The Distinct Lattice Spring Model (DLSM) is a newly developed numerical model for rock dynamics problems, i.e., dynamic failure and wave propagation. The current constitutive law for bond spring used in DLSM is the simplest brittle linear one, which is not enough to describe the complex mechanical behavior of rock material. An advanced micro constitutive law of bond springs is developed and implemented into DLSM. Using the damage variable principle, a tri-linear constitutive law and nonlinear micro constitutive law has been developed, which can fully represent the linear stage, the hardening stage and the softening stage of the bond spring. Two failure modes, tensile failure of the normal spring and shear failure of the shear spring, are considered. A preliminary study on failure behavior of DLSM model with the developed constitutive model is performed. Uniaxial tensile and compressive tests were performed to find influences of micro constitutive law on the macro mechanical response of DLSM. It is found that the ratio of compressive strength to tensile strength for DLSM is much lower than that for rock materials (typically around 10-12), e.g. it is around seven for the regular lattice model and three for the random lattice model. Therefore, further development of the constitutive model in DLSM is required.