PSI - Issue 79

Domenico Ammendolea et al. / Procedia Structural Integrity 79 (2026) 467–474

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where, ( ) R ∇ ⋅ are the gradient operator with respect to the material and referential system, respectivelly, and Ψ J is the Jacobian matrix of the transformation. Similarly, the material time derivative ( v  ) is related to the referential time derivative ( R v  ) by the equation: 1 R M R R R R − = −∇ = −∇ Ψ v v vX v vJ X      (2) in which R X  is a convective term that describes the mutual movement between R χ and R X . 2.2. Governing equations The proposed model consists of three sets of governing equations. The first set involves the governing equations of solid mechanics, where the weak form, with reference to Figure 1, assumes the following form (Greco et al. (2017), Gaetano et al. (2022), Ammendolea et al. (2023)): [ ] : : ( ) p h d h d h t dS h d δ δ δ ρ δ Ω Ω ∂Ω Ω ∇ ∇ Ω− ⋅ Ω− ⋅ =− ⋅ Ω ∫ ∫ ∫ ∫ C u u f u p u u u  (3) where, C is the fourth‐order elastic constitutive tensor of the material, h is the thickness of the domain, f and p (t) are the body force and the external tractions acting along the boundary of the domain, respectively. By projecting Eq.(3) into the referential frame, one obtains: ( ) { } { } { } { } 1 1 1 1 1 1 1 1 : : ( ) 2 R R R R R R R R S R R R R R R R R R R R R R R R R h J d h J d h t J dS h J d δ δ δ ρ δ − − Ω Ω Ω Ω ∂Ω − − − − − − Ω Ω   ∇ ∇ Ω − ⋅ Ω + ⋅ =       − − ∇ −∇ +∇ ∇ +∇ ∇ ⋅ Ω     ∫ ∫ ∫ ∫ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ C uJ uJ f u p u u u J X uJ X uJ JXX uJ XJX u         (4) The second set of equations comprises those associated with the moving mesh kinematics. The mesh node motion follows a rezoning method using a Laplace operator applied to the coordinates x of the mesh nodes: 0 R R ∇ ⋅∇ = x (5) These differential equations must be solved with boundary conditions that fix the nodes along the boundary of the computational domain and require the nodes associated with a crack tip to advance at the crack propagation velocity ( ) a t  in the direction of propagation c n . Considering the presence of n TIP crack tips in the computational domain, the boundary conditions assume the following form: where = − u x χ  . Finally, the third set of equations is those of fracture mechanics, specifically the criteria governing crack onset, propagation, and branching. According to the Maximum Hoop Stress (Erdogan and Sih (1963)), it is possible to define an equivalent stress intensity factor * K and the direction of propagation * θ using the following expressions: ( ) ( ) ( ) ( ) * * 3 * 2 * * , , cos 2 3 cos 2 sin 2 I II I II K K K K K θ θ θ θ = − (7) ( ) ( ) 2 * , 2arctan 4 ( ) 4 8 I II I II II I II K K K K sign K K K θ   = − +     (8) where, ( , I II K K ) are the mode I and mode II DSIFs at the crack front. The proposed method determines the DSIFs using the well-established M -integral method, a robust numerical method for quantifying these values. Specifically, our model uses the M -integral method adapted for the ALE formulation. For the sake of brevity, a detailed presentation of the ALE formulation of the M -integral is omitted here. However, a thorough discussion on this topic is available in (Greco et al. (2022), Ammendolea et al. (2025a)). By using * K the following fracture function is introduced to identify the conditions of crack onset: 0 * 0 0 0 (no initiation) 1 0 (initiation) F F Id F f K f K f  < = − ⇒  = (9) where, Id K is the dynamic crack initiation toughness for the material. , c i , c i on ; ( ) at For 1,.., i = i TIP a t  O n = u 0  ∂Ω ⋅  = u n  (6) ( ) M ∇ ⋅ and