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[[Image:Willam Warnke Yield Surface 3Db.png|300px|right|thumb|Three-parameter Willam-Warnke yield surface.]]
The '''Willam-Warnke [[yield (engineering)|yield criterion]]''' <ref>Willam, K. J. and Warnke, E. P. (1975). ''Constitutive models for the triaxial behavior of concrete.'' Proceedings of the International Assoc. for Bridge and Structural Engineering , vol 19, pp. 1- 30.</ref> is a function that is used to predict when failure will occur in [[concrete]] and other cohesive-frictional materials such as [[rock (geology)|rock]], [[soil]], and [[ceramic]]s. This yield criterion has the functional form
:<math>
  f(I_1, J_2, J_3) = 0 \,
</math>
where <math>I_1</math> is the first invariant of the Cauchy stress tensor, and <math>J_2, J_3</math> are the second and third invariants of the deviatoric part of the Cauchy stress tensor. There are three material parameters (<math>\sigma_c</math> - the uniaxial compressive strength, <math>\sigma_t</math> - the uniaxial tensile strength, <math>\sigma_b</math> - the equibiaxial compressive strength) that have to be determined before the Willam-Warnke yield criterion may be applied to predict failure.
 
In terms of <math>I_1, J_2, J_3</math>, the Willam-Warnke yield criterion can be expressed as
:<math>
  f := \sqrt{J_2} + \lambda(J_2,J_3)~(\tfrac{I_1}{3} - B) = 0
</math>
where <math>\lambda</math> is a function that depends on <math>J_2,J_3</math> and the three material parameters and <math>B</math> depends only on the material parameters. The function <math>\lambda</math> can be interpreted as the friction angle which depends on the Lode angle (<math>\theta</math>). The quantity <math>B</math> is interpreted as a cohesion pressure.  The Willam-Warnke yield criterion may therefore be viewed as a combination of the [[Mohr-Coulomb theory|Mohr-Coulomb]] and the [[Drucker Prager|Drucker-Prager]] yield criteria.
 
== Willam-Warnke yield function ==
[[Image:Willam Warnke Yield Surface 3Da.png|300px|right|thumb|View of three-parameter Willam-Warnke yield surface in 3D space of principal stresses for <math>\sigma_c=1, \sigma_t=0.3, \sigma_b=1.7</math>]]
[[Image:Willam Warnke Yield Surface sig1sig2.png|300px|right|thumb|Trace of the three-parameter Willam-Warnke yield surface in the <math>\sigma_1-\sigma_2</math>-plane for <math>\sigma_c=1, \sigma_t=0.3, \sigma_b=1.7</math>]]
In the original paper, the three-parameter Willam-Warnke yield function was expressed as
:<math>
  f := \cfrac{1}{3z}~\cfrac{I_1}{\sigma_c} + \sqrt{\cfrac{2}{5}}~\cfrac{1}{r(\theta)}\cfrac{\sqrt{J_2}}{\sigma_c} - 1 \le 0
</math>
where <math>I_1</math> is the first invariant of the stress tensor, <math>J_2</math> is the second invariant of the deviatoric part of the stress tensor, <math>\sigma_c</math> is the yield stress in uniaxial compression, and <math>\theta</math> is the Lode angle given by
:<math>
  \theta = \tfrac{1}{3}\cos^{-1}\left(\cfrac{3\sqrt{3}}{2}~\cfrac{J_3}{J_2^{3/2}}\right) ~.
</math>
The locus of the boundary of the stress surface in the deviatoric stress plane is expressed in polar coordinates by the quantity <math>r(\theta)</math> which is given by
:<math>
  r(\theta) := \cfrac{u(\theta)+v(\theta)}{w(\theta)}
</math>
where
:<math>
  \begin{align}
    u(\theta) := & 2~r_c~(r_c^2-r_t^2)~\cos\theta \\
    v(\theta) := & r_c~(2~r_t - r_c)\sqrt{4~(r_c^2 - r_t^2)~\cos^2\theta + 5~r_t^2 - 4~r_t~r_c} \\
    w(\theta) := & 4(r_c^2 - r_t^2)\cos^2\theta + (r_c-2~r_t)^2
  \end{align}
</math>
 
The quantities <math>r_t</math> and <math>r_c</math> describe the position vectors at the locations <math>\theta=0^\circ, 60^\circ</math> and can be expressed in terms of <math>\sigma_c, \sigma_b, \sigma_t</math> as
:<math>
  r_c :=  \sqrt{\cfrac{6}{5}}\left[\cfrac{\sigma_b\sigma_t}{3\sigma_b\sigma_t + \sigma_c(\sigma_b - \sigma_t)}\right] ~;~~
  r_t :=  \sqrt{\cfrac{6}{5}}\left[\cfrac{\sigma_b\sigma_t}{\sigma_c(2\sigma_b+\sigma_t)}\right]
</math>
The parameter <math>z</math> in the model is given by
:<math>
  z :=  \cfrac{\sigma_b\sigma_t}{\sigma_c(\sigma_b-\sigma_t)} ~.
</math>
 
