Conditions arise in many geoengineering applications where both tensile and compressive normal (effective) stresses act due to change in stress from excavation or pore pressure. However, testing of rock at these stress states associated with low mean stress, say p < C0/3, is rare because of experimental difficulties, where p = (σxx + σ¬yy + σzz)/3 and Co = uniaxial compressive strength. The objective of this research is to evaluate rock failure at low mean stress using dog-bone specimens of (dry) Dunnville sandstone. Results from these special triaxial extension tests were used in conjunction with conventional triaxial extension and compression experiments with right-circular cylinders to evaluate four failure criteria: (1) Mohr-Coulomb (MC) with a tension cut-off, (2) Paul-Mohr-Coulomb (PMC) with a tension cut-off, (3) Hoek-Brown (HB), and (4) Fairhurst (Fh). Results for the Dunnville sandstone show that the three failure criteria that either include a tension cut-off (MC and PMC) or have a “natural” tension cut-off (Fh) best capture failure in the low mean stress regime, -T/3 < p < C0/3, where T = uniaxial tensile strength. Of the four criteria considered, Fh provided the best overall fit because it is nonlinear and contains a tension cut-off. Fracture surfaces of the dog-bone specimens were evaluated for failure mode based on surface roughness and it was found that there is a transition of decreasing roughness from tensile failure to hybrid (opening and sliding) failure to shear failure.