Earth media are incomplete media. There exist many cracks in it. The achievements of fracture mechanics show that the strength of the incomplete materials will be much lower than that of the complete materials. We consider that earthquake occurrence is the result of unstable propagation of a crack in crust media in proper condition and the earthquake rupture is the phenomenon of a failure by fast fracture under applied low shear stress. It has already been explained by fracture mechanics. The occurrence of failure by fast fracture is necessarily associated with the presence of high level concentration of local stress and strain. The elastic/plastic stress analysis in cracked pieces by Dugdale indicates that the state of stress at the tip of a crack takes a very important role to crack propagation. A plastic zone has necessarily formed in the tip of a crack due to stress concentration. Therefore, the dislocations at the tip of a crack are naturally a plastic displacement, rather than elastic one. The plastic displacementδ=πaτ 0 2 /2µτ y , whereτ 0 is applied shear stress which is equivalent to initial or tectonic shear stress when the quake occurs,a is the half length of a crack,μ is the rigidity,τ y is the yield stresses in shear. The main seismic dislocations take place exactly at the ends of the crack where the plastic zone had been formed. So, a critical assumption is adopted, i. e. we assume the dislocationD(ξ 1,t) as formula (5) in text. The maximum earthquake dislocationD max=πLτ 0 2 /4πτ y , whereL is the fault length. Ifμ is taken the value in the upper crust,μ=33 GPa; andτ y is taken the average value given from laboratories,τ y =30 MPa. Thus, according to observation values ofD max andL, using the formula, one can estimate the initial shear stresses for large earthquakes. Computations show that the initial shear stresses for large earthquakes all over the world are about 5–20 MPa which have some differences between regions. We further research the characteristics of source spectra and have derived the dependent relation of body wave magnitudem b on the shear stressτ 0 and seismic momentM 0 as formula (11) in text. Thus, the formula provides a possibility of computation of large amount of tectonic shear stress values from seismic data. We consider that the tectonic shear stress field is a main factor which controls the earthquake occurrence. The regions with high tectonic shear stress values are considered to be prone to occur great earthquakes (M s⩾6) and called earthquake hazard regions. Based on this criterion,τ 0 values for all earthquakes withm b⩾3.8 all over China since 1987 have been computed, and the great earthquake hazard regions with magnitude ranges have been zoned in the Chinese mainland. During April 1992–January 31, 1994, there were 9M s⩾6 earthquakes which occurred in the Chinese mainland, 8 earthquakes of the 9 had fallen into the regions delineated by us prior to the earthquake occurrence, with only one failure. This new approach as a method for medium-term prediction of strong earthquakes has been proved by practice to be an efficient one. It has good physical bases and bright prospect and worth further research.