GRIFFITH'S EQUATION - AN INTRODUCTION TO FRACTURE MECHANICS

A simple explanation of the ground-breaking work of Aeronautical Engineer A. A. Griffith on the failures in brittle materials:

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Background:

Griffith's work began when he realised there was a large difference between the theoretical strength of a material and the experimental strength of materials. This problem had gone unanswered for a long time and was still causing problems:

WWII era T2 tanker fails by brittle fracture at a welded joint

The problem lay in the engineer's calculation of the theoretical strength of a material. They based the strength of materials on the cohesive force between the layers of atoms, Although this seems logical it crucially fails to consider the way in which a brittle material fails:

As shown above the material fails by a crack moving growing through the material at a right angle to the stress (this is a 'mode 1' fracture there are in fact others however for simplicity they have been left out). Griffith set out to find an equation to relate the stress and crack length at fracture.

The Equation:

the graph above shows the various energies in a material as the crack grows:

Surface Energy (S):

the surface of a material has energy - surface tension. As a crack grows its surface area increases. If there is more surface area there is more surface energy. Surface energy is constant for a material - that is everytime a square meter of surface is formed the same amount of energy is required.

Strain energy (U):

When the material is loaded it gains energy this is the only input of energy into the system so it follows that energy used to propagate the crack is strain energy. Therefore as the crack grows the strain energy decreases.

When a material is loaded it stretches slightly and work is done. Assuming 100% efficiency the work done is all converted to strain energy. Strain energy (per cubic metre) is therefore the integral of stress with respect to strain (the area under the stress strain curve): (1/2)*(stress)*(strain). The units are Nm^-2 meaning strain energy is proportional to (crack length^2) (think area) hence the curve.

The point of fracture:

at the point of fracture the strain energy (source of energy) must be greater than or equal to the energy required to create the new surfaces when the crack grows. This is the critical point - on the graph this is the stationary point on the total energy curve which occurs when (dS/da) = (dU/da). At this point the total energy begins to decrease as work is done fracturing the material.

the differentials mentioned above can be thought of as:

dS/da = resistance to crack growth (rate of surface energy change)

dU/da = rate of strain energy release

when the two are equal the critical value of strain energy release (Gcrit) is achieved. Deriving the Griffith equation is then a simple task of rearrangeing the equation:

(dS/da) = Gcrit

(Surface energy per square metre) * (area) .... for a full explanation of this scroll down to the bottom...

rearrange for stress....

Surface Energy:

strain energy per metre cubed is the area under a stress strain curve:

to find the strain energy at the very surface of a material (surface tension) this is multiplied by atomic seperation:

strain is then substituted for: stress / young'sModulus:

*this is surface energy per square metre of material.

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