The role of defects in the design of a space
elevator cable: From nanotube to megatube Latest research results
Nicola M. Pugno
Politecnico di Torino, Italy
2th Int. Conf. on Space Elevator Climber and Tether Design,
December 6-7, 2008, Luxembourg, Luxembourg
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).
Deterministic approach
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).
2.
Stress concentrations and intensifications
Linear Elastic Plate (infinitely large) with an hole under (remote) traction s.
(a) Circular hole:
stress concentration
s max  3s
(b) Elliptical hole:
stress concentration
s
s max
a

 s 1  2 
b

(c) Crack: infinite stress
concentration, i.e.,
“stress intensification”
s asy
KI

2r
K I  s a
K = stress-intensity factor
r = distance from the tip
Maximum stress criterion (2)
Maximum stress = material strength
s max s   s C
E.g., strength for a plate with a circular hole =
1/3 strength for the plate without the hole
…independently from the size of the hole!?
I cannot believe it!
Vanishing strength for a plate with a crack!?
I cannot believe it! (PARADOX)
Griffith’s energy balance criterion (2)
Energy release rate = fracture energy
d
W  W   0
dA
G   d W d A  GC
Stability (or instable if
larger than zero):
d G d AC  0
GC E
s
a
Criterion for fracture propagation
W = total potential energy
W = dissipated energy
A = crack surface area
GC = W/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)
3. Quantized fracture mechanics (QFM)

W  W   0
A
W  GC A

if G
G *   W A  GC
K
*
I , II , III

K
where K I2, II , III
2
I , II , III
A A
A
A A
A
1

A

A
if G * A C  0, stable
 K I , II , IIIC

if K
*
C
 0, unstable

A
if K I*,2II , III A C  0, stable
*2
I , II , III
C
 0, unstable
A A
2
K
 I ,II ,III dA
A
Stress intensity factors
from Handbooks
Very simple application
The Griffith case treated with QFM (3)
LEFM can treat only “large” and sharp cracks
QFM has no restrictions on defect size and shape
LEFM Q=0:
s QFM  K IC
1   2Q
1   2Q
sC
 a  Q 2
1  2a Q
This represents the link between
concentration and intensification factors!
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
1

W  T  Ω dt  0

t t  t A
t
Balance of action quanta
Time quantum t
Ω  GC A
KdIC/K IC
2.2
2
1.8
1.6
1.4
1.2
1
-5
-4
-3
-2
-1
Log (t f /s)
0
1
2
3
4. Fracture of nanotubes: nanocrack
Strength [GPa]
n=2
n=4
n=6
n=8
MM - (80,0)
64.1
50.3
42.1
36.9
QFM
64.1
49.6
42.0
37.0
r0
(n=2)
Q  3r0
= Interatomic distance
QFM: formula for blunt cracks with length 2a  nQ
s C  93.5GPa from MM;

  0.8Q  2.0 A
best fit (very reasonable);
Nanoholes (4)
MM
QFM
(50,0)
(100,0)
m=1
0.68
0.64
0.65
m=2
0.48
0.51
0.53
m=3
0.42
0.44
0.47
m=4
0.39
0.40
0.43
m=5
0.37
0.37
0.41
m=6
0.36
0.34
0.39
(m=1)
R  2m  1 r0
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!).
MM also in good agreement with fully
quantum mechanical calculations
Nanotensile tests on nanotubes (4)
Stretching of multi-walled carbon nanotubes between Atomic
Force Microscope opposite tips
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!
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!
Is the
strength
Ordine
delquantized?
Giorno (4)
s n  s C 1

2Q
1  n 
1 2
, n0
E.g., blunt cracks
2a  nQ
1
0.8
0.6
0.4
0.2
Quantized
Strength Levels
and
Forbidden bands
Observed Strength/ Ideal Strength
Experiments on ideal strength (4)
Experiments on b-SiC nanorods, a-Si3N4
whiskers and MWCNTs
Quantized Levels
Si3N4-59GPa
Si3N4-75GPa
SiC-53GPa
SiC-68GPa
MWCNT-115GPa
MWCNT-104GPa
1
0.8
0.6
0.4
0.2
0
1
2
3
4
n
5
6
7
Thus, the strength is quantized as a consequence
of the quantization of the defect size!
8
Nanoscale Weibull Statistics (NWS, 5)
Weibull distribution for the
strength of solids:probability of
failure for a specimen of volume
V under tension s
Number of “critical” defects assumed to be proportional
to the volume V of the specimen
s 0 , m material constants (m Weibull’s modulus)
  s m 
F s   1  exp  V   
  s 0  
Alternatively, V is substituted by the surface S of the
specimen (for surface predominant defects)
In contrast:
At nanoscale nearly defect free structures!
We substitute V with a fixed number n of defect (e.g., n=1)
Application to experimental data on nanotubes (5)
  s m 
F s   1  exp  n  
  s 0  
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 megacable
Multiscale simulations (5)
NANOTUBE
NANOTUBE BUNDLE
SPACE ELEVATOR CABLE
…
Level 0
Level 2
Level 1
Level N
Ny2 …
Ny1
Nx1
…
Nx2
NyN
NxN
Strength of nanotube-based megacable (5)
p(sf1)
p(sf2)
0.18
0.20
a)
simulation
0.16
0.18
weibull
0.14
0.16
b)
weibull
0.14
0.12
p( f)
simulation
0.12
0.10
0.10
0.08
0.08
0.06
0.06
0.04
0.04
0.02
0.02
0.00
0.00
11.05 11.08 11.10 11.13 11.16 11.19 11.21 11.24
1
3
5
7
9
11
s f (GPa)
p(sf3)
0.30
13
15
17
19
sf1 (GPa)
p(sf4)
sf2 (GPa)
p(sf5)
0.30
1.20
d)
c)
e)
0.25
0.25
1.00
0.20
0.20
0.80
0.15
0.15
0.60
0.10
0.10
0.40
0.05
0.05
0.20
0.00
10.80 10.84 10.87 10.91 10.94
0.00
10.20 10.24 10.28 10.33 10.37
0.00
sf3 (GPa)
sf4 (GPa)
10.16 10.18 10.21 10.23 10.25
sf5 (GPa)
Size-effect (5)
1.2
simulation
analytical 1
1.0
analytical 2
analytical 3
s f/s 0
0.8
0.6
0.4
0.2
0.0
1.0E-07
1.0E-05
1.0E-03
1.0E-01
1.0E+01 1.0E+03
L (m)
Strength of the megacable?
Multiscale approach: 10GPa
1.0E+05 1.0E+07 1.0E+09
Holes in the cables: 30GPa
Cracks <30GPa
Thermodynamic limit: 45GPa
, not 100GPa…
Elasticity of defective Nanotubes (6)
E
 1  afn
Eth
The increment in compliance could
result in a dynamic instability of
the megacable
Nanobiocomposites (7)
t
1
h
n
l
Fundamental roles of:
(i) Tough soft matrix, (ii) Strong hard inclusions and (iii) hierarchy,
for activating toughening mechanisms at all the size-scales
Example of bio-inspired nanomaterial (7)
“Super-nanotubes”
as hierarchical fiber
reinforcements
Example of bio-inspired nanomaterial (7)
Toughening
mechanism
= fibre pull-out
N-opt=2, to optimize the material with respect to both
strength and toughness, as Nature does in nacre
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)
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
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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.
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N. Pugno, R. Ruoff, Nanoscale Weibull statistics. J. OF APPLIED PHYSICS (2006), 99, 024301/1-4.
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