# Vacuum Balloon–A 350-Year-Old Dream

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{a}= 1.01×10

^{5}Pa, then $\frac{h}{R}\approx 1.6\times {10}^{-4},\sigma \approx 320\mathrm{MPa}$, i.e., the stress is of the same order of magnitude as the compressive strength of contemporary aluminum alloys. It is important to note that this result does not depend on the radius of the shell.

## 2. Methods and Results

#### 2.1. A Sandwich Vacuum Balloon and Its Buckling Analysis

^{−5}(very small values of the Poisson’s ratio and the in-plane components of the modulus and the shear modulus were used to avoid singularities, in accordance with [19], p. 20).

#### 2.2. Other Modes of Failure

#### 2.3. Towards a Prototype Vacuum Balloon

## 3. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

^{−3}. ${\rho}_{f}$ = density of the face skins of a sandwich shell. ${\rho}_{f}^{\prime}$ = relative density of the face skins of a sandwich shell, ${\rho}_{f}^{\prime}={\rho}_{f}/{\rho}_{a}$. ${\rho}_{s}$ = density of a thin homogeneous shell. $\sigma $ = compressive stress in a thin homogeneous shell. ${\sigma}_{dp}$ = critical uniaxial stress for intracell buckling. ${\sigma}_{f}$ = compressive stress in the face skins of a sandwich shell. ${\sigma}_{wr}$ = allowable uniaxial wrinkling stress (for face skin wrinkling analysis). ${\sigma}_{x},{\sigma}_{y}$ = stresses in the face skins in two orthogonal directions, ${\sigma}_{x}={\sigma}_{y}={\sigma}_{f}$. ${\tau}_{xy}$ = shear stress in the face skins, ${\tau}_{xy}=0$.

