1. Objectives a1 a2 b C1 C2 O1 O2 x y ∆ Figure 1: Two circular cracks. Let us consider two circular coplanar cracks embedded in an isotropic elastic body (such as depicted in figure 1). The aim of this paper is to predict the in-plane propagation of these cracks subjected to remote tensile stress σ ∞ at infinity in brittle fracture. The method consists in coupling the Bueckner-Rice formalism with a propagation law starting from a configuration for which the needed quantities, namely the stress intensity factor (SIF) along the front and a certain kernel, are known. The procedure consists of different steps: • Determination of the SIF for a given geometry: Knowing the geometry and the loading, how to calculate the SIF along the fronts? • Propagation problem with a threshold: In brittle fracture, it is assumed that the propagation law is given by Irwin’s criterion: ( K<Kc : no propagation K=Kc : possible propagation (1) For a given crack geometry, there is a critical loading σc ∞ such as: if σ ∞ < σc ∞ then K(M) < Kc, ∀Mand if σ ∞ =σc ∞ , there is at least one point Mof the front that verifies K(M) =Kc. We want to determine this stability thresholdσc ∞ all along the propagation. 2. Numerical approach 2.1. Adimensionalization Let us define the dimensionless problem for which the distance between cracks centers b is taken at 1. The quantities of interest become: a1/b and a2/b, with a loading unit σ ∞ =1, letting K(M) = √bσ ∞b K(M), where bKis a dimensionless quantity. 2.2. Rice incremental formulae Suppose that the crack geometry is slighty perturbed in its plane and consider a point Mi ∈ C =C1 SC2. Let us set α=(1, 2) and β =(1, 2) , β ,α. Then Rice’s first formula reads ([7]): δbKα(Mi) = 1 2π VPZ C W(Mk, Mi) D2(Mk, Mi) b K(Mk)[δa(Mk) −δ ∗ a(Mk)]ds(Mk) = 1 2π VPZ Cα Wαα(Mi, Mk) D2 αα(Mi, Mk) b Kα(Mk)δa (α) α (Mk)ds(Mk) + 1 2π Z Cβ Wαβ(Mi, Mk) D2 αβ (Mi, Mk) b Kβ(Mk)δa (β) α (Mk)ds(Mk) (2) 2
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