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:$f(I\_1,\; J\_2,\; J\_3)\; =\; 0\; ,$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:$f\; :=\; sqrt\{J\_2\}\; +\; lambda(J\_2,J\_3)~(\; frac\{I\_1\}\{3\}\; -\; B)\; =\; 0$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:$f\; :=\; cfrac\{1\}\{3z\}~cfrac\{I\_1\}\{sigma\_c\}\; +\; sqrt\{cfrac\{2\}\{5~cfrac\{1\}\{r(\; heta)\}cfrac\{sqrt\{J\_2\{sigma\_c\}\; -\; 1\; le\; 0$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:$heta\; =\; frac\{1\}\{3\}cos^\{-1\}left(cfrac\{3sqrt\{3\{2\}~cfrac\{J\_3\}\{J\_2^\{3/2\; ight)\; ~.$The locus of the boundary of the stress surface in the deviatoric stress plane is expressed in polar coordinates by the quantity $r(\; heta)$ which is given by :$r(\; heta)\; :=\; cfrac\{u(\; heta)+v(\; heta)\}\{w(\; heta)\}$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:$r\_c\; :=\; sqrt\{cfrac\{6\}\{5left\; [cfrac\{sigma\_bsigma\_t\}\{3sigma\_bsigma\_t\; +\; sigma\_c(sigma\_b\; -\; sigma\_t)\}\; ight]\; ~;~~\; r\_t\; :=\; sqrt\{cfrac\{6\}\{5left\; [cfrac\{sigma\_bsigma\_t\}\{sigma\_c(2sigma\_b+sigma\_t)\}\; ight]$The parameter $z$ in the model is given by:$z\; :=\; cfrac\{sigma\_bsigma\_t\}\{sigma\_c(sigma\_b-sigma\_t)\}\; ~.$

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 $0\; le\; heta\; le\; cfrac\{pi\}\{3\}$.

References

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

See also

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

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.