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(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: egin{align} u( heta) := & 2~r_c~(r_c^2-r_t^2)~cos heta \ v( heta) := & r_c~(2~r_t - r_c)sqrt{4~(r_c^2 - r_t^2)~cos^2 heta + 5~r_t^2 - 4~r_t~r_c} \ w( heta) := & 4(r_c^2 - r_t^2)cos^2 heta + (r_c-2~r_t)^2 end{align} 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: f(xi, ho, heta) = 0 , quad equiv quad f := ar{lambda}( heta)~ ho + ar{B}~xi - sigma_c le 0 where: ar{B} := cfrac{1}{sqrt{3}~z} ~;~~ ar{lambda} := cfrac{1}{sqrt{5}~r( heta)} ~.

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.] :: f(xi, ho, heta) = 0 , quad ext{or} quad f := ho + ar{lambda}( heta)~left(xi - ar{B} ight) = 0 where: ar{lambda} := sqrt{ frac{2}{3~cfrac{u( heta)+v( heta)}{w( heta)} ~;~~ ar{B} := frac{1}{sqrt{3~left [cfrac{sigma_bsigma_t}{sigma_b-sigma_t} ight] and : egin{align} r_t := & cfrac{sqrt{3}~(sigma_b-sigma_t)}{2sigma_b-sigma_t} \ r_c := & cfrac{sqrt{3}~sigma_c~(sigma_b-sigma_t)}{(sigma_c+sigma_t)sigma_b-sigma_csigma_t} end{align} 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.


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