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}.


* 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.

Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • Yield surface — A yield surface is a five dimensional surface in the six dimensional space of stresses. The state of stress of inside the yield surface is elastic. When the stress state lies on the surface the material is said to have reached its yield point and …   Wikipedia

  • Failure theory (material) — v · d · e Materials failure modes Buckling · Corro …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”