1. The role of defects in the design of a space elevator cable: From nanotube to megatube - Latest research results 2th Int. Conf. on Space Elevator Climber and Tether Design, December 6-7, 2008, Luxembourg, Luxembourg Nicola M. Pugno Politecnico di Torino, Italy

2. 1. Introduction - Griffith The father of Fracture Mechanics: Alan Arnold Griffith 1893-1963 The Phenomena of rupture and flow in solids; Philosophical Transactions of the Royal Society, 221A, 163 (1920). The father of Fracture Mechanics: Alan Arnold Griffith 1893-1963 The Phenomena of rupture and flow in solids; Philosophical Transactions of the Royal Society, 221A, 163 (1920). Deterministic approach

3. Weibull (1) The father of the statistical theory of the strength of solids Waloddi Weibull 1887-1979 A statistical theory of the strength of materials; Ingeniörsvetenskapsakademiens Handlingar 151 (1939). The father of the statistical theory of the strength of solids Waloddi Weibull 1887-1979 A statistical theory of the strength of materials; Ingeniörsvetenskapsakademiens Handlingar 151 (1939).

4. 2.Stress concentrations and intensifications Linear Elastic Plate (infinitely large) with an hole under (remote) traction . (a)Circular hole: stress concentration (b)Elliptical hole: stress concentration (c)Crack: infinite stress concentration, i.e., “stress intensification” K = stress-intensity factor r = distance from the tip

5. Maximum stress criterion (2) Maximum stress = material strength E.g., strength for a plate with a circular hole = 1/3 strength for the plate without the hole E.g., strength for a plate with a circular hole = 1/3 strength for the plate without the hole Vanishing strength for a plate with a crack!? I cannot believe it! (PARADOX) Vanishing strength for a plate with a crack!? I cannot believe it! (PARADOX) …independently from the size of the hole!? I cannot believe it! …independently from the size of the hole!? I cannot believe it!

6. Griffith’s energy balance criterion (2) Stability (or instable if larger than zero): Criterion for fracture propagation W = total potential energy  = dissipated energy A = crack surface area G C =  / A = fracture energy of the material (per unit area) G = energy release rate (per unit area) E.g., strength for the cracked plate E = Young modulus Improvement: not vanishing strength, but… infinite strength for defect free solids!? I cannot believe it! (PARADOX) Energy release rate = fracture energy

7. 3. Quantized fracture mechanics (QFM) Stress intensity factors from Handbooks Very simple application

8. The Griffith case treated with QFM (3) This represents the link between concentration and intensification factors! This represents the link between concentration and intensification factors! LEFM can treat only “large” and sharp cracks QFM has no restrictions on defect size and shape LEFM can treat only “large” and sharp cracks QFM has no restrictions on defect size and shape LEFM  Q=0:

9. Dynamic quantized fracture mechanics (DQFM, 3) Quantization not only in space but also in time (finite time required to generate a fracture quantum) Kinetic energy T included in the energy balance Quantization not only in space but also in time (finite time required to generate a fracture quantum) Kinetic energy T included in the energy balance Time quantum Balance of action quanta

10. 4. Fracture of nanotubes: nanocrack Strength [GPa]n=2n=4n=6n=8(n=2) MM - (80,0)64.150.342.136.9 QFM64.149.642.037.0 QFM: formula for blunt cracks with length from MM; = Interatomic distance best fit (very reasonable);

11. Nanoholes (4) m=1m=2m=3m=4m=5m=6 QFM0.680.480.420.390.370.36 (50,0)0.640.510.440.400.370.34 (100,0)0.650.530.470.430.410.39 (m=1) MM also in good agreement with fully quantum mechanical calculations MM Note in addition that by MM strength reductions due to one vacancy by factors of 0.81 for (10,0) and 0.74 for (5,5) nanotubes are again close to our QFM-based prediction, that yields 0.79 (not 1/3 or 0!).

12. Nanotensile tests on nanotubes (4) Stretching of multi-walled carbon nanotubes between Atomic Force Microscope opposite tips

13. Experiments on Strength of (C) Nanotubes (4) Measured strength (Ruoff’s group) of 64, 45, 43… GPa (against the theoretical (DFT) value of about 100 GPa) Defects!

14. Comparison between experiments (4) (A) Assuming an ideal strength for the multi-walled carbon nanotubes experimentally investigated of 93.5GPa, as numerically (MM) computed, and applying QFM: 1. the corresponding strength for a pinhole m=1 defect is 64GPa, against the measured value of 63GPa, 2. for an m=2 defect is 45GPa, against the measured value of 43 GPa, 3. for an m=3 defect is 39 GPa, as the measured value, and so on… Does a strength quantization exist? (B) For m tending to infinity (large holes) the strength reduction is predicted by QFM of a factor 1/3.36 (close to the classical 1/3!) (C) In addition note that, also with an exceptionally small defect - a single missing atom- a strength reduction by a factor of 20% is expected! (A) Assuming an ideal strength for the multi-walled carbon nanotubes experimentally investigated of 93.5GPa, as numerically (MM) computed, and applying QFM: 1. the corresponding strength for a pinhole m=1 defect is 64GPa, against the measured value of 63GPa, 2. for an m=2 defect is 45GPa, against the measured value of 43 GPa, 3. for an m=3 defect is 39 GPa, as the measured value, and so on… Does a strength quantization exist? (B) For m tending to infinity (large holes) the strength reduction is predicted by QFM of a factor 1/3.36 (close to the classical 1/3!) (C) In addition note that, also with an exceptionally small defect - a single missing atom- a strength reduction by a factor of 20% is expected!

