PSI - Issue 2_B

Pasquale Gallo et al. / Procedia Structural Integrity 2 (2016) 809–816 P. Gallo et al. / Structural Integrity Procedia 00 (2016) 000–000

812

4

The Lazzarin-Tovo equations, in the presence of a traction loading, along the bisector (x axis), can be expressed as follows, as a function of the maximum stress (see Fig. 1):

     

      

1 1   

0         r   r r r

















1 1       

3

1   

   

 

  

1

1

1

1

1

1

1

1 



   

0   r   r

r    

(1)

max

     

4

1 1   

















3

1       

3    

1   

 



1

1

1

1

1

1

0 Where ߪ ௠௔௫ can be expressed as a function of stress concentration factor ܭ ௧ (evaluated through linear elastic finite element analysis) and the applied load ߪ ௡௢௠ , max t nom K    (2) Employing the more general conformal mapping of Neuber (1958) that permit a unified analysis of sharp and blunt notches, the notch radius, ρ, and the origin of the coordinate system, r 0 , are related by the following equation on the basis of trigonometric considerations: . The main steps to extend the method to blunt V-Notches can be summarised as follows:  Assumption of Lazzarin-Tovo equations to describe the stress distribution ahead the notch tip instead of Creager-Paris equations;  Calculation of the origin of the coordinate system, r 0 , as a function of the opening angle and notch radius, as described by Eq. (3);  Re-definition of the plastic zone correction factor ܥ ௣ that is a function of plastic zone size ݎ ௣ and plastic zone increment ȟ ݎ ௣ ; The definition of the parameters ܥ ௣ , ݎ ௣ and ȟ ݎ ௣ is very similar to that clearly reported by Glinka (1985), except for the assumption of different elastic stress distribution equations. Definition of these variables is briefly reported hereafter. Referring to Fig. 2, considering the Von Mises (1913) yield criterion in polar coordinate: (4) and introducing Eqs. (1) into Eq. (4), a first approximation of ݎ ௣ that can be solved numerically is obtained. Once ݎ ௣ is known, the force F 1 can be evaluated as follows: ܨ ଵ ൌ න ߪ ఏ ݀ ݎ ௥ ೛ ௥ బ െ ߪ ఏ ൫ ݎ ௣ ൯ ή ൫ ݎ ௣ െ ݎ ଴ ൯ ൌ ௄ ೟ ఙ ೙೚೘ ସ ቐ൫ ݎ ଴ െ ݎ ௣ ൯ ቀ ௥ ೛ ௥ బ ቁ ఒ భ ିଵ ൤ሺ ߣ ଵ െ ͳሻ ൅ ߯ ଵ ሺͳ െ ߣ ଵ ሻ ൤ͳ െ ቀ ௥ ೛ ௥ బ ቁ ఓ భ ିఒ భ ൨ ൅ ሺ͵ െ ߣ ଵ ሻ ቀ ௥ ೛ ௥ బ ቁ ఓ భ ିఒ భ ൨ െ ሾሺఒ భ ାଵሻାఞ భ ሺଵିఒ భ ሻሿቈ௥ బ ି௥ ೛ ቀ ೝ ೝ ೛ బ ቁ ഊ భ షభ ቉ ఒ భ ൅ ሾఞ భ ሺଵିఒ భ ሻିሺଷିఒ భ ሻሿ൤௥ బ ି௥ ೛ ቀ ೝ ೝ ೛ బ ቁ ഋ భ షభ ൨ ఓ భ ቑ (5) 0 1 q r q     (3) where 2 2 q      2 2 ys           r r