The application of the described h-adaptive methodology is presented on a 2D simulation of the brazilian splitting test, which is a standard technique for determination of the tensile strength of concrete. In this test, a cylindrical specimen is loaded along its vertical diametral plane. The compressive load, transfered to the specimen via steel bearing plates at the top and bottom sides, induces tension stress in the horizontal direction leading finally to the rupture of the specimen along the loading plane. Due to the double symmetry, the analysis itself is performed only on the quarter of the specimen under plain strain conditions. 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 and the relevant material parameters are shown in Fig. 1 on the left.

The analysis was initially performed without the adaptive refinement using the discretization depicted in Fig. 1 on the right. The nonlinear problem was solved incrementally applying the cylindrical arc length method taking the horizontal displacement of point B (see Fig. 1) as the arc length control parameter. The highly nonlinear response of the specimen is displayed in terms of the load-displacement diagram of point A in Fig. 2. To assess quantitatively the quality of the obtained solution the error estimation has been carried on at each of the 63 steps of the nonlinear analysis. The resulting profile of the evolution of the relative error is displayed in Fig. 3.

The adaptive simulation of the brazilian splitting test was performed with the target relative error 10 % (taken as a common engineering tolerance and being similar to the initial relative error of the simulation accomplished without adaptive refinement). The evolution of the relative error, controlled by the adaptive analysis, is sketched in Fig. 5. The mesh used for the nonadaptive computation was employed as the initial mesh for the adaptivity. This mesh was simultaneously used as the coarsest allowable mesh in order to prevent derefinement of the localization zone (especially at the late stages of the analysis) to ensure the nonlocal model is working properly. The arc length procedure was controlled by the same parameter as in the nonadaptive simulation. The obtained structural response in terms of loading path of point A is depicted in Fig. 4. The individual curves correspond to the response evaluated for consequent discretizations. The overlap of subsequent pairs of curves in Figs 4 and 5 is related to the restart of the analysis from the previous step. Surprisingly, there is not much difference in the response (given by the loading paths of point A) between the adaptive and nonadaptive analysis. This verifies the objectivity of the applied nonlocal continuum approach giving the same global response (note that point A is actually outside of the fracture process zone) independent of the discretization, provided that the elements in the localization zone are smaller than the material characteristic length. The benefit of the adaptive solution consists in the higher resolution of the localization zone resulting in strain profile captured with higher accuracy.

*Daniel Rypl
2005-12-03*