The objective of this contribution is to study for the first time the growth-induced instabilities of an elastic film on a viscoelastic substrate using an analytical approach as well as computational simulations via eigenvalue analysis. The growth-induced instabilities of a thin film on a substrate is of particular interest in modeling living tissues such as skin, brain, and airways. The analytical solution is based on Airy’s stress function adopted to viscoelastic constitutive behavior. The computational simulations, on the other hand, are carried out using the finite deformation continuum theory accounting for growth via the multiplicative decomposition of the deformation gradient into elastic and growth parts. To capture the critical growth of elastic films and the associated folding pattern, eigenvalue analysis is utilized, in contrast to the commonly used perturbation strategy. The eigenvalue analysis provides accurate, reliable, and reproducible solutions as contrasted to the perturbation approach. The numerical results obtained from the finite element method show an excellent agreement between the computational simulations and the proposed analytical solution.

中文翻译：

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粘弹性基材上弹性薄膜的生长诱导不稳定性：通过特征值分析的解析解和计算方法

这项贡献的目的是首次使用分析方法以及通过特征值分析的计算模拟来研究粘弹性基材上弹性薄膜的生长引起的不稳定性。基板上薄膜的生长引起的不稳定性在模拟活组织（如皮肤、大脑和气道）时特别有趣。解析解基于用于粘弹性本构行为的艾里应力函数。另一方面，计算模拟是使用有限变形连续体理论进行的，通过将变形梯度乘法分解为弹性和增长部分来考虑增长。为了捕捉弹性薄膜的临界增长和相关的折叠模式，利用特征值分析，与常用的扰动策略相反。与扰动方法相比，特征值分析提供了准确、可靠和可重复的解决方案。从有限元方法获得的数值结果显示了计算模拟和建议的解析解之间的极好一致性。