Mixed-mode failure of a notched unreinforced concrete beam

Two finite element meshes are used to show the influence of mesh refinement on the load-displacement response of the concrete beam. The value of the Mode I fracture energy, $G_f^I$, can be specified directly for brittle cracking properties to define tension softening behavior that gives approximately mesh insensitive results. However, this is not done here for two reasons: first, this specification restricts the postcracking normal stress evolution to a linear variation, and we want to be more flexible than that in some of our studies; second, by specifying the postfailure stress-strain relationship directly, we show how Abaqus/Explicit converts fracture energy data into cracking stress versus cracking strain data.

If we specify the tension softening behavior in terms of stress versus cracking strain and we assume a linear dependence of stress on cracking strain, as shown in Figure 3, the cracking strain at which the stress reaches a zero value, $e_0^{ck}$, can be calculated as $2G_f^I$ / (${\sigma }_{tu}^{I}h$), where ${\sigma }_{tu}^I$ is the cracking failure stress and h is a characteristic element length. This characteristic length represents the size of the element that cracks and has values of 15 and 7.5 mm for the coarse and fine meshes, respectively. This method of calculating the cracking strain at which the stress reaches a zero value provides material data that will give approximately mesh insensitive results and is essentially what Abaqus/Explicit does when the parameter TYPE=GFI is used. This is discussed in more detail in Cracking Model for Concrete and A cracking model for concrete and other brittle materials.

The shear retention properties used for the two meshes are shown in Figure 4. The evolution of the shear retention factor, $\rho$, is chosen such that the shear resistance of the material is reduced drastically as soon as the crack initiates.

The response of the load transmitted at point B or D versus the crack mouth sliding displacement (CMSD) of the notched beam obtained with the two meshes is shown in Figure 5. This figure shows that the coarse and fine meshes give similar results. Based on this observation, all subsequent studies are performed using only the fine mesh. Displaced shapes and crack patterns obtained at the end of the analysis are shown for the two meshes in Figure 6 and Figure 7. The crack propagation path tends to curve away from the original crack tip and move toward point B. This behavior is typical for a crack subjected to mixed-mode loading.

The previous results were obtained using linear tension softening. The maximum load-carrying capacity of the beam compares well with the experimental observations of Arrea and Ingraffea. However, the postcracking behavior is somewhat stiff compared to the experiments. In the following study we use three different evolutions of the stress as a function of cracking strain. We compare the linear variation used previously to two tension softening functions where the stress is reduced more rapidly as the crack initiates. These functions are shown in Figure 8: one consists of a two-segment representation of softening, and the other is a four-segment representation. The area under the softening curve is the same in all cases so that the value of the Mode I fracture energy of the material is preserved.

The load-CMSD responses obtained for the three tension softening representations are shown in Figure 9. Although the analyses were performed over the same duration (0.38 seconds), the end value of the crack mouth sliding displacement increases as tension softening is lowered. This is to be expected, since the crack faces are likely to slide more with respect to each other as tension softening is lowered. The peculiar behavior observed at a CMSD value of about 0.15 mm in the case of the four-segment tension softening simply shows that the response is no longer quasi-static because the crack has propagated completely through the depth of the beam. It is clear that more rapid reductions of the stress after initial cracking lead to less stiff responses. Although the simulation predicts the trend of the experimental results, the decrease in the simulated load-carrying capacity in the softening region is not as great as the experimental results suggest. The effect of shear retention is, therefore, addressed next in an attempt to bring the numerical results closer to the experimental observations.

Two different evolutions of shear retention are used to show the influence of shear retention on the load-CMSD response of the beam. One is the evolution of shear retention that was used in all previous analyses. The other is a lower shear retention model, as shown in Figure 10. This lower shear retention model corresponds to practically no shear carrying capability in the cracked elements once cracking initiates.

The load-CMSD responses obtained for these two cases are shown in Figure 11 for the fine mesh with the two-segment tension softening model and in Figure 12 for the fine mesh with the four-segment tension softening model. Although we still apply the same linearly varying velocity at point C (0.75 mm/second at 0.38 seconds), the analyses for the lower shear retention model were stopped at 0.36 seconds and 0.34 seconds for the mesh with the two- and four-segment tension softening models, respectively. These times roughly correspond to times at which the crack has propagated across the entire depth of the beam. Responses obtained after these times are no longer meaningful in the context of this problem, since the beam no longer has any static load-carrying capacity, and the applied velocity loading causes the beam to respond dynamically.

The results show that, even using zero shear retention, the numerical simulation is not able to predict both a peak load of about 140 kN and the sharp reduction of that load observed in the experiments. This can be explained by the bias introduced when using a rectangular mesh, which tends to promote crack propagation along vertical lines of elements instead of the more curved crack path observed in the experiments. Rots et al. (1989) have indeed shown numerical results that match the softening response of the beam better by using a mesh designed with elements aligned along the experimentally observed curved crack path. This can be done in a case such as this one where good experimental data exist, but it is not possible in general. Results obtained for plain concrete should, therefore, be treated as only relatively coarse approximations of actual behavior.

Abaqus/Explicit provides a brittle failure criterion that allows elements to be removed when any local direct cracking strain (or displacement) reaches a failure strain (or displacement). This option is intended primarily to avoid analyses that end prematurely because cracked elements undergo too severe distortion. However, as discussed later, by setting the failure strain for element removal to a relatively low value, the removal of cracked elements can also create a significantly weaker postfailure behavior.

Figure 13 and Figure 14 show the effect of element removal. In Figure 13 the two- and four-segment tension softening curves of Figure 8 are used, respectively, and the failure strain is chosen as 0.4%. The load-CMSD responses obtained for these two simulations are plotted compared to the corresponding responses without element removal. In Figure 14 the two-segment tension softening curve is used. Two levels of failure strain—i.e., 0.2% and 0.4%, respectively—are considered. The resulting load-CMSD responses are plotted along with the corresponding responses without element removal. As expected, the use of this brittle failure model produces a large drop in the load after the peak load is reached.