Building the Space Elevator: Lessons from Biological Design
BBuilding the Space Elevator: Lessons from Biological Design
Dan M. Popescu ∗ Department of Applied Mathematics and Statistics,Johns Hopkins University, Baltimore, Maryland 21218, USA
Sean X. Sun † Department of Mechanical Engineering and Department of Biomedical Engineering,Johns Hopkins University, Baltimore, Maryland 21218, USA (Dated: April 19, 2018)
Abstract
One of the biggest perceived challenges in building megastructures, such as the space elevator,is the unavailability of materials with sufficient tensile strength. The presumed necessity of verystrong materials stems from a design paradigm which requires structures to operate at a small frac-tion of their maximum tensile strength (usually, 50% or less). This criterion limits the probabilityof failure by giving structures sufficient leeway in handling stochastic components, such as variabil-ity in material strength and/or external forces. While reasonable for typical engineering structures,low working stress ratios — defined as operating stress as a fraction of ultimate tensile strength —in the case of megastructures are both too stringent and unable to adequately control the failureprobability. We draw inspiration from natural biological structures, such as bones, tendons andligaments, which are made up of smaller substructures and exhibit self-repair, and suggest a designthat requires structures to operate at significantly higher stress ratios, while maintaining reliabilitythrough a continuous repair mechanism. We outline a mathematical framework for analysing thereliability of structures with components exhibiting probabilistic rupture and repair that dependon their time-in-use (age). Further, we predict time-to-failure distributions for the overall struc-ture. We then apply this framework to the space elevator and find that a high degree of reliabilityis achievable using currently existing materials, provided it operates at sufficiently high workingstress ratios, sustained through an autonomous repair mechanism, implemented via, e . g ., robots. Keywords: Space Elevator, biological design, age-structured dynamics ∗ [email protected] † [email protected] a r X i v : . [ phy s i c s . s p ace - ph ] A p r . INTRODUCTION Once an element of science fiction, the space elevator has become in recent years one ofthe most ambitious and grandiose engineering projects. Although the concept of a spaceelevator was introduced by Russian physicist Konstantin Tsiolkovsky in 1875 [1], the ideagoes back to biblical times when the attempt to create a tower to heaven (later named “TheTower of Babel”) ended in ruin. In the late 1990s, NASA considered the idea rigorously andconcluded that such a massive structure is not only feasible, but is a cost-efficient way totransport payloads into space [2]. A few years later, two NASA Institute of Advanced Science(NIAC) reports outlined various engineering considerations to building the megastructure[3, 4]. The reports emphasized the necessity of extremely strong materials, but the dawn ofcarbon nanotubes dispelled some of the scepticism in the scientific community. Currently,commercial companies planning on building the elevator are on hold, awaiting advancementsin materials science.In this manuscript, we argue that a key concept needed for building megastructures likethe space elevator can be borrowed from biology. On a much smaller scale, living organismscan be viewed as megastructures when compared to their building blocks ( e . g ., tendonscomposed of collagen fibres, bones made of osteons, etc .). So how does biological designcreate such stable structures? The answer is not only to maximize the strength of thematerials used, but also to cheaply repair by recycling material, while operating at veryhigh loads. Although it is a good rule of thumb in reliability engineering to have structureswith a maximum safety factor — that is, how much load the part can withstand vs. actual orexpected load — of 2, biological systems operate significantly below this value. For example,in humans, Achilles’ tendons experience safety factors well below 1.5, routinely withstandingmechanical stresses very close to their ultimate tensile strengths [5]. Similarly, lumbar spinesin humans can also sustain tremendous stresses, especially in athletes [6]. As Taylor et al .point out [7], the key to sustainability lies in the repair mechanism inherent in biologicalsystems.Incidentally, engineering has a long history of borrowing from biology dating back toclassic civilizations’ use of ballistae, which used twisted tendons to accelerate projectileson account of the little weight they would add to the machine [8]. In the same spirit, wesuggest a megastructure design that not only allows components to fail, but has a self-2epair mechanism to replace the broken components. This will allow structures to operateat significantly higher loads, without compromising their integrity, which, in turn, will makemegastructures built from existing materials a reality.The physics of the space elevator as a balanced tether extending from the Equator pastgeosynchronous height has been studied in previous works [9–11]. The tether is freestanding— that is, it exerts no force on the ground — if its weight and outward centrifugal forceare in balance, thus maintaining it under lengthwise tension. Using the notation in [11],Fig. 1(b) shows that each small, horizontal element of the tether experiences four forces:its weight W , the outward centrifugal force F C , and upward/downward forces F U and F D ,due to the part of the cable above/below the element (and a potential counterweight placedabove geosynchronous height to reduce the cable length needed). A balanced tether impliesthat each segment is in equilibrium, that is, F U + F C = W + F D . Note that, in equilibrium, W = F C (and F U = F D ) at geostationary height, W > F C (and F U > F D ) below, andthe reverse is true for an element above this height.Pearson suggested that a desirable design is to maintain a constant stress σ throughoutthe tether [10]. Then, for an element below geostationary orbit, we have F U − F D = σdA = W − F C , where A is the cross-sectional area of the cable. This results in an exponentialtapering of A shown schematically in Fig. 1(a): A increases from a small value at thebase to a large one at geostationary height and back to a small one thereafter. The taperratio — defined as area at geostationary height divided by area at the Earth’s surface —is given by T = exp ( K/L c ). Here, K is a constant that depends on Earth’s radius andgeostationary height and L c = σ/w is the characteristic length of the material, i . e ., the ratiobetween the constant stress in the tower σ and the specific weight w . It can be seen that,to avoid prohibitively large cross-sectional areas, one should use light (small w ) materialsable to sustain high stresses (large σ ). For reference, using a safety factor of 2, a steel cablerequires a taper ratio T = 2 . × , whereas for carbon nanotubes, assuming a maximumtensile strength of 130 GPa, the taper ratio is T = 2 . −
31 GPa [13], highlighting we are fast approaching the material strength3anges necessary for stable megastructures with self-repair mechanisms.
