By adopting gradually tetrahedral meshing technique for random distributed aggregates and embedding scheme of cohesive element, a high efficiency 3D aggregate-mortar-interface meso finite element model is established, in which the uniaxial tension and uniaxial compression experiments of concrete cube samples are carried out in single computer. The numerical result shows that the whole process from weak connection to separation between aggregates and mortar is readily captured, which reproduces the failure mode of concrete specimens. The inner mechanism of the macro-mechanical properties of concrete is thus revealed by comparing the computed stress-strain curves from uniaxial tension and compression experiments, which is supposed to provide a technical support to explore the inner mechanism of concrete like composite materials.