Pattern selection when a layer buckles on a soft substrate.
Pattern selection when a layer buckles on a soft substrate.
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DOI：
10.1039/c8sm02548g
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发表时间：
2019-05
期刊：
影响因子：
3.4
通讯作者：
N. Cheewaruangroj;J. Biggins
中科院分区：
文献类型：
--
作者：
N. Cheewaruangroj;J. Biggins
如果粘附到新胡克基底的新胡克弹性层等双轴地生长，则其将弯曲成地形图案。在这里，我们结合联合收割机高阶摄动理论和有限元数值预测的模式形成刚刚超过屈曲阈值。更准确地说，我们构建了一系列的解决方案，对应于六边形，正方形和条纹图案，并展开的弹性能量为每个图案作为一个朗道的能量系列的地形振幅。我们看到，对于正方形和条纹图案，弹性能量在地形反演下是不变的，使得不稳定性超临界。然而，由于六边形凹陷的图案在物理上不同于六边形凸起的图案，六边形能量缺乏这种不变性。这种缺乏引入了一个立方项，导致六边形图案形成亚临界，因此积极的青睐。我们的三次项的分析计算使我们能够确定，凹痕是有利的不可压缩系统，但颠簸是有利的充分可压缩系统。最后，我们考虑一个刚性层夹在一个相同的基板和superstrate对。该系统具有地形反演对称性，超临界地形成六边形，并有利于正方形图案。我们使用有限元计算来验证我们对每个模式的理论预测，并确认选择哪种模式。以前的工作使用了一个简化的弹性模型（一个板和一个线性弹性基板），具有地形反演下的不变性，因此错误地预测方形图案。我们的工作表明，大应变的几何形状是足以打破这种对称性，并解释在屈曲实验中观察到的六边形凹痕图案。
If a neo-Hookean elastic layer adhered to a neo-Hookean substrate grows equibiaxially, it will buckle into a topographic pattern. Here, we combine higher order perturbation theory and finite element numerics to predict the pattern formed just beyond the buckling threshold. More precisely, we construct a series of solutions corresponding to hexagonal, square and stripe patterns, and expand the elastic energy for each pattern as a Landau-like energy series in the topography amplitude. We see that, for square and stripe patterns, the elastic energy is invariant under topography inversion, making the instabilities supercritical. However, since patterns of hexagonal dents are physically different to patterns of hexagonal bumps, the hexagonal energy lacks this invariance. This lack introduces a cubic term which causes hexagonal patterns to be formed subcritically and are hence energetically favoured. Our analytic calculation of the cubic term allows us to determine that dents are favoured in incompressible systems, but bumps are favored in sufficiently compressible systems. Finally, we consider a stiff layer sandwiched between an identical substrate and superstrate pair. This system has topography inversion symmetry, so hexagons form supercritically, and square patterns are favoured. We use finite element calculations to verify our theoretical predictions for each pattern, and confirm which pattern is selected. Previous work has used a simplified elastic model (a plate & a linear elastic substrate) that possesses invariance under topography inversion, and hence incorrectly predicted square patterns. Our work demonstrates that large strain geometry is sufficient to break this symmetry and explain the hexagonal dent patterns observed in buckling experiments.