Effects of bending stiffness on localized bulging in a pressurized hyperelastic tube.

The problem of localised bulging in inflated thin-walled tubes has been studied by many authors. In all these studies, the strain-energy function is expressed only in term of principal stretches. However, there are some applications where the cylindrical tube concerned may have walls thick enough so that the membrane theory may become invalid. One such situation that motivates the present study is the mathematical modeling of aneurysm initiation; in that context the human arteries exhibit noticeable bending stiffness. The effects of bending stiffness on localized bulging are studied using two different approaches. The first approach is related to the continuum-mechanical theory for three-dimensional finite deformations of coated elastic solids formulated by Steigmann and Ogden (1997, 1999). Strain-energy function has been defined in terms of the curvature of the middle surface and the principal stretches. The elasticity of the coating incorporates bending stiffness and generalizes the theory of Gurtin and Murdoch (1975). A bifurcation condition is derived using a weakly non-linear analysis and the near-critical behaviour is determined analytically. A finite difference scheme and a shooting method are formulated to determine the fully non-linear bulging solutions numerically. The second approach is based on the exact theory of finite elasticity, and the tube concerned is assumed to have arbitrary thickness. The exact bifurcation condition is derived and used to quantify the effects of bending stiffness. A two-term asymptotic bifurcation condition that incorporates bending stiffness is also derived. Finally, it is shown that when the axial force is held fixed, the bifurcation pressure is equal to the maximum pressure in uniform inflation. However when the axial stretch is fixed, localized solution is possible even if the pressure does not have a maximum in uniform inflation. This last result is particularly relevant to the continuum-mechanical modelling of the initiation of aneurysms in human arteries