PSI - Issue 77

344 6 K Y = ∫ YG(A, t) 0 A dt =Y √ Wk ( W A ) [1 − h(A,a)] [7] K ∞ =K Y ⇒ σ ∞ =Y[1 − h(A,a)] Thus, the depth A comes from the function h(A, a) given by Eq.4 for the relative crack size ã = a/W: A= a ã+ ( 1−ã ) cos( π 2 σ Y∞ 1 ( 1−ã ) 2 ) [8] The field  (x) of crack opening displacements on the cohesive crack (Fig. 3a) is the difference between those given by Eq. (2) when K  (t) is respectively particularized for the stress intensity factors K ∞ and K Y , and the crack size a is for A: ω = ω (x)= E 2 ∫ G(t,x) x A ( K ∞ (t) − K Y (t) ) dt = 2W E Y ∫ k 2 ( W t ) A x h 2 (t, x)[h(t, a) − h(A,a)]dt [9] The respective values  M and  t of this field at the positions x = a and x = 0 are the crack opening displacements at the physical crack tip (CTOD) and at the mouth notch (CMOD): CTOD= ω t = 2W E Y ∫ k 2 ( W t ) A a h 2 (t, a)[h(t, a) − h(A,a)]dt [10] CMOD= ω M = 2W E Y ∫ k 2 ( W t ) A 0 h 2 (t, 0)[h(t, a) − h(A,a)]dt [11] Given that the cohesive crack does not entail any stress singularity, the integration to determine the J-integral can be infinitely close to the crack faces, in which case its value is the product Y ω t , and Eq. (10) provides the J/Y quotient. The integrals of Eq. (10) and Eq. (11) have been numerically solved and combined with Eq. (9) to determine the applied remote stress and the value of the J-integral as function of CMOD and of the relative crack size ã = a/W, with the elastic modulus and the cohesive resistance as generic constants. The graphs given in Fig. 4a and Fig. 4b shows these results; the dimensionless values of the J-integral are the same for the dimensionless CTOD ( ω t E)/(WY). P. Santos et al. / Procedia Structural Integrity 77 (2026) 339–347 P. Santos et al. / Structural Integrity Procedia 00 (2025) 000 – 000