Willam–Warnke yield criterion

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

Three-parameter Willam-Warnke yield surface.

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

In terms of , the Willam-Warnke yield criterion can be expressed as

where is a function that depends on and the three material parameters and depends only on the material parameters. The function can be interpreted as the friction angle which depends on the Lode angle (). The quantity 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

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View of three-parameter Willam-Warnke yield surface in 3D space of principal stresses for  
 
Trace of the three-parameter Willam-Warnke yield surface in the  -plane for  

In the original paper, the three-parameter Willam-Warnke yield function was expressed as

 

where   is the first invariant of the stress tensor,   is the second invariant of the deviatoric part of the stress tensor,   is the yield stress in uniaxial compression, and   is the Lode angle given by

 

The locus of the boundary of the stress surface in the deviatoric stress plane is expressed in polar coordinates by the quantity   which is given by

 

where

 

The quantities   and   describe the position vectors at the locations   and can be expressed in terms of   as (here   is the failure stress under equi-biaxial compression and   is the failure stress under uniaxial tension)

 

The parameter   in the model is given by

 

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

 

where

 

Modified forms of the Willam-Warnke yield criterion

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Ulm-Coussy-Bazant version of the three-parameter Willam-Warnke yield surface in the  -plane for  

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

 

where

 

and

 

The quantities   are interpreted as friction coefficients. For the yield surface to be convex, the Willam-Warnke yield criterion requires that   and  .

 
View of Ulm-Coussy-Bazant version of the three-parameter Willam-Warnke yield surface in 3D space of principal stresses for  
 
Trace of the Ulm-Coussy-Bazant version of the three-parameter Willam-Warnke yield surface in the  -plane for  

See also

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References

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