# Effect of Gravity on the Scale of Compliant Shells

^{*}

## Abstract

**:**

## 1. Introduction

_{i}), which is the ratio of the elastogravity length scale [20,21] to the characteristic dimension of the shell. The elastogravity length scale determines the limit at which bending deformations due to gravity appear in the shell. Finally, using this newly introduced parameter, we measure the scale at which compliant shells become highly susceptible to gravity induced deformations.

## 2. Methodology

_{FvK}, describes the type of deformation that dominates the mechanical behavior of a thin shell. The gravity impact number (G

_{i}) characterizes the influence of the gravitational force on the shell. The thin shells are considered isotropic elastic in this study.

#### 2.1. Sample of Fixed and Compliant Shells

- Engineered stiff shells [15]. 25 large scale reinforced concrete (high Young’s modulus) thin shells used in buildings and architecture, their shape is fixed and can carry external applied loads. Their characteristic dimension (R) is in the $\left[6\times {10}^{\xb0}\mathrm{m};8\times {10}^{1}\mathrm{m}\right]$ range, while their thickness H is in the $\left[5\times {10}^{-2}\mathrm{m};4\times {10}^{-1}\mathrm{m}\right]$ range
- Engineered compliant shells [12,16,17,18,24,25,26,27,28,29,30,31,32]. 18 shells designed for use as mechanisms, they are very flexible. Materials are varied but all have high Young’s modulii. $R$ in $\left[2\times {10}^{-2}\mathrm{m};8\times {10}^{0}\mathrm{m}\right]$ and $H\mathrm{in}\text{}\left[1.2\times {10}^{-4}\mathrm{m};9\times {10}^{-4}\mathrm{m}\right]$
- Plant compliant shells [33,34,35]. 8 plant structures that can be described as thin shells and exhibit fast and repeated motions. The material is a living tissue of low Young’s modulus ($~{10}^{6}\mathrm{N}/{\mathrm{m}}^{2}$). $R$ in $\left[1.5\times {10}^{-4}\text{}\mathrm{m};1\times {10}^{-2}\text{}\mathrm{m}\right]$ and $H\mathrm{in}\text{}\left[3\times {10}^{-5}\mathrm{m};4\times {10}^{-4}\mathrm{m}\right]$
- Avian egg stiff shells [36,37,38]. 8 stiff bird eggshells. The geometry is rigid and the material is carbon silicate of various mechanical properties detailed in [38]. $R$ in $\left[3\times {10}^{-2}\mathrm{m};1.55\times {10}^{-1}\text{}\mathrm{m}\right]$ and $H\mathrm{in}\text{}\left[2.2\times {10}^{-4}\text{}\mathrm{m};2.55\times {10}^{-3}\text{}\mathrm{m}\right]$
- Micro-scale compliant shells [8,19,39,40,41,42]. 5 types of shells from red blood cells to virus. They have been described mechanically as a shell and deform significantly in operation. They are highly flexible. $R$ in $\left[2\times {10}^{-8}\mathrm{m};5\times {10}^{-4}\text{}\mathrm{m}\right]$ and $H\mathrm{in}\text{}\left[2\times {10}^{-9}\mathrm{m};1\times {10}^{-6}\mathrm{m}\right]$

#### 2.2. Quantification of Bending versus Stretching Deformation

^{3}[45]. For equal energy levels, bending deformations can be much larger than stretching deformations. Therefore, bending allows the shell to deform with less impact on the overall elastic strain energy compared to stretching. Since isotropic thin shells are considered, the general form for the surface strain energy density $W$ is given by the equation [45]

_{FvK}, number [19] is used. This number measures the ratio of stretching to bending strain energy densities and is given by

_{FvK}number predicts the type of deformation a shell will experience. Very large values of γ

_{FvK}indicate that the shell behaves similarly to a membrane. Such shells accommodate elastic compressive strain by wrinkling and if very thin, crumpling [47]. The shells with high values of γ

_{FvK}exhibit large bending and low stretching. Lower values of γ

_{FvK}correspond to thicker shells that have a high bending stiffness. Such shells have both bending and stretching deformations and require large applied loading to be deformed. The ideal behavior for a thin shell used as a mechanism is characterized by a small actuation force that results in both bending deformations and preservation of the smoothness of the surface (i.e., no crumpling or wrinkling). This force can only be made small if it activates is low stiffness deformation mode of the shell [16]. This ideal behavior occurs if the shell is stiff enough to have a bending stiffness that preserve the continuity of the material under loading and flexible enough to allow large elastic out-of-plane deformation. The instances of compliant shells selected in this study fit this description, their γ

