PSI - Issue 12

Venanzio Giannella et al. / Procedia Structural Integrity 12 (2018) 404–415 V. Giannella/ Structural Integrity Procedia 00 (2018) 000 – 000

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Fig. 5: LC approach description.

3.3. J-integral formulation

An application of the J -integral to 3D crack problems can be found in Rigby et al. (1993, 1998) and its generalization to 3D thermoelastic c rack problems can be found in Dell’Erba et al. (2000) . The J -integral for 3D problems can be defined as: = ∫ (Wn 1 − ij 1 j ) dΓ Γ = = ∫ (Wn 1 − ij 1 j ) dΓ + − ∫ 3 ( 3 1 ) dΩ Ω( ) (3) where Γ is a contour identical to C but proceeding in an anti-clockwise direction (Fig. 6). The integral J is defined in the plane x 3 = 0 for any position on the crack front. Considering a traction free crack, the contour integral over the crack faces ω is zero, whereas, when considering loaded crack faces, as for the LC approach here presented, the contribution of ∫ − ij 1 j dω has to be added.

Figure 6. Closed path around the crack tip (Wilson, 1979).