Lyapunov-Schmidt bifurcation analysis of a supported compressible elastic beam
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910783759187968 |
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| author | Yong, Ee Hou Mahadevan, L. |
| author_facet | Yong, Ee Hou Mahadevan, L. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_08028 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lyapunov-Schmidt bifurcation analysis of a supported compressible elastic beam Yong, Ee Hou Mahadevan, L. Classical Physics Soft Condensed Matter 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. |
| title | Lyapunov-Schmidt bifurcation analysis of a supported compressible elastic beam |
| topic | Classical Physics Soft Condensed Matter |
| url | https://arxiv.org/abs/2501.08028 |