# Willam-Warnke yield criterion

Willam-Warnke yield criterion

The Willam-Warnke yield criterion [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.] is a function that is used to predict when failure will occur in concrete and other cohesive-frictional materials such as rock, soil, and ceramics. This yield criterion has the functional form:$f\left(I_1, J_2, J_3\right) = 0 ,$where $I_1$ is the first invariant of the Cauchy stress tensor, and $J_2, J_3$ are the second and third invariants of the deviatoric part of the Cauchy stress tensor. There are three material parameters ($sigma_c$ - the uniaxial compressive strength, $sigma_t$ - the uniaxial tensile strength, $sigma_b$ - the equibiaxial compressive strength) that have to be determined before the Willam-Warnke yield criterion may be applied to predict failure.

In terms of $I_1, J_2, J_3$, the Willam-Warnke yield criterion can be expressed as:$f := sqrt\left\{J_2\right\} + lambda\left(J_2,J_3\right)~\left( frac\left\{I_1\right\}\left\{3\right\} - B\right) = 0$where $lambda$ is a function that depends on $J_2,J_3$ and the three material parameters and $B$ depends only on the material parameters. The function $lambda$ can be interpreted as the friction angle which depends on the Lode angle ($heta$). The quantity $B$ is interpreted as a cohesion pressure. The Willam-Warnke yield criterion may therefore be viewed as a combination of the Mohr-Coulomb and the Drucker-Prager yield criteria.

Willam-Warnke yield function

In the original paper, the three-parameter Willam-Warnke yield function was expressed as:$f := cfrac\left\{1\right\}\left\{3z\right\}~cfrac\left\{I_1\right\}\left\{sigma_c\right\} + sqrt\left\{cfrac\left\{2\right\}\left\{5~cfrac\left\{1\right\}\left\{r\left( heta\right)\right\}cfrac\left\{sqrt\left\{J_2\left\{sigma_c\right\} - 1 le 0$where $I_1$ is the first invariant of the stress tensor, $J_2$ is the second invariant of the deviatoric part of the stress tensor, $sigma_c$ is the yield stress in uniaxial compression, and $heta$ is the Lode angle given by:$heta = frac\left\{1\right\}\left\{3\right\}cos^\left\{-1\right\}left\left(cfrac\left\{3sqrt\left\{3\left\{2\right\}~cfrac\left\{J_3\right\}\left\{J_2^\left\{3/2 ight\right) ~.$The locus of the boundary of the stress surface in the deviatoric stress plane is expressed in polar coordinates by the quantity $r\left( heta\right)$ which is given by :$r\left( heta\right) := cfrac\left\{u\left( heta\right)+v\left( heta\right)\right\}\left\{w\left( heta\right)\right\}$where:The quantities $r_t$ and $r_c$ describe the position vectors at the locations $heta=0^circ, 60^circ$ and can be expressed in terms of $sigma_c, sigma_b, sigma_t$ as:$r_c := sqrt\left\{cfrac\left\{6\right\}\left\{5left \left[cfrac\left\{sigma_bsigma_t\right\}\left\{3sigma_bsigma_t + sigma_c\left(sigma_b - sigma_t\right)\right\} ight\right] ~;~~ r_t := sqrt\left\{cfrac\left\{6\right\}\left\{5left \left[cfrac\left\{sigma_bsigma_t\right\}\left\{sigma_c\left(2sigma_b+sigma_t\right)\right\} ight\right]$The parameter $z$ in the model is given by:$z := cfrac\left\{sigma_bsigma_t\right\}\left\{sigma_c\left(sigma_b-sigma_t\right)\right\} ~.$

The Haigh-Westergaard representation of the Willam-Warnke yield condition can bewritten as:where:

Modified forms of the Willam-Warnke yield criterion

An alternative form of the Willam-Warnke yield criterion in Haigh-Westergaard coordinates is the Ulm-Coussy-Bazant form [ 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.] ::where:and :The quantities $r_c, r_t$ are interpreted as friction coefficients. For the yield surface to be convex, the Willam-Warnke yield criterion requires that $2~r_t ge r_c ge r_t/2$ and $0 le heta le cfrac\left\{pi\right\}\left\{3\right\}$.

References

* Chen, W. F. (1982). Plasticity in Reinforced Concrete. McGraw Hill. New York.

* Yield (engineering)
* Yield surface
* Plasticity (physics)

* 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.

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