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
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
;
Ingeni
ö
rsvetenskapsakademiens
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
(b)
Elliptical hole:
stress concentration
(c)
Crack: infinite stress
concentration, i.e.,
“stress intensification”
K
= stress

intensity factor
r
= distance from the tip
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
Vanishing strength for a plate with a crack!?
I cannot believe it! (PARADOX)
…independently from the size of the hole!?
I cannot believe it!
Griffith’s energy balance criterion (
2)
Stability
(or instable if
larger than zero):
Criterion for fracture propagation
W
= total potential energy
W
= dissipated energy
A
= crack surface area
G
C
=
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)
Energy release rate = fracture energy
3. Quantized fracture mechanics (QFM)
Stress intensity factors
from Handbooks
Very simple application
The Griffith case treated with QFM (3)
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
Q=0:
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
Time quantum
Balance of action quanta
4. 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
from
MM
;
=
Interatomic
distance
best
fit
(very
reasonable)
;
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
)
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)
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
64
GPa,
against the measured value of
63
GPa,
2. for an
m
=2
defect is
45
GPa,
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 Giorno
Is the strength quantized? (4)
E.g., blunt cracks
Experiments on
b

SiC nanorods,
a

Si3N4
whiskers and MWCNTs
0.2
0.4
0.6
0.8
1
0
1
2
3
4
5
6
7
8
n
Observed Strength/ Ideal Strength
Quantized Levels
Si3N4

59GPa
Si3N4

75GPa
SiC

53GPa
SiC

68GPa
MWCNT

115GPa
MWCNT

104GPa
Experiments on ideal strength (4)
Thus, the strength is quantized as a consequence
of the quantization of the defect size!
Nanoscale Weibull Statistics (NWS, 5)
Weibull distribution for the
strength of solids:probability of
failure for a specimen of volume
V
under tension
s
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
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
6. The Nanotube

based space elevator megacable
Multiscale simulations (5)
Strength of nanotube

based megacable (5)
Size

effect (5)
Strength of the megacable?
Multiscale approach: 10GPa
Holes in the cables: 30GPa
Cracks <30GPa
Thermodynamic limit: 45GPa
, not 100GPa…
Elasticity of defective Nanotubes (6)
The increment in compliance could
result in a dynamic instability of
the megacable
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
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)
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
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.
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