II. FILAMENT BUNDLE RUPTURE DYNAMICS WITH REPAIRA. Space elevator model
Although the finished space elevator may comprise of enough parallel tethers (cables)to meet cargo transport demands [3, 4], we focus here on the first cable. Specifically, wemodel each tether as a set of vertically stacked segments (see Fig. 1); each segment ismade up of identical, parallel, non-interacting filaments. The total number of segments isdetermined by the maximum filament length and the amount of stress variation permittedin the segment (gravitational forces acting on segments vary with height). To maintaina tapered shape of the cable, each segment’s cross-sectional area changes with height byvarying the numbers of filaments in the segment, effectively obtaining a step-wise discretisedversion of the continuous exponential tapering discussed above.We further restrict the analysis to a single segment shown schematically in Fig. 1(b).Filaments in the segment are active if supporting load and inactive if broken and not sus-taining load. Additionally, active filaments can fail and become inactive and, conversely,inactive filaments are repaired by replacing them with active ones. We assume the processesof rupture and repair do not significantly change the mass of the segment. Furthermore, thesegment height is considered small enough to ignore variability in gravity and centrifugalforces. Therefore, the net force on the segment is constant and, hence, segment dynamicsare independent of the dynamics of its neighbours When filaments are gained or lost, theresulting load is instantaneously divided among all active filaments. We ignore the inter-action between filaments ( e . g ., friction) and changes in the inter-filament platform angles.However, the model is flexible enough to incorporate aspects discussed in [4], such as aribbon pattern to protect against potential hazards ( e . g ., by changing segment orientation).We point out that the model also mirrors biological structures built with smaller subunits,for example, parallel arrays of collagen fibres, which form tendons.The segment-filament model proposed here is a simplified model for gaining intuitionabout the structure-substructure interaction, rather than a suggestion for a specific engi-neering design. In the case of no repair, our non-interacting filament model is known as the4 qual load sharing fibre bundle model . This has been studied extensively in the literature,beginning with Daniels [14], who analysed bundle strength in fast rupture and Coleman [15–17], who worked on fibre bundle lifetime in time-dependent creep-rupture, with further gen-eralizations by Phoenix [18, 19]. Past analytic work is restricted to the case where fibrerupture times are exponentially distributed, leading to a memoryless Markov process (seeSec. III F 1) and involves “mean field” approaches, as well as asymptotics for large number offibres, where fluctuations can be ignored. Newman and Phoenix’s more recent work [20, 21]explores simulation algorithms for large number of fibres in the case of local load sharingbreakage for more general underlying fibre lifetime distributions. The analytic approachused in our paper does not impose restrictions on the underlying filament lifetime distribu-tions, can be solved exactly, and, more importantly, extends to the case where filaments arerepaired, a case where the age-structure of the ensemble becomes crucial. We emphasize ouranalysis combines the deterministic aspect of ageing with the stochastic rupture/repair ofthe filaments. B. Dynamics of active filaments
As underlined in the model description, the number of cable segments is sufficiently largeto approximate the segment total force as constant. Then, changes in the single segmentstress are due solely to variations in its cross-sectional area. This area is the product between n ( t ) — the number of active filaments at time t — and the constant cross-sectional area ofa single filament. Equivalently, the product σ ( t ) × n ( t ) is constant, where σ is the stressin the segment at time t . The segment is considered operational if σ ( t ) < σ max , with σ max a constant representing the ultimate tensile strength (UTS) of the material. It is moreconvenient to view this inequality in terms of the working stress ratio , which we define as ω ( t ) := σ ( t ) /σ max . Then, the condition for reliability of the structure becomes ω ( t ) < ω .When considering the dynamics of ω ( t ), it is more direct to analyse n ( t ), the number ofactive filaments. We assume there are two stochastic effects which govern the kinetics of n ( t ):filament rupture and repair . Filament rupture times are, therefore, random variables drawnfrom a lifetime distribution, which depends on the stress (load history) σ ( t ) (or, equivalently,5n n ( t )). A typical choice for this distribution is Weibull[22–25]. Since new filaments areintroduced in the system through the repair process at various times, we denote by a i the i th active filament’s age — the time elapsed from the moment it begins bearing load. Eachfilament therefore has a probability rate of rupturing k n ( a i ). On the other hand, we assumethat filaments are autonomously repaired by robots with a constant probability per unittime ρ (see Sec. II C for a detailed discussion on the transition probability rates).The dynamics of n ( t ) are represented schematically in Fig. 2. During any small incrementof time τ , either an active filament ruptures ( n → n −
