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Nonlinear mechanics of a ring structure subjected to multi-pairs of evenly distributed equal radial forces

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Abstract

Combining the elastica theory, finite element (FE) analysis, and a geometrical topological experiment, we studied the mechanical behavior of a ring subjected to multi-pairs of evenly distributed equal radial forces by looking at its seven distinct states. The results showed that the theoretical predictions of the ring deformation and strain energy matched the FE results very well, and that the ring deformations were comparable to the topological experiment. Moreover, no matter whether the ring was compressed or tensioned by N-pairs of forces, the ring always tended to be regular polygons with 2N sides as the force increased, and a proper compressive force deformed the ring into exquisite flower-like patterns. The present study solves a basic mechanical problem of a ring subjected to lateral forces, which can be useful for studying the relevant mechanical behavior of ring structures from the nano- to the macro-scale.

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Acknowledgements

This work is partially supported by the National Natural Science Foundation of China (Grants 31300780, 11272091, 11422222, and 31470043), the Fundamental Research Funds for the Central Universities (Grant 2242016R30014), and ARC (Grant FT140101152). N.M.P. is supported by the European Research Council PoC 2015 “Silkene” (Grant 693670), by the European Commission H2020 under the Graphene Flagship Core 1 (Grant 696656) (WP14 “Polymer Nanocomposites”), and under the FET Proactive “Neurofibres” (Grant 732344).

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Appendices

Appendix A

The coordinate vector (10) of the point P can be solved as

$$\begin{aligned} {\varvec{r}}\left( \theta \right)= & {} -\frac{1}{k}\sqrt{\frac{\sin \varphi }{2}}\left[ \int \limits _0^\theta {\frac{\cos \theta }{\sqrt{K^{2}+\cos (\theta +\varphi )}}\hbox {d}\theta } ,\right. \nonumber \\&\left. \int \limits _0^\theta {\frac{\sin \theta }{\sqrt{K^{2}+\cos (\theta +\varphi )}}\hbox {d}\theta } \right] ^{\mathrm{T}}+\left( {0,\bar{{a}}} \right] ^{\mathrm{T}} \nonumber \\= & {} -\frac{1}{k}\sqrt{\frac{\sin \varphi }{2}}\left[ \int \limits _0^\theta {\frac{\cos (\theta +\varphi -\varphi )}{\sqrt{K^{2}+\cos (\theta +\varphi )}}\hbox {d}\theta } ,\right. \nonumber \\&\left. \int \limits _0^\theta {\frac{\sin (\theta +\varphi -\varphi )}{\sqrt{K^{2}+\cos (\theta +\varphi )}}\hbox {d}\theta } \right] ^{\mathrm{T}}+\left( {0,\bar{{a}}} \right] ^{\mathrm{T}} \nonumber \\= & {} -\frac{1}{k}\sqrt{\frac{\sin \varphi }{2}}\left\{ \left[ \sin \varphi \int \limits _0^\theta {\frac{\sin (\theta +\varphi )}{\sqrt{K^{2}+\cos (\theta +\varphi )}}\hbox {d}\theta }\right. \right. \nonumber \\&\left. \left. +\cos \varphi \int \limits _0^\theta {\frac{\cos (\theta +\varphi )}{\sqrt{K^{2}+\cos (\theta +\varphi )}}\hbox {d}\theta } \right] ,\right. \nonumber \\&\left[ ( \cos \varphi \int \limits _0^\theta {\frac{\sin (\theta +\varphi )}{\sqrt{K^{2}+\cos (\theta +\varphi )}}\hbox {d}\theta }\right. \nonumber \\&\left. \left. -\sin \varphi \int \limits _0^\theta {\frac{\cos (\theta +\varphi )}{\sqrt{K^{2}+\cos (\theta +\varphi )}}\hbox {d}\theta } \right] \right\} ^{\mathrm{T}}+\left( {0,\bar{{a}}} \right) ^{\mathrm{T}} \nonumber \\= & {} -\frac{1}{k}\sqrt{\frac{\sin \varphi }{2}}\left[ {{\begin{array}{cc} {\sin \varphi }&{} {\cos \varphi } \\ {\cos \varphi }&{} {-\sin \varphi } \\ \end{array} }} \right] \nonumber \\&\times \left[ \int \limits _0^\theta {\frac{\sin (\theta +\varphi )}{\sqrt{K^{2}+\cos (\theta +\varphi )}}\hbox {d}\theta },\right. \nonumber \\&\left. \int \limits _0^\theta {\frac{\cos (\theta +\varphi )}{\sqrt{K^{2}+\cos (\theta +\varphi )}}\hbox {d}\theta } \right] ^{\mathrm{T}} +\left( {0,\bar{{a}}} \right) ^{\mathrm{T}} \nonumber \\= & {} -\frac{1}{k}\sqrt{\frac{\sin \varphi }{2}}\left[ {{\begin{array}{cc} {\sin \varphi }&{} {\cos \varphi } \\ {\cos \varphi }&{} {-\sin \varphi } \\ \end{array} }} \right] \left[ {\begin{array}{l} A\left( \theta \right) \\ B\left( \theta \right) \\ \end{array}} \right] +\left( {\begin{array}{l} 0 \\ \bar{{a}} \\ \end{array}} \right) ,\nonumber \\ \end{aligned}$$
(A1)

