This work focuses on planar growth-induced instabilities in three-dimensional bilayer structures, i.e., thick stiff film on a compliant substrate. Growth-induced instabilities are examined for a different range of fiber stiffness with a five-field Hu-Washizu type mixed variational formulation. The quasi-incompressible and quasiinextensible limits of transversely isotropic materials were considered. A numerical example was solved by implementing the T2P0F0 element on an automated differential equation solver platform, FEniCS. It was shown that both the wavelength and critical growth parameter g decrease by increasing the fiber stiffness for the first instability, which is obtained along the stiff fiber direction. The effect of the fiber stiffness is minor on the secondary buckling, which was observed perpendicular to the fiber direction. For a range of fiber stiffnesses, bifurcation points of instabilities were also determined by monitoring displacements and energies. The energy contributions of layerswith different ranges of fiber stiffnesses were examined. It is concluded that the energy release mechanism at the initiation of the primary buckling is mainly due to isotropic and anisotropic contributions of the stiff filmlayer. For high fiber stiffnesses, the effect of the anisotropic energy on the first buckling becomes more dominant over other types. However, in the secondary instability, the isotropic energy of the film layer becomes the dominant one. Numerical outcomes of this study will help to understand the fiber stiffness effect on the buckling and post-buckling behavior of bilayer systems.

翻译：这项工作侧重于三维双层结构中由平面增长引发的不稳定性, 即, 在符合基底的厚薄膜上, 厚厚的胶片。 以五野Hu- Washizu 型混合变异配方对不同的纤维僵硬度进行检查。 考虑的是跨偏偏偏异的异热带材料的准不压缩和半伸缩极限。 通过在自动差异方程式解析器平台FENICS上实施 T2P0F0 元素, 解决了一个数字示例。 显示波长和关键增长参数都通过提高第一个不稳定的纤维直度来降低。 纤维僵硬度对二次重压的影响不大, 与纤维方向密切相关。 一系列纤维僵硬度, 不稳定的两极点点点点点也通过监测迁移和能量。 不同纤维硬度的层的能量贡献得到了研究。 其结论是, 第一层的能量释放机制是纤维的僵硬度, 开始的次层的硬度, 开始的硬度是 硬度的硬度 。