The Christensen failure criterion is a material failure theory for isotropic materials that attempts to span the range from ductile to brittle materials. It has a two-property form calibrated by the uniaxial tensile and compressive strengths T
\left(\sigmaT\right)
\left(\sigmaC\right)
The theory was developed by Stanford professor Richard. M. Christensen and first published in 1997.[1] [2]
The Christensen failure criterion is composed of two separate subcriteria representing competitive failure mechanisms. when expressed in principal stress components, it is given by :
For
| 0\le | T |
| C |
\le1
For
0\le
| T | |
| C |
\le
| 1 | |
| 2 |
The geometric form of is that of a paraboloid in principal stress space. The fracture criterion (applicable only over the partial range 0 ≤ T/C ≤ 1/2) cuts slices off the paraboloid, leaving three flattened elliptical surfaces on it. The fracture cutoff is vanishingly small at T/C=1/2 but it grows progressively larger as T/C diminishes.
The organizing principle underlying the theory is that all isotropic materials admit a distinct classification system based upon their T/C ratio. The comprehensive failure criterion and reduces to the Mises criterion at the ductile limit, T/C = 1. At the brittle limit, T/C = 0, it reduces to a form that cannot sustain any tensile components of stress.
Many cases of verification have been examined over the complete range of materials from extremely ductile to extremely brittle types.[3] Also, examples of applications have been given. Related criteria distinguishing ductile from brittle failure behaviors have been derived and interpreted.
Applications have been given by Ha[4] to the failure of the isotropic, polymeric matrix phase in fiber composite materials.