Vacuum balloon - a 350-year-old dream

The centuries-old idea of a lighter-than-air vacuum balloon has not materialized yet as such structure needs to be both light enough to float in the air and strong enough to withstand atmospheric pressure. We propose a design of a rigid spherical sandwich shell and demonstrate that it can satisfy these stringent conditions with commercially available materials, such as boron carbide ceramics and aluminum alloy honeycombs. A finite element analysis was employed to demonstrate that buckling can be prevented in such a structure. Other modes of failure were evaluated. Approaches to manufacturing are discussed briefly.


I. Introduction
HE idea of a lighter-than-air vacuum balloon is centuries old. In 1670, F. Lana di Terzi proposed a design of an airship where buoyancy was to be provided by evacuated copper spheres (Ref. [1], see also Ref. [2] containing historical information related to the design). However, this dream has not materialized so far, because it is very difficult to design and manufacture a shell that is light enough to float in the air and strong enough to reliably withstand the atmospheric pressure. For example, A.F. Zahm (Ref. [3]) calculated the stress in a thin homogeneous one-layer rigid shell with vacuum inside and zero buoyancy, so that its mass equals that of the displaced air: where is the radius of the shell, ℎ is the shell thickness, and are the densities of air and of the shell material, respectively (we use an approximation for a thin shell). Let us then consider the condition for equilibrium of a half of the shell (see Fig.1): where is the compressive stress in the shell and is the atmospheric pressure at normal conditions (we used a condition of equilibrium for a hemisphere of air in the atmosphere to calculate the right-hand side).
We obtain:  A.F. Zahm notes that, while the results of stress calculation are quite problematic, buckling is an even more dangerous mode of failure for such a structure. Let us perform a simple buckling analysis for the structure (Ref. [4]). The critical buckling pressure for a thin spherical shell is given by the well-known formula of the linear theory of stability (Ref. [5]): where and are the modulus of elasticity and the Poisson's ratio of the material of the shell, respectively. If = and, e.g., = 0.3, then Even if we use diamond as the shell material ( = 1.2 • 10 12 and = 3500 • 3 ), we obtain 2 ≈ 10 5 −1 5 −2 .

(6)
In other words, even the maximally optimized homogeneous diamond shell of ideally spherical shape would inevitably fail already at ~0.2 . Thus, one-layer shells made of any solid material in existence either cannot float in the air or have no chance of withstanding the atmospheric pressure. (It should be noted that we only considered the spherical shape in our analysis, as this shape is certainly the optimal one.) Problems of this kind are quite common in aircraft design, and typical solutions are multi-layer shells with light core or stiffened shells. In our patent application (Ref. [4]), we defined viable designs of a vacuum balloon based on threelayer shells made of commercially available materials. Numerous patents and articles on vacuum balloons had been published earlier (see, e.g., Ref. [6]), but, to the best of our knowledge, none of them properly addressed the crucial issue of buckling. More recently, other work addressing the issue of buckling for vacuum balloons was published (see, e.g., Refs. [7][8][9][10][11] and references there).
As our design (Ref. [4]) attracted some interest, it is advisable to describe it here, in a journal article format, after significant rework, providing details of the all-important buckling analysis. Detailed comparison with the widely differing designs of related work by others (Refs. [7][8][9][10][11]) is beyond the scope of this article. Let us just note that our design, unlike designs by others, is spherically symmetric and scalable (see some caveats below), so it has few parameters, which facilitates its analytical optimization. It is also noteworthy that the design does not contain any elements under tension.

II. A Sandwich Vacuum Balloon and Its Buckling Analysis
As an example, let us consider a three-layer spherical shell with face sheets of equal thickness ℎ 1 = ℎ 2 and a core of aluminum alloy honeycombs of thickness ℎ 3 (see Fig. 2

Fig 2 A fragment of a spherical sandwich shell a) before and b) after assembly (not to scale, imprecise colors).
In order to prove the feasibility, we used parameters of commercially available materials in our study. Boron carbide ceramics was chosen as the face sheet material (density If is the radius of the shell, we assume that ≫ ℎ 3 ≫ ℎ 1 . To assess the feasibility, let us anticipate also that the shell allows a small payload fraction = 0.1 (the ratio of the mass of the payload at zero buoyancy and the mass of the displaced air). Then, the condition of zero buoyancy has the following form: where The buckling stability condition that we used is described by the following semi-empirical formula for critical pressure obtained for three-layer domes (Ref. [14]): In this case, is the modulus of elasticity of the face sheet material, and is the maximum pressure at which the three-layer shell is stable. The requirements for core rigidity are discussed below, but let us first find the values of ℎ 1 ′ and ℎ 3 ′ that maximize . Using Eq. (8), let us eliminate ℎ 1 ′ from Eq. (9): The value of is maximal for In that case the optimal values of ℎ 1 ′ and are: This is a good indication of the design feasibility. However, we need to assess the buckling stability more accurately and consider other modes of failure. The results of the above approach were verified and adjusted by a finite element analysis (FEA) using ANSYS, which enabled us to compute the stress and strain in the shell components and to perform the eigenvalue buckling analysis (that is actually a classical Euler buckling analysis). In so doing, a 2D axisymmetric model in the spherical system of coordinates was found sufficient and adequate (Fig. 3). In this model, PLANE82 2D high-order 8-node elements, which are well suited for curved boundaries, were used for the finite element mesh in axisymmetric mode. The mesh was refined appropriately, and the elements' aspect ratios were adjusted.

