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Mechanical Stability of Eccentrically Stiffened Auxetic Truncated Conical Sandwich Shells Surrounded by Elastic Foundations

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Mechanics of Composite Materials Aims and scope

The static stability of auxetic truncated conical sandwich shells reinforced by stiffeners surrounded by elastic foundations is investigated. The shells are made from two isotropic outer layers and an auxetic core layer with a negative Poisson ratio. Based on the classical shell theory, combined with the displacement and Bubnov–Galerkin methods, the governing equations of the shells are derived and solved. The critical buckling load of the shells as a function of their geometrical parameters, the honeycomb structure, stiffeners, and types of elastic foundations are examined in detail.

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Acknowledgement

This research was funded by the Grant number CN.21.06 of VNU Hanoi — University of Engineering and Technology. The authors are grateful for this support. Pham Dinh Nguyen was funded by Vingroup Joint Stock Company and supported by the Domestic Master/ PhD Scholarship Programme of Vingroup Innovation Foundation (VINIF), Vingroup Big Data Institute (VINBIGDATA), code VINIF.2020.TS.17.

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Authors and Affiliations

  1. Department of Engineering and Technology of Constructions and Transportation, VNU – Hanoi – University of Engineering and Technology (UET), 144 – Xuan Thuy – Cau Giay, Hanoi, Vietnam

    Nguyen Dinh Duc, Duong Tuan Manh, Nguyen Dinh Khoa & Pham Dinh Nguyen

  2. NTT Institute of High Technology, Nguyen Tat Thanh University, Ho Chi Minh City, Vietnam

    Nguyen Dinh Duc

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  1. Nguyen Dinh Duc
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  2. Duong Tuan Manh
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  3. Nguyen Dinh Khoa
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Correspondence to Nguyen Dinh Duc.

Additional information

Translated from Mekhanika Kompozitnykh Materialov, Vol. 58, No. 3, pp. 521-544, May-June, 2022. Russian DOI: https://doi.org/10.22364/mkm.58.3.04.

