Asymptotic solutions on the circumferential wrinkling of growing tubular tissues

It is well-known that a growing tubular tissue with geometric constraint will buckle and the threshold value can be determined by solving a variable coefficient eigenvalue problem. In this paper, we study the growth induced wrinkling for both single- and bi-layer tubes and aim to deduce some asymptotic expressions for the critical growth factor and critical mode number. Explicit bifurcation conditions are derived by use of the WKB approach when the mode number n is large. For the single layer case, an iterative method is utilized to deduce an asymptotic solution for the critical growth factor gc and critical mode number nc under the thin layer limit. It is found that the critical mode number nc=O(lnH/H)1 and the leading-order term of gc is a constant 1.839. For a bilayer tube with each layer having its own shear modulus, we further assume that a scaled thickness η for the inner layer is small and the modulus ratio ξ between the inner and outer layers is large. An order analysis shows that the critical growth factor for the inner layer gc^=1+O(ξ−23) and ηnc=O(ξ23). Besides, we also provide some necessary higher-order corrections. All asymptotic results are validated by the corresponding numerical solutions.