_{FvK}values can be considered as characteristic for compliant shells. The range of γ

_{FvK}values in this study is 10

^{3}to 10

^{8}(see Section 3). However, physically, the maximum value of the γ

_{FvK}number is γ

_{FvK}$\approx {10}^{14}$. This value does not occur for shells since it describes the behavior of a 200 $\mathsf{\mu}\mathrm{m}$ square sheet of graphene [48]. Graphene is one-atom thick membrane with high in plane Young’s modulus ($Y=500\mathrm{GPa}$). It has no bending stiffness and therefore is not relevant for this study. Since there is nothing thinner than a single layer of atoms, graphene constitutes the limit of physically feasible structures.

#### 2.3. Influence of Gravity Body Forces on Shells

_{i}, number is introduced in this paper to quantify the gravitational impact on a shell’s behavior. The G

_{i}number is defined as the ratio of the elastogravity length scale ${l}_{eg}$ [20,21] to the characteristic dimension of the shell $R$.

_{i}number for a thin shell is

_{i}is larger than one, the characteristic dimension of a thin shell is smaller than ${l}_{eg}$: the gravity effect on the behavior of the compliant shell can be ignored. The nondimensional G

_{i}number determines the tendency of a compliant shell to be affected by the gravitational pull as a function of its scale. Values of G

_{i}lower than unity indicate that gravitational forces exert a large influence on the shell’s mechanical behavior. In contrast, G

_{i}values over one indicate gravitational forces are not of key importance in the deformation. The gravitational pull increases as the characteristic dimension of the shell increases. Compliant thin shells of large dimensions are rare but there are some examples of such shells where the characteristic dimension is in the order of magnitude of ${10}^{0}$ m or below. Therefore, the G

_{i}number is used in this study to detect and highlight the scaling limits of compliant thin shell.

## 3. Results

#### 3.1. Föppl–von Kármán Number Values Across Scales

_{FvK}describes whether stretching and/or bending deformations control the deformed state of the shell. Being a non-dimensional number, it applies to any shell, independent of the magnitude of its characteristic dimension R. The average values of γ

_{FvK}shown in Table 1 are within the range ${10}^{3}$ to ${10}^{8}$. To understand the variability observed in Table 1, we need to define precisely the subcategories of solids that appear on Figure 2. In this study the ratio R/H for a thin shell is adopted from [22] and given by

_{FvK}in Table 1 indicate a mechanical behavior dominated by bending deformation for both stiff and compliant shells. Overall, since thin shells have R/H ratios in the range of $[20;100,000$], their γ

_{FvK}values are bounded by lower $({\gamma}_{FvK}\text{}~{10}^{3}$) and upper bounds $({\gamma}_{FvK}\text{}~{10}^{8}$). This observation indicates that thin shells—whether they are engineered stiff or compliant, plant compliant, micro scale compliant, or egg stiff—exhibit similar mechanical behavior, which is dominated by bending deformations across scales.

_{FvK}between ${10}^{3}$ and ${10}^{8}$, as shown in Figure 2 and Figure 3. In the sample of shells selected for this study, only some of the compliant plant shells present values of γ

_{FvK}lower than 10

^{3}(Figure 3). Those same instances are on the border of the range of R/H ratios that characterizes shell structures (Figure 2). The main simplifying hypothesis of this study is that the material of the structures selected is isotropic elastic. In the case of the plant structures, the complex nature of the plant material (referred to as plant tissue) requires further justification for being included in this study. Biological tissues that constitute the moving organs of the plants instances included in the study are a hierarchized, non-homogeneous material [1]. As a living material, not all parts of tissue perform structural functions [49]. The structural layers of the tissue are thinner than the overall tissue [1] therefore in the cases presented in the study, the ratio R/H of the plants despite being loaer than other examples of shells are still accepted.

_{FvK}indicates a high in-plane stiffness compared to the out-of-plane bending stiffness. Therefore, bending deformations are more likely to occur than stretching for both structures. In theory, the structure of the Algerciras Market hall should be able to undergo similar reversible large shape changes as red blood cells. The market hall is a stiff concrete shell considered a model of shell design [15]. While in pure mechanical terms the concrete structure could be used as a compliant shell, the actual Algerciras Market Hall is dominated by dead-load’s vertical action and subjected to edge boundary conditions.