Appendix

Appendix

$$ {\displaystyle \begin{array}{c}{A}_{66}={Q}_{44}^T\left(\int_{\frac{h_2}{2}}^{\frac{h_2}{2}+{h}_1} dz+\int_{\frac{h_2}{2}{h}_3}^{\frac{h_2}{2}} dz\right)+{Q}_{44}^C\int_{-\frac{h_2}{2}}^{-\frac{h_2}{2}} dz,{B}_{66}={Q}_{44}^T\left(\int_{\frac{h_2}{2}}^{\frac{h_2}{2}+{h}_1} zdz+\int_{\frac{h_2}{2}{h}_3}^{\frac{h_2}{2}} zdz\right)+{Q}_{44}^C\int_{-\frac{h_2}{2}}^{-\frac{h_2}{2}} zdz,\\ {}{D}_{11}={Q}_{11}^T\left(\int_{\frac{h_2}{2}}^{\frac{h_2}{2}+{h}_1}{z}^2 dz+\int_{\frac{h_2}{2}{h}_3}^{\frac{h_2}{2}}{z}^2 dz\right)+\int_{-\frac{h_2}{2}}^{\frac{h_2}{2}}{z}^2 dz,{D}_{12}={Q}_{12}^T\left(\int_{\frac{h_2}{2}}^{\frac{h_2}{2}+{h}_1}{z}^2 dz+\int_{\frac{h_2}{2}{h}_3}^{\frac{h_2}{2}}{z}^2 dz\right)+{Q}_{12}^C\int_{-\frac{h_2}{2}}^{\frac{h_2}{2}}{z}^2 dz,\\ {}{D}_{22}={Q}_{22}^T\left(\int_{\frac{h_2}{2}}^{\frac{h_2}{2}+{h}_1}{z}^2 dz+\int_{-\frac{h_2}{2}{h}_3}^{-\frac{h_2}{2}}{z}^2 dz\right)+{Q}_{22}^C\int_{\frac{h_2}{2}}^{\frac{h_2}{2}}{z}^2 dz+{E}^S\frac{d_{\theta }}{s_{\theta }}\int_{\frac{h_2}{2}+{h}_1}^{\frac{h_2}{2}+{h}_1+h}{z}^2 dz,\\ {}{D}_{66}={Q}_{44}^T\left(\int_{\frac{h_2}{2}}^{\frac{h_2}{2}+{h}_1}{z}^2 dz+\int_{-\frac{h_2}{2}{h}_3}^{-\frac{h_2}{2}}{z}^2 dz\right)+{Q}_{44}^C\int_{-\frac{h_2}{2}}^{-\frac{h_2}{2}}{z}^2 dz.\\ {}{c}_{11}\left({u}_1\right)={A}_{11}\frac{\partial }{\partial x}\left({u}_1\right)-{A}_{22}\frac{\partial }{x}\left({u}_1\right)+x{A}_{11}\frac{\partial^2}{\partial {x}^2}\left({u}_1\right)+{A}_{66}\frac{1}{x\sin^2}\frac{\partial^2}{\partial {\theta}^2}\left({u}_1\right),\\ {}{C}_{12}\left({v}_1\right)=\left({A}_{66}+{A}_{12}\right)\frac{1}{\sin \alpha}\frac{\partial^2}{\partial x\partial \theta}\left({v}_1\right)-\left({A}_{66}+{A}_{22}\right)\frac{1}{x\sin \alpha}\frac{\partial }{\partial \theta}\left({v}_1\right),\\ {}{C}_{13}\left({w}_1\right)=\left({B}_{22}\frac{1}{x}+{A}_{12}\cot \alpha \right)\frac{\partial }{\partial x}\left({w}_1\right)-{A}_{22}\frac{1}{x}\cot \alpha \left({w}_1\right)-{B}_{11}x\frac{\partial^3}{\partial {x}^3}\left({w}_1\right)\\ {}+\left({B}_{12}+2{B}_{66}+{B}_{22}\right)\frac{1}{x^2{\sin}^2\alpha}\frac{\partial^2}{\partial {\theta}^2}\left({w}_1\right)-\left({B}_{12}+2{B}_{66}\right)\frac{1}{x\sin^2\alpha}\frac{\partial^3}{\partial x\partial {\theta}^2}\left({w}_1\right)-{B}_{11}\frac{\partial^2}{\partial {x}^2}\left({w}_1\right),\\ {}{C}_{21}\left({u}_1\right)=\left({A}_{66}+{A}_{12}\right)\frac{1}{\sin \alpha}\frac{\partial^2}{\partial x\partial \theta}\left({u}_1\right)+\left({A}_{66}+{A}_{22}\right)\frac{1}{x\sin \alpha}\frac{\partial }{\partial \theta}\left({u}_1\right),\\ {}{C}_{22}\left({v}_1\right)=\frac{A_{22}}{x\sin \alpha}\frac{\partial^2}{\partial {\theta}^2}\left({v}_1\right)+x{A}_{66}\frac{\partial^2}{\partial {x}^2}\left({v}_1\right)+{A}_{66}\frac{\partial }{\partial x}\left({v}_1\right)-{A}_{66}\frac{1}{x}\left({v}_1\right),\\ {}{C}_{23}\left({w}_1\right)={A}_{22}\frac{\cot \alpha }{x\sin \alpha}\frac{\partial }{\partial \theta}\left({w}_1\right)-\left({B}_{12}+2{B}_{66}\right)\frac{1}{\sin \alpha}\frac{\partial^3}{\partial {x}^2\partial \theta}\left({w}_1\right)\\ {}-{B}_{22}\frac{1}{x^2{\sin}^3\alpha}\frac{\partial^3}{\partial {\theta}^3}\left({w}_1\right)-{B}_{22}\frac{1}{x\sin \alpha}\frac{\partial^2}{\partial x\partial \theta}\left({w}_1\right).\\ {}{C}_{31}\left({u}_1\right)=\left(\frac{B_{22}}{x^2}-\frac{A_{22}}{x}\cot \alpha \right)\left({u}_1\right)-\left(\frac{B_{22}}{x}+{A}_{12}\cot \alpha \right)\frac{\partial }{\partial x}\left({u}_1\right)+2{B}_{11}\frac{\partial^2}{\partial {x}^2}\left({u}_1\right)\\ {}+\frac{1}{x^2{\sin}^2\alpha }{B}_{22}\frac{\partial^2}{\partial {\theta}^2}\left({u}_1\right)+\frac{1}{x\sin^2\alpha}\left(2{B}_{66}+{B}_{12}\right)\frac{\partial^3}{\partial x\partial {\theta}^2}\left({u}_1\right)+x{B}_{11}\frac{\partial^3}{\partial {x}^3}\left({u}_1\right),\\ {}{C}_{32}\left({v}_1\right)=\frac{1}{x\sin \alpha}\left(\frac{B_{22}}{x}-{A}_{22}\cot \alpha \right)\frac{\partial }{\partial \theta}\left({v}_1\right)-{B}_{22}\frac{1}{x\sin \alpha}\frac{\partial^2}{\partial x\partial \theta}\left({v}_1\right)\\ {}+{B}_{22}\frac{1}{x^2{\sin}^3\alpha}\frac{\partial^3}{\partial {\theta}^3}\left({v}_1\right)+\frac{1}{\sin \alpha}\left(2{B}_{66}+{B}_{12}\right)\frac{\partial^3}{\partial {x}^2\partial \theta}\left({v}_1\right),\\ {}{C}_{33}\left({w}_1\right)=\left({B}_{22}\frac{\cot \alpha }{x^2}-{A}_{22}\frac{\cot^2\alpha }{x}-x{K}_1\right)\left({w}_1\right)+\left({K}_2-{D}_{22}\frac{1}{x^2}\right)\frac{\partial \left({w}_1\right)}{\partial x}\\ {}+\left(x{K}_2+{D}_{22}\frac{1}{x}+2{B}_{12}\cot \alpha \right)\frac{\partial^2\left({w}_1\right)}{\partial {x}^2}\\ {}+\left(2{B}_{22}\frac{\cot \alpha }{x^2{\sin}^2\alpha }-\left({D}_{12}+{D}_{22}+2{D}_{66}\right)\frac{2}{x^3{\sin}^2\alpha }+\frac{K_2}{x\sin^2\alpha}\right)\frac{\partial \left({w}_1\right)}{\partial {\theta}^2}\\ {}-2{D}_{11}\frac{\partial^3\left({w}_1\right)}{\partial {x}^3}+\left({D}_{12}+2{D}_{66}\right)\frac{2}{x^2{\sin}^2\alpha}\frac{\partial^3\left({w}_1\right)}{\partial x\partial {\theta}^2}-x{D}_{11}\frac{\partial^3\left({w}_1\right)}{\partial {x}^4}-{D}_{22}\frac{1}{x^3{\sin}^4\alpha}\frac{\partial^4\left({w}_1\right)}{\partial {\theta}^4}\\ {}-D\left({D}_{12}+2{D}_{66}\right)\frac{2}{x\sin^2\alpha}\frac{\partial^4\left({w}_1\right)}{\partial {x}^2\partial {\theta}^2},{C}_{34}\left({w}_1\right)=-\frac{1}{\pi \sin \left(2\alpha \right)}\frac{\partial^2\left({w}_1\right)}{\partial {x}^2}.