Compressed elastic films on soft substrates release part of their strain energy by wrinkling, which represents a loss of symmetry, characterized by a pitchfork bifurcation. Its development is well understood at the onset of supercritical bifurcation, but not beyond, or in the case of subcritical bifurcation. This is mainly due to nonlinearities and the extreme imperfection sensitivity. In both types of bifurcations, the energy–displacement diagrams that can characterize an energy landscape are non-convex, which is notoriously difficult to determine numerically or experimentally, let alone analytically. To gain an elementary understanding of such potential energy landscapes, we take a thin beam theory suitable for analyzing large displacements under small strains and significantly reduce its complexity by reformulating it in terms of the tangent rotation angle. This enables a comprehensive analytical and numerical analysis of wrinkling elastic films on planar substrates, which are effective stiffening and/or softening due to either geometric or material nonlinearities. We also validate our findings experimentally. We explicitly show how effective stiffening nonlinear behavior (e.g., due to substrate or membrane deformations) leads to a supercritical post-bifurcation response, makes the energy landscape non-convex through energy barriers causing multistability, which is extremely problematic for numerical computation. Moreover, this type of nonlinearity promotes uni-modal, uniformly distributed, periodic deformation patterns. In contrast, nonlinear effective softening behavior leads to subcritical post-bifurcation behavior, similarly divides the energy landscape by energy barriers and conversely promotes localization of deformations. With our theoretical model we can thus explain an experimentally observed phenomenon that in structures with effective softening followed by an effective stiffening behavior, the symmetry is initially broken by localizing the deformation and later restored by forming periodic, distributed deformation patterns as the load is increased. Finally, we show that initial imperfections can significantly alter the local or global energy-minimizing deformation pattern and completely remove some energy barriers. We envision that this knowledge can be extrapolated and exploited to convexify extremely divergent energy landscapes of more sophisticated systems, such as wrinkling compressed films on curved substrates (e.g., on cylinders and spheres) and that it will enable elementary analysis and the development of specialized numerical tools.

中文翻译：

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非线性和几何缺陷对平面基底上皱纹薄膜的多稳定性和变形局部化的影响


软基材上的压缩弹性薄膜通过起皱释放部分应变能，这代表对称性的丧失，其特征是干草叉分叉。它的发展在超临界分岔开始时已得到很好的理解，但在超临界分岔之后或在亚临界分岔的情况下则不然。这主要是由于非线性和极端缺陷敏感性。在这两种类型的分岔中，能够表征能量景观的能量-位移图都是非凸的，这很难通过数值或实验来确定，更不用说分析了。为了对这种势能景观有一个基本的了解，我们采用了适合分析小应变下大位移的薄梁理论，并通过根据切线旋转角重新表述它来显着降低其复杂性。这使得能够对平面基底上的起皱弹性薄膜进行全面的分析和数值分析，这些薄膜由于几何或材料的非线性而有效地硬化和/或软化。我们还通过实验验证了我们的发现。我们明确地展示了有效的硬化非线性行为（例如，由于基底或膜变形）如何导致超临界后分叉响应，通过能量势垒使能量景观非凸，从而导致多稳定性，这对于数值计算来说是极其成问题的。此外，这种类型的非线性促进了单模态、均匀分布、周期性变形模式。相反，非线性有效软化行为导致亚临界后分岔行为，类似地通过能量势垒划分能量景观，并反过来促进变形的局部化。 因此，利用我们的理论模型，我们可以解释实验观察到的现象，即在具有有效软化和有效硬化行为的结构中，对称性最初通过局部变形而被打破，随后随着载荷的增加而通过形成周期性的分布变形模式来恢复。最后，我们表明初始缺陷可以显着改变局部或全局能量最小化变形模式并完全消除一些能量障碍。我们设想，可以推断和利用这些知识来凸化更复杂系统的极其不同的能量景观，例如弯曲基底（例如圆柱体和球体）上的起皱压缩薄膜，并且它将使得基本分析和专门数值计算的开发成为可能。工具。