PSI - Issue 79

Umberto De Maio et al. / Procedia Structural Integrity 79 (2026) 386–393

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2. Numerical modeling The numerical framework presented is built upon a synergistic coupling of an Arbitrary Lagrangian-Eulerian (ALE) formulation and an adaptive cohesive interface strategy. This procedure simulates fracture propagation. When a crack initiation criterion is met, the ALE-based moving mesh capability is activated to reorient a segment of the mesh, aligning an element boundary with the predicted fracture direction. Immediately following this mesh adjustment, a zero-thickness cohesive element, governed by a traction-separation law, is inserted along this newly aligned boundary element to model the nonlinear failure process. Given the ALE kinematics, the variational formulation is established in the referential (mesh) configuration rather than the standard material configuration. This necessitates a transformation of the spatial derivatives within the weak form of the boundary value problem (BVP). By employing the Jacobian matrix of the transformation (Donea et al., 2004; Amini and Shahani, 2013; Ponthot and Belytschko, 1998), the quasi-static equilibrium problem is expressed as: ( ) ( )   ( ) ( ) ( ) R R R R R R R 1 R R 1 R R R R R R : d d d d c N c R R J J J J V − − Ω Γ Ω Γ Ω Γ Ω Γ    ∇ ∇ Ω + ⋅ Γ =    ⋅ Ω + ⋅ Γ ∀ ∈ ∫ ∫ ∫ ∫ C u J v J t v f v p v v χ χ χ χ χ (1) where Ω = Ω / Ω and Γ = Γ / Γ are the Jacobian related to the volume and area (representing the ratio of differential length), respectively, while Ω , Γ , Γ and are the volume, the cohesive and Neumann boundary in the referential configuration. Moreover, the vectors and are unknown displacement field and arbitrary virtual displacement field, respectively, which belong to the set of kinematically admissible displacements U and to the set of kinematically admissible variations of the approximated displacement field V , respectively. The analysis considers a stress-based criterion for crack initiation; in particular, a crack starts to propagate when the normal stress across any mesh boundary exceeds the predefined tensile strength of the material. Once initiated, the subsequent crack growth direction is calculated by the Maximum Energy Release Rate (MERR) criterion (Kumar et al., 2017). To implement MERR in a multiphase context, we employ a revised J-integral approach. The standard J integral must be enhanced to remain valid for heterogeneous media, which is achieved by explicitly incorporating the discontinuities in the stress and strain fields at the matrix-inclusion interfaces. Furthermore, a second challenge arises: the J-integral becomes path-dependent for curved or non-straight cracks. To resolve this, we adopt the modified integration path proposed in (Eischen, 1987) and depicted in Figure 1.

Figure 1. Scheme of the integration paths and notation.

This approach excludes a small circular region D r (with radius r ) around the singular crack tip and explicitly integrates along the crack faces. This dual-modification strategy leads to the following robust expression for the energy release rate: