圆柱壳在轴压作用下的屈曲载荷取决于其截面曲率
The buckling load of cylindrical shells under axial compression depends on the cross-section curvature
摘要：
众所周知，Koiter关于圆柱壳轴压屈曲临界载荷的著名理论公式与实验数据并不相符。也就是说，虽然Koiter公式预测了屈曲载荷$λ(h)$与壳体厚度$h$的线性相关性（$h>0$是一个小参数），但在实验中观察到了相关性$λ(h)\sim h^{3/2}$;即，对于小厚度，壳体在小得多的载荷下弯曲。这种理论预测失败被认为是由所谓的缺陷现象（形状和负载）的敏感性。Grabovsky和第一作者在[ \textit{J. Nonl. Sci.,} Vol. 26，Iss. 1，pp. 83- 119，2016年2月]，在轴向压缩下的圆柱壳屈曲问题中，小载荷扭转导致屈曲载荷缩放$λ(h)\sim h^{5/4},$，而形状缺陷可能导致缩放$λ(h)\sim h^{3/2}.$。在这项工作中，我们证明，实际上圆柱形的屈曲载荷$λ(h)$（不一定是圆形）壳体在垂直压力下的变形取决于横截面曲线的曲率。当横截面是具有均匀正曲率的凸曲线时，则为$λ(h)\sim h,$，而当横截面曲线除了在许多点处具有正曲率时，则为$C_1h^{8/5}\leq λ(h)\leq C_2h^{3/2}$，对于$h$小厚度$h>0.$
英文摘要：
It is known that the famous theoretical formula by Koiter for the critical buckling load of circular cylindrical shells under axial compression does not coincide with the experimental data. Namely, while Koiter's formula predicts linear dependence of the buckling load $λ(h)$ of the shell thickness $h$ ($h>0$ is a small parameter), one observes the dependence $λ(h)\sim h^{3/2}$ in experiments; i.e., the shell buckles at much smaller loads for small thickness. This theoretical prediction failure is believed to be caused by the so-called sensitivity to imperfections phenomenon (both, shape and load). Grabovsky and the first author have rigorously proven in [\textit{J. Nonl. Sci.,} Vol. 26, Iss. 1, pp. 83--119, Feb. 2016], that in the problem of circular cylindrical shells buckling under axial compression, a small load twist leads to the buckling load scaling $λ(h)\sim h^{5/4},$ while shape imperfections are likely to result in the scaling $λ(h)\sim h^{3/2}.$ In this work we prove, that in fact the buckling load $λ(h)$ of cylindrical (not necessarily circular) shells under vertical compression depends on the curvature of the cross section curve. When the cross section is a convex curve with uniformly positive curvature, then $λ(h)\sim h,$ and when the the cross section curve has positive curvature except at finitely many points, then $C_1h^{8/5}\leq λ(h)\leq C_2h^{3/2}$ for $h$ small thickness $h>0.$
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