1) according to k n ( a ), or an inactive oneis repaired according to ρ ( n → n +1), or neither. In either case, all loaded filaments will agedeterministically, shifting the age structure of active filaments. We can describe this processmathematically in the formalism of Chou and Greenman [26, 27]. We let p n ( a n ; t ) d a n bethe probability that out of n randomly selected active filaments, the i th one has age in theinterval [ a i , a i + da i ], where a n = ( a , a , ..., a n ) is the vector of ages. We can then write thehierarchy of coupled integro-differential equations as: ∂p n ( a n ; t ) ∂t + n (cid:88) i =1 ∂p n ( a n ; t ) ∂a i = − p n ( a n ; t ) n (cid:88) i =1 k n ( a i )+ ( n + 1) (cid:90) ∞ k n +1 ( α ) p n +1 ( a n , α ; t ) dα. (1)with the associated boundary condition np n ( a n − , t ) = ρp n − ( a n − ; t ). In addition to theboundary conditions, one needs to also provide an initial condition p n ( a n ; t = 0) to fullyspecify the system. Integrating over all ages a n , one gets the probability of having n ( t )active filaments at time t , that is, p ( n, t ) = (cid:82) p n ( a n ; t ) d a n . This hierarchy leads to an exactanalytic solution for the probability density, albeit an unwieldy one [27]. C. Derivation of the transition probabilities
1. Rupture
There are various modes in which mechanical structures can fail ( e . g ., ductile fracture,brittle fracture, fatigue, etc .) [28]. In this manuscript, we focus exclusively on creep-rupture— the time-dependent deformation process under moderate to high stresses. Our decision isjustified given the tapered design of the space elevator cable, which implies a high constantstress throughout the structure. It is interesting to note that creep-rupture data turns out6o be far from abundant for low temperatures. This is somewhat expected given that thestresses involved in obtaining reasonable times to rupture in relevant materials are typicallysignificantly above 50% of the their ultimate tensile strengths. Since most engineeringstructures are designed to operate below these stress ratios, research in this area is somewhatscarce.To obtain the probability of failure due to creep-rupture, it is reasonable to assume thatfilament rupture time is distributed according to a Weibull distribution [22–25]. We highlightthat the inferences drawn regarding the trade-off between repair rates and sustaining higherstresses do not change meaningfully depending on the choice of distributions; we are limitingthe analysis to Weibull for the sake of definiteness. We seek the conditional probability thata filament ruptures in an interval of time τ , given that it has been in use a time of a i , i . e ., hasage a i . We let F W ( a i ) be the Weibull probability of rupture in the interval [0 , a i ] in Eq. (12)and f W ( a i ) = F (cid:48) W ( a i ) its associated probability density function. If τ is small, the probabilityof rupturing during [ a i , a i + τ ] is f W ( a i ) τ . The probability that the filament reached age a i unruptured is 1 − F W ( a i ). The conditional probability per unit time (transition probabilityrate) is then k ( a ; λ, s ) = lim τ → τ f W ( a ) τ − F W ( a ) = sλ s a s − . (2)As shown in Fig. 8, we use the relationship ln( λ ) = ˆ α ln( σ )+ ˆ β to express the scale parameter λ in terms of the stress σ and take the shape parameter s as constant, using the average (cid:104) ˆ s (cid:105) . The rupture rate becomes k n ( a ) = c [ σ ( n )] c a c , (3)where the Kevlar-specific constants are c = 2 . × − , c = 7 . c = 0 .
2. Repair
The repair mechanism in this manuscript is independent of the filament number or agedistribution; during every small time increment τ , there is a probability ρτ for the entiresegment to be repaired. The repair amounts to adding an active filament and removing aninactive one, thus leaving mass unchanged. Therefore, in this simplified case, the probabilityrate per unit time is a constant ρ . Alternatively, ρ filaments will be added on average perunit time. To continue the biological analogy, we can envision a mechanism that performs7epairs automatically ( e . g ., autonomous robots). Given robots’ arbitrary positions along thecable, each segment has a certain probability of getting repaired. The trade-off in addingmore repairing robots comes from the added mass associated with them. However, we canalso consider the control problem associated with picking more complex functional formsfor the repair rate to potentially minimize material flux and total robot mass. It turns outthat, despite being overly-conservative and choosing ρ as constant, the repair rate value isreasonable and structures can operate reliably at higher stresses. D. Age-dependent stochastic simulation
In the case in which the rupture rates k n ( a i ) depend on the number of active fila-ments n , the hierarchy in (1) leads to a somewhat unwieldy analytic solution. We usean age-dependent stochastic simulation method based on the time-dependent Gillespie al-gorithm [29], which takes into account the age-structure of the population. Starting with N filaments (see Sec. III D for the choice of N ), the algorithm generates a transition atevery step of the iteration either until a passage condition is reached ( e . g ., the number offilaments drops below a critical value corresponding to ω = 100%) or a maximum numberof iterations condition is reached. Each transition is broken down into two steps: findingthe time to the first transition and determining which transition occurs.Tackling the first step requires knowing the distribution of jump times. Let τ be theinterval of time such that given a jump occurs at t , then the next jump will occur at t + τ .Assume there are n filaments after the jump at t with ages a n . We are interested in thecumulative distribution of τ denoted F n → n ± ( τ | a n ; t ). First, focus on the probability thatin the interval [ t, t + τ ] there occur no jumps. To derive this, we break up the interval τ into q small sub-intervals of size ∆ τ . Using the definition of transition probabilities, we canwrite the probability that no transitions occur in [ t + l ∆ τ, t + ( l + 1)∆ τ ] for l = 0 , . . . , q − − [ ρ + (cid:80) ni =1 k n ( a i + l ∆ τ )] ∆ τ . Since ∆ τ is chosen sufficiently small, we can write theprobability as exp (cid:110) − (cid:82) ( l +1)∆ τl ∆ τ [ ρ + (cid:80) ni =1 k n ( a i + τ (cid:48) )] dτ (cid:48) (cid:111) . Taking the product over all l =0 , . . . , q , we get the probability that no transition occurs on any of the sub-intervals. Then, F n → n ± ( τ | a n ; t ) = 1 − exp (cid:40) − (cid:34) ρτ + n (cid:88) i =1 (cid:90) τ k n ( a i + τ (cid:48) ) dτ (cid:48) (cid:35)(cid:41) . (4)We draw R , a uniform random number on [0 ,
1] and find the jump time τ ∗ as the solution8o the equation F n → n ± ( τ ∗ | a n ; t ) = R via the Newton-Raphson method.The second step of the transition is to determine whether one of the n filaments rupturesor the segment is repaired. To accomplish this, we sample the categorical (multinomialwith one trial) distribution, where each category has (unscaled) probability ρ , k n ( a + τ ∗ ), k n ( a + τ ∗ ), ..., k n ( a n + τ ∗ ) (only include a category for repair if n < N ).Once a transition occurs, the vector of ages a n is incremented by τ ∗ component-wise.If the filament is broken, it leaves the pool and is no longer tracked. If the segment isrepaired, a new filament with age a min enters the pool. If no stopping conditions are met( e . g ., barriers, maximum time), the algorithm continues to generate transitions. III. RESULTSA. The need for autonomous repair
In classic reliability engineering, a typical way of ensuring structure integrity is by de-signing it to operate at low working stress ratios ω (or, conversely, at high safety factors).This is a good rule of thumb when the distributions of material properties are well studiedand stresses in the structure are low enough to allow for high safety factors. In the spaceelevator, however, high safety factors are unrealistic, as these would lead to exponentialincreases in the taper ratio [3]. Furthermore, while ductile materials, such as steel, havewell-understood tensile properties, carbon nanotubes (most realistic material to be used forthe space elevator) were shown to have considerably variable strengths [30]. Their brittlenature [31], coupled with the practical limits imposed on the safety factor, led us to suggesta paradigm shift from low working stress ratios to higher ones and continuous repairs. Froma practical standpoint, this could be done by enhancing the climbers in [3, 4] through robotscapable of autonomous repair.Currently, much of the focus in carbon nanotube technology research revolves aroundenhancing their strength, with little emphasis on exploring their creep-rupture time distri-butions. Data is much more readily available for a similarly brittle fibre, namely aramid(Kevlar R (cid:13) , manufactured by DuPont). The comparison is warranted in light of [31]. We arenot suggesting that the space elevator ought to be built using Kevlar; rather, we are aimingto draw inferences on the effects of repair on the dynamics of the tether using real-world9ata. Encouraging results for Kevlar, a material significantly weaker than the currentlyavailable carbon nanotubes [30], suggest that one should opt for a design which incorporatesan autonomous repair mechanism.For the sake of concreteness, we analyse the dynamics of a cable segment constructedusing Kevlar fibres relying on the data in [25] (see Appendix for data analysis). It was foundthat creep-rupture lifetime data for Kevlar fibres is well described by a Weibull distribution[22, 25]. Following a derivation in [32], we obtain the explicit form of the rupture probabilityper unit time for a filament of age a i as outlined in Sec.II C: k n ( a i ) = γ n γ a γ i , (5)where the γ j are fitted constants specific to Kevlar.Starting with a fixed number of active filaments, corresponding to a targeted workingstress ratio ω , we use the stochastic simulation scheme for age-structured dynamics de-scribed in Sec. II D to predict the probability that the system is reliable over time. If thereis no repair mechanism in the system, not only is failure inevitable, but the distributionof times to failure has a large spread (Fig. 3). The only way one can improve reliabilitywithout repair in this framework is to decrease the operating ratio to a low enough valueto delay the inevitable. This is not tenable in the space elevator, since this would eitherrequire lowering the operating stress by increasing the taper ratio to extreme values or byusing materials much stronger than those currently available. B. The effects of an autonomous repair mechanism
As previously mentioned, operating the space elevator segment in the absence of a repairmechanism will lead to eventual segment failures in time. We now introduce an autonomousrepair mechanism, which amounts to repairing inactive filaments with a probability per unittime ρ (incidentally, an interesting optimal control problem is how to modulate ρ with thenumber of active filaments n most efficiently from a cost perspective). We consider thesimple case of constant repair rates with the understanding that this is not optimal. Asshown in Fig. 4, the segment dynamics in Fig. 3 improve dramatically with modest repairrates (1-4 filaments every 10 hours) by creating a bifurcation in behaviour: either filamentsrupture quickly and the system fails or they last long enough for the repair rate to take over10nd stabilize the system. Note that, to ensure the segment mass does not increase, we do notallow the number of filaments to go above the initial value, i . e ., we have a reflective barrier.This guarantees that the system is stabilized at a number of filaments corresponding to theinitially targeted working stress ratio. We see that with higher repair rates, not only do weeliminate trajectories ending in failure, but we also speed up the time to reach the stableregime.With the introduction of a repair mechanism, Fig. 5(a)-(d) show that the system canbe stabilized at significantly higher working stress ratios. This is crucial, because it impliesthat one can use materials with a lower ultimate tensile strength. The trade-off comes in theform of higher repair rates, but the scaling of repair with working stress ratio is encouraging(Fig. 5(e)). An additional benefit to operating at higher working stress ratios is that thesystem stabilizes much faster, at which point repair could be modulated down (insets ofFig. 