Why a Cable Will Break

CNT alone as a solid cylinder will break at 1 meter, correctly pointed out by a study by GoogleX, as gravity versus centrifugal force is too great for a cable where center of mass (COM) is at GEO, or 35,786 km (22,236 mi). The total gravitational field on all points when the cable is tight is:

g = – G x M/r^2 + ω^2 x r

g is the acceleration of apparent gravity, pointing down (negative) or up (positive) along the vertical cable (m s−2),

a is the centrifugal acceleration, pointing up (positive) along the vertical cable (m s−2),

G is the gravitational constant (m3 s−2 kg−1)

M is the mass of the Earth (kg)

r is the distance from that point to Earth’s center (m),

ω is Earth’s rotation speed (radian/s).

At some point up the cable, the two terms (downward gravity and upward centrifugal force) are equal and opposite. Objects fixed to the cable at that point put no weight on the cable. This altitude (r1) depends on the mass of the planet and its rotation rate. Setting actual gravity equal to centrifugal acceleration gives:

r 1 = ( G ⋅ M / ω ^ 2 ) ^ 1 / 3

See https://en.wikipedia.org/wiki/Space_elevator.

Single-walled carbon nanotube, the most sought-after human-made material for highest possible specific strength
taper ratio
Only CNT provide hope of a tapered solid cylinder that can withstand high tension for something manufactured thin enough without high production costs and too many rocket launches. However, one must consider elevator stress and hysteresis, or limited specific strength that changes over time

 

 

 

 

 

 

 

See https://pdfs.semanticscholar.org/d402/ba5f97884b7398ae2a1ff79136f9c1a03993.pdf for the best computation and analysis for how to construct a tapered space cable (solid), where theoretically, a thickening of only 1.6 times ESS/CW of a CNT tube at COM is possible, the same breaking length regardless of average width, and protected by Kevlar (see below) because of wear and fatigue caused by elevator, meteorites, lightning, acid rain, wind, vibrations from Coriolis effect, etc. The protective Kevlar has a melting point of 930 degrees Fahrenheit, much higher than the 250 degrees Fahrenheit of outer space. The combined force of gravity and the centrifugal force is 30-100 Mega-Newtons for prospective materials as a solid cable. The “weakest link” is the middle, or COM, where it’s likely to break first. A solid cable would have to exponentially expand in diameter toward COM and exponentially contract toward a smaller diameter away from COM, best positioned at GEO. But regardless of the material composition (stronger than CNT, silk, cat whiskers, diamond nanotubes, or single crystal graphene) the number of rocket launches to unfurl solid, non-elastic cable would be too expensive, unsafe, too time consuming, and ultimately a bad insurance policy for a probable short lifespan. Some charge the thickness of a solid cable at COM can never be less than 1,024 times that of the cable at ground level and CW (see https://space.stackexchange.com/questions/18129/space-elevator-cord-thickness-why-is-the-strength-to-weight-ratio-a-problem) and therefore elevator climbing would be impossible, trying to grip something that starts at one cm, mean 10 meters near and at COM. The above calculation, universal for all untapered solid cylinders, makes it clear breaking length for the necessary 60,000 mile cable is 6,385 km, and you would have to thicken it to something 2,441 times Earth ground level and CW end-point thickness at COM. That’s about 24 meters for a 1 cm cylinder to begin with, which requires too many rocket launches. For a hose, the S is one or two thin rings for cross-sectional area, not a circle, so S is much lower, and dS is much higher because of z-elasticity, so dS/S is much higher, and density (ρ) is much lower because of a 25%-75% PVP/silicone composition, so σ = dS/(ρ * S) * constant => σ, which is the maximum amount of stress the SH can handle before stretched to its maximum length, and potential breaking, and yields a breaking length at well beyond COM at GEO, about 58,000 miles (see Weight Calculation).

CNT L/d
Ideal length to diameter ratio of CNT tubes, about 200, means you capitalize on minimizing effort to not have to create longer tubes and benefit from the same approximate resistance to elasticity, or Young’s modulus. However, as a helix, the average strength of the composite hose may fragment unless CNT’s are stretched many meters, from one hose connector to the next, as most filler polymers like HDPE is not as elastic
space hose
Visualize spring as MW CNT woven into high elasticity, high temperature tolerant polymer (eg. vulcanized rubber). An SH that can stretch far enough to overcome apogee versus perigee difference in length will not break, plus stiffness in perpendicular direction is great, hence non-isotropic, allowing greater life span of a hose not likely to break. Any hose, like surgical tubing, is more elastic (eg. sling-shot).

With so great a challenge for a large variation in cylinder thickness means, in my opinion and 80% of the scientists I talked to, the death of the solid cylinder or ribbon. A hose has more elasticity, with CNT having a high elasticity as well when compared to the competition, with more leverage, like with a ribbon, because of the pinching effect of rollers on them, so a hose that could hold up against 100 Mega-Newtons is not impossible to construct, and can still be used for vital material transport even if rocket propellant at end of hose for shooting outward, to draw CW inward under Newton’s Third Law, is not necessary, but still an excellent added insurance policy. While a more elastic hose will be pulled on in equal and opposite directions with greater stress and strain, the tolerance of more stress and strain is greater, and therefore offsetting. But higher elasticity because polymer fillers like PVP and silicone that have very low densities means lower tension. The main thing is, the elastic hose, versus stiff, brittle solid cylinder, has a longer lifespan because of elastic historesis being less “square” for long periods of time (see graphs below), the dissipation of heat through a greater stretching effect to alleviate over-stretching of ductile and brittle components, causing a longer life span, also helps. The hose is also more elastic, therefore less dense, because of 24 hours of 250° sunlight making the PVP or silicone more “putty like”, but well under the melting point, for less tension, but then cooled 1 hour a day when shadowed by the Earth, then re-solidified.

