论文标题
弹性曲线的第一个积分：旋转螺旋的不稳定性
First integrals for elastic curves: twisting instabilities of helices
论文作者
论文摘要
我们提出了一个适用于研究曲线的差异框架，其能量取决于它们的弯曲和扭曲程度。通过使用材料曲率描述这种弹性变形模式，我们得出了代表曲线力和扭矩平衡的平衡方程。曲线上力和扭矩的保护定律是源于能量的欧几里得不变性的，使我们能够获得平衡方程的第一积分。为了说明这一框架，我们将其应用于确定各向同性和各向异性的Kirchhoff弹性杆的第一个积分，其能量在材料曲率中是二次的。我们使用它们来分析螺旋的扭曲导致螺旋的变形。我们在无法伸展的螺旋上检查了三种扭曲的不稳定性，其特征是它们的波数，具体取决于其边界是固定的，沿着径向方向或正交向其移动。我们还扰动地分析了弯曲各向异性对变形状态的影响，该状态引入了不同波数的变形模式之间的耦合。
We put forward a variational framework suitable for the study of curves whose energies depend on their bend and twist degrees of freedom. By employing the material curvatures to describe such elastic deformation modes, we derive the equilibrium equations representing the balance of forces and torques on the curve. The conservation laws of the force and torque on the curve, stemming from the Euclidean invariance of the energy, allow us to obtain first integrals of the equilibrium equations. To illustrate this framework, we apply it to determine the first integrals for isotropic and anisotropic Kirchhoff elastic rods, whose energies are quadratic in the material curvatures. We use them to analyze perturbatively the deformations of helices resulting from their twisting. We examine three kinds of twisting instabilities on unstretchable helices, characterized by their wavenumbers, depending on whether their boundaries are fixed, displaced along the radial direction or orthogonally to it. We also analyze perturbatively the effect of the bending anisotropy on the deformed states, which introduces a coupling between deformation modes with different wavenumbers.
