AbstractA realistic criterion was introduced in the 3D principal stress space for describing failure of transversely isotropic bodies taking also into account the strength differential effect. This criterion is represented by an elliptic paraboloid failure surface (EPFS). The criterion stipulates that in the compression–compression–compression octant these materials may sustain very high loadings. To remedy this ambiguous definition of failure in this zone, an improved version of the EPFS was also introduced as a complement of the EPFS criterion. Based on the same principles as the EPFS, a symmetric ellipsoid failure surface is defined whose symmetry axis is parallel to the hydrostatic axis, which is also passing though the three pairs of failure strengths in tension and compression along each of the principal stress axes. Both criteria are completely defined by stipulating that their symmetry axes are parallel to the hydrostatic axis and that the fourth rank tensors H defining failure should have a zero eigenvalue associated with the spherical tensor 1. Extensive experimental evidence with different orthotropic materials has shown that their failure results corroborate both criteria, depending on the form of their struct