13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- the influence of the resolution finesse on the numerical results, the cell was discretized using different grids of regularly spaced Fourier points. The resolutions used for discretization are 100×100, 200×200, 400×400 and 600×600 pixels. The pure mode-I load is prescribed by imposing an average strain { } 12 22 11 E E E = E { } 1 0.3 0 11 − =E . 0 0,2 0,4 0,6 0,8 1 1,2 0 10 20 30 40 a/R 100*100 100*100 200*200 200*200 400*400 400*400 600*600 600*600 Normalized Normalized KI remote stress Figure 1: Normalized stress intensity factor Ic IK K and normalized remote stresses c σ σ∞ as function of normalized semi-crack length a R for mode-I loaded cracks at fracture We first calculated the remote tensile loads ∞σ for crack growth according to the damage criterion (7). The calculations were carried out for different crack length and with different discretization resolutions. The results of these calculations are reported in Figure 1. In this figure, the stress intensity factors are normalized by the critical value Ic K ; the remote stresses are normalized by the ultimate stress cσof the material and the crack length is normalized by the non-local action radius R. From this figure, several remarks can be made: 1. When the crack is sufficiently long with respect to the non-local radius R (a R3> ), the stress intensity factors evaluated from the present non-local damage model for crack growth equal correctly the critical stress intensity factor of the material. This result confirms that the proposed damage model is equivalent to the criterion Ic IK K≥ for mode-I cracks; 2. When the FFT discretization is sufficiently fine, the proposed damage model is independent of the FFT grid resolution. 3. The remote stress at fracture tends to the ultimate stress of material as the crack length tends to zero. In this case, the proposed crack growth criterion degenerates to the maximum stress criterion for non-singular stresses. 4.2: Cracking in concrete The second example deals with a recurrent problem in concrete mechanics. The concrete is often modeled by a 3-phase particle composite where stiff and strong aggregate particles are embedded in a weaker and softer cement matrix. A third phase exists between these two phases, namely a thin
RkJQdWJsaXNoZXIy MjM0NDE=