At constant shear rate, the process of deformation of the paste slurry is divided into two stages: one is the initial structural failure process with increasing shear stress; the other is the thixotropic process with decreasing shear stress after yielding. Based on experiments, the mechanical response characteristics of the paste slurry in the initial structural failure process under different shear rate conditions were studied in this paper. At the same time, according to the Maxwell model, the stress-time model equation describing the initial structure failure stage of the paste was deduced and the constant shearing test was carried out on the paste slurry at different mass concentrations; the model equation was used to fit the test data of the initial stress increment stage. The results showed that the model equation had higher prediction accuracy and better popularity. In the initial structural failure stage, the paste had a nonlinear stress-time relationship. At different shear rates (0.05, 0.5, and 1 s−1), the lower the rotation speed, the smoother the curve, and the slurry at various stages in the yielding process could be more clearly reflected; in the range of low constant shear rate (0.03, 0.05, and 0.07 s−1), the initial stress and yield stress of the paste increased with the increase of shear rate at the same mass concentration, and the time to yield was shorter. The yield stress increased exponentially with mass concentration.