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 3. Control volumes in the new approach (a) and in the modified version of the new approach (b)

where I 1,s , I 2,s and I 12,s are the integrals of the angular stress functions (Treifi and Oyadiji, 2013), which depend on the notch opening angle, 2α, the angle defined by the semi -circular sector, γ, and the Poisson's ratio ν. In this case also the contribution of the mixed term ( K 1 ∙ K 2 ) is present because the integration for the strain energy evaluation is not performed on a control volume symmetrical with respect to the notch bisector line (Fig. 2b). Due to the presence of the mixed term it is possible to decouple the contributions of the loading modes, obtaining a solution of the system even in the crack case. Solving the system of equations, the values of the NSIFs can be determined: 1 = √ ( ̅ , + ̅ , ) ± √ ( ̅ , + ̅ , ) 2 − 4 ∙ ∙ ∙ ( ̅ , − ̅ , ) 2 2 4 ∙ (7) 2 = ̅ , − ̅ , 2 ∙ ∙ 1 (8) 2.3. New approach based on the deviatoric SED In the present contribution a new method is proposed. It is based on the evaluation of the total and deviatoric SED averaged in a single control volume (circular sector) centred at the notch tip and characterized by a radius (Fig. 3a). Two independent equations can be obtained: one linked to the total SED and the other to the deviatoric one. In this way it is possible to obtain a solution of the system even in the crack case. As previously, knowing the SED values ( ̅ and ̅ ), by means of a FE analysis, and defining the control radius ( R ), 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 ̅ , = 1 + 3 [ 1 , 2 1 ∙ 1 2 2 ( 1− 1 ) + 2 , 2 2 ∙ 22 2 ( 1− 2 ) ] = ∙ 1 2 + ∙ 22 (9) where 1 , and 2 , are the integrals of the angular stress functions related to the deviatoric strain energy density (Lazzarin and Zambardi, 2001), which depend on the notch opening angle, 2 , and the Poisson's ratio . Solving the system of equations, the values of the NSIFs can be determined: 1 = √ ∙ ̅ , − ∙ ̅ ∙ − ∙ (10) 2 = √ ̅ , − ∙ 1 2 (11)