Pattern formation driven by differential strain in constrained elastic systems is a common motif in many technological and biological systems. Here we introduce a biologically motivated case of elastic patterning that allows us to explore the conditions for the existence of local puckering and global wrinkling patterns: a soft growing composite ring adhered elastically to a constraining rigid ring. We explore how differential growth of the soft ring and the elastic resistance to shear and stretching deformations induced by soft adherence lead to a range of phenomena that include uniform aperture-like modes, localized puckers that are Nambu–Goldstone-like modes and global wrinkles in the system. Our analysis combines computer simulations of a discrete rod model with a nonlinear stability analysis of the differential equations in the continuum limit. We provide phase diagrams and scaling relations that reveal the nature and extent of the deformation patterns. Overall, our study reveals how geometry and mechanics conspire to yield a rich phenomenology that could serve as a guide to the design of programmable localized elastic deformations while being relevant for the mechanical basis of biological morphogenesis.

受约束的弹性系统中由差异应变驱动的图案形成是许多技术和生物系统中的常见图案。在这里，我们介绍了一种具有生物学动机的弹性图案案例，该案例使我们能够探索存在局部褶皱和整体起皱图案的条件：一个柔软生长的复合环，弹性地粘附在一个受约束的刚性环上。我们探讨了软环的差异性增长以及由软粘附引起的对剪切和拉伸变形的弹性抵抗如何导致一系列现象，包括均匀的孔状模态，局部褶皱（类似Nambu–Goldstone状模态）和整体皱纹。系统。我们的分析将离散杆模型的计算机模拟与连续极限内的微分方程的非线性稳定性分析相结合。我们提供相位图和比例关系，以揭示变形模式的性质和程度。总的来说，我们的研究揭示了几何和力学如何共同产生丰富的现象学，可以作为可编程局部弹性变形设计的指南，同时与生物学形态发生的机械基础有关。