Willam Warnke yield criterion

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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 I1 is the first invariant of the Cauchy stress tensor, and J2,J3 are the second and third invariants of the deviatoric part of the Cauchy stress tensor. There are three material parameters (σc - the uniaxial compressive strength, σt - the uniaxial tensile strength, σb - the equibiaxial compressive strength) that have to be determined before the Willam-Warnke yield criterion may be applied to predict failure.

In terms of I1,J2,J3, 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 λ is a function that depends on J2,J3 and the three material parameters and B depends only on the material parameters. The function λ can be interpreted as the friction angle which depends on the Lode angle (θ). 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.

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[edit] 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(\theta)}\cfrac{\sqrt{J_2}}{\sigma_c} - 1 \le 0

where I1 is the first invariant of the stress tensor, J2 is the second invariant of the deviatoric part of the stress tensor, σc is the yield stress in uniaxial compression, and θ 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(θ) 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 rt and rc describe the position vectors at the locations \theta=0^\circ, 60^\circ and can be expressed in terms of σc,σb,σt 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)} ~.
Figure 1: View of three-parameter Willam-Warnke yield surface in 3D space of principal stresses for σc = 1,σt = 0.3,σb = 1.7
Figure 1: View of three-parameter Willam-Warnke yield surface in 3D space of principal stresses for σc = 1,σt = 0.3,σb = 1.7
Figure 2: Three-parameter Willam-Warnke yield surface in the π-plane for σc = 1,σt = 0.3,σb = 1.7
Figure 2: Three-parameter Willam-Warnke yield surface in the π-plane for σc = 1,σt = 0.3,σb = 1.7
Figure 3: Trace of the three-parameter Willam-Warnke yield surface in the σ1 − σ2-plane for σc = 1,σt = 0.3,σb = 1.7
Figure 3: Trace of the three-parameter Willam-Warnke yield surface in the σ1 − σ2-plane for σc = 1,σt = 0.3,σb = 1.7

[edit] 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 [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 rc,rt 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}.

Figure 4: View of Ulm-Coussy-Bazant version of the three-parameter Willam-Warnke yield surface in 3D space of principal stresses for σc = 1,σt = 0.3,σb = 1.7
Figure 4: View of Ulm-Coussy-Bazant version of the three-parameter Willam-Warnke yield surface in 3D space of principal stresses for σc = 1,σt = 0.3,σb = 1.7
Figure 5: Ulm-Coussy-Bazant version of the three-parameter Willam-Warnke yield surface in the π-plane for σc = 1,σt = 0.3,σb = 1.7
Figure 5: Ulm-Coussy-Bazant version of the three-parameter Willam-Warnke yield surface in the π-plane for σc = 1,σt = 0.3,σb = 1.7
Figure 6: Trace of the Ulm-Coussy-Bazant version of the three-parameter Willam-Warnke yield surface in the σ1 − σ2-plane for σc = 1,σt = 0.3,σb = 1.7
Figure 6: Trace of the Ulm-Coussy-Bazant version of the three-parameter Willam-Warnke yield surface in the σ1 − σ2-plane for σc = 1,σt = 0.3,σb = 1.7

[edit] 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 Enginering Mechanics, vol. 125, no. 3, pp. 272-282.
  • Chen, W. F. (1982). Plasticity in Reinforced Concrete. McGraw Hill. New York.

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