L18: Quantized fracture mechanics of nanostructures with atomic vacancies

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1 L18: Quantized fracture mechanics of nanostructures with atomic vacancies 198 estimate: $10B : 4% of GDP

http://www.nlm.nih.gov/medlineplus/enc y/article/000001.htm http://www.animatedteeth.

htm Arrows show where failure occurred: the remnant cracks On 16 January 1943, Schenectady, a Liberty tanker, split

$mile, the deck and sides of the vessel fractured just aft of the bridge superstructure.$ $The fracture extended almost instantaneously to the turn of the bilge port and starboard.$ The vessel jack-knifed and the center portion rose so that no water entered.

The vessel jack-knifed and the center portion rose so that no water entered.

$The official oast Guard report on the Schenectady incident attributed the fracture to welds in critical seams that$ " Avalanche Accident on Mt Hood, Oregon-5/31/98 Photo: The photo below is courtesy GW-Television in Portland,

" Avalanche Accident on Mt Hood, Oregon-5/31/98 Photo: The photo below is courtesy GW-Television in Portland,

$The large fracture appearing crack in the right middle of the picture is the Hogsback bergschrund (crack produced$

2 y/article/ htm ns_what_are.htm Arrows show where failure occurred: the remnant cracks On 16 January 1943, Schenectady, a Liberty tanker, split in two while moored in calm water at the outfitting dock at Swan Island, Oregon. A US oast Guard report described the incident: Without warning and with a report which was heard for at least a mile, the deck and sides of the vessel fractured just aft of the bridge superstructure. The fracture extended almost instantaneously to the turn of the bilge port and starboard. The deck side shell, longitudinal bulkhead and bottom girders fractured. Only the bottom plating held. The vessel jack-knifed and the center portion rose so that no water entered. The bow and stern settled into the silt of the river bottom. The ship was twenty-four hours old. The official oast Guard report on the Schenectady incident attributed the fracture to welds in critical seams that "were found to be defective." Avalanche Accident on Mt Hood, Oregon-5/31/98 Photo: The photo below is courtesy GW-Television in Portland, Oregon, and was available from the GW-TV Sky am at Timberline Lodge on Mt Hood. This particular picture was zoomed in on the mountain after the event-and was taken ~4 hours after the event. The large fracture appearing crack in the right middle of the picture is the Hogsback bergschrund (crack produced when a moving mass of snow slides away from an anchored part of the snowpack) which cuts across the shallow snow ridge above the saddle near rater Rock. The actual avalanche for this incident appears as a long crack in the upper left of the photo, cutting across the top of rater Rock; the deepest part of the fracture lies just below the rocks where the fracture line begins to extend downhill to the left.

Specimen: 80 80 1.8 m Generation of a big crack!

Ruoff Northwestern University! Nanotechnology " # $%!

3 Scaling of Sea Ice Fracture. ourtesy, Zdenek Bazant How the load was applied Notched Specimen: m Generation of a big crack! ONR Project, Director & Organizer: John Dempsey Rod Ruoff Northwestern University! Nanotechnology " # $%!$ " & $% ' $( ) * +, ' - * / 0 1 +( !0( " + + Northwestern University Rod Ruoff Nanotechnology

3 4+4+ 5 -//&! " #6 $ 7++3 8%+%9 49&!/1" $%& :;+7+!+%+*"'+!"σ!

4 //&! " #6 $ %+%9 49&!/1" $%& :;+7+!+%+*"'+!"σ!"++ σ max = 3σ!";+++ a σ = + max σ 1 b!" <9 9=> σ asy I = πr a I = σ

5 ' (%&)$* ' (+, ( σ ) σ σ = max ;*9*+'++@ A*+'+ B +%+B 4 9'+<'9,**+'< -,./ )$* d d A G dw d A = ( d G d ) < 0 σ = 0,. +,. ( W + Ω) = 0 A G πa G * W@++*% Ω@ *% " +%(or instable if larger than zero) ;*9*< +@ * + "11,, /3, 3

6 %4. 51 )$* G = I / G = ;,+ I = I G / = 7+ / = ( 1 v ) 7+,@7 I I G / % + D <9*9 % <9 <!/-&" ;*9+ I = σ πa *+ %4. +, 6!", ')$* G = I / + II / 1+ν + III * + * 3*!*9" I, II, III = I, II, III ( I d A) 0 d, II, III < 7* +%<*!++*"