#### 3.2. Impact of Gravity on Shell Mechanical Behavior Across Scales

_{i}number and geometry is shown in Figure 4. The Figure shows that stiff engineered thin shells have the largest values of G

_{i}, while micro-scale compliant shells have the lowest values.

_{i}such as for example façade shading shells [16,17,28]. Shells with a characteristic dimension R lower than 0.1 m tend to have $Gi>1$. For these shells, large deformation caused by gravity does not occur. The relationship $Gi>1$ only occurs for one-third of compliant engineered shells, which means that most engineered shells must deal with the influence of gravity. All studied stiff engineered shells have an elastogravity length scale shorter than their characteristic dimension R. This observation indicates that for these shells the gravitational forces due to self-weight dominate the elastic bending resistance. The average value of G

_{i}is found to be 0.109 for stiff engineered shells, 0.610 for compliant engineered shells, 2.465 for plant compliant shells, 0.822 for the egg shells, and 7.739 for the micro-scale compliant shells.

_{i}lower than one (Figure 4). This scale is displayed by the red dotted line on Figure 4. No structure to the right of this line has a gravity impact number larger than one.

_{i}between compliant and stiff thin shells. Some engineered compliant thin shells are used as mechanisms but have a lower G

_{i}value than the one of stiff shells. A high value of G

_{i}can also indicate a shell with large thickness H with a corresponding low γ

_{FvK}value. The plant compliant shells have relatively high G

_{i}values, which means the shell does not deform under the influence of gravity. The larger plant compliant shells have G

_{i}values comparable to those of stiff engineered shells, indicating that the shell would be susceptible to the influence of gravity. However, for the living tissues, the ratio of volumetric mass density ρ to Young’s modulus Y is $~{10}^{3}$ times lower than that for engineered shells, which explains some of the low values of G

_{i}despite the small characteristic dimensions R.

## 4. Discussion

_{FvK}between ${10}^{3}$ and ${10}^{8}$. This non-dimensional number is significant because it unifies the behavior of shells across scales. This outcome is in line with the bio-inspiration approach that distills geometries of a plant or micro-scale shell and scales them up for engineering applications [1,16]. As long as the ratio of characteristic dimension R over thickness H is kept high, the mechanical behavior of the compliant shell is similar at the large engineered scale and the observed biological scale. The five categories of thin shells presented in this paper (i.e., engineered stiff, engineered compliant, plant compliant, micro-scale compliant, and egg stiff) have instances with ${\gamma}_{FvK}$ in the range of ${10}^{4}$ to ${10}^{5}$. This observation exemplifies the that shells used as mechanisms appear at all scales Figure 5. Thin shell can have a similar mechanical behavior dominated by bending deformation across 10 orders of magnitude of their characteristic dimension R.

_{eg}scale is large compared to their characteristic dimension R. In the genus Stylidium for example, the characteristic dimension R of the mechanism is 4.3 times larger than the elastogravity length scale l

_{eg}, which indicates that the plant’s movement is quasi unaffected by gravity. In general, plants can move without having the deformed geometry being influenced too much by gravity. The orientation of their mobile parts with respect to the gravitational pull does not obstruct or favor the shell movement.

## 5. Conclusions

_{i}was introduced in this paper. This non dimensional number determines at what scale gravity becomes relevant in the study of shell mechanics (third contribution). In particular, G

_{i}is defined as the ratio of the elastogravity length scale to the characteristic dimension of the shell and measures whether the scale at which bending deformation due to self-weight appears in a shell is larger or smaller than the actual size of the shell. The fourth contribution is the identification of the scale at which shells become influenced by gravity. Based on the characteristics of the 64 listed shells and using G

_{i}, it is shown that the effect of gravity on compliant shells sets on at a scale of ~0.1 m. Compliant shells at larger scales (R > 0.1 m) are prone to self-weight deformation under gravity load. This deformation can hinder their function depending on the nature of the application. A mechanism based on compliant shells that needs to perform reliably under varying orientation (e.g., airplane wing) will not be able to be scaled to large scales. However, if the application does not demand a change of orientation, the structure can be scaled up providing that the orientation of gravity is taken into account in the design of the compliant shell.

_{FvK}.

