The Christensen failure criterion is a
material failure theory
Material failure theory is an interdisciplinary field of materials science and solid mechanics which attempts to predict the conditions under which solid materials fail under the action of external loads. The failure of a material is usuall ...
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
In physics, tension is described as the pulling force transmitted axially by the means of a string, a rope, chain, or similar object, or by each end of a rod, truss member, or similar three-dimensional object; tension might also be described as t ...
and
compressive strengths T
and C
.
The theory was developed by
Stanford
Stanford University, officially Leland Stanford Junior University, is a private research university in Stanford, California. The campus occupies , among the largest in the United States, and enrolls over 17,000 students. Stanford is considere ...
professor Richard. M. Christensen and first published in 1997.
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
;Coordinated Fracture Criterion
For
The geometric form of () is that of a paraboloid in principal stress space. The
fracture
Fracture is the separation of an object or material into two or more pieces under the action of stress. The fracture of a solid usually occurs due to the development of certain displacement discontinuity surfaces within the solid. If a displa ...
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.
[Christensen, R. M., (2010), http://www.failurecriteria.com.] 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
[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.] to the failure of the isotropic, polymeric matrix phase in fiber
composite material
A composite material (also called a composition material or shortened to composite, which is the common name) is a material which is produced from two or more constituent materials. These constituent materials have notably dissimilar chemical or ...
s.
See also
*
Strength of materials
*
material failure theory
Material failure theory is an interdisciplinary field of materials science and solid mechanics which attempts to predict the conditions under which solid materials fail under the action of external loads. The failure of a material is usuall ...
*
Von Mises yield criterion
The maximum distortion criterion (also von Mises yield criterion) states that yielding of a ductile material begins when the second invariant of deviatoric stress J_2 reaches a critical value. It is a part of plasticity theory that mostly applie ...
*
Mohr–Coulomb theory
Mohr–Coulomb theory is a mathematical model (see yield surface) describing the response of brittle materials such as concrete, or rubble piles, to shear stress as well as normal stress. Most of the classical engineering materials follow this r ...
References
{{Reflist
Mechanical failure
Plasticity (physics)
Solid mechanics
Mechanics