Christensen published a series of papers between 1997 and 2007 developing a three dimensional failure criterion suitable for isotropic materials, bridging the gap between failure criteria for ductile and brittle materials. The results are summarized in www.FailureCriteria.com. This criterion has a quadratic form similar to the von Mises criterion and, at the same time, is intended to be applicable across a wide range of materials, similar to the CoulombMohr criterion.
The stressbased Christensen criterion for isotropic material states that failure occurs when combined normalized stress effect exceeds 1. In addition, for brittle materials, no principal stress shall exceed the tensile strength .
For all materials ():
For brittle materials (), add the following fracture criterion:
where and are the tensile and compressive strengths of the isotropic material, respectively.


In terms of principal stresses:
