  We study the initiation and evolution of aneurysmal morphology in a
pressurized soft tube where the elastic modulus is non-uniform in the radial
direction. The primary deformation prior to instability is characterized within
the framework of nonlinear elasticity for a general material constitution and a
generic modulus gradient. To unravel the influence of modulus gradient on
aneurysm formation, we employ the incompressible Gent model and select three
representative modulus gradients, including a linear, an exponential, and a
sinusoidal function. In particular, the sinusoidal distribution can be used to
model actual artery structure. In addition, two prototypical loading conditions
are considered, namely, either the resultant axial force or the axial length
can be fixed. Based on an explicit bifurcation condition in terms of the
internal pressure and the resultant axial force for aneurysm formation or
localized bulging, an exhaustive theoretical analysis on bulge initiation is
carried out and the effect of geometric and material parameters and modulus
gradient on the critical stretch generating localized bulging is revealed. It
turns out that the modulus mismatch, as well as the position of maximum
modulus, can dramatically affect the onset of localized bulging. Then we
analytically elucidate the influence of modulus gradient on bulge propagation
and conduct a finite element analysis of bulge evolution based on a robust
finite element model established in Abaqus by UHYPER subroutine coding.
Interestingly, it is found that a sinusoidally distributed modulus has
negligible influence on the critical stretch of bulge initiation, the
deformation process of bugle growth, and the maximum size of a bulge. The
current analysis can provide useful insight into the biological evolution of
human artery and into localized instabilities in graded structures.
