A numerical approach to the stability analysis of growing cylindrical tubes and spherical tissues

Abstract

In this paper, the effect of growth on the stability of elastic materials is examined through a numerical approach. Growth and resorption are considered to have two main effects from the stability standpoint. Corresponding to the change in mass, the geometry of a system changes, and the critical length of the system can be modified. In addition, the growth of the material may be affected by the stress, yet it may also impose residual stresses. As a result, the material is either stabilized or destabilized by these stresses. As a general framework for the description of elastic properties, the theory of finite elasticity is employed to investigate the growth of elastic materials. Growth is taken into account through matrix multiplication of the deformation gradient. The formalism of incremental deformation is adopted to consider growth effects. Using this formalism, the stability of growing neo-Hookean incompressible cylindrical and spherical shells under external pressure is considered. Firstly, the incremental equilibrium equations and the corresponding boundary conditions for the incompressible growing shells are summarized and afterward employed in order to analyze the behavior of spherical and cylindrical shells subjected to external pressure. The generalized differential quadrature method (GDQ) is utilized to solve the eigenvalue problem that results from a linear bifurcation analysis. It is shown that this numerical method can be efficiently utilized to solve the considered problem. The results are in full agreement with the previously obtained results. In the presence of external radial pressure, an elastic shell will buckle circumferentially to a noncircular cross-section. A change in thickness due to the growth can significantly affect buckling, both in terms of the critical pressure and the buckling pattern. Finally, the effects of the thickness ratio A1/A2, mode number n, and the growth parameter γ are studied on the shell’s stability.