at each numeric step such as: max M∈C K(M)/Kc =1. This condition ensures that K(M) < Kc, ∀M ∈ Cand that there is always an “active” part of the front. The loading is thus recalculated at each numerical step: σ ∞ √b = Kc maxbK M∈C (7) We can define here the real nondimensionalized loading as follows: √b σ ∞ Kc = 1 max M∈Cb K(M) (8) 3. Determination of the SIF along two coplanar circular cracks of same radius a1 = a2 = a In this section, the SIF values obtained for two coplanar penny-shaped cracks in interaction are presented. Unfortunately, to our knowledge, no 3D analytical solution exists for this problem that should serve as benchmarks. For weakly interacting cracks (that is the SIF remains close to their value for one single crack), Isida et al. [4], Fabrikant [3], Kachanov and Laures [5], Chen and Lee [2] and Zhan and Wang [8] provide numerical results that are in agreement with each other. We thus believe that those values must be correct and shall serve to validate our code and test the influence of our numerical parameters a0/b, N and δa/a (section 3.1). For closely spaced cracks, the numerical approximations are more questionable and few studies exist. Among them those of Fabrikant [3], Kachanov and Laures [5] and Zhan and Wang [8] will serve to compare with our simulations (section 3.2). 3.1. Weak interaction Figures 2 reflect the influence of different parameters: the number of nodes N on each front, the initial adimensionless radius a0/band the crack advanceδa/a. In this section, cracks are subjected to a uniform advance defined as follows: δa a =γ min a, ∆ 2! . All the figures 2 represent Kmax/K0 as a function of ∆/2ain the same y-range to make easier comparisons. Moreover it shall be noticed that the SIF of all points within the frame are less diffent than 10 % from the ones for a single isolated crack. The initial cracks should not be too small because of the incremental nature of the method, numerical errors would accumulate. Typically, one shall choose a0/bbetween0.05and0.1. There is a few dependence on the number of points, provided that N >100. Moreover we notice that K(s) presents some irregularities for N < 160. Thus values of N ≥ 160 shall be used. Since the CPU depends on N, we shall be reasonable. Typically N =160 seems a good compromise. Once again due to the incremental nature of the method, we notice on figure (2c) that δa/ashall be not too small but enough to use the first order perturbation formulae. Typically, δa/a ∈ (0.025−0.1) is acceptable. We shall use a0/b =0.1, N=160, δa/a =0.025 in the sequel. 3.2. Strong interaction In the sequel, let’s define: l =min(a, ∆). It shall be noticed that the method is unstable for some set of parameters. It is linked to the incremental nature of the method and to the amplification of Kas soon as some angular points appear along the crack front. For instance, one can notice on figure (3c) that for N = 160, (a0/b = 0.1, δa/l = 0.01 the value of Kmax diverges. In the sequel, we consider those calculations as ill and arrange to find well suited set of parameters. A systematic study of numerical stability is under consideration and will be published in the future. From those results, we can conclude that the method is able to give qualitatively correct values of K but quantitatively, is quiet sensitive to the numerical parameters. In particular for cracks as close as ∆/2a < 10−4, a relative dispersion (standard deviation/mean value) can be observed of approximately 100 % by choosing reasonable parameters (N =100−200, a0/b =0.1−0.2, δa/l =0.01−0.05). For higher values of ∆/2a, the dispersion decreases. It is of 50%for ∆/2a ∼10− 2, 10%for ∆/2a ∼10− 1, 1%for ∆/2a ∼0.5, 0.1%for ∆/2a ∼1. 4
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