where

$$\begin{aligned} {\begin{array}{l} A\left( \theta \right) =\int \limits _0^\theta {\frac{\sin (\theta +\varphi )}{\sqrt{K^{2}+\cos (\theta +\varphi )}}\hbox {d}\theta } \\ \qquad \quad \, =-2\left[ {\sqrt{K^{2}+\cos \left( {\theta +\varphi } \right) }-\sqrt{K^{2}+\cos \varphi }} \right] , \\ B\left( \theta \right) =\int \limits _0^\theta {\frac{\cos (\theta +\varphi )}{\sqrt{K^{2}+\cos (\theta +\varphi )}}\hbox {d}\theta }\\ \qquad \quad \,=-\sqrt{2}p\left\{ K^{2}\left[ {F\left( {\frac{\theta }{2}+\frac{\varphi }{2},p} \right) -F\left( {\frac{\varphi }{2},p} \right) } \right] \right. \\ \quad \,\,\quad \qquad \left. -\frac{2}{p^{2}}\left[ {E\left( {\frac{\theta }{2}+\frac{\varphi }{2},p} \right) -E\left( {\frac{\varphi }{2},p} \right) } \right] \right\} . \end{array}} \end{aligned}$$
(A2)

Appendix B

Replacing \(\theta \) in \(A\left( \theta \right) \) of Eq. (12) by \(-\varphi \) and inserting \(A\left( {-\varphi } \right) \) into the first expression of Eq. (14), we obtain

$$\begin{aligned}&-\frac{1}{k}\sqrt{\frac{\sin \varphi }{2}}\left[ {-2\left( {\sqrt{K^{2}+1}-\sqrt{K^{2}+\cos \varphi }} \right) } \right] \nonumber \\&\quad =\bar{{b}}-\bar{{a}}\cos \varphi . \end{aligned}$$
(B1)

Re-arranging Eq. (B1), the following expression arrives

$$\begin{aligned} \sqrt{K^{2}+1}-\sqrt{\frac{\sin \varphi }{2}}k\bar{{b}}\xi (\bar{{a}},\bar{{b}})=\sqrt{K^{2}+\cos \varphi }. \end{aligned}$$
(B2)

Squaring both sides of the Eq. (B2), it is re-expressed as

$$\begin{aligned}&1-\cos \varphi -2\sqrt{K^{2}+1}\sqrt{\frac{\sin \varphi }{2}}k\bar{{b}}\xi (\bar{{a}},\bar{{b}})\nonumber \\&\quad +\frac{\sin \varphi }{2}k^{2}\bar{{b}}^{2}\xi ^{2}(\bar{{a}},\bar{{b}})=0. \end{aligned}$$
(B3)