\\ {}{d}_{11}=\frac{A_{11}\pi \sin \left(\alpha \right)L\left(L+2{x}_0\right)}{4}\left[{A}_{22}\sin \left(\alpha \right)+\frac{A_{66}{n}^2}{4\sin \left(\alpha \right)}\right]\left[\frac{\pi L\left(L+2{x}_0\right)}{4}\right]\\ {}-\frac{A_{11}{m}^2{\pi}^3}{8{L}^2}\sin \left(\alpha \right)\left[{\left(L+{x}_0\right)}^4-{x_0}^4+\frac{3{L}^3\left(L+2{x}_0\right)}{m^2{\pi}^2}\right],\\ {}{d}_{12}=-\frac{\left({A}_{66}+{A}_{12}\right) mn{\pi}^2}{12L}\left[{\left(L+{x}_0\right)}^3-{x_0}^3+\frac{3{L}^3}{2{m}^2{\pi}^2}\right]-\frac{\left({A}_{66}+{A}_{22}\right)n{L}^2}{8m},\\ {}{d}_{13}=\left[{B}_{22} m\pi \sin \left(\alpha \right)+\frac{\left({B}_{12}+2{B}_{66}\right)m{n}^2\pi }{4\sin \left(\alpha \right)}\right]\frac{\pi \left(L+2{x}_0\right)}{4}\\ {}+\frac{A_{12}m{\pi}^2\cos \left(\alpha \right)}{6L}\left[{\left(L+{x}_0\right)}^3-{x_0}^3+\frac{3{L}^3}{2{m}^2{\pi}^2}\right]+\frac{A_{22}\cos \left(\alpha \right){L}^2}{4m}\\ {}-\frac{B_{11}m{\pi}^2\sin \left(\alpha \right)\left(L+2{x}_0\right)}{4},\\ {}{d}_{21}=-\frac{\left({A}_{66}+{A}_{12}\right) mn{\pi}^2}{12L}\left[{\left(L+{x}_0\right)}^3-{x_0}^3-\frac{3{L}^3}{2{m}^2{\pi}^2}\right]-\frac{\left({A}_{66}+{A}_{22}\right)n{L}^2}{8m},\\ {}{d}_{22}=-\frac{A_{22}{n}^2 L\pi \left(L+2{x}_0\right)}{16\sin \left(\alpha \right)}-\frac{A_{22}\sin \left(\alpha \right) L\pi \left(L+2{x}_0\right)}{2}\\ {}-\frac{A_{66}{m}^2{\pi}^3\sin \left(\alpha \right)}{8{L}^2}\left[{\left(L+{x}_0\right)}^4-{x_0}^4\right]+\frac{3\pi {A}_{22}\sin \left(\alpha \right)L\left(L+2{x}_0\right)}{8},\\ {}\begin{array}{c}{d}_{23}=\frac{A_{22}\cot \left(\alpha \right) n L\pi \left(L+2{x}_0\right)}{8}+\frac{B_{22}{n}^3 L\pi}{16\sin^2\left(\alpha \right)}\\ {}+\frac{\left({B}_{12}+2{B}_{66}\right){m}^2{\pi}^3n}{12{L}^2}\left[{\left(L+{x}_0\right)}^3-{x_0}^3-\frac{3{L}^3}{2{m}^2{\pi}^2}\right]+\frac{B_{22} n\pi L}{8},\\ {}{d}_{31}={B}_{22}\left[\frac{n^2}{4\sin \left(\alpha \right)}-\sin \left(\alpha \right)\right]\frac{L^2}{4m}+\frac{A_{22}\cos \left(\alpha \right){L}^2\left(L+2{x}_0\right)}{4m}\\ {}+\left[\frac{B_{22}m{\pi}^2\sin \left(\alpha \right)}{6L}+\frac{\left({B}_{12}+2{B}_{66}\right)m{\pi}^2{n}^2}{24\sin \left(\alpha \right)L}\right]\left[{\left(L+{x}_0\right)}^3-{x_0}^3-\frac{3{L}^3}{2{m}^2{\pi}^2}\right]\\ {}+\frac{A_{12}\cos \left(\alpha \right)m{\pi}^2}{8L}\left[{\left(L+{x}_0\right)}^4-{x_0}^4-\frac{3{L}^3\left(L+2{x}_0\right)}{m^2{\pi}^2}\right]\\ {}+\frac{B_{11}m{\pi}^2\sin \left(\alpha \right)}{2L}\left[{\left(L+{x}_0\right)}^3-{x_0}^3-\frac{3{L}^3}{2{m}^2{\pi}^2}\right]\\ {}+\frac{B_{11}{m}^3{\pi}^4\sin \left(\alpha \right)}{L^3}\left[\frac{{\left(L+{x}_0\right)}^5-{x_0}^5}{10}+\frac{3{L}^3}{4{m}^4{\pi}^4}+{L}^2\frac{{x_0}^3-{\left(L+{x}_0\right)}^3}{2{m}^2{\pi}^2}\right],\\ {}{d}_{32}={B}_{22}n\left[\frac{n^2}{8\sin^2\left(\alpha \right)}-1\right]\frac{L\pi \left(L+2{x}_0\right)}{4}+\frac{A_{22}\cot \left(\alpha \right) n\pi}{12}\left[{\left(L+{x}_0\right)}^3-{x_0}^3-\frac{3{L}^3}{2{m}^2{\pi}^2}\right]\\ {}+\frac{\left({B}_{12}+2{B}_{66}\right){m}^2{\pi}^3n}{16{L}^2}\left[{\left(L+{x}_0\right)}^4-{x_0}^4-\frac{3{L}^3\left(L+{x}_0\right)}{m^2{\pi}^2}\right],\\ {}{d}_{33}={B}_{22}\cos \left(\alpha \right)\left[1-\frac{n^2}{2\sin^2\left(\alpha \right)}\right]\left[\frac{L\pi \left(L+2{x}_0\right)}{4}\right]\\ {}-\left[\frac{A_{22}{\cot}^2\left(\alpha \right)\sin \left(\alpha \right)\pi }{6}+\frac{D_{22}{m}^2{\pi}^3\sin \left(\alpha \right)}{6{L}^2}+\frac{\left({D}_{12}+2{D}_{66}\right){m}^2{n}^2{\pi}^3}{12{L}^2\sin \left(\alpha \right)}\right]\left[{\left(L+{x}_0\right)}^3-{x_0}^3-\frac{3{L}^3}{2{m}^2{\pi}^2}\right]\\ {}+\left[{D}_{22}\sin \left(\alpha \right)+\frac{\left({D}_{12}+2{D}_{66}\right){n}^2}{2\sin \left(\alpha \right)}\right]\frac{\pi L}{4}\\ {}-\frac{B_{12}{m}^2{\pi}^3\cos \left(\alpha \right)}{4{L}^2}\left[{\left(L+{x}_0\right)}^4-{x_0}^4-\frac{3{L}^3\left(L+2{x}_0\right)}{m^2{\pi}^2}\right]\\ {}+\frac{L\pi {n}^2}{4\sin \left(\alpha \right)}\left[\left({D}_{12}+{D}_{22}+2{D}_{66}\right)-\frac{D_{22}{n}^2}{8\sin^2\left(\alpha \right)}\right]\\ {}-\frac{D_{11}{m}^2{\pi}^3\sin \left(\alpha \right)}{2{L}^2}\left[{\left(L+{x}_0\right)}^3-{x_0}^3-\frac{3{L}^3}{2{m}^2{\pi}^2}\right]\\ {}-\frac{D_{11}{m}^4{\pi}^5\sin \left(\alpha \right)}{L^4}\left[\frac{{\left(L+{x}_0\right)}^5-{x_0}^5}{10}+\frac{3{L}^5}{4{m}^4{\pi}^4}+\frac{{x_0}^3-{\left(L+{x}_0\right)}^3}{2{m}^2{\pi}^2}{L}^2\right],\\ {}{d}_{34}=\frac{m^2{\pi}^2}{16{L}^2\cos \left(\alpha \right)}\left[{\left(L+{x}_0\right)}^4-{x_0}^4-\frac{3{L}^3\left(L+2{x}_0\right)}{m^2{\pi}^2}\right],\\ {}{d}_{35}=-\sin \left(\alpha \right)\pi \left[\frac{{\left(L+{x}_0\right)}^5-{x_0}^5}{10}+\frac{3{L}^5}{4{m}^4{\pi}^4}+\frac{{x_0}^3-{\left(L+{x}_0\right)}^3}{2{m}^2{\pi}^2}{L}^3\right],\\ {}{d}_{36}=-\frac{\pi \sin \left(\alpha \right)}{4}\left[\left(L+{x_0}^3\right)-{x_0}^3\frac{3{L}^3}{2{m}^2{\pi}^2}\right]\\ {}\begin{array}{c}-\frac{m^2{\pi}^3\sin \left(\alpha \right)}{L^2}\left[\frac{{\left(L+{x}_0\right)}^5-{x_0}^5}{10}+\frac{3{L}^5}{4{m}^4{\pi}^4}+{L}^2\frac{{x_0}^3-{\left(L+{x}_0\right)}^3}{2{m}^2{\pi}^2}\right]\\ {}-\frac{n^2\pi }{24\sin \left(\alpha \right)}\left[{\left(L+{x}_0\right)}^3-{x_0}^3-\frac{3{L}^3}{2{m}^2{\pi}^2}\right].\end{array}\end{array}\end{array}} $$

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Duc, N.D., Manh, D.T., Khoa, N.D. et al. Mechanical Stability of Eccentrically Stiffened Auxetic Truncated Conical Sandwich Shells Surrounded by Elastic Foundations. Mech Compos Mater 58, 365–382 (2022). https://doi.org/10.1007/s11029-022-10035-0

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