5(a)-(d)). For example, we see that, for Kevlar, operating the segment at ω = 90%requires a repair rate ρ = 30 filaments per hour. Although this number may seem high, itis worth pointing out that the material flux is 3% of the segment mass every hour and thatthe system stabilizes in just 20 hours.We have thus found that, by adding an autonomous repair mechanism, one can ensurereliability at higher working stress ratios, which, in turn, allows for reasonable taper ratiosand construction using weaker materials. In his report, Edwards [3] considers a workingstress ratio of 50% and claims carbon nanotubes with σ max = 130 GPa would be sufficientfor the cable specifications he suggests. Using recent measurements of carbon nanotubestrength [12, 33] of >
100 GPa and operating at the stress Edwards suggests implies aworking stress ratio of ω = 65%. At ω = 65%, the repair rate needed for a reliable Kevlarsegment would be less than ρ = 1 filament per hour. C. Moment equations for the number of filaments
In the special case in which the rupture rates do not depend on the number of activefilaments n , a hierarchy for the moments can be written explicitly and solved analyticallystarting with (1). Let k n ( a i ) ≡ k ( a i ) and define the marginal j-dimensional distributionfunction as in [27]: p ( j ) n ( a j ; t ) ≡ (cid:90) ∞ da j +1 ... (cid:90) ∞ da n p n ( a n ; t ) (6)11nd the factorial moments X ( j ) ( a j ; t ) ≡ ∞ (cid:88) n = j n !( n − j )! p ( j ) n ( a j ; t ) , (7)for j ≥ X (0) ≡
1. We can now write and solve the moment equations: ∂X ( j ) ( a j ; t ) ∂t + j (cid:88) i =1 ∂X ( j ) ( a j ; t ) ∂a i + X ( j ) ( a j ; t ) j (cid:88) i =1 k ( a i ) = 0 (8a) X ( j ) ( a j − , t ) = ρX ( j − ( a j − ; t ) (8b) X ( j ) ( a j ; t = 0) = g ( j ) ( a j ) (8c)To make the problem concrete, we derive explicit forms for the initial conditions g j ( a j ).Assume the cable segment starts off with N initial number of filaments, all with age 0.Then, p n ( a n ; t = 0) = δ n,N (cid:81) nl =1 δ ( a l ), where δ i,j is the Kronecker delta and δ ( · ) is the Diracdelta function. From (6) and (7), we find g ( j ) ( a j ) = N !( N − j )! j (cid:89) l =1 δ ( a l ) . (9)We note that, for k = 1, (8a) reduces to the classic McKendrick–von Foerster equation [34,35], which can easily be solved via the method of characteristics (see [27]). We find for thefirst two moments: X (1) ( a ; t ) = N δ ( a − t ) U( a − t, a ) ( a ≥ t ) ρ U(0 , a ) , ( a < t ) X (2) ( a , a ; t ) = N ( N − (cid:81) l =1 δ ( a l − t ) U( a l − t, a l ) ( t < a < a ) N ρδ ( a − t ) U(0 , a ) U( a − t, a ) ( a < t < a ) ρ U(0 , a ) U(0 , a ) ( a < a < t )where the propagator is U( a, b ) ≡ exp (cid:104) − (cid:82) ba k ( α ) dα (cid:105) and only the cases a < a wereconsidered, given that the moments are invariant in the ordering of the age arguments. Asshown in [36], if we let n [ a ,a ] ( t ) be the random variable representing the number of particleswith ages in the interval [ a , a ], we have (cid:10) n [ a ,a ] ( t ) (cid:11) = (cid:82) a a X (1) ( u ; t ) du and (cid:68) n a ,a ] ( t ) (cid:69) =12 a a X (1) ( u ; t ) du + (cid:82) a a (cid:82) a a X (2) ( u, v ; t ) dudv , and for a < t < a , we get for the expectationand variance: E n [ a ,a ] ( t ) = N U(0 , t ) + ρ W( a , t )Var n [ a ,a ] ( t ) = N U(0 , t ) [1 − U(0 , t )]+ ρ W( a , t ) , (10)where the integral of the propagator is W( a, b ) = (cid:82) ba U(0 , α ) dα . If we now let a → a → ∞ , we get the total expected number of filaments and their fluctuations. D. Choosing minimum filament age and initial number of filaments
It is worthwhile mentioning a subtle, but consequential point regarding filament aging.We have established that Weibull-distributed times to rupture lead to age-dependent tran-sition probabilities per unit time of the form (2). If s < a = 0. Inderiving the analytic result, we assumed that a newly added filaments start off with ageexactly a min = 0 hours. Fig. 6(a) shows that the statistics obtained from the simulationare sensitive to the minimum age at small ages, but the dependency is much weaker aftera few hours. Since filaments can already be stretched by the time they are installed in thesegment (either as part of quality assurance, or through process of installation itself), it isreasonable to assume they will have a non-zero initial age. In all simulations, we assumed a min = 12 hours.Another constant in the simulation is the initial number of active filaments N . Theactual choice of N depends on the material used, as well as on the position along the cableof the segment analysed Our results, however, are not overly-sensitive to the numerical valueof N as evidenced by Fig. 6(b), so we will choose an arbitrary value N = 1000. E. Comparison to analytic result
