Engineering Rock Mechanics

Failure criteria 109

6.5.2 The plane Griffith criterion

The essence of the Griffith criterion is that for a material to break in tension

owing to the presence of an existing microcrack, sufficient energy must be

released to provide the necessary new surface energy as the crack prop-

agates. The rate of strain energy release must be equal to or greater than

the required surface energy increase. This results in the expression shown

in Fig. 6.19 for the uniaxially loaded plate shown. It is possible to extend this

criterion from the plane stress case shown to plane strain in both tension

and compression, as the figure shows. The basic concept of supplying

sufficient surface energy during fracturing also applies during crack prop-

agation, However, the formulae refer only to the onset of cracking because

the geometry changes during crack propagation. In the case of a tensile test

for enpeering purposes, fracture initiation and specimen collapse may be

considered as synonymous; in the case of compression, however, we have

already noted that microstructural cracking occurs throughout the complete

stress-strain curve and that the compressive strength is an arbitrary stage

in the microstructural breakdown process. Thus, whilst it is interesting to

utilize the Griffith criterion for studying microcrack intitiation under

compressive loading, it is unlikely that the formula can provide a useful

estimate of the engineering compressive strength.

The formula for tensile failure is

ot = (kaE/~)”~

where ct is the tensile stress applied to the specimen at failure, k is a
parameter that varies with the testing conditions, i.e. k = ~TC for plane
stress and k = 2( 1 - v2 )/nfor plane strain, ais the unit crack surface energy,
E isthe Young’s modulus, and c is half the initial crack length.
Thus, for a given rock and testing configuration, the tensile strength will
vary inversely as the square root of the initial crack length. This provides

Material fractures when sufficient strain energy is released

u unit thickness

fractures when sufficient strain energy is released

unit thickness

to enable cracks to propagate

k = i for plane stress

= (I - v2) for plane strain

I CY = unit surface energy of the crack

In compression:

(a, - ai)’ = 8T, (a, + cr3) when u, + 3u3 > 0

u3 = -To when u, + 3u3 < 0

Note: compression positive, To positive (-To = ut)

Figure 6.19 The plane Criffith failure criterion