- The mechanism is oriented to limit the increase of cantilevered length during the movement. For example, the façade of the Yoesu Expo 2012 Pavilion was designed so that the flexible shell elements do not create large overhangs during the out-of-plane buckling deformation [28]. The longest elements are 8 m tall and still able to be elastically deformed repeatedly.
- The final strategy to create large-scale compliant thin shells is to operate in outer space. The behavior of shells is similar across scales. Bending deformation modes dominate stretching modes when shells are thin enough. Being able to remove gravity forces could lead to large shells being used as compliant mechanisms.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

R | Characteristic dimension of shells |

H | Thickness |

Y | Young’s Modulus |

${\gamma}_{FvK}$ | Föppl–von Kármán number |

Gi | Gravity impact number |

$W$ | Strain energy density |

${W}_{stretching}$ | Stretching strain energy density |

${W}_{bending}$ | Bending strain energy density |

$\nu $ | Poisson’s ratio |

$\u03f5$ | Average in-plane strain |

$\kappa $ | Average variation of shell curvature |

${l}_{eg}$ | Elastogravity length scale |

$\delta $ | Deformation due to gravity |

$g$ | Acceleration due to gravity |

$\rho $ | Volumetric mass density |

$D$ | Bending modulus (or flexural stiffness) |

## Appendix A. Sample of Cross-Scale Stiff and Compliant Thin Shells Instances

#### Appendix A.1. Stiff Engineered Thin Shells

_{i}numbers are Young’s modulus, volumetric mass density and Poisson’s Ratio. Those values are taken from the Eurocode 2 [50] and presented in Table A1. The values reflect commonly used values of concrete in design. They are indicative of order-of-magnitude for the parameters considered.

**Table A1.**Material properties used in the calculation of ${\gamma}_{FvK}$ and G

_{i}numbers for the engineered thin shells.

Material Property | Value |
---|---|

Volumetric mass density (kg·m^{−3}) | 2500 |

Modulus of Elasticity (GPa) | 35 |

Poisson’s ratio | 0.20 |

**Table A2.**Dimensions of large scale engineered thin shells included in the study. The structures are described in [15].

Id | Name/Location | Designer | Ref. | Span (m) | Thickness (m) | ||||
---|---|---|---|---|---|---|---|---|---|

Min. | Max. | Avg. | Min. | Max. | Avg. | ||||

1 | Aichtal | Balz, Isler | [15] | 42.0 | 42.0 | 42.0 | 0.090 | 0.120 | 0.105 |

2 | Algeciras | Sanchew Arcas, Torroja | [15] | 47.5 | 47.5 | 47.5 | 0.089 | 0.457 | 0.273 |

3 | Bacardi | Candela | [15] | 36.8 | 36.8 | 36.8 | 0.040 | 0.040 | 0.040 |

4 | Bundesgartenschau | SBP | [15] | 10.0 | 26.0 | 18.0 | 0.012 | 0.015 | 0.014 |

5 | Lomas De Cuernavaca | Candela | [15] | 18.0 | 31.0 | 24.5 | 0.040 | 0.040 | 0.040 |

6 | Milagrosa | Candela | [15] | 11.0 | 21.0 | 16.0 | 0.040 | 0.040 | 0.040 |

7 | San Jose Obrero | Candela | [15] | 30.0 | 30.0 | 30.0 | 0.040 | 0.040 | 0.040 |

8 | Cosmic Rays | Candela | [15] | 12.0 | 12.0 | 12.0 | 0.015 | 0.050 | 0.033 |

9 | Deitingen | Isler | [15] | 31.6 | 31.6 | 31.6 | 0.090 | 0.090 | 0.090 |

10 | Florelite | Isler | [15] | 41.0 | 41.0 | 41.0 | 0.080 | 0.080 | 0.080 |

11 | GiessHauss | Henschel | [15] | 16.0 | 16.0 | 16.0 | 0.175 | 0.320 | 0.248 |

12 | Gringrin | Sasaki | [15] | 70.0 | 70.0 | 70.0 | 0.400 | 0.400 | 0.400 |

13 | Heimberg | Isler | [15] | 48.5 | 48.5 | 48.5 | 0.090 | 0.100 | 0.095 |

14 | Hippo | SBP | [15] | 29.0 | 29.0 | 29.0 | 0.040 | 0.060 | 0.050 |

15 | Hyperthreads | Zaha Hadid | [15] | 6.0 | 6.0 | 6.0 | 0.080 | 0.080 | 0.080 |

16 | Jeronimo | De Castillo, de Boitaca | [15] | 10.0 | 10.0 | 10.0 | 0.070 | 0.100 | 0.085 |

