PSI - Issue 47

Domenico Ammendolea et al. / Procedia Structural Integrity 47 (2023) 488–502 Author name / Structural Integrity Procedia 00 (2019) 000–000

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where x and  x represent the position vector of the mesh nodes in the current configuration of the computational domain and the associated test function, while λ is the LMM vector. 2.3. ALE formulation of the M-integral The proposed method extracts the Stress Intensity Factors (SIFs) at the crack front using the M -integral method initially proposed by Yau and coauthors ((Yau et al., 1980)). The M -integral expression is derived starting from the application of the classical form of the J -integral to a superimposed state, formed by the addition to the stress, strain, and displacement fields of the problem under investigation ( , , act act act u ε σ ), commonly denoted as “actual state”, those associated with the analytic solutions of the fracture problem of a straight crack in infinity and remotely loaded plate (next denoted as , , aux aux aux u ε σ ), and referred to as “auxiliary states”. Typically, the auxiliary states are expressed through the asymptotic analytic solutions for homogeneous materials proposed in (Williams, 1957). The Equivalent Domain Integral (EDI) form of the M -integral can be expressed as follows (see (Ammendolea et al., 2020, 2023) for additional details):     ,1 ,1 1 , with , , 1,2 act aux aux act act aux ij j ij j jk jk i i A M u u q dA i j k               (9)

Figure 3. On the Equivalent Domain Integral form of the M-integral and a schematic of the arbitrary function 1 2 ( , ) C C q X X .

1 2 ( , ) C C q X X represents a scalar weighting function that is zero on the

in which 1 i  is the Kronecker operator and

contour on the domain integral area and it assumes value one at the crack tip (see Figure 3). Because the use of the ALE formulation, the actual fields must be expressed with respect to the Referential frame R  . Then, by means of Eq. (2), the ALE formulation of the M -integral assumes the following expression:                         , ,1 , , , , , , , , ,1 , , , 1 , , , , , R R R M C M C M C M C R aux aux R ijkl k h h n j h h n A R ij j h h n h h n A R n l n i n n i A A M C M C R aux jkpq p h h n jk i h h n A R n q n i A M C u X u q X J dA u X q X J dA C u X q X J dA                                   (10) in which , , , ( ) ( ) , ( ) ( ) , ( ) ( ) R M C C i i i i i i X X              . Besides, R A represents the domain interaction area projected in the referential system and A J is the Jacobian of the transformation. The SIFs of the actual state can be evaluated through the following expression that relate the M -integral and mixed-mode SIFs ( , I II K K ):   2 2 act aux act aux I I II II K K K K M E    (11) where, E E   for plane stress and 2 (1 ) E E    for plane strain conditions. In particular, in order to evaluate the