Surface instabilities in graded tubular tissues induced by volumetric growth

Growth-induced pattern formation in tubular tissues is intimately correlated to normal physiological functions. Moreover, either the microstructure or certain diseases can give rise to material inhomogeneity, which can lead to a change of shape in the tissue. Therefore, it is of fundamental importance to understand surface instabilities and pattern transitions of graded tubular tissues. In this paper we perform such analysis by the use of a mechanical model of a graded tube which grows with a fixed outer boundary by focusing on a planestrain problem within the framework of nonlinear elasticity. A theoretical model is established to determine the uniform growth state, the critical growth factor, and the critical wavenumber for a general material model and for a general material gradient. For a case study, the material is specified by the neo-Hookean model, and the shear modulus is assumed to decay linearly or exponentially from the inner surface. Then, a parametric study is carried out to unravel the effects of material and geometrical parameters on the bifurcation threshold and the associated wrinkled pattern. In addition, a finite element model, which is validated by the theoretical one, is developed to trace the post-buckling evolution. It is found that wrinkled pattern will evolve into an arch mode and then into a creasing mode if the modulus decays linearly. However, the typical creasing mode may give way to a period-doubling mode when applying an exponentially decaying modulus, and there is a co-existence of the creasing mode and the wrinkling mode. As a result, different modulus gradients can generate diverse pattern formations. The obtained results are useful to supply insight into the effects of material inhomogeneity and different modulus gradients on surface instabilities and morphology evolutions in graded tubular tissues.