A three invariant model of failure in true triaxial tests on Castlegate sandstone

By far the most common rock testing configuration is axisymmetric compression in which the most compressive principal stress (σ1) is axial and the other two are equal and provide the lateral confinement. Somewhat less common is axisymmetric extension in which the least compressive stress (σ3) is axial and the other two are again equal and provide the lateral confinement. Although these tests have provided an enormous amount of information about rock deformation and failure, they are limited in that the intermediate principal stress (σ2) is always equal to the most or least compressive stress. Consequently, these tests are able to cover only a restricted range of possible stress states. Furthermore, axisymmetric stress states are not typical of those in the Earth's crust. Even when the tectonic stress field is nearly axisymmetric, the stress state around inhomogeneities, such as aquifers, reservoirs, mines or faults, will be fully three-dimensional. In the 60's Mogi noted that the differences in failure stress between axisymmetric compression and extension tests indicated a dependence on the intermediate principal stress. As a result, he began to conduct true triaxial tests on cubical specimens in which all three principal stresses differed. Until recently, these tests were conducted primarily on low porosity rocks. Now, however, true triaxial tests have been done on higher porosity sandstones. The conventional way in which true triaxial tests are done is that the specimen is first loaded in hydrostatic compression, then stresses are increased on two of the faces until the desired level of the intermediate principal stress, σ2, is reached, and finally the stress on one of the faces is increased to failure. Consequently, two of the three principal stresses are constant. Although this loading suffices to study the role of σ2 on failure, it has disadvantages for understanding the physical processes of failure. In particular, both the hydrostatic (mean normal) stress and the deviatoric stress state (defined more precisely below) change during loading to failure. An exception to this conventional loading is plane strain loading. Because the intermediate stress is controlled to maintain approximately plane strain conditions, it is changing during loading to failure. Consequently, both the hydrostatic stress and the deviatoric stress state are also changing for this loading. Recent true triaxial tests have been conducted in which the deviatoric stress state is held constant. In tests on two porous sandstones, Coconino and Bentheim, Ma et al. maintain σ3 constant and raise the two other principal stresses in a fixed ratio to keep the deviatoric stress state constant. The hydrostatic stress is not, however, constant. Ingraham et al. have conducted unique tests on Castlegate sandstone in which they control changes in all three principal stresses in order to keep both the deviatoric stress state and the hydrostatic stress constant. Then the magnitude of the deviatoric stress is increased to failure. In this note, a simple failure condition introduced by Rudnicki is applied to the data of Ingraham et al. on Castlegate sandstone. Application of this condition provides additional interpretation of the results and a means of generalizing them to arbitrary stress states.