17 | Kakamigara | Ito, Sasaki | [15] | 20.0 | 20.0 | 20.0 | 0.200 | 0.200 | 0.200 |

18 | Kitagata | Isozaki, Sasaki | [15] | 25.0 | 25.0 | 25.0 | 0.150 | 0.150 | 0.150 |

19 | Kresge | Saarinen, B&H, A&W | [15] | 48.8 | 48.8 | 48.8 | 0.075 | 0.455 | 0.265 |

20 | Los Manantiales | Candela | [15] | 42.5 | 42.5 | 42.5 | 0.040 | 0.040 | 0.040 |

21 | Mapungubwe | Rich, Ochsendorf, Ramage | [15] | 5.0 | 14.0 | 9.5 | 0.300 | 0.300 | 0.300 |

22 | Rolex | SANAA, Sasaki | [15] | 80.0 | 80.0 | 80.0 | 0.040 | 0.080 | 0.060 |

23 | Rio Warehouse | Candela | [15] | 15.3 | 15.3 | 15.3 | 0.040 | 0.040 | 0.040 |

24 | Sicli | Hiberer, Isler | [15] | 58.0 | 58.0 | 58.0 | 0.100 | 0.100 | 0.100 |

25 | Teshima | Nishizawa, Sasaki | [15] | 43.0 | 60.0 | 51.5 | 0.250 | 0.250 | 0.250 |

#### Appendix A.2. Compliant Engineered Thin Shells

Id | Description | Ref. | Material | Poisson’s Ratio | Young’s Modulus (N/m^{−2}) | Volumetric Mass Density (kg/m^{−3}) |
---|---|---|---|---|---|---|

1 | Aldrovanda Half Sphere | [16] | CFRP | 0.3 | 7.60 × 10^{10} | 1800 |

2 | Snap Curved Helicoid | [24] | Polycaprolactone | 0.4 | 3.53 × 10^{8} | 1145 |

3 | Snap Curved Cylinder | [24] | PET | 0.4 | 5.00 × 10^{9} | 1380 |

4 | Flectofin | [18] | GFRP | 0.4 | 2.50 × 10^{10} | 1800 |

5 | Flectofold | [17] | GFRP | 0.4 | 1.15 × 10^{10} | 1100 |

6 | Gravity Compliant Shell | [26] | PETG | 0.4 | 2.35 × 10^{9} | 1300 |

7 | Multistable-Corrugated Shells | [27] | copper–beryllium | 0.3 | 1.31 × 10^{11} | 8950 |

8 | Multistable Inlet | [12] | CFRP | 0.3 | 7.60 × 10^{10} | 1800 |

9 | Yoesu One Ocean | [28] | GFRP | 0.4 | 2.50 × 10^{10} | 1800 |

10 | Scoliosis Brace Helix | [29] | CFRP | 0.3 | 7.60 × 10^{10} | 1800 |

11 | Scoliosis Brace Cantilever | [29] | Polycarbonate | 0.4 | 2.90 × 10^{9} | 1270 |

12 | Tape Spring | [31] | Steel | 0.3 | 2.10 × 10^{11} | 7800 |

13 | Stiffness Study Shell 1 | [32] | Acrylic | 0.4 | 3.20 × 10^{9} | 1180 |

14 | Stiffness Study Shell 2 | [32] | PETG | 0.4 | 2.06 × 10^{9} | 1270 |

15 | Antenna Tape Spring | [30] | CFRP | 0.3 | 3.56 × 10^{10} | 1440 |

16 | Collapsible Booms | [25] | CFRP | 0.3 | 7.60 × 10^{10} | 1800 |

17 | Deformable Mirrors | [51] | CFRP | 0.3 | 7.60 × 10^{10} | 1800 |

Id | Description | Ref. | Span (m) | Thickness (m) | ||||
---|---|---|---|---|---|---|---|---|

Min. | Max. | Avg. | Min. | Max. | Avg. | |||

1 | Aldrovanda Half Sphere | [16] | 0.800 | 1.000 | 0.900 | 5.00 × 10^{−4} | 8.00 × 10^{−4} | 6.50 × 10^{−4} |

2 | Snap Curved Helicoid | [24] | 0.025 | 0.035 | 0.030 | 1.00 × 10^{−3} | 1.00 × 10^{−3} | 1.00 × 10^{−3} |

3 | Snap Curved Cylinder | [24] | 0.025 | 0.035 | 0.030 | 1.20 × 10^{−4} | 1.20 × 10^{−4} | 1.20 × 10^{−4} |

