CNT/Single Crystal Graphene:
Density = 1.4 g / cm ^ 3;
Density = 1.1 g / cm ^ 3;
=> Avg. Density = 1.25 g / cm ^ 3;
1 cm = 10 mm;
1 cm ^ 3 = 1,000 mm ^ 3;
1.3 g / cm ^ 3 x 1 kg / 1,000 g x 1 cm ^ 3 / (0.01 m ) ^ 3 = 1,300 kg / m^3;
(0.005 ^ 2 – 0.00475 ^ 2) x 3.1415 = 0.00076574 m ^ 2 x 60,000 mi x 1.62 km / mi x 1,000 m / km = volume of SH = 744.3 m ^ 3;
0.125 kg / m ^ 3 x 744.3 m ^ 3 = 973,000 kg;
Cross-sectional area = 0.0000076574 m ^ 2;
Maximum stress = F / Cross-sectional area = 123,066,000 kg / m x s ^ 2 = 123 MPa;
Minimum stress to break SW CNT = 25 GPa;
g is the acceleration along the radius (m·s−2);
S is the cross-section area of the cable at any given point r, (m2) and dS its variation (m2 as well);
ρ is the density of the material used for the cable (kg·m−3).
σ is the stress the cross-section area can bear without yielding (N·m−2=kg·m−1·s−2), its elastic limit.
ω is Earth’s rotation speed (radian/s).
To compare materials, the specific strength of the material for the space elevator can be expressed in terms of the characteristic length, or “free breaking length”: the length of an un-tapered cylindrical cable at which it will break under its own weight under constant gravity. For a given material, that
S(1) = S(0) * e ^ ρ * g(0) * r (0) / σ (1 + ½ * x – 3/2 * x ^ 1/3);
x = ω ^ 2 * r(0) / g(0) = 0.0035;
L( c ) = σ / ρ * g (0);
Δ Ln (S) = r (0) / x g(0) * (1 + ½ * x – 3/2 * x ^ 1/3);
L ( c ) = 123,066,000 kg / m * s ^ 2 / 0.13 kg / m ^ 3 * 9.8 m / s ^ 2 = 96,598,000 m = 58,925 miles (this is beyond the COM and almost equal to maximum length of SH, so for lightest and smallest possible SH, anticipate no breaking with un-tapered hollow cylinder, only one rocket launch is required). 3 to 6 SH’s can easily be transported, along with material for SS, in one rocket mission.
Dragging missions can also be cut to a 1/3 of the work if safety/feasibility testing shows the same smaller/lighter hoses can be used. The 96.76 kg figure does not include weight of SS, CW (but CW can be brought to 37,000 mile mark, whose constituents come from the Moon where the escape velocity on the Moon is many times less than the Earth, to cut SH length to 2/3 total length, and wall thickness to 0.5 mm, means a best-case total weight of about 973,000 kg, with much lighter Moon mining equipment sent to Moon for much cheaper rocket launches back to CW for CW construction). This means a maximum total tension of 123 MPa (when adding counterweight, total weight plus equal and opposite centrifugal force), where 30 to 50 GPa are usually needed to break a SW CNT tube, and the element added that is z-axis (radial direction) elasticity, very low elasticity or high tensile strength in x-y (N-S-E-W) directions, and designing a hose that thickens toward the COM, without effecting the above tension figures, means a stand-alone hose may not need rocket emissions or Ferris Wheel torque to avoid breaking, but it must be a hose instead of a cable just from the benefit of radial elasticity being a good thing, along with needed transportation of materials for more active astronauts.