# Physics:Christensen failure criterion

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 R. 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 $\displaystyle{ 0\le\frac{T}{C}\le1 }$

$\displaystyle{ \left (\frac{1}{T}-\frac{1}{C} \right )\left (\sigma_1+\sigma_2+\sigma_3\right )+\frac{1}{2TC}\left [\left (\sigma_1-\sigma_2\right )^2+\left (\sigma_2-\sigma_3\right )^2+\left (\sigma_3-\sigma_1\right )^2\right ]\le 1 }$

(1)

Coordinated Fracture Criterion

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

$\displaystyle{ \begin{array}{lcl} \sigma_1 & \le & T \\ \sigma_2 & \le & T \\ \sigma_3 & \le & T \end{array} }$

(2)

For plane stresses,$\displaystyle{ \sigma_3 = 0 }$ and T/C=0.3(brittle materials). Blue line is polynomial invariants failure criterion (1). Red line is 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.