## References

- Lana, F. Prodromo. Ouero Saggio di Alcune Inuentioni Nuoue Premesso All’arte Maestra, Rizzardi, Brescia, 1670, Chapter 6. Available online: https://books.google.ru/books?id=o7AGGIKz0_wC&pg=PP9&hl=ru&source=gbs_selected_pages&cad=2#v=onepage&q&f=false (accessed on 8 July 2018). (In Italian).
- Shikhovtsev, E. Available online: http://mir.k156.ru/aeroplan/de_bausset_aeroplane-03-1.html#a03-1-16 (accessed on 5 February 2019). (In Russian and In English).
- Zahm, A.F. Aërial Navigation: A Popular Treatise on the Growth of Air Craft and on Aeronautical Meteorology; D. Appleton and Company: New York, NY, USA; London, UK, 1911; p. 443. Available online: https://books.google.com/books?id=hRdDAAAAIAAJ&pg=PA443&lpg=PA443&dq=%22zahm%22+vacuum+balloon&source=bl&ots=HO8PwEw0M5&sig=AQGKWimBz3oa9Y32TwTnEgAu-UQ&hl=en&sa=X&ved=2ahUKEwiFroi-s4LfAhUq9YMKHWFHBtwQ6AEwAXoECAkQAQ#v=onepage&q=%22zahm%22%20vacuum%20balloon&f=false (accessed on 2 December 2018).
- Akhmeteli, A.M.; Gavrilin, A.V. Layered Shell Vacuum Balloons. U.S. Patent 11/517,915, 8 September 2006. [Google Scholar]
- Timoshenko, S.P.; Gere, J.M.; Prager, W. Theory of Elastic Stability, Second Edition. J. Appl. Mech.
**1962**, 29, 220–221. [Google Scholar] [CrossRef] - Osserman, R. The Isoperimetric Inequality. Bull. Am. Math. Soc.
**1978**, 84, 1182. [Google Scholar] [CrossRef] [Green Version] - Alcocer, A.; Forès, P.; Giuffré, G.P.; Parareda, C.; Roca, A.; Roca, J. Pressure Hull Design and Construction of the Manned Submersible Ictineu 3. Instrum. Viewpoint
**2009**, 8, m3. [Google Scholar] - Armstrong, L.M.; Peoria, I.L. Aircraft of the Lighter-Than-Air Type. U.S. Patent 1,390,745, 13 September 1921. [Google Scholar]
- Barton, S.A. Florida State University Research Foundation (Tallahassee, FL, US), U.S. Patent 7,708,161 for “Light-Weight Vacuum Chamber and Applications Thereof”. U.S. Patent 7,708,161, 4 May 2010. [Google Scholar]
- Snyder, J.W.; Palazotto, A. Finite Element Design and Modal Analysis of a Hexakis Icosahedron Frame for Use in a Vacuum Lighter-Than-Air Vehicle. J. Eng. Mech.
**2018**, 144, 04018042. [Google Scholar] [CrossRef] - Adorno-Rodriguez, R.; Palazotto, A. Nonlinear Structural Analysis of an Icosahedron under an Internal Vacuum. J. Aircr.
**2015**, 52, 878–883. [Google Scholar] [CrossRef] - Rapport, N.; Middleton, W.I. U.S. Patent Application for “Lighter-Than-Air Fractal Tensegrity Structures”. U.S. Patent 14/807118, 23 July 2015. [Google Scholar]
- Jenett, B.; Gregg, C.; Cheung, K. Discrete Lattice Material Vacuum Airship. In Proceedings of the AIAA Scitech 2019 Forum, San Diego, CA, USA, 7–11 January 2019; p. 815. [Google Scholar] [CrossRef] [Green Version]
- Ball, P. Flying on empty. New Sci.
**2019**, 244, 68–69. [Google Scholar] [CrossRef] - Surcouf, O. Dirigeables: Le Miracle du Vide, Science&Vie, No. 1233. June 2020, pp. 90–93. Available online: https://www.science-et-vie.com/technos-et-futur/dirigeables-le-miracle-du-vide-56281 (accessed on 29 July 2020). (In French).
- Available online: http://www.skylinecomponents.com/B4C.html (accessed on 25 December 2018).
- Available online: https://www.plascore.com/honeycomb/honeycomb-cores/aluminum/pamg-xr1-5056-aluminum-honeycomb-core/ (accessed on 25 December 2018).
- Brix, G. Durchschlagen von GFP-Sandwichkuppeln bei gleichförmigem Außendruck, IfL-Mitt. Mitteilung aus dem Institut für Leichtbau und Ökonomische Verwendung von Werkstoffen Dresden
**1968**, 7, 408–413. (In German) [Google Scholar] - Available online: https://www.hexcel.com/user_area/content_media/raw/Honeycomb_Sandwich_Design_Technology.pdf (accessed on 25 December 2018).
- Sullins, R.T.; Smith, G.W.; Spier, E.E. Manual for Structural Stability Analysis of Sandwich Plates and Shells, NASA CR-1457. 1969. Available online: https://apps.dtic.mil/dtic/tr/fulltext/u2/a310684.pdf (accessed on 1 August 2020).
- Yao, J.C. Buckling of Sandwich Sphere under Normal Pressure. J. Aerosp. Sci.
**1962**, 29, 264–268. [Google Scholar] [CrossRef] - Plantema, F.J. Sandwich Construction: The Bending and Buckling of Sandwich Beams, Plates and Shells; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 1966. [Google Scholar]
- Krenzke, M.A.; Kiernan, T.J. Elastic Stability of Near-Perfect Shallow Spherical Shells. AIAA J.
**1963**, 1, 2855–2857. [Google Scholar] [CrossRef] - Krenzke, M.A.; Kiernan, T.J. Erratum: Elastic Stability of Near-Perfect Shallow Spherical Shells. AIAA J.
**1964**, 2, 0784b. [Google Scholar] [CrossRef] - Błażejewski, P.; Marcinowski, J.; Rotter, M. 04.21: Buckling of externally pressurised spherical shells: Experimental results compared with recent design recommendations. Ce/Papers
**2017**, 1, 1010–1018. [Google Scholar] [CrossRef] - Collier, C. Consistent Structural Integrity and Efficient Certification with Analysis, Volume 3, AFRL-VA-WP-TR-2005-3035. 2005. Available online: https://apps.dtic.mil/dtic/tr/fulltext/u2/a444085.pdf (accessed on 27 December 2018).
- Kaiser, A. Hydraulic Pressing of Advanced Ceramics, cfi/Berichte der DKG. 2007, 84, No. 6, pp. 27–32. Available online: http://www.alpha-ceramics.de/system/00/01/52/15245/633855139353281250_1.pdf (accessed on 10 August 2020).
- Kaiser, A.; Lutz, R. Uniaxial Hydraulic Pressing as Shaping Technology for Advanced Ceramic Products of Larger Size, Interceram. 2011. No. 03–04. pp. 230–234. Available online: http://www.laeis.eu/System/00/01/95/19513/634559894557055155_1.pdf (accessed on 10 August 2020).
- Lu, R.; Chandrasekaran, S.; Du Frane, W.L.; Landingham, R.L.; Worsley, M.A.; Kuntz, J.D. Complex shaped boron carbides from negative additive manufacturing. Mater. Des.
**2018**, 148, 8–16. [Google Scholar] [CrossRef] - Chen, R.; Qi, J.; Su, L.; Shi, Q.; Guo, X.; Wu, D.; Lu, T.; Liao, Z. Rapid preparation and uniformity control of B4C ceramic double-curvature shells: Aim to advance its applications as ICF capsules. J. Alloy. Compd.
**2018**, 762, 67–72. [Google Scholar] [CrossRef] - Lee, A.; Brun, P.-T.; Marthelot, J.; Balestra, G.; Gallaire, F.; Reis, P.M. Fabrication of slender elastic shells by the coating of curved surfaces. Nat. Commun.
**2016**, 7, 11155. [Google Scholar] [CrossRef] [PubMed] - Sandwich Panel Fabrication Technology, Hexcel LTU 018. 2001. Available online: https://studylib.net/doc/18103284/sandwich-panel-fabrication-technology (accessed on 21 July 2019).
- Ning, X.; Pellegrino, S. Searching for imperfection insensitive externally pressurized near-spherical thin shells. J. Mech. Phys. Solids
**2018**, 120, 49–67. [Google Scholar] [CrossRef] [Green Version] - Rion, J.; Leterrier, Y.; Månson, J.A.E. Prediction of the adhesive fillet size for skin to honeycomb core bonding in ultra-light sandwich structures. Compos. Part A Appl. Sci. Manuf.
**2008**, 39, 1547–1555. [Google Scholar] [CrossRef]

