Tensile and compressive strengths of quasi-brittle materials such as rocks and concretes are often obtained indirectly from laboratory tests, for example, the Brazilian test and three-point bending for tensile strength and four-point bending and the scratch test for the compressive strength. Compared with the direct uniaxial tests, these tests have distinct advantages from the experimental standpoint. Nevertheless, these tests are indirect since the stress states at the locations, where failure initiates and progresses, could be complex and non-uniform. Depending on the material properties and sample size, both the ductile mode, associated with crushing and decohesion of grains, and the brittle mode, associated with crack initiation and propagation, could be involved. Interpretation of the experimental results is therefore complicated by the progressive failure, manifested through the development of fracture process zones, and the existence of the dual failure modes. Over the last two decades, discrete element method (DEM) has become an indispensable numerical tool to model the failure behaviors in quasi-brittle materials. Calibration of the material properties is a necessary step for DEM modeling of any engineering problems. Perhaps out of necessity for matching the experimental data, strength calibration in the literature is also sometime performed with the indirect test configurations, even though the direct tests are easier to interpret and to perform numerically. A long-standing well-known issue in DEM modeling with bonded spherical particles in random dense packing is that the ratio between the compressive and tensile strengths, as determined from the unconfined tests, are much lower than those of realistic values of rocks and concretes. Consequently, the highly nonlinear failure envelope at the low confining range cannot be captured. We should then ask the questions: How are the failure mechanisms in the indirect tests affected by the low strength ratio? What are the differences in calibrating the material strengths of a particle assembly by using the indirect and direct tests? A comprehensive numerical study is conducted to examine the failure behaviors in the intact Brazilian test. Specifically, we examine how the micro-scale material parameters and the geometrical configurations affect the macro-scale failure mechanisms, the transition between the brittle and ductile modes and consequently the material strengths. The numerical analysis is conducted using the DEM code PFC2D/3D. A novel displacement-softening contact model is formulated and implemented in PFC2D/3D to solve the issue of the low strength ratio. Dependence of the failure mechanisms on the material properties as well as the sample size, as consistent with experimental evidences, can now be reproduced in the numerical simulations. Outcome from this research could not only help better interpret the laboratory tests but also improve the material strength calibration for DEM modeling in general.