As shown in III C, if transition probabilities of rupture and repair do not depend on thenumber of active filaments n , we can obtain analytic results for first and second moments ofthe distribution of active filaments with ages in a given interval. We can then use the results13n Eq. (10) to ensure that the stochastic simulation scheme agrees with the analytic results.In our analysis, the repair rate is a constant, but the rupture probability rate depends onstress and, therefore, analytic solutions are not straightforward. For the sake of comparingthe simulation results with he analytic solutions, we will assume in this section only that thestress stays constant as filaments rupture. Physically, this would be equivalent to losing thefilament when it ruptures, thus decreasing the mass and force on the segment in a mannercommensurate to the loss of cross-sectional area.We examine the dynamics of a segment starting with N = 100 Kevlar filaments subjectedto a constant stress of 3 . ω ≈ a min = 10 − hours. Fig. 6(b) shows the comparisonbetween the analytic expected value/standard deviation of the number of active filaments inEq. (10) and what was obtained based on the stochastic simulation. The repair probabilityrate is constant, at ρ = 10 filaments per hour. We show 30 sample trajectories out of the10 generated and used in obtaining statistics. Each trajectory was assigned a maximumnumber of transitions (here, 400) as stopping conditions. The maximum time plotted waschosen as a predefined constant. One can see that the analytic result and the simulationsare in perfect agreement. F. Segment dynamics sensitivity to filament lifetime distribution
The model used in this manuscript to characterize an individual segment of the spaceelevator can be generalized in a few different ways. The main question the model addressesis how the stochastic lifetime of individual components translates into that of the structurebuilt by the substructures. An important feature of the model is that the rupture proba-bility rates of the substructures is age-dependent; that is, we combine the stochasticity ofrupture times with the deterministic aspect of ageing It turns out that this is a reasonablemodel for a wide-range of applications ( e . g ., cell division times). For the space elevator, weassume Weibull-distributed rupture times for the substructures. Additionally, we assumethat the filaments building up the segment do not interact directly, i . e ., they are statisticallyindependent.In this section, we relax the assumptions made about the lifetime distribution of thesub-components and explore the response in the lifetime distribution of the entire structure.14he intention here is not to exhaust the possible distributions, but to highlight the wideapplicability of the model. Alwis and Burgoyne [22] provide a comprehensive comparison ofvarious Kevlar fibre lifetime distributions. They consider lognormal vs. Weibull, as well asdifferent functional forms for the shape and scale parameter dependency on applied stress.It was found that out of the 120 models considered, the difference between best and worstwas only 1%. Therefore, we will only focus here on varying the Kevlar-specific constants,rather than changing functional forms. That is, we start with the shape and scale parametersestimated based on [25] and seek to understand how results change when these parametersare “shocked”.For this analysis, we will continue to assume that each filament has an age-dependentprobability of rupture given by a Weibull distribution with shape and scale parameters s and λ . We consider s a constant and λ a function of stress applied, given by ln( λ ) = α ln( σ ) + β ,where α and β are material constants. We varied the three parameters s , α and β from −
10% to −
10% of the original fitted value and analysed the response in failure time of thesegment.Fig. 7(a) shows the cumulative Weibull distribution for individual filaments rupture timesunder different parameter shocks and values. To see how these changes impact the failuretime distribution of the entire segment, one can look at Fig. 7(b). We point out that changesin the shape parameter of the distribution have a significantly smaller influence than changesto the scale parameter. Since it is the latter we would expect to be different for a strongermaterial (being the only parameter in the model which depends on stress), this furtherhighlights the importance of lifetime data for carbon nanotubes.
1. The exponential case
In the special case in which the shape parameter of a Weibull distribution is equal to 1,the distribution becomes exponential. This is particularly important when considering thefilament rupture probability rate given in (2), which takes the form k ( n ) = 1 λ ( σ ( n )) , (11)and, therefore, independent of the filament age. In other words, we are dealing withexponentially-distributed “jump” times and one can write a master equation for the number15f active filaments. Letting P ( n, t ) be the probability that at time t the segment has n activefilaments, one can write the familiar ∂P ( n, t ) ∂t = ρP ( n − , t ) + ( n + 1) k ( n ) P ( n + 1 , t ) − [ ρ + nk ( n )] P ( n, t ) , where ρ is the constant repair rate. The complicated dependency of k on n does not allow forstraightforward analytic solutions, but one can easily perform simulations using essentiallythe same method described in this manuscript. IV. CREEP-RUPTURE LIFETIME DATA FOR KEVLAR
In our analysis, we chose Kevlar as an example material for the space elevator segment.Our choice is justified by the material’s brittle nature and the extensive study of creep-rupture lifetime data [22–25]. We are not suggesting the space elevator be built out ofKevlar, but wanted to show concretely that even a material 10 times weaker than carbonnanotubes leads to reliable segments, given a reasonable repair mechanism. To estimate γ , γ and γ in Eq. (5), we use the data in Wagner et al . [25] The authors find that thelifetime distribution of aramid fibres under various constant stress levels is best describedby a Weibull distribution with cumulative function: F W ( a ; λ, s ) = 1 − exp (cid:104)(cid:16) aλ (cid:17) s (cid:105) , (12)where a is the age of the fibre and λ , s are the scale and shape parameters. In one of the datasets analysed, they measure rupture times of 46-48 aramid fibres subjected to stresses rangingfrom 2 . . s is constant, the scale parameter ln ( λ ) is linear in ln ( σ ). This is consistentwith the recent analysis in [22]. We find an explicit dependence by fitting a line of the formln( λ ) = ˆ α ln( σ ) + ˆ β to the data in Table I and find ˆ α = − .
283 [ln(hours) / ln(GPa)] andˆ β = − .
