the role of defects in the design of a space elevator cable: from nanotube to megatube - latest...
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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/
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),
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