主动薄壳中曲率驱动的不稳定性
Curvature-driven instabilities in thin active shells
摘要：
自发的材料形状变化，例如膨胀、生长或热膨胀，可用于触发薄壳中的显著弹性不稳定性。这些不稳定性起源于壳体的优选的外在曲率和内在曲率之间的几何不相容性，这可以通过分别穿过厚度和平面内的主动变形来修改。在这里，我们解决了这种不稳定性的最简单的可能模型，它假设壳是浅的，薄到足以弯曲，但不拉伸，并受到均匀的首选曲率。我们分别考虑高斯曲率为零、正、负的情况。我们识别出两种类型的超临界对称性破缺失稳，其中壳体的主曲率自发地分别破缺离散上下对称性和连续平面各向同性。这些然后被反转不稳定性和旋转不稳定性增强，反转不稳定性中，壳在上/下对称性破缺状态之间亚临界地跳跃，旋转不稳定性中，曲率在各向同性破缺状态之间旋转90度而不释放能量。每种不稳定性都有一个与厚度无关的阈值，用于与高斯曲率的平方根成比例的优选非本征曲率。最后，我们表明，各向同性破坏不稳定性的阈值是相同的深球冠，最近发表的数据吻合得很好。
英文摘要：
Spontaneous material shape changes, such as swelling, growth or thermal expansion, can be used to trigger dramatic elastic instabilities in thin shells. These instabilities originate in geometric incompatibility between the preferred extrinsic and intrinsic curvature of the shell, which may be modified by active deformations through the thickness and in plane respectively. Here, we solve the simplest possible model of such instabilities, which assumes the shells are shallow, thin enough to bend but not stretch, and subject to homogeneous preferred curvatures. We consider separately the cases of zero, positive and negative Gaussian curvature. We identify two types of super-critical symmetry breaking instability, in which the shell's principal curvature spontaneously breaks discrete up-down symmetry and continuous planar isotropy respectively. These are then augmented by inversion instabilities, in which the shell jumps sub-critically between up/down broken symmetry states, and rotation instabilities, in which the curvatures rotate by 90 degrees between states of broken isotropy without release of energy. Each instability has a thickness independent threshold value for the preferred extrinsic curvature proportional to the square-root of Gauss curvature. Finally, we show that the threshold for the isotropy-breaking instability is the same for deep spherical caps, in good agreement with recently published data.
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