4 | Flectofin | [18] | 0.250 | 0.250 | 0.250 | 2.00 × 10^{−3} | 2.00 × 10^{−3} | 2.00 × 10^{−3} |

5 | Flectofold | [17] | 1.100 | 1.100 | 1.100 | 1.25 × 10^{−3} | 1.25 × 10^{−3} | 1.25 × 10^{−3} |

6 | Gravity Compliant Shell | [26] | 0.050 | 0.100 | 0.075 | 9.00 × 10^{−4} | 9.00 × 10^{−4} | 9.00 × 10^{−4} |

7 | Multistable-Corrugated Shells | [27] | 0.100 | 0.250 | 0.175 | 1.25 × 10^{−4} | 1.25 × 10^{−4} | 1.25 × 10^{−4} |

8 | Multistable Inlet | [12] | 0.040 | 0.100 | 0.070 | 2.50 × 10^{−4} | 2.50 × 10^{−4} | 2.50 × 10^{−4} |

9 | Yoesu One Ocean | [28] | 1.300 | 8.000 | 4.650 | 9.00 × 10^{−3} | 9.00 × 10^{−3} | 9.00 × 10^{−3} |

10 | Scoliosis Brace Helix | [29] | 0.050 | 0.050 | 0.050 | 3.50 × 10^{−3} | 3.50 × 10^{−3} | 3.50 × 10^{−3} |

11 | Scoliosis Brace Cantilever | [29] | 0.070 | 0.100 | 0.085 | 3.00 × 10^{−3} | 3.00 × 10^{−3} | 3.00 × 10^{−3} |

12 | Tape Spring | [31] | 0.021 | 0.050 | 0.036 | 2.00 × 10^{−4} | 2.00 × 10^{−4} | 2.00 × 10^{−4} |

13 | Stiffness Study Shell 1 | [32] | 0.100 | 0.150 | 0.125 | 2.00 × 10^{−3} | 2.00 × 10^{−3} | 2.00 × 10^{−3} |

14 | Stiffness Study Shell 2 | [32] | 0.015 | 0.075 | 0.045 | 5.00 × 10^{−4} | 5.00 × 10^{−4} | 5.00 × 10^{−4} |

15 | Antenna Tape Spring | [30] | 0.050 | 0.050 | 0.050 | 2.25 × 10^{−4} | 3.00 × 10^{−4} | 2.63 × 10^{−4} |

16 | Collapsible Booms | [25] | 0.011 | 0.036 | 0.023 | 2.00 × 10^{−4} | 2.00 × 10^{−4} | 2.00 × 10^{−4} |

17 | Deformable Mirrors | [51] | 1.000 | 1.000 | 1.000 | 2.00 × 10^{−4} | 3.00 × 10^{−4} | 2.50 × 10^{−4} |

#### Appendix A.3. Compliant Plant Thin Shells

**Table A5.**Material properties used in the calculation of ${\gamma}_{FvK}$ and G

_{i}numbers for the compliant plant thin shells [34].

Material Property | Value |
---|---|

Volumetric mass density (kg·m^{−3}) | 1300 |

Modulus of Elasticity (MPa) | 5 |

Poisson’s ratio | 0.5 |

id | Name | Ref | Span (m) | Thickness (m) | ||||
---|---|---|---|---|---|---|---|---|

Min. | Max. | Avg. | Min. | Max. | Avg. | |||

1 | Stylidium crossocephalum | [33] | 1.00 × 10^{−3} | 1.00 × 10^{−3} | 1.00 × 10^{−3} | 5.00 × 10^{−4} | 5.00 × 10^{−4} | 5.00 × 10^{−4} |

2 | Stylidium graminifolium | [33] | 1.00 × 10^{−3} | 1.00 × 10^{−3} | 1.00 × 10^{−3} | 5.00 × 10^{−4} | 5.00 × 10^{−4} | 5.00 × 10^{−4} |

3 | Stylidium piliferum | [33] | 1.00 × 10^{−3} | 1.00 × 10^{−3} | 1.00 × 10^{−3} | 5.00 × 10^{−4} | 5.00 × 10^{−4} | 5.00 × 10^{−4} |

4 | Aldrovanda vesiculosa | [34] | 2.60 × 10^{−3} | 2.60 × 10^{−3} | 2.60 × 10^{−3} | 4.00 × 10^{−5} | 7.00 × 10^{−5} | 5.50 × 10^{−5} |

