Empirical rock failure criteria generally relates lateral stresses to axial strengths by a particular equation in which uniaxial tensile and/or compressive strengths are to be estimated from the failure criterion by applying the obtained parameters from triaxial datasets into the criterion equation. In order to satisfy the exact amounts of uniaxial tensile and compressive strengths in a failure criterion, a rational function is proposed as a failure criterion for isotropic, dry and intact rocks to be used in both tensile and compressive regions. The proposed criterion is validated by using 59 series of accurate triaxial test datasets which are belong to 20 different rock types selected from the literature. It has been shown that the proposed criterion is capable of predicting the triaxial behavior of rocks in both tensile and compressive regions. There are two coefficients for the proposed criterion (‘c’ and ‘d’) which have been estimated for used rock datasets by applying brittle-ductile transition as an initial boundary condition. The most proper brittle into ductile boundary, the confinement in which ∂ τ/ ∂ σ n=0 or in other words when ∂ σ1/∂ σ3=1, has been determined (σ3=η c) in such a way that the best triaxial predictions have been obtained. Also, the values of the criterion coefficients are discussed in detail.
Empirical rock failure criteria are expressed as an equation in terms of some parameters. The parameters of a failure criterion are dependent on rock type and its engineering properties . The parameters are predicted by applying laboratory tests and the criterion should satisfy the certain boundary conditions such as tensile strength, uniaxial compressive strength, concavity requirement and brittle-ductile transition (BDT) properly predicting failure conditions . Jaeger and Cook  indicated that a different type of behavior is shown by some rocks in high confinements, notably carbonates and some sediments, which was first studied in detail in the work of von Karman  on Carrara marble. For this type of rock, in confining pressures of up to about 50 MPa, brittle fracture was occurred with an increase of strength and a small increase in permanent set, but the behavior of rock for σ3 = 68.5 MPa was completely different, since the material could undergo strains of over 7 percent with no loss of strength. This is generally known as ductile behavior. The conclusion is that there is a rather ill-defined value of the confining pressure at which there is a transition from typical brittle behavior to fully ductile behavior. This is called the BDT. At higher confining pressures, 165 and 326 MPa, 1 increases steadily with increasing strain after the yield point has been passed (work-hardening). Similar and more detailed results have been obtained by Heard  for Solenhofen limestone.
It is well recognized that the behavior of most rocks changes from brittle to ductile at some elevated confining pressure, defined by Byerlee  as the BDT pressure.