PSI - Issue 39

Arturo Pascuzzo et al. / Procedia Structural Integrity 39 (2022) 649–662 Author name / Structural Integrity Procedia 00 (2019) 000–000

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for plane strain conditions. In the proposed model, in addition to Eq. (9), according to the approach proposed by Yu et al. (Yu et al. (2009)), an extra and compatible strain field ( 0 aux ij ε ) is considered: ( ) 0 0 , , 1 and 2 aux tip aux aux aux aux ij ijkl kl ij i j j i S u u ε σ ε = = + (10) Let consider the classic J -integral expression gives as: ( ) 0 1 ,1 with , 1, 2 i ij j i S J W u n dS i j δ σ = − = ∫ (11) where 0 S is an arbitrary path surrounding the crack tip, 1 2 1 2 ij ij ijkl ij kl W C σ ε ε ε = = is the strain energy density, 1 i δ is the Kronecker delta, i n is the unit outward normal vector to 0 S . From a computational point of view, the curvilinear expression of the J -integral represented in Eq. (11) is usually converted into an Equivalent Domain Integral form. By referring to Figure 3, Eq. (11) is firstly expressed by considering a closed contour 1 0 S S S S S + − = + + − and a weighting function ( ) 1 2 , q x x , which assume values of the unity on 0 S ( i.e. , the inner boundary of S) and zero on 1 S ( i.e. , the outer boundary of S), thus achieving: ( ) ( ) 0 1 ,1 ,1 1 with , 1, 2 i ij j i ij j i i S S J W u n dS u W m qdS i j δ σ σ δ = − = − = ∫ ∫ (12) act aux + (14) in which, the mutual term M is the M -integral expression. In particular, according to Eq. (9), the M -integral expression assumes the following expression: ( ) ( ) ,1 ,1 1 , , 1 ,1 aux aux aux aux aux ij j ij j ij ij i i ij j i ij ij A A M u u q dA u qdA σ σ σ ε δ σ σ ε = + − + − ∫ ∫ (15) Starting from Eq. (15), by introducing the extra strain field defined by Eq. (10), Eq. (15) becomes as follows: ( ) ( ) ,1 ,1 1 , ,1 aux aux aux tip aux ij j ij j ij ij i i ij ijkl ijkl kl A A M u u q dA S S X qdA σ σ σ ε δ σ σ   = + − +  −  ∫ ∫ % (16) which is valid for heterogeneous materials with smoothly continuous properties. By using the relationship between the J -integral and SIFs, one arrives to: ( ) 2 2 act aux act aux I I II II K K K K M E + = ′ (17) where, E E ′ = for plane stress and ( ) 2 1 E E ν ′ = − for plane strain. The SIFs of the actual state ( i.e. , , act act I II K K ) are computed using two interaction integrals, derived by using a pure mode-I and a pure mode-II auxiliary fields, thus achieving: act J J M aux = + + J being i i m n = − on 0 S , and 1 0 m = , 2 1 m = ± on S + and S − . Next, by applying the divergence theorem to Eq. (12), it results: ( ) ( ) ,1 1 , i u W q dA δ − ij j i σ ,1 1 , u W qdA δ − ij j i i σ A A J = + ∫ ∫ (13) The application of the J -integral to the superimposed state leads to the following expression: