Post-buckling analysis on growing tubular tissues: A semi-analytical approach and imperfection sensitivity

Abstract

Deriving the amplitude equation for a buckling mode is an important issue in non-linear elasticity. Focusing on pattern formations in growing tubular tissues, many existing literature often adopted the numerical methods. In this paper, we propose a semi-analytical approach for the bilayer tubular structures under growth to derive the amplitude equation of a single wrinkling mode, from which a transition between supercritical and subcritical bifurcations can be determined. In the framework of finite elasticity, a weakly non-linear analysis is carried out and the amplitude equation is deduced by the virtual work method. A semi-analytical solution is obtained, with an analytical expression whose exact coefficients are determined numerically. Then a parametric study is carried out by use of the semi-analytical solution. When the total growth factor is prescribed, the critical mode number governs the amplitude, and a lower mode corresponds to a higher amplitude. When the incremental growth factor after bifurcation is fixed, it turns out that the dependence of the amplitude on the modulus ratio is non-monotonic if the thicknesses of the two layers are specified. For a given geometry, when the modulus ratio ξ>5, the wrinkled amplitude is mainly dominated by the critical mode number, and a smaller critical mode number deepens the wrinkle. However, the amplitude is a decreasing function of ξ when ξ < 5. The obtained analytical solutions are also validated by the corresponding numerical solutions based on the finite element method. The proposed semi-analytical approach is applicable for most variable coefficient problems arising from cylindrical and spherical structures.