PSI - Issue 35

Kai Friebertshauser et al. / Procedia Structural Integrity 35 (2022) 159–167 K. Friebertsha¨user and M. Werner and K. Weinberg / Structural Integrity Procedia 00 (2021) 000–000

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Fig. 6. Discretization of the pressure pad for application of the pressure load (left) and mid-surface of the entire concrete cylinder model (right)

4.2. Peridynamics simulation

The peridynamic model of the cylindrical block consists of a uniformly distributed material point cloud with 90 × 90 × 50 points for (i) and (ii) and 100 × 100 × 36 points for (iii). Each point is assigned with the same volume V k = V 1 = V 2 = · · · = V N , which is calculated as ratio of total cylinder volume by N . The material parameters used for the simulation are shown in Table 2. In the center of the block the pressure pad is represented by two layers of material points, one above the other. The horizontal and the inclined pad of configuration (i) and (ii) are straight lines of the respective size. There are no material points between these layers. For the curved pad configuration (iii), the upwardly bent arms of the pressure pad are located on a parabola whose vertex lies in the center of the cylinder (see Fig. 6). Since the period of simulation is only a few milliseconds, a linear increase in pressure has be selected as an approximation of the experimental situation. The pressure load inside the cylinder is modeled in the same way as in the penny shaped crack, Eq. (15), with a external force density b k ( t ) for each material point k in normal direction n k of the parabola and an internal pressure p ( t ) = q 0 · t , where q 0 = 7 · 10 3 bar s − 1 . To apply boundary conditions, the bottom of the specimen is complectly fixed in x , y and z direction in the lowest material point layer of the cylinder. Additionally, the degrees of freedom in the x and y directions of all material points in the lateral surface of the concrete cylinder are fixed in order to account for the plastic vessel around the concrete block. From Fig. 6 we see that the initial crack is notably apart from the boundaries. Therefore, surface correction factors are negligible. Volume correction factors are not applied due to the slightly nonuniform point positions of the pressure pad. We do not use a no-fail zone throughout the model, as no unintentional cracks occur. The results of our simulations are shown in Fig. 7-8. A comparison of the simulated crack paths and the upper fragments of the plane configurations (i) and (ii) are shown in Fig. 7. We see here a good agreement for both cases. For the horizontal pad configuration, it can be seen that in the experiment as well as in the simulation the cracks extend with a slope in positive z direction. For the inclined pressure pad configuration (ii), the simulation and the experiment are similar but deviate more. In the computation, the crack is branched in the edge area of the pressurized cylinder. This di ff erence may be attributed to a higher local pressure or simply to the non-homogenous concrete material. It is preferable to compare the e ff ects and results of configuration (iii) than configuration (i) and (ii). In Fig. 8, the damage D k of the curved configuration at time t = 1 . 38 ms and internal pressure p = 9 . 66 bar is shown. The inserted bell curve has nearly the same slope and evidently a very good agreement to the fragment of Fig. 5. In total, we state that the peridynamic computation can realistically map the pressure-initiated crack formation inside the concrete cylinder. For all pressure pad configurations, the computed crack pattern is very close to reality.

5. Summary

In this study, we present a peridynamic approach to pneumatic fracture. The model is based on classical state-based peridynamics with a linear solid material and a damage definition resting on critical bond straining. In a simulation of a penny shaped crack subjected to internal pressure, we showed that our peridynamic pneumatic fracture model captures the crack evolution in a quantitative correct manner. The crack tip opening displacement as a function of pressure agrees well with the analytical solution. Additional simulations of pneumatic cracking of concrete cylinders