Understanding the mechanical controls on shear strength decrease due to preexisting weakness is a fundamental problem in tectonic studies. In this study, applying Reactivation Tendency Analysis theory, we developed a theoretical framework and defined Shear-strength Coefficient (f_d) for evaluating the shear strength decrease and anisotropies due to preexisting weakness(es). Our analysis overcomes the restrictions of the early work that assumes the weakness plane containing the intermediate stress (σ₂) and vertical or horizontal orientations of principal stresses. Using a coordination transformation, we developed a new graphical technique (Mohr-space), which may be much easier to understand than that of equations for most geologists, to predict the shear strength decrease and anisotropies caused by preexisting weakness(es). Applying Mohr-space, we build quantitative and intuitive relationship between Shear-strength Coefficient (f_d) and weakness relative-orientation (q ', φ'), weakness mechanical properties (C_w and μ_w) and relative σ₂ in any uniform tri-axial stress state. The results of theoretical analysis show that (1) Weakness relative-orientation (q ', φ'), which is determined jointly by orientations of weakness and three axes of principal stress, is the predominant factor to lead shear strength anisotropies. (2) Weakness mechanical properties (C_w and μ_w) are the predominant factors to lead shear strength decrease. (3) The affection of relative σ₂ to shear strength is a little complicated, and is related to θ’ and φ’, particularly φ’. The change of σ₂ should also not be ignored in the evaluation of shear strength decrease due to the pre-existing weakness.

In a region with a progressive increase in the magnitude of differential stresses while the directions of the principal stresses maintain the same and with multiple pre-existing weaknesses, we can also get that multiple phases of fault initiation with different trends can be generated with predicted sequence according to their Shear–strength coefficient (from small to large). It was verified by a simple sandbox experiment. Our work provides critical information on the mechanical properties of deforming lithosphere.