On prismatic and bending bifurcations of fiber-reinforced elastic membranes under swelling with application to aortic aneurysms

Abstract

The influence of swelling on prismatic and bending bifurcation modes of inflated thin-walled cylinders under axial loading is examined. The bifurcation criteria for a membrane cylinder subjected to combined axial loading, internal pressure, and swelling is provided. We consider orthotropic materials with two preferred directions which are mechanically equivalent and symmetrically disposed. The mechanical behavior of the matrix is described by a swellable isotropic model. The isotropic material is augmented with two functions that are equal, each one of them accounting for the existence of a unidirectional reinforcement. Two reinforcing models that depend only on the stretch in the fiber direction are considered: the so-called standard reinforcing model and an exponential one. The analysis of bifurcation modes for these models under the conditions at hand may establish the connection with modeling of the normal and diseased aorta in arterial wall tissue. The effects of the axial stretch, the strength of the fiber reinforcement and the fiber winding angle on the onset of prismatic and bending bifurcations are investigated. It is shown that for membranes without fibers, prismatic bifurcation is not feasible. On the other hand, bending bifurcation is more likely to occur for swollen cylinders. However, for a particular model of fiber-reinforced membranes, the standard model, there exists a domain of deformation values together with material constant values that may trigger prismatic bifurcation. The exponential model does not allow prismatic bifurcations. Both models allow bending bifurcation and may or may not trigger it depending on the deformation together with material parameters.

Bifurcation of fiber reinforced inflated membranes with different natural configurations of the constituents

“…The inflated and loaded tube may face various instability modes, for example, bulging, bending, and prismatic bifurcation [1,16]. The initiation and relevance of these modes depend, for instance, on the geometry of the considered problems and the distribution of the material parameter of the constituents.…”

“…Due to the large water content in soft tissue, the material is taken to be incompressible, so that detF = 1. Considerations that may include volume changes in the incompressible material, for example, due to swelling [1,6], are ignored.…”

“…Pressurized and axially stretched cylinders may face various instability modes, for example bulging, necking, and bending (see, e.g. [1][2][3][4][5][6]). The initiation of a bulge is a nonlinear phenomenon, which has attracted the attention of various researchers.…”

Remarks on bifurcation of an inflated and extended swellable isotropic tube

“…Swellable neo-Hookean membranes have been extensively studied for bulging bifurcation, and we refer to Demirkoparan and Merodio [22,23], Topol et al [25], Al-Chlaihawi et al [24] that treat this type of material with and without fiber reinforcement.…”

“…The changes in both the cylinder geometry and the elastic properties due to swelling will affect the bifurcation characteristics. Such a swelling effect has been studied in Demirkoparan and Merodio [22,23] and Al-Chlaihawi et al [24] for bulging bifurcation of fibrous materials, and in most cases, the isotropic constituent has been taken to be a neo-Hookean or Mooney-Rivlin type of material [25]. The study of fibrous material is often inspired by the composition of arterial soft tissue, in which multiple collagen fiber families reinforce a ground substance material [26].…”