The Christensen failure criterion is a material failure theory for isotropic materials that attempts to span the range from ductile to brittle materials.^[1] It has a two-property form calibrated by the uniaxial tensile and compressive strengths T ${\displaystyle \left(\sigma _{T}\right)}$ and C ${\displaystyle \left(\sigma _{C}\right)}$.

The theory was developed by Stanford professor Richard. M. Christensen and first published in 1997.^[2][3]

Description

The Christensen failure criterion is composed of two separate subcriteria representing competitive failure mechanisms. when expressed in principal stress components, it is given by :

Polynomial invariants failure criterion

For ${\displaystyle 0\leq {\frac {T}{C}}\leq 1}$

${\displaystyle \left({\frac {1}{T}}-{\frac {1}{C}}\right)\left(\sigma _{1}+\sigma _{2}+\sigma _{3}\right)+{\frac {1}{2TC}}\left[\left(\sigma _{1}-\sigma _{2}\right)^{2}+\left(\sigma _{2}-\sigma _{3}\right)^{2}+\left(\sigma _{3}-\sigma _{1}\right)^{2}\right]\leq 1}$

(1)

Coordinated Fracture Criterion

For ${\displaystyle 0\leq {\frac {T}{C}}\leq {\frac {1}{2}}}$

${\displaystyle {\begin{array}{lcl}\sigma _{1}&\leq &T\\\sigma _{2}&\leq &T\\\sigma _{3}&\leq &T\end{array}}}$

(2)

For plane stresses,${\displaystyle \sigma _{3}=0}$ and T/C=0.3(brittle materials). Blue line is polynomial invariants failure criterion (1). Red lines are coordinated fracture criterion(2).

The geometric form of (1) is that of a paraboloid in principal stress space. The fracture criterion (2) (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 (1) and (2) 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.^[1] 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.

See also

• Strength of materials
• material failure theory
• Von Mises yield criterion
• Mohr–Coulomb theory

References