A multiplicity of localized buckling modes for twisted rod equations
A multiplicity of localized buckling modes for twisted rod equations
复制标题
扭杆方程的多种局部屈曲模式
DOI：
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发表时间：
1996
期刊：
影响因子：
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通讯作者：
J. M. Thompson
中科院分区：
文献类型：
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作者：
A. Champneys;J. M. Thompson
本文回顾了受端部拉力和力矩作用的细长弹性杆的空间平衡问题的Kirchhoff-Love方程，并利用该方程研究了局部屈曲解的存在性。剪切和轴向延伸的影响不被考虑，但该模型也允许非线性本构关系。在无限长的假设下，动力学相空间类比允许人们使用动力学系统理论的技术来描述许多可能的平衡路径。局部化解对应于动力系统的同宿轨道。在无量纲化的扭杆方程取决于一个单一的负载参数，并使用分析和数值技术研究这个问题的局部化解决方案的分叉行为。首先，在具有相等的主弯曲刚度的杆的情况下，平衡方程是完全可积的，一个已知的单参数家庭的局部化解决方案计算各种亚临界载荷。载荷-挠度图计算为这个家庭和某些材料的非线性本构关系示出的定性图片的差别不大。几何圆对称的破坏破坏完全可积性，特别是，打破了平凡稳态的稳定和不稳定流形的非横截相交。由此产生的横向交叉，这是已知的，导致空间混乱，明确证明意味着大量的本地化屈曲模式。一个样本的主要和多模态的解决方案进行数值计算，辅助的微分方程的可逆性。最后，平行画与概念上更简单的问题的支柱休息（非线性）弹性基础上，更多的信息是已知的局部屈曲模式的全球行为。
The Kirchhoff-Love equations governing the spatial equilibria of long thin elastic rods subject to end tension and moment are reviewed and used to examine the existence of localized buckling solutions. The effects of shear and axial extension are not considered, but the model does additionally allow for nonlinear constitutive laws. Under the assumption of infinite length, the dynamical phase space analogy allows one to use techniques from dynamical systems theory to characterize many possible equilibrium paths. Localizing solutions correspond to homoclinic orbits of the dynamical system. Under non-dimensionalization the twisted rod equations are shown to depend on a single load parameter, and the bifurcation behaviour of localizing solutions of this problem is investigated using analytical and numerical techniques. First, in the case of a rod with equal principal bending stiffnesses, where the equilibrium equations are completely integrable, a known one-parameter family of localizing solutions is computed for a variety of subcritical loads. Load-deflection diagrams are computed for this family and certain materially nonlinear constitutive laws are shown to make little difference to the qualitative picture. The breaking of the geometrical circular symmetry destroys complete integrability and, in particular, breaks the non-transverse intersection of the stable and unstable manifolds of the trivial steady state. The resulting transverse intersection, which is already known to lead to spatial chaos, is explicitly demonstrated to imply a multitude of localized buckling modes. A sample of primary and multi-modal solutions are computed numerically, aided by the reversibility of the differential equations. Finally, parallels are drawn with the conceptually simpler problem of a strut resting on a (nonlinear) elastic foundation, for which much more information is known about the global behaviour of localized buckling modes.