The fracture strength (or micro-crack nucleation stress) of a  material was first theoretically estimated by Alan Arnold Griffith in 1921:

  
    
      
        
          σ
          
            
              t
              h
              e
              o
              r
              e
              t
              i
              c
              a
              l
            
          
        
        =
        
          
            
              
                E
                γ
              
              
                r
                
                  o
                
              
            
          
        
      
    
    {\displaystyle \sigma _{\mathrm {theoretical} }={\sqrt {\frac {E\gamma }{r_{o}}}}}
  where: –

  
    
      
        E
      
    
    {\displaystyle E}
   is the Youngs modulus of the material,

  
    
      
        γ
      
    
    {\displaystyle \gamma }
   is the surface energy, and

  
    
      
        
          r
          
            o
          
        
      
    
    {\displaystyle r_{o}}
   is the micro-crack length (or equilibrium distance between atomic centers in a crystalline solid).On the other hand, a crack introduces a stress concentration modeled by

  
    
      
        
          σ
          
            
              e
              l
              l
              i
              p
              t
              i
              c
              a
              l
               
              c
              r
              a
              c
              k
            
          
        
        =
        
          σ
          
            
              a
              p
              p
              l
              i
              e
              d
            
          
        
        
          (
          
            1
            +
            2
            
              
                
                  a
                  ρ
                
              
            
          
          )
        
        =
        2
        
          σ
          
            
              a
              p
              p
              l
              i
              e
              d
            
          
        
        
          
            
              a
              ρ
            
          
        
      
    
    {\displaystyle \sigma _{\mathrm {elliptical\ crack} }=\sigma _{\mathrm {applied} }\left(1+2{\sqrt {\frac {a}{\rho }}}\right)=2\sigma _{\mathrm {applied} }{\sqrt {\frac {a}{\rho }}}}
   (For sharp cracks)where: –

  
    
      
        
          σ
          
            
              a
              p
              p
              l
              i
              e
              d
            
          
        
      
    
    {\displaystyle \sigma _{\mathrm {applied} }}
   is the loading stress,

  
    
      
        a
      
    
    {\displaystyle a}
   is half the length of the crack, and

  
    
      
        ρ
      
    
    {\displaystyle \rho }
   is the radius of curvature at the crack tip.Putting these two equations together gets

  
    
      
        
          σ
          
            
              f
              r
              a
              c
              t
              u
              r
              e
            
          
        
        =
        
          
            
              
                E
                γ
                ρ
              
              
                4
                a
                
                  r
                  
                    o
                  
                
              
            
          
        
        .
This is done by taking the test piece with its fabricated notch of length 
  
    
      
        
          c
          ′
        
      
    
    {\textstyle \mathrm {c\prime } }
   and sharpening this notch to better emulate a crack tip found in real-world materials. Cyclical prestressing the sample can then induce a fatigue crack which extends the crack from the fabricated notch length of 
  
    
      
        
          c
          ′
        
      
    
    {\textstyle \mathrm {c\prime } }
    to 
  
    
      
        
          c
        
      
    
    {\textstyle \mathrm {c} }
  .