We propose a theoretical approach to predict the onset of cracks in arterial wall. The arterial wall is a soft composite made up of hydrated ground matrix of proteoglycans reinforced by elastin and collagen fibers spatially dispersed in the matrix. Like any other material, the arterial tissue cannot store and dissipate strain energy above a certain threshold. This threshold value is introduced in the constitutive theory via energy limiters. The limiters naturally constrain reachable stresses and enable analysis of mathematical condition of strong ellipticity. Loss of the strong ellipticity corresponds to the juncture when superimposed waves cease to propagate due to localization of material failure into cracks perpendicular to a possible wave direction. Thus, the direction in which crack starts to appear can be analyzed in addition to its inception. We enrich the recently developed constitutive theories that account for fiber dispersion of the arterial wall by including 8 and 16 structure tensors with energy limiters. We analyze the loss of strong ellipticity in uniaxial tension in circumferential and axial directions of the arterial wall. We find that cracks appear in the direction perpendicular to tension, when the speed of the superimposed longitudinal wave vanishes. We also find that the appearance of cracks is predicted in the direction inclined (non-perpendicular) to tension, when the speed of the superimposed transverse wave vanishes.