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Christensen Criterion

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 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 Coulomb-Mohr criterion.

The stress-based 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:

Two-Dimensional Christensen Criterion

For plane stresses, . The Christensen criterion becomes:

For all materials ():

For brittle materials (), add the following fracture constraint.

This criteron can be illustrated as follows:

For "ideal" ductile materials, , the Cristensen criterion is reduced to

exactly the same as the von Mises criterion.

For nominally ductile materials, , the ellipse is shifted toward the compression directions (down and left):

For brittle materials, , the Christensen criterion is represented by an ellipse that is further shifted,

and two fracture lines:

Failure Criteria Comparison

The Christensen criterion covers a broader range of materials. The following illustrates how it compares with other commonly used failure criteria:

The Cristensen criterion is essentially the von Mises criterion for very ductile materials. Both criteria are more generous than the Maximum Shear Stress criterion.
For brittle materials, the Christensen criterion is more conservative for pure tensile stress (the 1st quadrant) than both the Maximum Normal Stress and the Coulomb-Mohr criterion. It is between these two criteria when under mixed (tension-compression) loadings (the 2nd and 4th quadrants). The fracture cutoff lines in the 2nd and 4th quadrants become more pronounced as the tensile to compressive strength ratio diminishes. Finally, it is more aggressive than the other two criteria in compression loadings (the 3rd quadrant).