13th International Conference on Fracture June 16–21, 2013, Beijing, China -8- Figure 6 shows that the failure process is better described with the interaction-based model since the distance between its crack opening profile and the one corresponding to a strong discontinuity COD tends rapidly to zero. At complete failure, the crack opening computed according to the same technique should be independent of the element size. In a simple 1D setting, for instance, and assuming that the crack opening is smeared over the finite element that contains the discontinuous displacement at complete failure, the crack opening is equal to the strain distribution times the element size. Therefore, after complete failure, the strain in the cracked element should evolve in inverse proportion of the element size (for constant strain element). Figure 6 shows that the complete failure is better described with the interaction-based model since the strain versus adimensional element size curve follows a linear trend in a logarithmic plot. Moreover the slope is coherent with the CMOD estimated at complete failure. Note that in the original model, the element size is dimensioned by the internal length lc whereas in the new model, it is dimensioned by the characteristic length a0 . For the integration-based model, a peak discontinuity is observed when the element size is approximately equal to the characteristics length a0. It means that several elements are needed inside the inclusion where the perturbation is produced to well reconstruct the interaction-based weight function. 3.3. Spalling test A second 1D example is used to test the response of the new model close to a boundary. This 1D example consists of a spalling test presented by [12] based on a split Hopkinson pressure bar test primarily developed by [27] for material dynamic behavior characterization, but often adapted for dynamic fracture testing [28, 29]. A striker bar generates a square compressive wave that then propagates along the bar in the linear elastic regime. When this compressive wave reaches the free extremity of the bar, it is converted into a tensile wave and added to the incoming compressive wave (see Fig. 7 and Table 2). The resulting wave stays equal to zero until the tensile one reaches a distance from the boundary equal to half the initial signal length. Failure is initiated at this point if the amplitude is greater than the tensile strength, generating a spall at a controlled distance from the boundary that depends on the initial compressive signal duration. For all numerical studies, the time step is chosen to be equal to the critical time step of the corresponding element size. Figure 7 shows that the spalling failure is better described with the interaction-based model since the spall location is predicted inside the bar whereas the damage is maximum on the boundary with the original model. Figure 7. Spalling test: test description, time evolution of the strain amplitude repartition along the rod (left) and damage repartition along the bar after failure (right).
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