A considerable difference between the fracture stresses theoretically calculated and those actually observed is usually explained by the presence of minute flaws, the so-called Griffith cracks, in the material.
Assuming that the fracture is caused by the stress concentration around Griffith cracks, a statistical theory of brittle fracture is given which could be applied to arbitrary stress field. In this theory, the distribution of strength of test piece is related to the distribution of shapes of Griffith cracks.
In this paper, it is supposed that the probability density function of the shape of Griffith cracks is
f(ξ0)=δkδ/ξ0(δ+1)·exp{-(k/ξ0)δ}
where ξ0 is the index of the shape or flatness of Griffith cracks.
Based on this probability density function of the shape of Griffith cracks and some assumption concerning the stress concentration around cracks, the size effects of the uniaxial tensile and compressive strengths are calculated and given in the equations which coincide with the formula derived from Weibull's distribution function, that is,
St=Ct·k·s·n-1/m
for the uniaxial tensile strength of the test piece which contains n cracks, where Ct is a constant, and s is the theoretically calculated fracture stress.
In the calculation of the uniaxial compressive strength, besides Griffith's theory, a modified Griffith theory by McClintock and Walsh is considered.
As the result of these calculations, the relations between the parameter of the distribution function of the shape of Griffith cracks δ and Weibull's coefficient of uniformity m are given in graphs. And the relations between the parameter δ and the“brittleness index”(the ratio between the uniaxial compressive and tensile strength) is given in another graph.
Using these graphs, the dispersions of the actually observed uniaxial compressive and tensile strengths of rocks are discussed.
To conclude, the following suggestions are given:
(1) The brittleness index is not always a constant, but is a function of the distribution of shapes of Griffith cracks.
(2) Besides the“inherent”dispersion discussed in this paper, there may be some experimental errors. Based on the theoretical aspect given in this paper, it is possible to evaluate the difference of experimental errors between the uniaxial compressive and tensile strength.
