Abstract: The distribution and evolution of fractures in coal roof under mining influence is the basis for preventing water inrush disasters and realizing the protection of groundwater resources in coal mines. The key to the accuracy of nu⁃ merical results is the coupling mechanical response characteristics of continuous⁃discrete media of roof rock in mining advance. The elastic⁃plastic damage mechanics was introduced into the pure tension and pure shear fracture modes， and combined with the Benzeggagh⁃Kenane fracture criterion under arbitrary mixing ratio， the mixed ductile fracture(MD)constitutive equations of quasi⁃brittle rock under complex tension and shear stress were obtained. The contact state criterion of discrete blocks was constructed，combined with the direct shear experimental data and the shear⁃friction relationship proposed by Sargin， the constitutive equations of compression⁃shear friction ( SF ) applicable to the rough discontinuities under various normal stresses were derived. On this basis，the finite ⁃ discrete element(FDEM)numerical calculation program was developed to realize the “continuous” and“discrete”medium con⁃ version process of the overburden rock. In the numerical calculation，the friction and fracture properties were respec⁃ tively assigned to the discontinuities(or contact pairs)and the zero⁃thickness cohesive elements，and the fracture⁃fric⁃ tion coupling was realized by sharing the nodal force. Fracture mechanics experiment and direct shear test were used to identify material parameters and verify the rationality of MD-SF constitutive equations. Based on this，the evolution law of mining⁃induced water⁃conducting fractures in the Jurassic and Carboniferous coal seams with mining advances was simulated. The results show that1 The MD-SF constitutive equations and the corresponding FDEM numerical calcula⁃ tion method can realize the transformation process of rock mass from continuous medium to discrete medium，and re⁃ flect the response of mixed ductile fracture and shear friction of rough discontinuities of quasi⁃brittle rock. 2 Un⁃ der the simulated conditions，the smaller the spacing between stratum and coal，the larger the total width of wa⁃ ter⁃conducting fracture，and the total width of fracture in the mining process was 2.26-7.11 times that after the move⁃ ment of overburden was stable，indicating that water inrush was more likely to occur in the mining process. 3 The re⁃ markable ductile failure characteristics of kaolin sandstone can effectively control the height of water⁃conducting frac⁃ tures，but under the disturbance of repeated mining of the Jurassic and Carboniferous coal seams，the ratio of the height of water⁃conducting fracture in overburden to the coal seam height is 20.8，which is far higher than the results obtained by empirical formula. The simulation results of water⁃conducting fracture distribution and evolution have been tested in the Datong mining area， which provides theoretical support for further mining overburden movement， safe coal mining under water and water retaining coal mining design.