[<< wikibooks] Advanced Structural Analysis/Part I - Theory/Failure Modes/Fatigue/Crack Initiation/The Wöhler Curve
The Wöhler curve, also referred to as the S-N curve, describes the function 
  
    
      
        
          σ
          
            a
          
        
        (
        
          N
          
            f
          
        
        )
      
    
    {\displaystyle \sigma _{a}(N_{f})}
   or 
  
    
      
        
          σ
          
            m
          
        
        (
        
          N
          
            f
          
        
        )
      
    
    {\displaystyle \sigma _{m}(N_{f})}
  . It is based on empirical results and often represents the median of the data scatter.
A significant interval of the curve can be approximated by the Basquin relation

  
    
      
        
          σ
          
            a
          
          
            m
          
        
        N
        =
        C
      
    
    {\displaystyle \sigma _{a}^{m}N=C}
  
Where 
  
    
      
        C
      
    
    {\displaystyle C}
   is a constant specific to the test case.
The Basquin relation is often presented in the form

  
    
      
        Δ
        σ
        =
        Δ
        
          σ
          
            C
          
        
        (
        
          
            
              N
              
                C
              
            
            N
          
        
        
          )
          
            
              1
              m
            
          
        
      
    
    {\displaystyle \Delta \sigma =\Delta \sigma _{C}({\frac {N_{C}}{N}})^{\frac {1}{m}}}
  
where

  
    
      
        
          N
          
            C
          
        
        =
        2
        ⋅
        
          10
          
            6
          
        
      
    
    {\displaystyle N_{C}=2\cdot 10^{6}}
   cycles

  
    
      
        Δ
        
          σ
          
            C
          
        
        =
      
    
    {\displaystyle \Delta \sigma _{C}=}
   is the so called detail category number.


== Detail Category Number ==
The fatigue detail number defines the Basquin relation and specifies a Wöhler curve. The property is often denoted FAT, C, or in mathematical expressions: 
  
    
      
        Δ
        
          σ
          
            C
          
        
      
    
    {\displaystyle \Delta \sigma _{C}}
  .
If there are more than one FAT tied to a geometry-type, they refer to different specifics such as weld quality, loading direction etc. Moreover, the FAT normally corresponds to a fatigue life of 
  
    
      
        N
        =
        2
        ⋅
        
          10
          
            6
          
        
      
    
    {\displaystyle N=2\cdot 10^{6}}
   cycles.


== Statistical Properties ==