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The '''Christensen failure criterion''' is a [[material failure theory]] for isotropic materials that attempts to span the range from [[ductility|ductile]] to [[Brittleness|brittle]] materials. <ref name="chri"/> It has a two-property form calibrated by the uniaxial [[tensile strength|tensile]] and [[compressive strengths]] T <math>\left (\sigma_T\right )</math> and C <math>\left (\sigma_C\right )</math>.
 
The theory was developed by R. M. Christensen and first published in 1997.<ref>Christensen, R.M. (1997).''Yield Functions/Failure Criteria for Isotropic Materials'', Pro. Royal Soc. London, Vol. 453, No. 1962, pp. 1473–1491</ref><ref>Christensen, R.M. (2007), ''A Comprehensive Theory of Yielding and Failure for Isotropic Materials'', J. Engr. Mater. and Technol., 129, 173–181</ref>
 
==Description==
The Christensen failure criterion is composed of two separate subcriteria representing competitive failure mechanisms. when expressed in principal [[Stress (mechanics)|stress]] components, it is given by :
 
;Polynomial invariants failure criterion
 
For      <math> 0\le\frac{T}{C}\le1</math>
 
{{NumBlk|:|<math>\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 </math>|{{EquationRef|1}}}}
 
;Coordinated Fracture Criterion
 
For      <math> 0\le \frac{T}{C}\le \frac{1}{2} </math>
{{NumBlk|:|<math>\begin{array}{lcl} \sigma_1 & \le & T \\ \sigma_2 & \le & T \\ \sigma_3 & \le & T \end{array}</math>|{{EquationRef|2}}}}
[[Image:Christensen fig 1.JPG|thumb|alt=A criteron illustrated|For plane stresses,<math>\sigma_3 = 0</math> and T/C=0.3(brittle materials). Blue line is polynomial invariants failure criterion ({{EquationNote|1}}). Red line is coordinated fracture criterion({{EquationNote|2}}).]]
 
The geometric form of ({{EquationNote|1}}) is that of a paraboloid in principal stress space. The [[fracture]] criterion ({{EquationNote|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 ({{EquationNote|1}}) and ({{EquationNote|2}}) reduces to the [[Von Mises yield criterion|Mises criterion]] at the [[Ductility|ductile]] limit, T/C = 1. At the [[Brittleness|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.<ref name="chri">Christensen, R. M.,(2010),http://www.failurecriteria.com.</ref> 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<ref>S. K. Ha, K. K. Jin and Y. C. Huang,(2008), ''Micro-Mechanics of Failure (MMF) for Continuous Fiber Reinforced Composites.'' Journal of Composite Materials, vol. 42, no. 18, pp. 1873–1895.</ref> to the failure of the isotropic, polymeric matrix phase in fiber [[composite material]]s.
 
== See also ==
* [[Strength of materials]]
* [[material failure theory]]
* [[Von Mises yield criterion]]
* [[Mohr–Coulomb theory]]
 
== References ==
{{Reflist}}
 
[[Category:Mechanical failure]]
[[Category:Plasticity]]
[[Category:Solid mechanics]]
[[Category:Mechanics]]

Revision as of 12:13, 20 August 2013

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 and C .

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

Template:NumBlk

Coordinated Fracture Criterion

For Template:NumBlk

A criteron illustrated
For plane stresses, and T/C=0.3(brittle materials). Blue line is polynomial invariants failure criterion (Template:EquationNote). Red line is coordinated fracture criterion(Template:EquationNote).

The geometric form of (Template:EquationNote) is that of a paraboloid in principal stress space. The fracture criterion (Template:EquationNote) (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 (Template:EquationNote) and (Template:EquationNote) 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

References

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  1. ↑ 1.0 1.1 Christensen, R. M.,(2010),http://www.failurecriteria.com.
  2. ↑ Christensen, R.M. (1997).Yield Functions/Failure Criteria for Isotropic Materials, Pro. Royal Soc. London, Vol. 453, No. 1962, pp. 1473–1491
  3. ↑ Christensen, R.M. (2007), A Comprehensive Theory of Yielding and Failure for Isotropic Materials, J. Engr. Mater. and Technol., 129, 173–181
  4. ↑ S. K. Ha, K. K. Jin and Y. C. Huang,(2008), Micro-Mechanics of Failure (MMF) for Continuous Fiber Reinforced Composites. Journal of Composite Materials, vol. 42, no. 18, pp. 1873–1895.
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