Nucleation of creases and folds in hyperelastic solids is not a local bifurcation
Nucleation of creases and folds in hyperelastic solids is not a local bifurcation
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DOI：
10.1016/j.jmps.2021.104749
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发表时间：
2022-01
影响因子：
5.3
通讯作者：
Shrinidhi S. Pandurangi;Andrew Akerson;R. Elliott;T. Healey;N. Triantafyllidis
中科院分区：
文献类型：
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作者：
Shrinidhi S. Pandurangi;Andrew Akerson;R. Elliott;T. Healey;N. Triantafyllidis
从分岔理论的角度考虑压缩超弹性固体中的折痕和褶皱。它们指的是高度局部化的表面变形，这种变形发生在明显低于众所周知的Biot不稳定性值的压缩载荷下。文献中的许多工作试图证明这种现象对应于不同于生物不稳定性的“局部分岔”。局部分岔是从平凡解分支上的一个（分岔）点出发的平衡解的路径，该分支存在于该分岔点的所有足够小的邻域中。推论通常是先引入一个小的表面缺陷；然后得到一个看似接近完美分岔图的解曲线。然而，不完备理论仅在一个分岔点的足够小的邻域内成立。因此，在没有将这些解与平凡解连接起来的平衡路径的情况下，就没有理由得出折痕和折叠是完美系统的局部分岔的结论。在这项工作中，我们直接处理这些溶液在完美，不完美的情况下的成核。我们证明了功能梯度和双层弹性半空间中的表面不稳定性，对应于从均匀状态出发的局部分岔，必然是光滑和振荡的；折痕/折叠最终沿着全局分岔解分支发展，尽管与平凡解“相去甚远”，正如相应的分岔图所证明的那样。此外，我们发现它们的稳定实现发生在远低于初始表面不稳定的载荷水平。此外，我们通过将连续参数从宏观侧向应变转换为薄膜-衬底剪切模量比，获得了完美均匀半空间的结果。当该比值达到统一时，我们得到了期望的局部变形解，避免了在Biot失稳高度简并均匀状态附近进行分析的需要。
We consider creases and folds in compressed hyperelastic solids from the point of view of bifurcation theory. They refer to highly localized surface deformations that occur at compressive loads significantly below the value of the well-known Biot instability. Much work from the literature attempts to make the case that this phenomenon corresponds to a “local bifurcation” distinct from the Biot instability. A local bifurcation is a path of equilibrium solutions emanating from a (bifurcation) point on the trivial solution branch that exists in all sufficiently small neighborhoods of that bifurcation point. The inference is usually made by first introducing a small surface imperfection; a solution curve is then obtained that is seemingly close to a perfect bifurcation diagram. However, imperfection theory is valid only in some sufficiently small neighborhood of a bifurcation point. Thus, in the absence of an equilibrium path connecting these solutions to the trivial one, there is no justification for concluding that creasing and folding arelocal bifurcationsof the perfect system.In this work, we directly address the nucleation of these solutions in the perfect, imperfection-free case. We demonstrate that surface instabilities in functionally graded and bilayer elastic halfspaces, corresponding to local bifurcations from the homogeneous state, are necessarily smooth and oscillatory; creases/folds eventually do develop along the global bifurcating solution branches, albeit “far” from the trivial solution, as evidenced by the corresponding bifurcation diagrams. In addition, we find that their stable realization occurs at load levels well below that of the initial surface instability. Moreover, we obtain such results for theperfect homogeneous halfspace, by switching the continuation parameter from macroscopic lateral strain to the film-to-substrate shear modulus ratio. When this ratio reaches unity, we obtain the desired localized deformation solution, avoiding the need for analysis near the highly degenerate homogeneous state at the Biot instability.