受支撑可压缩弹性梁的Lyapunov-Schmidt分岔分析
Lyapunov-Schmidt bifurcation analysis of a supported compressible elastic beam
AI总结：
针对软质可压缩弹性梁的亚临界失稳问题，采用Lyapunov-Schmidt约化法结合数值计算，研究弹性地基上可伸长梁的轴向受压稳定性，发现双临界载荷与多样屈曲形态，可通过基底刚度调控屈曲模态。
摘要：
结构的典型失稳与同名的Euler梁相关，该梁被建模为不可伸长曲线，在临界压缩载荷下会出现超临界分岔。相比之下，软质可压缩梁可能发生亚临界失稳，这一问题的研究要少得多，尽管它在软材料与软结构领域的相关性正日益提升。本文研究了弹性地基上的软质可伸长弹性梁在轴向压力作用下的稳定性，采用Lyapunov-Schmidt约化方法开展分析，并通过数值计算对结果进行验证。我们计算得到的分岔图与经典Euler-Bernoulli梁的分岔图不同，对于每个屈曲模态n，均存在两个临界载荷$p^\text{±}_{\text{cr}}(n)$。对于所有模态n和长细比，梁在$p^+_{\text{cr}}(n)$处均会发生超临界叉形分岔。由于弹性地基的存在，低阶模态在$p^-_{\text{cr}}(n)$处呈现亚临界叉形分岔，且或许令人意外的是，首个超临界叉形分岔点出现在更高的临界载荷下。地基的存在使得弹性梁更难发生屈曲，但一旦发生屈曲，梁往往会屈曲成更具波纹状的形态。总体而言，我们发现弹性支撑可使经典弹性线产生多种多样的屈曲形态，人们可以调节基底刚度来控制所需的屈曲模态——这是一项可通过实验验证的预测。
英文摘要：
The archetypal instability of a structure is associated with the eponymous Euler beam, modeled as an inextensible curve which exhibits a supercritical bifurcation at a critical compressive load. In contrast, a soft compressible beam is capable of a subcritical instability, a problem that is far less studied, even though it is increasingly relevant in the context of soft materials and structures. Here, we study the stability of a soft extensible elastic beam on an elastic foundation under the action of a compressive axial force, using the Lyapunov-Schmidt reduction method which we corroborate with numerical calculations. Our calculated bifurcation diagram differs from those associated with the classical Euler-Bernoulli beam, and shows two critical loads, $p^\pm_{\text{cr}}(n)$, for each buckling mode $n$. The beam undergoes a supercritical pitchfork bifurcation at $p^+_{\text{cr}}(n)$ for all $n$ and slenderness. Due to the elastic foundation, the lower order modes at $p^-_{\text{cr}}(n)$ exhibit subcritical pitchfork bifurcations, and perhaps surprisingly, the first supercritical pitchfork bifurcation point occurs at a higher critical load. The presence of the foundation makes it harder to buckle the elastic beam, but when it does so, it tends to buckle into a more undulated shape. Overall, we find that an elastic support can lead to a myraid of buckled shapes for the classical elastica and one can tune the substrate stiffness to control desired buckled modes -- an experimentally testable prediction.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
