Morphogenesis is a result of complex biological, chemical, and physical processes in which differential growth in biological systems is a common phenomenon, especially notable in plant organs such as petals and leaves. Mechanisms of these biologic structures have been studied in recent years with a growing focus from the mechanics point of view. However, understanding differential-growth-induced shape formation quantitatively in plant organs remains largely unknown. In this study, we conduct quantitative experimental measurement, theoretical analysis, and sufficient finite element analysis of constrained differential growth of a thin membrane-like structure. By deriving the corresponding strain energy expression of a buckled growing sample, we can calculate the shape function of such membrane structures explicitly. The results of this work demonstrate the effect of growth function, geometry characteristics, and material property. Our research points to potential approaches to novel geometrical design and inspirations on the microscale and the macroscale for items such as soft robotics and flexible electronics.

形态发生是复杂的生物、化学和物理过程的结果，其中生物系统中的差异生长是一种常见现象，尤其是在花瓣和叶子等植物器官中。近年来，人们对这些生物结构的机制进行了研究，并且越来越多地从力学角度进行研究。然而，定量理解植物器官中差异生长诱导的形状形成仍然很大程度上未知。在这项研究中，我们对薄膜状结构的约束差异生长进行了定量实验测量、理论分析和充分的有限元分析。通过推导屈曲生长样品相应的应变能表达式，我们可以明确计算此类膜结构的形函数。这项工作的结果证明了生长函数、几何特征和材料特性的影响。我们的研究指出了新颖几何设计的潜在方法以及软机器人和柔性电子产品等微观和宏观尺度上的灵感。