with: δa (γ) α (Mk) =δa γ(Mk) −(δaα(Mi)−→nα(Mi)).→−nγ(Mk) , γ =1, 2 andwhereWαγ(Mi, Mk) is a kernel expressing the effect of the advance of Mi ∈ Cα over the SIF at point Mk ∈ Cγ. Rice’s second formula can be written of the form: • if Mi ∈ Cα and Mk ∈ Cα (points belong to the same crack front): δWαα(Mi, Mk) = D2(Mi, Mk) 2π VPZ C W(Mi, M)W(M; Mk) D2(Mi; M)D2(M; Mk) [δa(M) −δ ∗∗ a(M)]ds(M) = D2 αα(Mi, Mk) 2π VPZ Cα Wαα(Mi, M)Wαα(Mk, M) D2 αα(Mi, M)D2 αα(Mk, M) δa (α) αα(M)ds(M) + D2 αα(Mi, Mk) 2π Z Cβ Wαβ(Mi, M)Wαβ(Mk, M) D2 αβ (Mi, M)D2 αβ (Mk, M) δa (β) αα(M)ds(M) (3) • if Mi ∈ Cα and Mk ∈ Cβ (points belong to different crack fronts): δWαβ(Mi, Mk) = D2 αβ (Mi, Mk) 2π VPZ Cα Wαα(Mi, M)Wαβ(M, Mk) D2 αα(Mi, M)D2 αβ (M, Mk) δa (α) αβ (M)ds(M) + D2 αβ(Mi , Mk) 2π VPZ Cβ Wαβ(Mi, M)Wββ(Mk, M) D2 αβ (Mi, M)D2 ββ (Mk, M) δa (β) αβ (M)ds(M) (4) with: δa (γ) αβ (M) =δa γ(M) − −→Vi,k(M).→−nγ(M), where −→Vi,k is a geometric transformation such as: δa (γ) αβ (Mi) =δa (γ) αβ (Mk) =0. These formulae give us the first order perturbation of the SIF and kernels, knowing the perturbation δa and the initial SIF and W. Here comes the natural idea of an iterative procedure to predict the propagation. For this purpose, it’s necessary to start from a configuration for which the quantities bKα and Wαγ are known. We assume that for two circular cracks which are distant enough, the SIF and functions Wαγ are those for single crack that is : bKα(M) =2 pa/π Wαα =1 Wαβ =0 (5) This situation will serve as starting point of our method, corresponding to two circular cracks of size a0/b <<1. 2.3. Propagation Assumption is made that propagation is governed by the SIF so that we have: δa(M) =δamax " K(M) Kmax # β (6) where δamax is a small given quantity. It is presumed here that this law simulates brutal fracture if β ≫1 and fatigue propagation otherwise. Subjected to Irwin’s criteria, cracks are supposed to propagate in a quasistatic way under a remote loadingσ ∞ , varying 3
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