893 [ln(hours)] [Fig. 8(b)]. 16 . DISCUSSION
In this manuscript, we contrasted the biological and engineering paradigms of designingcomplex structures. While the latter design is based on operating structures at very con-servative loads compared to the strength of the materials used, thus ensuring reliability, theformer allows for loads significantly closer to the maximum, but utilizes an autonomous andcontinuous repair mechanism to make up the potential loss of reliability. In megastructures,traditional engineering approaches are hampered by the necessity of prohibitively strong ma-terials. We argue that one approach to circumvent this problem is to draw inspiration frombiological structures and introduce self-repair mechanisms. In essence, this shifts the focusfrom requiring very strong — possibly unavailable — materials to repairing with weakermaterials at the necessary rate to maintain the structure’s integrity. We analysed the spaceelevator as an example of a megastructure and used an age-dependent stochastic model forits underlying components, which allowed us to quantitatively describe its reliability by look-ing at probabilities of segment failure. Although current materials are not strong enoughto support the stresses required, a built-in self-repair mechanism exhibiting low repair rateswas enough to maintain reliability in a cable made of Kevlar.The model in this manuscript focuses primarily on the dynamics of the non-interactingsub-components (in this case, filaments) and describes how fluctuations in their number,due to rupture and repair, translate into the reliability probability of the larger structure.We have avoided suggesting specific designs for the cable, as this was not in scope of themanuscript. Similarly, other potential stochastic effects ( e . g ., meteors, winds, erosion, etc .)were not included in the analysis, but can be incorporated. Additionally, although Kevlarwas found to be strong enough to maintain reliability, its density remains prohibitivelylarge to make it practical, given the massive volume of material which would need to betransported. On the other hand, carbon nanotubes already have the necessary strength,provided a repair mechanism can be incorporated to operate at higher working stress ratios.Estimating the repair rates for carbon nanotubes remains an open question, contingenton the availability of data regarding their creep-rupture lifetime distribution, which has notyet been thoroughly studied to our knowledge. More research in this direction is necessaryto quantify the exact requirements, but it is very encouraging to see that Kevlar, a materialweaker by an order of magnitude compared to the theoretically predicted strength of carbon17anotubes, can operate reliably without much material turnover. Incidentally, the inferencesdrawn from our model have biological applications: while healing, tendons remain under ten-sion due to cells exerting active forces to stretch the collagen, similar to how repairing robotswould stretch the fibres in the space elevator. This allows for a better understanding of thedynamics of biological repair, with possible applications to many different structures ( e . g .,bones, tendons, muscle, etc .). Furthermore, our analysis provides the necessary frameworkto consider more complex models in which filaments can interact, material strengths arestochastic and external noise on the cable is present. We also emphasize that constant re-pair probability rates are overly-conservative. More complicated control theory approachescan significantly increase feasibility by lowering the amount of repair needed as structuresstabilize. Data, code and materials:
All data needed to evaluate the conclusions in the paper arepresent in the manuscript. Additional data and code related to this paper may be requestedfrom the authors.
Competing interests:
We have no competing interests.
Authors’ contributions:
DMP and SXS conceived the research. DMP and SXS designedthe analyses. DMP and SXS conducted the analyses. DMP and SXS wrote the manuscript.
Acknowledgements:
The authors thank Benjamin W. Schafer for helpful conversations.The authors also thank Jenna Powell-Malloy for illustrating the space elevator cartoon.
Funding:
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FIG. 1.
Space elevator diagram. (a)
The space elevator tether is anchored at the Equator,extends past geostationary orbit and is balanced by a counterweight. The tether is made up ofindependent horizontal segments stacked vertically. Each segment is made up of filaments. Thenumber of filaments for each segment varies exponentially with height. (b)
A tether segmentexperiences four forces: its weight W , the outward centrifugal force F C , and upward/downwardforces F U and F D , due to the part of the cable above/below the element. At equilibrium, F U + F C = W + F D , leading to tension in the bundle. (c) Segment filaments are active if they carry load.Otherwise, they are inactive. Active segments can become inactive through rupture and inactivecables can become active through repair. u m b e r o f A c ti v e F il a m e n t s n + 1 n − 1 n Time t t + τ
Age Structure o f F il a m e n t s Age Structure o f F il a m e n t s ττ o f F il a m e n t s Age Structure τ Age Structure o f F il a m e n t s R e p a i r R up t u r e AgingAgingAging
FIG. 2.
Stochastic bundle model with aging.
At time t , there are n active filaments. The i th active filament has age a i , measured from the time of its loading. Ages can differ among filamentsdue to the repair process, according to which inactive filaments are replaced with active ones. Eachfilament has a rupture probability rate k n ( a i ), which depends on the specific filament’s age a i . Thewhole system has a probability rate of repair given by ρ . During each small increment of time τ , the system ages deterministically by τ , shifting the overall age distribution ( a i → a i + τ ) andalso jumps stochastically to one of three states: (i) n − (cid:80) ni =1 (cid:82) τ k n ( a i + τ (cid:48) ) dτ (cid:48) , (ii) n + 1 (repair, blue) with probability ρτ , or (iii) n filaments (grey) withprobability 1 − (cid:0) ρτ + (cid:80) ni =1 (cid:82) τ k n ( a i + τ (cid:48) ) dτ (cid:48) ) (cid:1) .
100 200 300100%90%80%70%60%50% 50060070080090010000 100 200 30000.020.040.060.080.100.12 P r ob a b ilit y o f F a il u r e (a)(b) W o r k i ng S t r e ss R a ti o , ω ( t ) Time (years) N u m b e r o f A c ti v e F il a m e n t s , n ( t ) FIG. 3.
