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 plane-strain 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.

肾小管组织中生长诱导的模式形成与正常的生理功能密切相关。而且，无论是微观结构还是某些疾病都可能引起物质的不均匀性，从而导致组织形状的改变。因此，了解渐变的管状组织的表面不稳定性和图案转变至关重要。在本文中，我们通过使用梯度管的机械模型进行这种分析，该梯度管通过关注非线性弹性框架内的平面应变问题而在固定的外边界处生长。建立理论模型以确定通用材料模型和通用材料梯度的均匀生长状态，临界生长因子和临界波数。对于案例研究 材料由新霍克模型指定，并且剪切模量假定从内表面线性或指数衰减。然后，进行了参数研究，以揭示材料和几何参数对分叉阈值和相关的起皱图案的影响。另外，开发了一种通过理论验证的有限元模型，以追踪屈曲后的演变。已经发现，如果模量线性衰减，则起皱的图案将演变成拱形模式，然后演变成折痕模式。但是，当采用指数衰减模量时，典型的折痕模式可能会换成周期倍增模式，并且折痕模式与起皱模式并存。结果，不同的模量梯度可以产生不同的图案形成。