Using a newly-developed Material Failure Process Analysis code (MFPA3D), the micro-fracturing process and the avalanche behavior characterization of brittle disordered materials such as rock or concrete is numerically studied under uniaxial compression and tension. It is found that, due to the heterogeneity of the disordered material, there is an avalanche behavior in the microcrack coalescence process. Meanwhile, a hierarchy of avalanche events also numerically observed though a study of numerically obtained acoustic emissions or seismic events. Numerical simulations indicate that macro-crack nucleation starts well before the peak stress is reached and the crack propagation and coalescence can be traced, which can be taken as a precursory to predict the macro-fracture of the brittle disordered materials. In addition, the numerically obtained results also reveal the presence of residual strength in the post-peak region and the resemblance in the stress-strain curves between uniaxial compression and tension.