The role of defects in the design of a The role of defects in the design of a space elevator cable: From nanotube to space elevator cable: From nanotube to

megatube - Latest research resultsmegatube - Latest research results

2th Int. Conf. on Space Elevator Climber and Tether Design, 2th Int. Conf. on Space Elevator Climber and Tether Design, December 6-7, 2008, Luxembourg, LuxembourgDecember 6-7, 2008, Luxembourg, Luxembourg

Nicola M. PugnoNicola M. PugnoPolitecnico di Torino, Italy

1.1. IntroductionIntroduction - Griffith- Griffith1.1. IntroductionIntroduction - Griffith- Griffith

The father ofThe father ofFracture Mechanics:Fracture Mechanics:Alan Arnold Griffith Alan Arnold Griffith 1893-19631893-1963

The Phenomena of The Phenomena of rupture and flow in rupture and flow in solidssolids; ;

Philosophical Philosophical Transactions of the Transactions of the Royal Society,Royal Society,221A, 163 (1920).221A, 163 (1920).

The father ofThe father ofFracture Mechanics:Fracture Mechanics:Alan Arnold Griffith Alan Arnold Griffith 1893-19631893-1963

The Phenomena of The Phenomena of rupture and flow in rupture and flow in solidssolids; ;

Philosophical Philosophical Transactions of the Transactions of the Royal Society,Royal Society,221A, 163 (1920).221A, 163 (1920).Deterministic approachDeterministic approachDeterministic approachDeterministic approach

Weibull (1)Weibull (1)Weibull (1)Weibull (1)

The father of the statistical The father of the statistical theorytheoryof the strength of solids of the strength of solids Waloddi Weibull Waloddi Weibull 1887-19791887-1979

A statistical theory of the A statistical theory of the strength of materialsstrength of materials;;

IngeniIngeniöörsvetenskapsakademiens rsvetenskapsakademiens Handlingar 151 (1939).Handlingar 151 (1939).

The father of the statistical The father of the statistical theorytheoryof the strength of solids of the strength of solids Waloddi Weibull Waloddi Weibull 1887-19791887-1979

A statistical theory of the A statistical theory of the strength of materialsstrength of materials;;

IngeniIngeniöörsvetenskapsakademiens rsvetenskapsakademiens Handlingar 151 (1939).Handlingar 151 (1939).

2.2. Stress concentrations and Stress concentrations and intensificationsintensifications

2.2. Stress concentrations and Stress concentrations and intensificationsintensifications

Linear Elastic Plate (infinitely large) with an hole under (remote) traction .

(a) Circular hole: stress concentration 3max

(b) Elliptical hole: stress concentration

b

a21max

(c) Crack: infinite stress concentration, i.e., “stress intensification”

r

K Iasy

2

aK I

K = stress-intensity factorr = distance from the tip

Maximum stress criterion (2)Maximum stress criterion (2)Maximum stress criterion (2)Maximum stress criterion (2)

Maximum stress = material strengthMaximum stress = material strengthMaximum stress = material strengthMaximum stress = material strength

E.g., strength for a plate with a circular hole = E.g., strength for a plate with a circular hole = 1/3 strength for the plate without the hole1/3 strength for the plate without the holeE.g., strength for a plate with a circular hole = E.g., strength for a plate with a circular hole = 1/3 strength for the plate without the hole1/3 strength for the plate without the hole

Vanishing strength for a plate with a crack!?Vanishing strength for a plate with a crack!?I cannot believe it! (PARADOX)I cannot believe it! (PARADOX)Vanishing strength for a plate with a crack!?Vanishing strength for a plate with a crack!?I cannot believe it! (PARADOX)I cannot believe it! (PARADOX)

……independently from the size of the hole!?independently from the size of the hole!?I cannot believe it!I cannot believe it! ……independently from the size of the hole!?independently from the size of the hole!?I cannot believe it!I cannot believe it!

C max

Griffith’s energy balance criterion (Griffith’s energy balance criterion (2)2) Griffith’s energy balance criterion (Griffith’s energy balance criterion (2)2)

CGAWG dd

0dd CAG

Stability (or instable if larger than zero):

Criterion for fracture propagation W = total potential energy = dissipated energy A = crack surface areaGC = /A = fracture energy of the material (per unit area)G = energy release rate (per unit area)

a

EGC

E.g., strength for the cracked plate E = Young modulusImprovement: not vanishing strength, but… infinite strength for defect free solids!? I cannot believe it! (PARADOX)

Energy release rate = fracture Energy release rate = fracture energyenergyEnergy release rate = fracture Energy release rate = fracture energyenergy

0d

dW

A

3. Quantized fracture mechanics (QFM)3. Quantized fracture mechanics (QFM)3. Quantized fracture mechanics (QFM)3. Quantized fracture mechanics (QFM)