5 | Dionea muscipula | [34] | 1.00 × 10^{−2} | 1.00 × 10^{−2} | 1.00 × 10^{−2} | 4.00 × 10^{−4} | 4.00 × 10^{−4} | 4.00 × 10^{−4} |

6 | Utricularia Sp. | [35] | 1.00 × 10^{−4} | 2.00 × 10^{−4} | 1.50 × 10^{−4} | 2.00 × 10^{−5} | 4.00 × 10^{−5} | 3.00 × 10^{−5} |

7 | Utricularia vulgaris | [35] | 1.00 × 10^{−4} | 3.00 × 10^{−4} | 2.00 × 10^{−4} | 2.00 × 10^{−5} | 4.00 × 10^{−5} | 3.00 × 10^{−5} |

8 | Utricularia australis | [35] | 3.30 × 10^{−4} | 7.20 × 10^{−4} | 5.25 × 10^{−4} | 2.00 × 10^{−5} | 4.00 × 10^{−5} | 3.00 × 10^{−5} |

#### Appendix A.4. Compliant Micro-Scale Thin Shells

Id | Description | Ref. | Poisson’s Ratio | Young’s Modulus (N/m^{−2} | Volumetric Mass Density (kg/m^{−3}) |
---|---|---|---|---|---|

1 | Red Blood Cell | [40] | 0.5 | 3.10 × 10^{6} | 1000 |

2 | Artificial Capsules | [19,39] | 0.5 | 1.00 × 10^{9} | 1000 |

3 | Virus | [8,19] | 0.5 | 3.10 × 10^{6} | 1000 |

4 | Vesicle 1 | [41] | 0.5 | 1.00 × 10^{9} | 1000 |

5 | Vesicle 2 | [41] | 0.5 | 1.00 × 10^{9} | 1000 |

Id | Description | Ref. | Span (m) | Thickness (m) | ||||
---|---|---|---|---|---|---|---|---|

Min. | Max. | Avg. | Min. | Max. | Avg. | |||

1 | Red Blood Cell | [40] | 4.00 × 10^{−6} | 1.00 × 10-5 | 7.00 × 10^{−6} | 9.00 × 10^{−8} | 9.00 × 10^{−8} | 9.00 × 10^{−8} |

2 | Artificial Capsules | [19,39] | 1.00 × 10^{−6} | 1.00 × 10^{−3} | 5.01 × 10^{−4} | 1.00 × 10^{−6} | 1.00 × 10^{−6} | 1.00 × 10^{−6} |

3 | Viruses | [8,19] | 1.50 × 10^{−8} | 3.00 × 10^{−8} | 2.25 × 10^{−8} | 2.00 × 10^{−9} | 2.00 × 10^{−9} | 2.00 × 10^{−9} |

4 | Vesicle 1 | [41] | 2.40 × 10^{−5} | 3.00 × 10^{−5} | 2.70 × 10^{−5} | 5.00 × 10^{−7} | 5.00 × 10^{−7} | 5.00 × 10^{−7} |

5 | Vesicle 2 | [41] | 3.20 × 10^{−5} | 4.00 × 10^{−5} | 3.60 × 10^{−5} | 5.00 × 10^{−7} | 5.00 × 10^{−7} | 5.00 × 10^{−7} |

#### Appendix A.5. Stiff Eggshells

Id | Description | Ref | Poisson’s Ratio | Young’s Modulus (N/m^{−2}) | Volumetric Mass Density (kg/m^{−3}) |
---|---|---|---|---|---|

1 | Hen’s Egg | [36,37] | 0.3 | 7.24 × 10^{10} | 2710 |

2 | Quail Egg | [38] | 0.3 | 1.05 × 10^{10} | 2710 |

3 | Chicken Pullet Egg | [38] | 0.3 | 1.48 × 10^{10} | 2710 |

4 | Chicken White Egg | [38] | 0.3 | 2.75 × 10^{10} | 2710 |

5 | Chicken Organic Egg | [38] | 0.3 | 1.80 × 10^{10} | 2710 |

6 | Chicken Jumbo Egg | [38] | 0.3 | 2.46 × 10^{10} | 2710 |

7 | Goose Egg | [38] | 0.3 | 1.04 × 10^{10} | 2710 |

8 | Ostrich Egg | [38] | 0.3 | 6.60 × 10^{10} | 2710 |

Id | Description | Ref. | Span (m) | Thickness (m) | ||||
---|---|---|---|---|---|---|---|---|

