PSI - Issue 68

Cainã Bemfica et al. / Procedia Structural Integrity 68 (2025) 1188 – 1195 Ludovic Vincent et al. / Structural Integrity Procedia 00 (2025) 000–000

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4. Brittle fracture model The MIBF model has been developed at CEA as a modelling strategy that incorporates the micromechanics of brittle failure within a local approach to fracture framework (see (Forget et al., 2016) for a detailed description of the model). In its original version, two microstructural effects are explicitly incorporated into the model: the effect of the carbide population (since the microcrack at the origin of brittle failure is considered to be due to carbide cracking) and the effect of the local stress heterogeneity inside a representative volume V 0 . In the present work, due to the similarities between crystallographic microstructures, the effect of local stress heterogeneities is not expected to play a major role and is, therefore, neglected. The stress inside V 0 is thus considered homogeneous. The critical event for brittle failure is the propagation of a microcrack into the matrix, which is assumed to take place when a microcrack is greater than a critical size r c . In the original version of the MIBF model, r c is given by a Griffith criterion. To incorporate the influence of the ferritic grain size into the MIBF model, the critical size r c is now supposed to follow the expression proposed by Petch [14]: * = F2 + − (1 − % ) " √ $ 2√2 K + LF2 + − (1 − % ) " √ $ 2√2 K % − (1 − % ) % " % $ % 4 (1 − % ) $ % (6) in which E , ν, γ f , σ 1 and d are the Young’s modulus, Poisson’s ratio, effective surface energy, maximum principal stress and average ferritic grain size respectively. k y is the Hall&Petch material parameter that relates the evolution of yield stress with the inverse of the square root of the grain size. According to (Petch, 1986), 21 N/mm 3/2 is an appropriate value for this type of material. Since microcracks are supposed to propagate from carbides broken by strain incompatibility with the surrounding matrix plastically deformed, the probability P r to find a microcrack of initial radius r greater than r c is identical to the probability to find a carbide greater than r c , which reads: , = ( > * ) = Q d d d -. , ! (7) With dF/dr given by Eq. (5). Within a weakest link framework, the failure probability of the representative volume element V 0 is the probability to find at least one microcrack greater than r c inside V 0 . Assuming a carbide density per volume * = / 0/% , the failure probability of V 0 can be written as 2 " =1−T(1− , ) 3 ! 2 " 45$ (8) Assuming again a weakest link hypothesis, this time at the scale of the complete CT specimen, the failure is assumed to take place whenever a single V 0 fails. The failure probability of the structure can thus analogously be written as 2 # = 1 − TU1 − 2 " V 2 $ (9) in which V p is the combination of all representative volumes in which plastic deformation takes place. 5. Modeling results and discussions Finite elements analysis was carried out to simulated fracture toughness tests using the same methodology presented in (Forget et al., 2016). At the vicinity of the crack tip, 50 x 50 x 50 µm 3 quadratic hexaedral elements were used for a better description of the stress gradients. The constitutive relation, illustrated in Figure 1, is described by