**Figure 1.**Stress and atmospheric pressure acting on one half of an evacuated spherical shell (not to scale).

**Figure 2.**Dimensions of a spherical sandwich shell (not to scale; note that, e.g., ${h}_{3}\gg {h}_{1}$ for the optimal design).

**Figure 3.**A fragment of a spherical sandwich shell (

**a**) before and (

**b**) after assembly (not to scale).

**Figure 4.**The ANSYS FEA: (

**a**) a 2D axisymmetric model and (

**b**) an enlarged fragment of the 3-layer sandwich shell’s solid model.

**Figure 5.**A fragment of the mesh used for the FEA (portions of the upper face skin and honeycomb are shown). The face skin has 6 divisions radially (the upper six “layers” of elements in the Figure). The honeycomb core has 32 divisions radially.

**Figure 6.**Von Mises stress in the structure (fragment) from the FEA analysis (buckling safety factor of 2.65 and payload fraction of 0.1). The maximum stress in the face skins does not exceed 645 MPa, and the stress in the honeycomb is comparable to the atmospheric pressure.

**Figure 7.**The 3 least eigenvalues ${\lambda}_{min}$, ${\lambda}_{2}$, and ${\lambda}_{3}$ from the ANSYS eigenvalue buckling analysis versus the relative core thickness ${h}_{3}^{\prime}$ for payload fraction $q=0.1$.

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**MDPI and ACS Style**

Akhmeteli, A.; Gavrilin, A.V.
Vacuum Balloon–A 350-Year-Old Dream. *Eng* **2021**, *2*, 480-491.
https://doi.org/10.3390/eng2040030

**AMA Style**

Akhmeteli A, Gavrilin AV.
Vacuum Balloon–A 350-Year-Old Dream. *Eng*. 2021; 2(4):480-491.
https://doi.org/10.3390/eng2040030

**Chicago/Turabian Style**

Akhmeteli, Andrey, and Andrew V. Gavrilin.
2021. "Vacuum Balloon–A 350-Year-Old Dream" *Eng* 2, no. 4: 480-491.
https://doi.org/10.3390/eng2040030