Min. | Mix. | Average | Min. | Mix. | Average | |||

1 | Hen’s Egg | [36,37] | 4.54 × 10^{−2} | 5.50 × 10^{−2} | 5.02 × 10^{−2} | 3.50 × 10^{−4} | 5.00 × 10^{−4} | 4.25 × 10^{−4} |

2 | Quail Egg | [38] | 3.00 × 10^{−2} | 3.00 × 10^{−2} | 3.00 × 10^{−2} | 2.20 × 10^{−4} | 2.20 × 10^{−4} | 2.20 × 10^{−4} |

3 | Chicken Pullet Egg | [38] | 5.45 × 10^{−2} | 5.45 × 10^{−2} | 5.45 × 10^{−2} | 4.40 × 10^{−4} | 4.40 × 10^{−4} | 4.40 × 10^{−4} |

4 | Chicken White Egg | [38] | 6.04 × 10^{−2} | 6.04 × 10^{−2} | 6.04 × 10^{−2} | 3.50 × 10^{−4} | 3.50 × 10^{−4} | 3.50 × 10^{−4} |

5 | Chicken Organic Egg | [38] | 6.04 × 10^{−2} | 6.04 × 10^{−2} | 6.04 × 10^{−2} | 4.10 × 10^{−4} | 4.10 × 10^{−4} | 4.10 × 10^{−4} |

6 | Chicken Jumbo Egg | [38] | 6.31 × 10^{−2} | 6.31 × 10^{−2} | 6.31 × 10^{−2} | 4.00 × 10^{−4} | 4.00 × 10^{−4} | 4.00 × 10^{−4} |

7 | Goose Egg | [38] | 8.74 × 10^{−2} | 8.74 × 10^{−2} | 8.74 × 10^{−2} | 6.70 × 10^{−4} | 6.70 × 10^{−4} | 6.70 × 10^{−4} |

8 | Ostrich Egg | [38] | 1.55 × 10^{−1} | 1.55 × 10^{−1} | 1.55 × 10^{−1} | 2.55 × 10^{−3} | 2.55 × 10^{−3} | 2.55 × 10^{−3} |

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**Figure 1.**Varying size/scale (R) of the five types of shells included in the study. The number of typologies for each type of shell is shown in blue.

**Figure 2.**Geometric properties and γ

_{FvK}values for stiff and compliant thin shells, plant compliant thin shells, and compliant micro-scale shells. The scale for both axes is logarithmic.

**Figure 3.**Föppl–von Kármán number, ${\gamma}_{FvK}$, in thin shells as a function of the characteristic dimension. The scale for both axes is logarithmic.

**Figure 4.**Gravitational force density impact in thin shells as a function of the characteristic dimension. The scale for both axes is logarithmic. The horizontal dotted line indicates values G

_{i}= 1 for which the gravitational force becomes predominant in the equilibrium of the shell. The red dotted line at R = 0.1 m represent the approximate limit at which thin shells start to be constrained by gravity.

**Figure 5.**Classification of compliant and stiff thin shells. The nondimensional gravitational force density G

_{i}is plotted as a function of the Föppl–von Kármán number γ

_{FvK}. The dotted line indicates values G

_{i}= 1 for which the gravitational force becomes predominant in the equilibrium of the shell.

Shell Type | $\mathbf{Average}\text{}{\mathit{\gamma}}_{\mathit{F}\mathit{v}\mathit{K}}$ |
---|---|

Stiff Engineered | $3.95\times {10}^{6}$ |

Compliant Engineered | $1.33\times {10}^{7}$ |

Stiff Avian Egg | $1.98\times {10}^{5}$ |

Compliant Plant | $3.84\times {10}^{3}$ |

Compliant Micro-Scale | $2.54\times {10}^{4}$ |

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Charpentier, V.; Adriaenssens, S.
Effect of Gravity on the Scale of Compliant Shells. *Biomimetics* **2020**, *5*, 4.
https://doi.org/10.3390/biomimetics5010004

**AMA Style**

Charpentier V, Adriaenssens S.
Effect of Gravity on the Scale of Compliant Shells. *Biomimetics*. 2020; 5(1):4.
https://doi.org/10.3390/biomimetics5010004

**Chicago/Turabian Style**

Charpentier, Victor, and Sigrid Adriaenssens.
2020. "Effect of Gravity on the Scale of Compliant Shells" *Biomimetics* 5, no. 1: 4.
https://doi.org/10.3390/biomimetics5010004