CGAWG *

IIICIII

AA

AIIIIIIIIIIII KKK ,,2

,,*

,,

AA

A

IIIIII

AA

AIIIIII AKA

K d1

where 2,,

2,,

unstable,0if

stable,0if*

*

C

C

AG

AG

unstable,0if

stable,0if2*

,,

2*,,

CIIIIII

CIIIIII

AK

AK

Stress intensity factors from Handbooks

Very simple application

0

WA

AGC

The Griffith case treated with QFM (3)The Griffith case treated with QFM (3)The Griffith case treated with QFM (3)The Griffith case treated with QFM (3)

Qa

Q

Qa

QK CICQFM 21

21

2

21

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 shapeLEFM can treat only “large” and sharp cracks QFM has no restrictions on defect size and shape

LEFM Q=0:

Dynamic quantized fracture mechanics Dynamic quantized fracture mechanics (DQFM, 3)(DQFM, 3)

Dynamic quantized fracture mechanics Dynamic quantized fracture mechanics (DQFM, 3)(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

0d1

t

tt

tΩTWAt

Time quantum tBalance of action

quanta

AGΩ C

1

1.2

1.4

1.6

1.8

2

2.2

-5 -4 -3 -2 -1 0 1 2 3

Log (t f /s)

KdI

C/K

IC

4. Fracture of nanotubes: nanocrack4. Fracture of nanotubes: nanocrack4. Fracture of nanotubes: nanocrack4. Fracture of nanotubes: nanocrack  

Strength [GPa] n=2 n=4 n=6 n=8 (n=2) 

MM - (80,0) 64.1 50.3 42.1 36.9

QFM 64.1 49.6 42.0 37.0

QFM: formula for blunt cracks with length nQa 2

A0.28.0 QGPa5.93C from MM;

03rQ = Interatomic distance

0r

best fit (very reasonable);

Nanoholes (4)Nanoholes (4)Nanoholes (4)Nanoholes (4)

m=1 m=2 m=3 m=4 m=5 m=6   QFM 0.68 0.48 0.42 0.39 0.37 0.36

(50,0) 0.64 0.51 0.44 0.40 0.37 0.34

(100,0) 0.65 0.53 0.47 0.43 0.41 0.39

(m=1)

012 rmR

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!).

Nanotensile tests on nanotubes (4)Nanotensile tests on nanotubes (4)Nanotensile tests on nanotubes (4)Nanotensile tests on nanotubes (4)

Stretching of multi-walled carbon nanotubes between Atomic Force Microscope opposite tips 

Experiments on Strength of (C) Nanotubes Experiments on Strength of (C) Nanotubes (4) (4)

Experiments on Strength of (C) Nanotubes Experiments on Strength of (C) Nanotubes (4) (4)

Measured strength (Ruoff’s group) of 64, 45, 43… GPa (against the theoretical (DFT) value of about 100 GPa) Defects!

Measured strength (Ruoff’s group) of 64, 45, 43… GPa (against the theoretical (DFT) value of about 100 GPa) Defects!

Comparison between experiments (4)Comparison between experiments (4)Comparison between experiments (4)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!

Ordine del GiornoOrdine del GiornoOrdine del GiornoOrdine del GiornoIs the strength quantized? (4)Is the strength quantized? (4)Is the strength quantized? (4)Is the strength quantized? (4)

0,12

1 21 nnQCn

Quantized Strength Levels

andForbidden bands

0.2

0.4

0.6

0.8

1

E.g., blunt cracksnQa 2

Experiments on -SiC nanorods, -Si3N4 whiskers and MWCNTs

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8n

Ob

se

rve

d S

tre

ng

th/

Ide

al

Str

en

gth

Quantized LevelsSi3N4-59GPaSi3N4-75GPaSiC-53GPaSiC-68GPaMWCNT-115GPaMWCNT-104GPa

Experiments on ideal strength (4) Experiments on ideal strength (4) Experiments on ideal strength (4) Experiments on ideal strength (4)

Thus, the strength is quantized as a consequenceof the quantization of the defect size!

Nanoscale Weibull Statistics (NWS, 5)Nanoscale Weibull Statistics (NWS, 5)Nanoscale Weibull Statistics (NWS, 5)Nanoscale Weibull Statistics (NWS, 5)

m

VF0

exp1

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)

m,0 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

Application to experimental data on Application to experimental data on nanotubes (5) nanotubes (5)

Application to experimental data on Application to experimental data on nanotubes (5) nanotubes (5)

y = 3.0905x - 1.5623

R2 = 0.6727

4

6

8

10

12

2 2.5 3 3.5 4 4.5

ln( GPa)

ln(l

n(1

/(1

F))

ln(V

/(n

m)3 )

Weibull statistics on nanotubes

y = 2.7179x - 9.3546

R2 = 0.9326

-4

-2

0

2

2 2.5 3 3.5 4 4.5

ln( /GPa)

ln(l

n(1

/(1

F)

