This study presents the results of the triaxial loading experiments performed to determine the fracture strength characteristics of granite subjected to truly three dimensional loading. These measurements were carried out on thin walled hollow cores which were subjected to various tension-torsion, compression-torsion loading paths, in the presence of confining pressures of up to 7000 psi, at ambient temperature and at 300(DEGREES)F. These results are compared with the classical failure theories of Coulomb-Mohr and Drucker and Prager. It is observed that none of the above criteria are able to predict the failure over the entire range of stresses, and at different temperatures. A new failure criterion based on the strain energy density and first stress invariant (J(, 1)) is proposed, to predict the failure envelope over the entire range of stresses and at different temperatures. Firstly, the results of the work done by three independent investigators on the failure of granite are compared with the Coulomb-Mohr, Drucker and Prager and the proposed new criterion. The improved effectiveness of the proposed new criterion in predicting the failure of granite relative to the other two criteria is then demonstrated. Also, failure results of four other commonly studied rocks (Solenhofen Limestone, sandstone, marble, and shale) are compared with the Coulomb-Mohr, Drucker and Prager, and the proposed new criterion. The improved or equal effectiveness of the proposed new criterion in predicting the failure of these common rocks relative to the other two criteria is also demonstrated.