Arterial tissue delamination, manifested as the fracture failure between arterial layers, is an important process of the atherosclerotic plaque rupture, leading to potential life-threatening clinical consequences. Numerous models have been used to characterize the arterial tissue delamination fracture failure. However, only a few have investigated the effect of cohesive zone model (CZM) shapes on predicting the delamination behavior of the arterial wall. In this study, four types of CZMs (triangular, trapezoidal, linear–exponential, and exponential–linear) were investigated to compare their prediction of the arterial wall fracture failure. The Holzapfel–Gasser–Ogden (HGO) model was adopted for modeling the mechanical behavior of the aortic bulk material. The CZMs optimized during the comparison of the aortic media delamination simulations were also used to perform the comparative study of the mouse plaque delamination and human fibrous cap delamination. The results show that: (1) the numerical predicted the relationships of force–displacement in the delamination behaviors based on the triangular, trapezoidal, linear–exponential, and exponential–linear CZMs match well with the experimental measurements. (2) The traction–separation relationship results simulated by the four types of CZMs could react well as the corresponding CZM shapes. (3) The predicted load–load point displacement curves using the triangular and exponential–linear CZMs are in good agreement with the experimental data, relative to the other two shapes of CZMs. All these provide a new method combined with the factor of shape in the cohesive models to simulate the crack propagation behaviors and can capture the arterial tissue failure response well.