Considering the expression of \(K^{2}\) in Eq. (7), we find

$$\begin{aligned} K^{2}+1= & {} \frac{\sin \varphi }{2k^{2}\bar{{b}}^{2}}\left[ {\frac{k^{2}\bar{{b}}^{2}}{2}\xi (\bar{{a}},\bar{{b}})-\zeta (\bar{{a}},\bar{{b}})} \right] ^{2}+1-\cos \varphi \nonumber \\= & {} \frac{\sin \varphi }{2k^{2}\bar{{b}}^{2}}\left\{ \left[ {\frac{k^{2}\bar{{b}}^{2}}{2}\xi (\bar{{a}},\bar{{b}})-\zeta (\bar{{a}},\bar{{b}})} \right] ^{2}\right. \nonumber \\&\left. +2k^{2}\bar{{b}}^{2}\frac{1-\cos \varphi }{\sin \varphi } \right\} \nonumber \\= & {} \frac{\sin \varphi }{2k^{2}\bar{{b}}^{2}}\left\{ \left[ {\frac{k^{2}\bar{{b}}^{2}}{2}\xi (\bar{{a}},\bar{{b}})-\zeta (\bar{{a}},\bar{{b}})} \right] ^{2}\right. \nonumber \\&\left. +2k^{2}\bar{{b}}^{2}\xi (\bar{{a}},\bar{{b}})\zeta (\bar{{a}},\bar{{b}}) \right\} \nonumber \\= & {} \frac{\sin \varphi }{2k^{2}\bar{{b}}^{2}}\left[ {\frac{k^{2}\bar{{b}}^{2}}{2}\xi (\bar{{a}},\bar{{b}})+\zeta (\bar{{a}},\bar{{b}})} \right] ^{2}{. } \end{aligned}$$
(B4)

Substituting Eq. (B4) into Eq. (B3), the left side of Eq. (B3) is expressed as

$$\begin{aligned}&1-\cos \varphi -2\sqrt{\frac{\sin \varphi }{2}}\frac{1}{k\bar{{b}}}\left[ {\frac{k^{2}\bar{{b}}^{2}}{2}\xi (\bar{{a}},\bar{{b}})+\zeta (\bar{{a}},\bar{{b}})} \right] \nonumber \\&\qquad \times \sqrt{\frac{\sin \varphi }{2}}k\bar{{b}}\xi (\bar{{a}},\bar{{b}})+\frac{\sin \varphi }{2}k^{2}\bar{{b}}^{2}\xi ^{2}(\bar{{a}},\bar{{b}}) \nonumber \\&\quad =1-\cos \varphi -\sin \varphi \left[ {\frac{k^{2}\bar{{b}}^{2}}{2}\xi ^{2}(\bar{{a}},\bar{{b}})+\xi (\bar{{a}},\bar{{b}})\zeta (\bar{{a}},\bar{{b}})} \right] \nonumber \\&\qquad +\frac{\sin \varphi }{2}k^{2}\bar{{b}}^{2}\xi ^{2}(\bar{{a}},\bar{{b}}) \nonumber \\&\quad =1-\cos \varphi -\sin \varphi \xi (\bar{{a}},\bar{{b}})\zeta (\bar{{a}},\bar{{b}}) \nonumber \\&\quad =1-\cos \varphi -\left( {1-\cos \varphi } \right) \nonumber \\&\quad =0. \end{aligned}$$
(B5)

Thus, the derivation of Eqs. (B1) – (B5) proves that the first expression in Eq. (14) always holds.

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Chen, Q., Sun, F., Li, Z.Y. et al. Nonlinear mechanics of a ring structure subjected to multi-pairs of evenly distributed equal radial forces. Acta Mech. Sin. 33, 942–953 (2017). https://doi.org/10.1007/s10409-017-0665-8

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