Returning to basics, breaking length is something that occurs because of too much stress or strain on a presumed uniform material, and the hose/cable is most vulnerable in the middle, at the COM:

Stress (σ) = Force / Cross-sectional Area;

Young’s Modulus (∈) = Stress (σ) /  Strain (σ), where Strain = ΔL/L;

MW CNT
Multi-walled CNT. More tensile strength by 50% versus single-walled CNT. 30-60 nm outside diameter, 20-50 nm inside diameter, so ideal length = 1.2 um

A hose is a “ring” in two dimensions, versus a circle for a solid cylinder, so the lower force (gravitational force plus centrifugal force) is lower, but canceled out by the lesser cross-sectional area, so stress is the same for using the same material, but as always, tensile strength / density is the most important factor, and the need, for a number or reasons, of high CNT concentrations impregnated into a non-H2-permeable polymer. If there is a need to make SH stronger at COM, there is a way to do it without making hose significantly wider, by increasing the percentage and number of long, thin spiral sheets of single crystal graphene, with a wall thicker than 0.5 mm.

Single-walled CNT
Single-walled CNT. 50% less tensile strength versus multi-walled CNT, but higher elasticity. The larger diameters reported are 3 nm, so ideal length = 600 nm, or 0.6 um.
CC with Braided CNT in the Middle
A less expensive alternative using 20-50 cm long CNT tubes available now, in a pure, easily accessible Kevlar coaxial cable, which has about the same density, but a lot more tensile strength. However, if reinforced in the very center with braided long CNTs, it will never be stretched to its limit. Here, the chain is as strong as the strongest link, not the weakest

When looking at data already produced for CNT, the difference between single walled CNT (tensile strength = 100 GPa) and multi-walled CNT (tensile strength = 150 GPa) may seem as an important enough 50% difference, the spiral helix, from one rotation around hose wall to another (new rung), means less of a bend of the MW CNT, especially for a larger diameter SH maintaining a fixed hose wall thickness. As the diameter of SH goes up, more rocket launches are required to deploy it, but a larger inner diameter means more hydrogen and oxygen gas, possibly other materials, can be pumped to SS and CW faster, and a linear increase in diameter means an exponential drop in increasing weight because of a fixed wall thickness, and even greater elasticity. So the hose that averages 50+% higher a diameter does not need additional rocket launches, as the 50+% increase in tensile strength, therefore, means higher specific strength from drop in density with MW CNT displacing polymer filler and therefore is off-setting. More concentric tubes from the multi-walled CNT means more displacement of a smaller specific strength polymer, so the composite specific strength could be much higher than a 50% increase. But elasticity remains high the further away from the COM you are, to guarantee, from hysteresis studies, a longer life span before SH becomes too brittle, the low density for lower tension, and more heat dissipation to prevent break-down of PVP/silicone into something more brittle.

stress vs strain
Stress vs strain for a ductile material, like copper. High tolerance of stress, but not strain. SH “spring”, with spiral helix coils, elastic filler polymer, means high tolerance of delta L / L, and much higher breaking point (higher force from apogee means less cross-sectional area, so much higher stress in addition to strain)

More computer simulations, starting with basic computations, are needed to prove viability of hose and complete non-viability of any solid cable, even with higher production costs. But the 75% concentration of MW CNT/single crystal graphene in polymers now produced with 3D printers represents a major savings with less human labor at a much faster pace, with CNT tubes 100 times as long as they are thick in diameter for best elasticity, but a hypothetically much longer tube would not be overkill but needed to keep helix solid and non-brittle from polymer filler plastic deformation. With larger diameter hosing, the MW CNT is more likely to produce the bend needed for a spiral helix effect.

  • L = the limit of proportionality, Hooke’s law applies up to this point.
  • E = elastic limit, beyond this point the material is permanently stretch and it will not go back to its original length. Elastic behavior is when a material returns to its original length, plastic behavior is when the stretched material does not return to its original length.
  • Y = yield point, beyond this point small increases in force give much big increases in length.
  • B = breaking point / breaking stress, the material breaks at this point.
Force vs Delta L
E = 1/2 x F x delta L, which is elastic strain energy in joules. Something elastic like this will dissipate heat to counteract deformation of non-elastic substance, and cool-off during the brief period the SH is shadowed by the Earth once a day

Δ

Δ

Four examples of stress versus strain for the stretching of different CNT cylinders, best case scenario is a MW CNT that breaks at 130% strain, and 40 GPa stress. The difference between semi-major and semi-minor axis is less than 130%, with stress = F / area going up during apogee, for a hose, but the above graph is for solid cylinders.
Elasticity Historesis
Hysteresis, or elastic history for different materials for the effects of wavering stress-strain increases and decreases. No substance is 100% or elastic, everything is slightly ductile. The more brittle, the more “square” the curve. Ductile is in the middle. Brittle should break right before first apogee, a hose that is elastic in z direction, stiff in x-y direction, will travel back and forth along straight line for a long time, then polymer and single crystal graphene will break down, and the hysteresis becomes more square, and only backup from rockets and torque from Ferris Wheels extend life span of cable/hose