Fig. 3 A 2D axisymmetric model used in the ANSYS FEA: a quarter of the spherical shell with corresponding boundary conditions at the edges (left). An enlarged fragment of the 3-layer sandwich shell's solid model (right).
In the FEA, the honeycombs were treated in accordance with recommendations of a honeycomb manufacturer (Ref. [15], p, 20). For the sake of simplicity, we assume that the honeycombs are a transversally isotropic material, so the lesser of the two values of shear modulus from Ref.
[13] was used (this makes the results more conservative).
There is the following relation between the minimum eigenvalue obtained in the eigenvalue buckling analysis and the critical pressure: The optimized parameters of the analytical approach were used as initial values for optimization through FEA.
The eigenvalue , regarded as a function of ℎ 3 ′ , has a rather sharp maximum of ≈ 2.65 (see Fig. 4) for a value of ℎ 3 ′ ≈ 3.53 • 10 −3 , which is close to the value we arrived at using the simplified method. The corresponding value of ℎ 1 ′ approximates 4.23 • 10 −5 . For comparison, the eigenvalue is approximately 3.21 for an optimized design with zero payload. Fig. 4 The three least eigenvalues , , and (modes 1, 2, and 3 in ANSYS) from the eigenvalue buckling analysis depending on the relative core thickness: ′ coincides with ′ up to a factor depending on the shell material properties.
The safety factor of 2.65 is not very high, as empirical knock-down factors are typically applied to the results of small-deflection analysis for externally-pressurized thin-walled spherical shells to take into account initial imperfections and other factors. For example, the knock-down factor of 0.2 is recommended in Ref. [16, p. 6-9] for hemispherical sandwich domes (there is a very good agreement between our FEA results and the results obtained with the use of formulas in Ref. [16, p. 6-6] with a knock-down factor of 1 for buckling critical pressure/stress). The formulas of Ref. [16] are based on the solution from Refs. [17,18]). If we perform linear (and, if required, non-linear) buckling analysis with due regard for imperfections of manufacturing, the safety factors will decrease, but the obtained results are high enough to reasonably expect that the safety factors will still be quite sufficient for the state-of-the-art manufacturing accuracy, as thoroughly manufactured thin spherical shells were shown to withstand external pressure of up to 80-90% of the critical one (Ref. [19,20]). Taking into account the imperfections is beyond the scope of this work as they depend on the peculiarities of specific technologies.

III. Other Modes of Failure
Now let us verify that other modes of failure are not problematic for the design. Let us first check that the compressive stress in the face sheets does not exceed the compressive strength of the face sheet material. Instead of Eq. (2) we have: which is much less than the compressive strength of boron carbide (3.2 ).
where is the allowable force per unit length of a sandwich plate in some direction and 3 = 20 ≈ 138 is the honeycomb shear modulus. The actual force per unit length is much less: The design was checked for face sheet wrinkling using the following formula (Ref. [21, p. 234]):  The design is scalable with respect to all of the above modes of failure (an equally successful design can be obtained by multiplying all linear dimensions by the same factor). However, this is not true for another mode of failure -intracell buckling (also known as dimpling). We use the following formula (Ref. [21], p. 241): where is the critical stress for intracell buckling and = 0.17 is the Poisson's ratio of the face sheets, and = where we can assume for our purposes that = 3 (see Ref. [21], p. 243) and obtain the condition of stability for

IV. Towards a Prototype Vacuum Balloon
The design of this article requires large and very thin face sheets made of boron carbide (the thickness of the face sheets is 0.1 mm by order of magnitude, unless the radius of the vacuum balloon is dozens of meters or larger). On the one hand, producing such parts is technologically challenging, on the other hand, the parts may be too fragile.
Detailed treatment of these issues is beyond the scope of this article, but a preliminary discussion is clearly necessary.
The face sheets can be produced either by deposition on a sacrificial substrate (this can be time-consuming if the process is to yield high elasticity modulus) or using gelcasting, which can provide "fine features down to 100 scale" (Ref. [22]).
To circumvent the issue of fragility, one can first bond the face sheets to the honeycomb and then remove the face sheet supports (the substrates or parts of the molds).
To fabricate the entire vacuum balloon, one will need to join several sandwich panels using some standard approach, such as bonded butt joints using H sections (Ref. [23]).
We did not discuss issues related to adhesives here (mass requirements, modes of failure, etc.) (these issues are less significant for shells of larger radius). Neither did we discuss potential use of more exotic materials (such as CVD (chemical vapor deposition) diamond for the face sheets or architected cellular materials (Ref. [11]) for the core) to increase safety factors and/or the payload fraction.

V. Conclusion
We showed that a lighter-than-air rigid vacuum balloon can be built using commercially available materials. A prototype vacuum balloon would also become the first lighter-than-air solid (as, for example, aerogels are actually not lighter than air due to the air inside).
It took mathematicians 357 years to prove Fermat's Last Theorem. Will it take us more to build the first vacuum balloon?