The objective of this contribution is the numerical investigation of growth-induced instabilities of an elastic film on a microstructured soft substrate. A nonlinear multiscale simulation framework is developed based on the FE² method, and numerical results are compared against simplified analytical approaches, which are also derived. Living tissues like brain, skin, and airways are often bilayered structures, consisting of a growing film on a substrate. Their modeling is of particular interest in understanding biological phenomena such as brain development and dysfunction. While in similar studies the substrate is assumed as a homogeneous material, this contribution considers the heterogeneity of the substrate and studies the effect of microstructure on the instabilities of a growing film. The computational approach is based on the mechanical modeling of finite deformation growth using a multiplicative decomposition of the deformation gradient into elastic and growth parts. Within the nonlinear, concurrent multiscale finite element framework, on the macroscale a nonlinear eigenvalue analysis is utilized to capture the occurrence of instabilities and corresponding folding patterns. The microstructure of the substrate is considered within the large deformation regime, and various unit cell topologies and parameters are studied to investigate the influence of the microstructure of the substrate on the macroscopic instabilities. Furthermore, an analytical approach is developed based on Airy’s stress function and Hashin–Shtrikman bounds. The wavelengths and critical growth factors from the analytical solution are compared with numerical results. In addition, the folding patterns are examined for two-phase microstructures and the influence of the parameters of the unit cell on the folding pattern is studied.

该贡献的目的是对微结构化软基底上弹性膜的生长诱导的不稳定性进行数值研究。基于FE ^2的非线性多尺度仿真框架方法，然后将数值结果与简化分析方法进行比较，简化分析方法也可以得出。诸如大脑，皮肤和气道之类的活组织通常是双层结构，由在基底上生长的薄膜组成。他们的模型对于理解生物学现象（例如大脑发育和功能障碍）特别感兴趣。在类似的研究中，假定基材为均质材料，但这种贡献考虑了基材的异质性，并研究了微结构对生长膜不稳定性的影响。该计算方法基于有限变形增长的机械模型，该模型使用了变形梯度到弹性和增长部分的乘法分解。在非线性并发多尺度有限元框架内，在宏观尺度上，利用非线性特征值分析来捕获不稳定性的发生和相应的折叠模式。在大变形范围内考虑了衬底的微观结构，并研究了各种晶胞拓扑结构和参数，以研究衬底的微观结构对宏观不稳定性的影响。此外，基于艾里的应力函数和Hashin–Shtrikman边界开发了一种分析方法。将分析溶液的波长和临界生长因子与数值结果进行比较。此外，检查了折叠模式的两相微观结构，并研究了晶胞参数对折叠模式的影响。