper explanation of the above theory is given as below by Wikipedia:

Fracture mechanics was developed during World War I by English aeronautical engineer, A. A. Griffith, to explain the failure of brittle materials. Griffith's work was motivated by two contradictory facts:

• The stress needed to fracture bulk glass is around 100 MPa (15,000 psi).
• The theoretical stress needed for breaking atomic bonds is approximately 10,000 MPa (1,500,000 psi).

        A theory was needed to reconcile these conflicting observations. Also, experiments on glass fibers that Griffith himself conducted suggested that the fracture stress increases as the fiber diameter decreases. Hence the uniaxial tensile strength, which had been used extensively to predict material failure before Griffith, could not be a specimen-independent material property. Griffith suggested that the low fracture strength observed in experiments, as well as the size-dependence of strength, was due to the presence of microscopic flaws in the bulk material.

      To verify the flaw hypothesis, Griffith introduced an artificial flaw in his experimental specimens. The artificial flaw was in the form of a surface crack which was much larger than other flaws in a specimen. The experiments showed that the product of the square root of the flaw length (a) and the stress at fracture (σ_f) was nearly constant, which is expressed by the equation:

An explanation of this relation in terms of linear elasticity theory is problematic. Linear elasticity theory predicts that stress (and hence the strain) at the tip of a sharp flaw in a linear elastic material is infinite. To avoid that problem, Griffith developed a thermodynamic approach to explain the relation that he observed.

      The growth of a crack requires the creation of two new surfaces and hence an increase in the surface energy. Griffith found an expression for the constant C in terms of the surface energy of the crack by solving the elasticity problem of a finite crack in an elastic plate. Briefly, the approach was:

• Compute the potential energy stored in a perfect specimen under an uni-axial tensile load.
• Fix the boundary so that the applied load does no work and then introduce a crack into the specimen. The crack relaxes the stress and hence reduces the elastic energy near the crack faces. On the other hand, the crack increases the total surface energy of the specimen.
• Compute the change in the free energy (surface energy − elastic energy) as a function of the crack length. Failure occurs when the free energy attains a peak value at a critical crack length, beyond which the free energy decreases by increasing the crack length, i.e. by causing fracture. Using this procedure, Griffith found that

where E is the Young's modulus of the material and γ is the surface energy density of the material. Assuming Eγ = 1 J/m² gives excellent agreement of Griffith's predicted fracture stress with experimental results for glass. = 62 GPa and

The above information is taken from Wikipedia. Please do refer to them for more info. Don't forget to review this copy of Material Science and Engineering book, which has info for all the syllabus of AMIE material science.

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