Willam-Warnke yield criterion

Three-parameter Willam-Warnke yield surface.

The Willam-Warnke yield criterion [1] 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)~(\tfrac{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 (\theta). 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

View of three-parameter Willam-Warnke yield surface in 3D space of principal stresses for \sigma_c=1, \sigma_t=0.3, \sigma_b=1.7
Trace of the three-parameter Willam-Warnke yield surface in the \sigma_1-\sigma_2-plane for \sigma_c=1, \sigma_t=0.3, \sigma_b=1.7

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(\theta)}\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 \theta is the Lode angle given by

\theta = \tfrac{1}{3}\cos^{-1}\left(\cfrac{3\sqrt{3}}{2}~\cfrac{J_3}{J_2^{3/2}}\right) ~.

The locus of the boundary of the stress surface in the deviatoric stress plane is expressed in polar coordinates by the quantity r(\theta) which is given by

r(\theta) := \cfrac{u(\theta)+v(\theta)}{w(\theta)}

where

\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}

The quantities r_t and r_c describe the position vectors at the locations \theta=0^\circ, 60^\circ and can be expressed in terms of \sigma_c, \sigma_b, \sigma_t as

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]

The parameter z in the model is given by

z := \cfrac{\sigma_b\sigma_t}{\sigma_c(\sigma_b-\sigma_t)} ~.

The Haigh-Westergaard representation of the Willam-Warnke yield condition can be written as

f(\xi, \rho, \theta) = 0 \, \quad \equiv \quad f := \bar{\lambda}(\theta)~\rho + \bar{B}~\xi - \sigma_c \le 0

where

\bar{B} := \cfrac{1}{\sqrt{3}~z} ~;~~ \bar{\lambda} := \cfrac{1}{\sqrt{5}~r(\theta)} ~.

Modified forms of the Willam-Warnke yield criterion

Ulm-Coussy-Bazant version of the three-parameter Willam-Warnke yield surface in the \pi-plane for \sigma_c=1, \sigma_t=0.3, \sigma_b=1.7

An alternative form of the Willam-Warnke yield criterion in Haigh-Westergaard coordinates is the Ulm-Coussy-Bazant form:[2]

f(\xi, \rho, \theta) = 0 \, \quad \text{or} \quad f := \rho + \bar{\lambda}(\theta)~\left(\xi - \bar{B}\right) = 0

where

\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]

and

\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}

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 \theta \le \cfrac{\pi}{3}.

View of Ulm-Coussy-Bazant version of the three-parameter Willam-Warnke yield surface in 3D space of principal stresses for \sigma_c=1, \sigma_t=0.3, \sigma_b=1.7
Trace of the Ulm-Coussy-Bazant version of the three-parameter Willam-Warnke yield surface in the \sigma_1-\sigma_2-plane for \sigma_c=1, \sigma_t=0.3, \sigma_b=1.7

See also

References

  1. ↑ 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.
  2. ↑ 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.

External links

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