page_title: Willam–Warnke yield criterion
section_title: Modified forms of the Willam-Warnke yield criterion
hierarchical_section_title: Willam–Warnke yield criterion / Modified forms of the Willam-Warnke yield criterion
context_page_description: The Willam–Warnke yield criterion 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
  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 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.
context_section_description: An alternative form of the Willam-Warnke yield criterion in Haigh-Westergaard coordinates is the Ulm-Coussy-Bazant form:
  where
  and 
  The quantities  are interpreted as friction coefficients.  For the yield surface to be convex, the Willam-Warnke yield criterion requires that  and .
caption: Ulm-Coussy-Bazant version of the three-parameter Willam-Warnke yield surface in the 
    
      
        
          π
        
      
      {\displaystyle \pi }
    
  -plane for 
    
      
        
          
            σ
            
              c
            
          
          =
          1
          ,
          
            σ
            
              t
            
          
          =
          0.3
          ,
          
            σ
            
              b
            
          
          =
          1.7
        
      
      {\displaystyle \sigma _{c}=1,\sigma _{t}=0.3,\sigma _{b}=1.7}
exif: {}