Giudici Andrea, Biggins John S
Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB21PZ, UK.
R Soc Open Sci. 2022 Oct 12;9(10):220487. doi: 10.1098/rsos.220487. eCollection 2022 Oct.
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 Gauss curvature. We identify two types of supercritical instability, in which the shell's principal curvature spontaneously breaks discrete up/down symmetry and continuous planar isotropy. These are then augmented by instabilities, in which the shell jumps subcritically between up/down broken symmetry states and instabilities, in which the curvatures rotate by 90° 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.
自发的材料形状变化，如膨胀、生长或热膨胀，可用于触发薄壳中的剧烈弹性失稳。这些失稳源于壳的优先外在曲率和内在曲率之间的几何不相容性，这可分别通过厚度方向和平面内的主动变形来改变。在此，我们求解这种失稳的最简单可能模型，该模型假设壳是浅的，薄到足以弯曲但不能拉伸，并受到均匀的优先曲率作用。我们分别考虑高斯曲率为零、正和负的情况。我们识别出两种超临界失稳类型，其中壳的主曲率自发地打破离散的上/下对称性和连续的平面各向同性。然后这些又因失稳而增强，其中壳在上下破缺对称状态之间亚临界跳跃，以及因失稳而增强，其中曲率在各向同性破缺状态之间旋转90°而不释放能量。每种失稳对于优先外在曲率都有一个与厚度无关的阈值，该阈值与高斯曲率的平方根成正比。最后，我们表明对于深球形帽，各向同性破缺失稳的阈值是相同的，这与最近发表的数据非常吻合。
