Abstract We report the full analytical solution of the large deformations of a cantilevered elastica loaded by a uniformly distributed follower pressure. We consider an unshearable, inextensible and linear elastic rod. We obtain a spatial nonlinear differential equation in the curvatures, analogous to the undamped Duffing oscillator with a constant driving force. We solve such differential equation, obtaining the curvature, although in implicit form, for arbitrarily large values of the load. We are then able to obtain the rotations owing to a change of variables from the curvilinear abscissa to the curvature. This step is somewhat mandatory due to the implicit nature of the solution. Finally, with the same change of variables, it is possible to obtain a closed-form solution for the deformation in Cartesian coordinates. The solutions show that the rod deforms into drop-like shapes. The number of drops is equal to the number of spatial periods of the solution, which goes with q ∗ 1 / 3 , with q ∗ a dimensionless load normalised to the bending stiffness. Interestingly, we find that for q ∗ ⩾ 3094.2 , the number of drop-like shapes does not increase, but remains three.

中文翻译：

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悬臂弹性体在法向、均匀分布的随动载荷作用下的解析解

摘要 我们报告了由均匀分布的随动压力加载的悬臂弹性体的大变形的完整解析解。我们考虑不可剪切、不可伸长的线弹性杆。我们获得了曲率中的空间非线性微分方程，类似于具有恒定驱动力的无阻尼 Duffing 振荡器。我们求解这样的微分方程，获得任意大载荷值的曲率，尽管是隐式形式。由于从曲线横坐标到曲率的变量变化，我们然后能够获得旋转。由于解决方案的隐含性质，此步骤在某种程度上是强制性的。最后，在相同的变量变化下，可以得到笛卡尔坐标下变形的闭式解。解表明棒变形为水滴状。滴数等于解的空间周期数，与 q ∗ 1 / 3 一致，q ∗ 是归一化为弯曲刚度的无量纲载荷。有趣的是，我们发现对于 q ∗ ⩾ 3094.2 ，水滴状形状的数量没有增加，但仍然是三个。