On the Onset of Cracks in Arteries

We present a theoretical approach to study the onset of failure localization into cracks in arterial wall. The arterial wall is a soft composite comprising hydrated ground matrix of proteoglycans reinforced by spatially dispersed elastin and collagen fibers. As any material, the arterial tissue cannot accumulate and dissipate strain energy beyond a critical value. This critical value is enforced in the constitutive theory via energy limiters. The limiters automatically bound reachable stresses and allow examining the mathematical condition of strong ellipticity. Loss of the strong ellipticity physically means inability of material to propagate superimposed waves. The waves cannot propagate because material failure localizes into cracks perpendicular to a possible wave direction. Thus, not only the onset of a crack can be analyzed but also its direction. We use the recently developed constitutive theories of the arterial wall including 8 and 16 structure tensors to account for the fiber dispersion. We enhance these theories with energy limiters. We examine the loss of strong ellipticity in uniaxial tension and pure shear in circumferential and axial directions of the arterial wall. We find that the vanishing longitudinal wave speed predicts the appearance of cracks in the direction perpendicular to tension. We also find that the vanishing transverse wave speed predicts the appearance of cracks in the the direction inclined (non-perpendicular) to tension. The latter result is counter-intuitive yet it is supported by recent experimental observations.