Based on a new elastic clump model, a flexible membrane is proposed for the discrete element numerical simulations of triaxial tests. Conversional triaxial tests of sandstone under the confining pressures of 2 MPa and 8 MPa were carried out, in order to validate the effectiveness of the proposed numerical simulation method. The numerical model is validated by comparing the numerical results with the test results. The deformation and failure process of numerical model is analyzed by stress-strain curves, micro fractures, displacement fields, stress fields and energy fields. The model shows an X-shape shear failure zone, of which the angle is very close to that of the test; the dip angle of most shear fractures is close to the angle of the internal friction; and there is a large amount of slipping frictional heat generated on the failure surface. During the loading process, the stress chain and stress concentration appear in the middle of the model, which lead to displacement zoning in the model. The failure of the model is associated with the growth of the micro tensile- and shear fractures. This study provides an effective tool for the macro-micro investigation of rock failure processes.