15. Ordine del Giorno Is the strength quantized? (4) E.g., blunt cracks

16. Experiments on ideal strength (4) Thus, the strength is quantized as a consequence of the quantization of the defect size!

17. Nanoscale Weibull Statistics (NWS, 5) Weibull distribution for the strength of solids:probability of failure for a specimen of volume V under tension  Alternatively, V is substituted by the surface S of the specimen (for surface predominant defects) material constants ( m Weibull ’ s modulus) In contrast: At nanoscale nearly defect free structures! We substitute V with a fixed number n of defect (e.g., n=1) Number of “critical” defects assumed to be proportional to the volume V of the specimen

18. Application to experimental data on nanotubes (5) n=1; Thus, for nanotubes m around 3 Again, it seems that few defects were responsible for fracture of that nanotubes

19. 6. The Nanotube-based space elevator megacable

20. Multiscale simulations (5)

21. Strength of nanotube-based megacable (5)

22. Size-effect (5) Strength of the megacable? Multiscale approach: 10GPa Holes in the cables: 30GPa Cracks <30GPa Thermodynamic limit: 45GPa, not 100GPa… Strength of the megacable? Multiscale approach: 10GPa Holes in the cables: 30GPa Cracks <30GPa Thermodynamic limit: 45GPa, not 100GPa…

23. Elasticity of defective Nanotubes (6) The increment in compliance could result in a dynamic instability of the megacable

24. Nanobiocomposites (7) Fundamental roles of: (i)Tough soft matrix, (ii) Strong hard inclusions and (iii) hierarchy, for activating toughening mechanisms at all the size-scales

25. Example of bio-inspired nanomaterial (7) “Super-nanotubes” as hierarchical fiber reinforcements

26. N-opt=2, to optimize the material with respect to both strength and toughness, as Nature does in nacre Toughening mechanism = fibre pull-out Example of bio-inspired nanomaterial (7)

27. Optimizing Nano-composites (7) Optimization maps. Iso-hardness lines are drawn in blue and iso-toughness lines in red. Numbers along the curves indicate hardness and fracture toughness increments % (of a PCD material). Theory fitted to experiments.

28. Nano-armors (7)

29. Conclusions “All models are wrong, but some are useful” (George Box) is valid also in the context of the space elevator cable design! I would like to thank: Drs. M. Klettner and B. Edwards for the kind invitation The European Spaceward Association, for supporting my visit here & you for your attention http://staff.polito.it/nicola.pugno/ nicola.pugno@polito.it

30. Main References N. Pugno, On the strength of the nanotube-based space elevator cable: from nanomechanics to megamechanics. J. OF PHYSICS -CONDENSED MATTER, (2006) 18, S1971-1990. N. Pugno. The role of defects in the design of the space elevator cable: from nanotube to megatube. ACTA MATERIALIA (2007), 55, 5269-5279. N. Pugno, Space Elevator: out of order?. NANO TODAY (2007), 2, 44-47. N. Pugno, F. Bosia, A. Carpinteri, Multiscale stochastic simulations for tensile testing of nanotube-based macroscopic cables. SMALL (2008), 4/8, 1044-1052. N. Pugno, M. Schwarzbart, A. Steindl, H. Troger, On the stability of the track of the space elevator. ACTA ASTRONAUTICA (2008). In Print. A. Carpinteri, N. Pugno, Are the scaling laws on strength of solids related to mechanics or to geometry? NATURE MATERIALS, June (2005), 4, 421-423. N. Pugno, R. Ruoff, Quantized Fracture Mechanics, PHILOSOPHICAL MAGAZINE (2004), 84/27, 2829-2845. N. Pugno, Dynamic Quantized Fracture Mechanics. INT. J. OF FRACTURE (2006), 140, 159-168. N. Pugno, New Quantized Failure Criteria: Application To Nanotubes And Nanowires. INT. J. OF FRACTURE (2006), 141, 311-323. N. Pugno, R. Ruoff, Nanoscale Weibull statistics. J. OF APPLIED PHYSICS (2006), 99, 024301/1-4. N. Pugno, R. Ruoff, Nanoscale Weibull Statistics for nanofibers and nanotubes. J. OF AEROSPACE ENGINEERING (2007), 20, 97-101. N. Pugno, Young’s modulus reduction of defective nanotubes. APPLIED PHYSICS LETTERS (2007), 90, 043106-1/3 N. Pugno, Mimicking Nacre With Super-nanotubes For Producing Optimized Super-composites. NANOTECHNOLOGY (2006), 17, 5480-5484. V.R. Coluci, N. Pugno, S.O. Dantas, D.S. Galvao, A. Jorio, Determination of the mechanical properties of “super” carbon nanotubes through atomistic simulations. NANOTECHNOLOGY (2007), 18, 335702 (7pp). N. Pugno, The strongest matter: Einsteinon could be one billion times stronger than carbon nanotubes. ACTA ASTRONAUTICA (2008), 63, 687-689.