# The role of defects in the design of a space

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29 Οκτ 2013 (πριν από 4 χρόνια και 6 μήνες)

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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

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.