PSI - Issue 2_B

L. Pittarello et al. / Procedia Structural Integrity 2 (2016) 1829–1836 Author name / Structural Integrity Procedia 00 (2016) 000 – 000

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Figure 2. Control volumes in the Lazzarin et al. approach (a) and in the Treifi and Oyadiji approach (b)

2. Approximate methods

2.1. Lazzarin et al. approach The first method has been proposed by Lazzarin et al. (Lazzarin et al., 2010) and it is based on the evaluation of the averaged SED on two different control volumes (circular sectors) centred at the notch tip and characterized by the radii and (Fig. 2a). Known the SED values ( ̅ and ̅ ), by means of a FE analysis, and defined the control radii ( and ), it is possible to obtain a system of two equations in two unknowns ( 1 and 2 ): { ̅ , = 2 1 [ 1 2 1 ∙ 1 2 2 ( 1− 1 ) + 2 2 2 ∙ 22 2 ( 1− 2 ) ] = ∙ 1 2 + ∙ 22 ̅ , = 2 1 [ 1 2 1 ∙ 1 2 2 ( 1− 1 ) + 2 2 2 ∙ 22 2 ( 1− 2 ) ] = ∙ 1 2 + ∙ 22 (3) where is the Young’s modulus of the material while 1 and 2 are the integrals of the angular stress functions (Lazzarin and Zambardi, 2001), which depend on the notch opening angle, 2 = 2 − 2 , and the Poisson's ratio . This method cannot be applied to a crack subjected to mixed mode loading, since an indeterminate system of equations would be obtained. Solving the system of equations, the values of the NSIFs can be determined: 1 = √ ∙ ̅ , − ∙ ̅ , ∙ − ∙ (4) 2 = √ ̅ , − ∙ 1 2 (5) 2.2. Treifi and Oyadiji approach The second method has been proposed by Treifi and Oyadiji (Treifi and Oyadiji, 2013) and it is based on the evaluation of the averaged SED on two different control volumes (semi-circular sectors with a central angle equal to ) centred at the notch tip and characterized by a radius (Fig. 2b). Known the SED values ( ̅ and ̅ ) by means of a FE analysis, and defined the control radius ( ), it is possible to obtain a system of two equations in two unknowns ( 1 and 2 ): { ̅ , = 2 1 [ 1 , 1 ∙ 1 2 2 ( 1− 1 ) + 2 , 2 ∙ 22 2 ( 1− 2 ) + 2 ∙ 12 , ( 1 + 2 ) ∙ 1 ∙ 2 2− 1 − 2 ] = ∙ 1 2 + ∙ 22 + ∙ 1 ∙ 2 ̅ , = 2 1 [ 1 , 1 ∙ 1 2 2 ( 1− 1 ) + 2 , 2 ∙ 22 2 ( 1− 2 ) − 2 ∙ 12 , ( 1 + 2 ) ∙ 1 ∙ 2 2− 1 − 2 ] = ∙ 1 2 + ∙ 22 − ∙ 1 ∙ 2 (6)