Dynamics without filament repair. (a)
A sample of 100 paths (grey) are shown for thenumber of filaments n ( t ) (right) and corresponding working stress ratio ω ( t ) := σ ( t ) /σ max (left).The blue and red dashed lines show the initial working stress ratio and the maximum stress ratio atwhich failure occurs. The shading highlights 90% of the distribution, while the black lines are the5 th and 95 th percentile paths (dashed) and the median path (solid) computed using a horizontalslice at n = 500 filaments or ω = 100%. (b) The histogram of times to failure shows a medianrupture time of approximately 125 years. o r k i ng S t r e ss R a ti o , ω ( t ) N u m b e r o f A c ti v e F il a m e n t s , n ( t ) Time (years) (a) (b)(c) (d)
FIG. 4.
Effects of repair on filament dynamics and on bundle stability.
A sample of100 paths (grey) are shown for the number of filaments n ( t ) (right) and corresponding workingstress ratio ω ( t ) := σ ( t ) /σ max (left). The blue and red dashed lines show the initial working stressratio and the maximum stress ratio at which failure occurs. The shading highlights 90% of thedistribution, while the black lines are the 5 th and 95 th percentile paths (dashed) and the medianpath (solid) computed using a vertical slice t = 100 years or at time of system stability, whicheveris sooner. The repair rates are (a) ρ = 10 − per hour, (b) ρ = 2 × − per hour, (c) ρ = 3 × − per hour, (d) ρ = 4 × − per hour. P r ob a b ilit y Time (years) P r ob a b ilit y Time (years)
Time (years) −3 P r ob a b ilit y Time (years × 10 −3 ) P r ob a b ilit y Time (years × 10 −3 ) × 10 −3 R e p a i r R a t e , ρ ( / h ) W o r k i ng S t r e ss R a ti o , ω ( t ) ( % ) Target Working Stress Ratio, ω (t) N o . o f A c ti v e F il a m e n t s , n ( t ) ( × ) (a) (c)(b) (d)(e)
60% 70% 80% 90%10 -2 FIG. 5.
Target working stress ratio vs. repair rate trade-off.
Sample paths (grey) for num-ber of filaments n ( t ) (right) and working stress ratio ω ( t ) := σ ( t ) /σ max (left) with corresponding5 th and 95 th percentile paths (dashed) and the median path (solid), as well as shading for 90%of the distribution and stabilizing time histograms (insets) are shown for different target workingstress ratios ans repair rates (a) ω = 60% and ρ = 0 .
08 per hour, (b) ω = 70% and ρ = 2 perhour, (c) ω = 80% and ρ = 10 per hour, (d) ω = 90% and ρ = 30 per hour. (e) Summary ofthe repair rates needed to sustain higher working stress ratios. Time (years) N u m b e r o f A c ti v e F il a m e n t s , n ( t ) −3 A v e r a g e T i m e t o F a il u r e ( y ea r s ) Initial Age of Filaments, a min (hours) Initial Number of Active Filaments, N (× 100) (a)(b) FIG. 6.
Age-dependent stochastic simulation. (a)
Using the simulation scheme presentedin the manuscript, we explore the sensitivity of the average time to failure to number of initialfilaments (top, blue, filament minimum age a min = 12 hours) and minimum age of a newly-addedfilament (bottom, black, N = 1000 initial filaments). (b) In the case where the stress σ is aconstant, we obtain analytic results for the mean (solid black line), one standard deviation aroundthe mean (purple shading) and two standard deviations around the mean (blue shading). Wesuperimpose the corresponding simulated values (dashed red lines). We show 30 sample trials(grey) out of the total 10000. The repair rate used was ρ = 10 filaments per hour and minimumfilament age of a min = 10 − hours.
10 -5 0 5 10-2000200400600800 -10 -5 0 5 10-2000200400600800 C u m u l a ti v e P r ob a b ilit y o f F a il u r e Δ i n M ea n F a il u r e T i m e ( % ) Change in Parameter Value (%) −10% +10%+0% s βα Δ i n S t d . D e v . o f F a il u r e T i m e ( % ) Age (hours) (a)(b)
FIG. 7.
Dynamics sensitivity to lifetime distributions. (a)
The shape parameter s (left) andthe two scale parameters – α (middle) and β (right) – are varied to assess the resulting Weibullcumulative probability distribution. The base distribution (dashed blue line) is the one for Kevlarat σ = 75% × σ max . (b) We analyze the response of mean and standard deviation for failuretimes of the segment as a percentage of their original value when the scale and shape parametersare shocked in increments of 5% of their initial value. Again, their initial values are given by theWeibull probability of rupture for Kevlar filaments at stress σ = 75% × σ max . .100.300.600.900.99 0.95 1 1.05 1.1 1.150369 C u m u l a ti v e P r ob a b ilit y o f F a il u r e Age (hours) –3 –2 –1 Logarithm of Working Stress, ln( σ ) (ln(GPa)) L og a r it h m o f S ca l e P a r a m e t e r , l n ( ) ( l n ( h )) (a) (b) FIG. 8.
Creep-rupture lifetime statistics for Kevlar.
Data sourced from Wagner et al . [25]. (a)
Lifetime data for Kevlar is shown in a Weibull plot at various stress levels: 2 . . . (b) A linear fit is performed to obtain the dependency of the scale parameter λ on thestress level σ .TABLE I. Shape and scale parameter estimates.
Maximum Likelihood Estimators for Weibullscale ˆ λ and shape ˆ s parameters for filament lifetime σ (GPa) ˆ λ (h) ˆ s2.6122 2902 0.1572.7887 518.3 0.1832.9652 11.46 0.1463.1417 1.156 0.212