The [[Yield surface|Haigh-Westergaard representation]] of the Willam-Warnke yield condition can be
written as
:<math>
  f(\xi, \rho, \theta) = 0 \, \quad \equiv \quad
  f := \bar{\lambda}(\theta)~\rho + \bar{B}~\xi - \sigma_c \le 0
</math>
where
:<math>
  \bar{B} := \cfrac{1}{\sqrt{3}~z} ~;~~ \bar{\lambda} := \cfrac{1}{\sqrt{5}~r(\theta)} ~.
</math>
<!--
{| border="0"
|-
| valign="bottom"|
[[Image:Willam Warnke Yield Surface 3Da.png|240px|none|thumb|Figure 1: View of three-parameter Willam-Warnke yield surface in 3D space of principal stresses for <math>\sigma_c=1, \sigma_t=0.3, \sigma_b=1.7</math>]]
| valign="bottom"|
[[Image:Willam Warnke Yield Surface 3Db.png|250px|none|thumb|Figure 2: Three-parameter Willam-Warnke yield surface in the <math>\pi</math>-plane for <math>\sigma_c=1, \sigma_t=0.3, \sigma_b=1.7</math>]]
| valign="bottom"|
[[Image:Willam Warnke Yield Surface sig1sig2.png|240px|none|thumb|Figure 3: Trace of the three-parameter Willam-Warnke yield surface in the <math>\sigma_1-\sigma_2</math>-plane for <math>\sigma_c=1, \sigma_t=0.3, \sigma_b=1.7</math>]]
|-
|}
-->
 
== Modified forms of the Willam-Warnke yield criterion ==
[[Image:UCB Willam Warnke Yield Surface 3Db.png|300px|right|thumb|Ulm-Coussy-Bazant version of the three-parameter Willam-Warnke yield surface in the <math>\pi</math>-plane for <math>\sigma_c=1, \sigma_t=0.3, \sigma_b=1.7</math>]]
An alternative form of the Willam-Warnke yield criterion in [[Yield surface|Haigh-Westergaard coordinates]] is the Ulm-Coussy-Bazant form:<ref>Ulm, F-J., Coussy, O., Bazant, Z. (1999) ''The ‘‘Chunnel’’ Fire. I: Chemoplastic softening in rapidly heated concrete.'' ASCE Journal of Engineering Mechanics, vol. 125, no. 3, pp. 272-282.</ref>
:<math>
  f(\xi, \rho, \theta) = 0 \, \quad \text{or} \quad
  f := \rho + \bar{\lambda}(\theta)~\left(\xi - \bar{B}\right) = 0
</math>
where
:<math>
  \bar{\lambda} :=  \sqrt{\tfrac{2}{3}}~\cfrac{u(\theta)+v(\theta)}{w(\theta)} ~;~~
  \bar{B} :=  \tfrac{1}{\sqrt{3}}~\left[\cfrac{\sigma_b\sigma_t}{\sigma_b-\sigma_t}\right]
</math>
and
:<math>
  \begin{align}
    r_t := & \cfrac{\sqrt{3}~(\sigma_b-\sigma_t)}{2\sigma_b-\sigma_t} \\
    r_c := & \cfrac{\sqrt{3}~\sigma_c~(\sigma_b-\sigma_t)}{(\sigma_c+\sigma_t)\sigma_b-\sigma_c\sigma_t}
  \end{align}
</math>
The quantities <math>r_c, r_t</math> are interpreted as friction coefficients. For the yield surface to be convex, the Willam-Warnke yield criterion requires that <math>2~r_t \ge r_c \ge r_t/2</math> and <math>0 \le \theta \le \cfrac{\pi}{3}</math>.
{| border="0"
|-
| valign="bottom"|
[[Image:UCB Willam Warnke Yield Surface 3Da.png|340px|none|thumb|View of Ulm-Coussy-Bazant version of the three-parameter Willam-Warnke yield surface in 3D space of principal stresses for <math>\sigma_c=1, \sigma_t=0.3, \sigma_b=1.7</math>]]
| valign="bottom"|
[[Image:UCB Willam Warnke Yield Surface sig1sig2.png|240px|none|thumb|Trace of the Ulm-Coussy-Bazant version of the three-parameter Willam-Warnke yield surface in the <math>\sigma_1-\sigma_2</math>-plane for <math>\sigma_c=1, \sigma_t=0.3, \sigma_b=1.7</math>]]
|-
|}
 
== See also ==
* [[Yield (engineering)]]
* [[Yield surface]]
* [[Plasticity (physics)]]
 
== References ==
<references/>
* Chen, W. F. (1982). '''Plasticity in Reinforced Concrete'''. McGraw Hill. New York.
 
== External links ==
* Kaspar Willam and E.P. Warnke (1974). [http://bechtel.colorado.edu/~willam/constitutivemodel.pdf Constitutive model for the triaxial behavior of concrete]
* Palko, J. L. (1993). [http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19930017817_1993017817.pdf Interactive reliability model for whisker-toughened ceramics]
* [http://www.civil.northwestern.edu/people/bazant/PDFs/Papers/379.pdf The ‘‘Chunnel’’ Fire. I: Chemoplastic softening in rapidly heated concrete] by Franz-Josef Ulm, Olivier Coussy, and Zdeneˇk P. Bazˇant.
 
[[Category:Plasticity]]
[[Category:Solid mechanics]]
[[Category:Yield criteria]]

Latest revision as of 15:49, 23 November 2014

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