The application of the described h-adaptive methodology is presented on a 2D simulation of a single edge notched plain concrete beam, known also as 4-point shear test. In this test, a simply supported beam with a notch at the midspan of the supported side is loaded by two point forces the locations of which mirror the position of the supports with respect to the beam center. The magnitude of the forces is then inversely proportional to their distance from to center of the side on which they are applied. The load (including reactions) is transferred to the specimen via steel plates. This test is characteristic by evolution of a curved crack starting at the notch tip and ending behind the load on the opposite face of the specimen. The concrete behaviour is described by the nonlocal scalar damage model, while the steel bearing plates are assumed to be linearly elastic. The considered dimensions of the specimen (Iosipescu geometry) and the relevant material parameters are shown in Figure 3.
The adaptive simulation of the 4-point test was performed with the target relative error 10 % taken as a common engineering tolerance. The nonlinear problem was solved incrementally applying the cylindrical arc length method taking the horizontal displacement of point B (see Figure 3) as the arc length control parameter. The highly nonlinear response of the specimen is displayed in terms of the load-displacement diagrams of point A in Figure 4. Note the severe snap-back on the loading path of point A, which is in qualitative agreement with the experimental observations. The individual curves in Figure 4 correspond to the response evaluated for consequent discretizations (linear elements were used) that are summarized in Figure 6. The discontinuities between the consequent curves are related to a slightly different equilibrium recovered on the new mesh on which the solution from the old configuration was mapped. The big jump between the response on meshes 4 and 5 can be attributed to too large elements in mesh 4 in front of the crack head, thus the nonlocal interactions did not develop properly. The damage evolution is illustrated in Figure 7. The damage initiates at the tip of the notch a short while before the peak load is reached. The full damage at this location is observed just after the peak load. The damage then propagates sideway from the notch tip and turns towards the right edge of the loading plate. The evolution of the relative error is sketched in Figure 5. The overlap of subsequent pairs of curves in Figure 5 is related to the restart of the analysis from the previous step. Note that the remeshing in the late stages of the analysis was triggered rather by the size of the damaged elements than by the global relative error, that remains far below the value of 10 %.