Nanoscale Weibull statistics on nanotubes

m

nF0

exp1

n=1; Thus, for nanotubes m around 3

Again, it seems that few defects were responsible for fracture of that nanotubes

6. The Nanotube-based space elevator 6. The Nanotube-based space elevator megacablemegacable

6. The Nanotube-based space elevator 6. The Nanotube-based space elevator megacablemegacable

Multiscale simulations (5)Multiscale simulations (5)Multiscale simulations (5)Multiscale simulations (5)

Level 0 Level 1 Level N

Nx1

Ny1

Nx2

Ny2

…

…

NxN

NyN

NANOTUBE NANOTUBE BUNDLE SPACE ELEVATOR CABLE

Level 2 …Level 0 Level 1 Level N

Nx1

Ny1

Nx2

Ny2

…

…

NxN

NyN

NANOTUBE NANOTUBE BUNDLE SPACE ELEVATOR CABLE

Level 2 …

Strength of nanotube-based megacable (5)Strength of nanotube-based megacable (5)Strength of nanotube-based megacable (5)Strength of nanotube-based megacable (5)

0.00

0.20

0.40

0.60

0.80

1.00

1.20

10.16 10.18 10.21 10.23 10.250.00

0.05

0.10

0.15

0.20

0.25

0.30

10.20 10.24 10.28 10.33 10.37

0.00

0.05

0.10

0.15

0.20

0.25

0.30

10.80 10.84 10.87 10.91 10.94

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

11.05 11.08 11.10 11.13 11.16 11.19 11.21 11.24

simulation

weibull

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

1 3 5 7 9 11 13 15 17 19

f (GPa)

p(

f)

simulation

weibull

f2 (GPa)

p(f2)p(f1)

f1 (GPa)

f4 (GPa)

p(f4)p(f3)

f3 (GPa) f5 (GPa)

p(f5)

a) b)

c) d) e)

0.00

0.20

0.40

0.60

0.80

1.00

1.20

10.16 10.18 10.21 10.23 10.250.00

0.05

0.10

0.15

0.20

0.25

0.30

10.20 10.24 10.28 10.33 10.37

0.00

0.05

0.10

0.15

0.20

0.25

0.30

10.80 10.84 10.87 10.91 10.94

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

11.05 11.08 11.10 11.13 11.16 11.19 11.21 11.24

simulation

weibull

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

1 3 5 7 9 11 13 15 17 19

f (GPa)

p(

f)

simulation

weibull

f2 (GPa)

p(f2)p(f1)

f1 (GPa)

f4 (GPa)

p(f4)p(f3)

f3 (GPa) f5 (GPa)

p(f5)

0.00

0.20

0.40

0.60

0.80

1.00

1.20

10.16 10.18 10.21 10.23 10.250.00

0.05

0.10

0.15

0.20

0.25

0.30

10.20 10.24 10.28 10.33 10.37

0.00

0.05

0.10

0.15

0.20

0.25

0.30

10.80 10.84 10.87 10.91 10.94

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

11.05 11.08 11.10 11.13 11.16 11.19 11.21 11.24

simulation

weibull

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

1 3 5 7 9 11 13 15 17 19

f (GPa)

p(

f)

simulation

weibull

f2 (GPa)

p(f2)p(f1)

f1 (GPa)

f4 (GPa)

p(f4)p(f3)

f3 (GPa) f5 (GPa)

p(f5)

a) b)

c) d) e)

Size-effect (5)Size-effect (5)Size-effect (5)Size-effect (5)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.0E-07 1.0E-05 1.0E-03 1.0E-01 1.0E+01 1.0E+03 1.0E+05 1.0E+07 1.0E+09

L (m)

f/

0

simulation

analytical 1

analytical 2

analytical 3

Strength of the megacable?Multiscale approach: 10GPaHoles in the cables: 30GPaCracks <30GPa Thermodynamic limit: 45GPa, not 100GPa…

Strength of the megacable?Multiscale approach: 10GPaHoles in the cables: 30GPaCracks <30GPa Thermodynamic limit: 45GPa, not 100GPa…

Elasticity of defective Nanotubes (6)Elasticity of defective Nanotubes (6)Elasticity of defective Nanotubes (6)Elasticity of defective Nanotubes (6)

fnE

E

th

1The increment in compliance could result in a dynamic instability of the megacable

The increment in compliance could result in a dynamic instability of the megacable

Nanobiocomposites (7)Nanobiocomposites (7)Nanobiocomposites (7)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

n

1

l

h

t

Example of bio-inspired nanomaterial (7)Example of bio-inspired nanomaterial (7)Example of bio-inspired nanomaterial (7)Example of bio-inspired nanomaterial (7)

“Super-nanotubes”

as hierarchical fiber

reinforcements

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)Example of bio-inspired nanomaterial (7)Example of bio-inspired nanomaterial (7)Example of bio-inspired nanomaterial (7)

Optimizing Nano-composites (7)Optimizing Nano-composites (7)Optimizing Nano-composites (7)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.

Nano-armors (7)Nano-armors (7)Nano-armors (7)Nano-armors (7)

Conclusions 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

Main References Main References Main References Main References

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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, 043106-1/3N. 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.