The microstructure design satisfying the mass constraint can reduce the structure weight more directly and effectively in comparison with the volume constraint. This paper is devoted to the topology optimization of microstructures with multiphase materials under the mass upper limitation constraint for maximizing the equivalent elastic tensors and their combinations. Firstly, the strain energy method is applied to compute the effective elastic properties of microstructures. In order to make sure that the formulation of mass constraint is linear with separable design variables, DMO (Discrete Material Optimization) model is adopted for the element density interpolation. Therefore, this optimization problem can be solved efficiently by means of mathematical programming approaches, especially the convex programming methods. Besides, the filtering technique is adopted to avoid the checkerboard pattern. There are two categories of numerical examples. In the first category, the modulus and the stiffness ratio of the solid material phase 1 are smaller than the solid material phase 2. In the second category, the modulus of the solid material phase 1 is still smaller than the solid material phase 2, but its stiffness ratio is bigger than the solid material phase 2.