7 7 8' (%& <* Q ;*9! 9 ' +" Q * 1 σ = σ y = Q d 0 ( θ = 0) x σ σ y F+%+% %6+%<' 0,+,97<9 <99!/&/" ' (), 5 * +, 7 8 )76'* 7* 897+*- 9-/!11 " W A ( + Ω) = 0 * G W A = G * A+ A I, II, III I, II, III = A I, II, III if if if if Ω = G A * ( G A) * ( G A) * ( I, II, III A) * ( A) I, II, III < 0, stable > 0, unstable < 0, stable > 0, unstable where I, II, III A+ A A 1 A A+ A A I, II, III da % D < G% : +,

8 0( "&! )* σ QFM = π I ( a + Q ) σ = LFM I πa '!" :;+!:;" *< + Q 0 4+%* ' +H a 0 +* ' ++'I ; )* <06'.=,> " 76' 8 " " + +<!ρ: " σ QFM = I 1+ ρ Q π = σ 1+ ρ Q ( a + Q ) 1+ a Q 3+<' H

9 4( *0 %+%*!*9" J %D < ' '< + 9H $ '+ ; ' Note that QFM can be successfully applied also at larger size scales!, see: $3% *9;050;; F8; ;D409!11#"?$9 11- Quantized Fracture Mechanics N. Pugno* and R. Ruoff *Politecnico di Torino, Italy Philosophical Magazine, 84 (004) The role of vacancy defects and holes in the fracture of carbon nanotubes Steven L. Mielke, Diego Troya, Sulin Zhang, Je-Luen Li, Shaoping Xiao, Roberto ar, Rodney S. Ruoff, George. Schatz, and Ted Belytschko, hemical Physics Letters, 390 (004)

β-si : 600 GPa σ ideal : 53 GPa (51 GPa <111>) Unit

approach used to probe mechanical properties of β-si

) Dai, H. et al., Nature, 375, p769, (1995). Wong,. W. et al., Science, 77, p1971, (1997).

α-si 3 N 4 : 345 GPa σ ideal : 59 GPa (54 GPa <1 0-1

$3 N 4 whisker (b) corresponding electron diffraction$ ) 1. Iwanaga, H. et al., J. Am. eram. Soc.

10 β-si : 600 GPa σ ideal : 53 GPa (51 GPa <111>) Unit cell of β-si (zinc blende structure) Overview of the approach used to probe mechanical properties of β-si nanorods TM image of β-si nanorods, 3. g/cc Ref.) Dai, H. et al., Nature, 375, p769, (1995). Wong,. W. et al., Science, 77, p1971, (1997). Li and Wang, Phys Rev B 59(6), 3993 (1999). α-si 3 N 4 : 345 GPa σ ideal : 59 GPa (54 GPa < >) Unit cell of α-si 3 N 4 (a) TM image of one α-si 3 N 4 whisker (b) corresponding electron diffraction pattern SM image of α-si 3 N 4 whiskers; 3. g/cc Ref.) 1. Iwanaga, H. et al., J. Am. eram. Soc. 81(3), p773, (1998).. Shigenobu Ogata, unpublished.

Multi-walled carbon nanotube Iijima, Nature 1991 Single-walled carbon nanotube

diameter and several microns in length ~1100 GPa;σ ideal : ~100-130 GPa ( 63 GPa)

More detailed view of cross section of a single walled carbon nanotube bundle,

11 Multi-walled carbon nanotube Iijima, Nature 1991 Single-walled carbon nanotube Iijima et al; Bethune et al; Nature 1993 SM image of carbon ropes with ~10-0 nm diameter and several microns in length ~1100 GPa;σ ideal : ~ GPa ( 63 GPa) Space elevator? More detailed view of cross section of a single walled carbon nanotube bundle, which is comprised of single-walled nanotubes with diameter ~ 1.4 nm. Thess et al, Science. Table 1: lustering of experimental strength [GPa] of β-si nanorods, α-si3n4 whiskers, and MWNTs. β-si 53.4± ± ±., 11.6±.9, 10.±.1 α-si 3 N MWNT* ,37,37,35,34 8,6,4,4 1,0,0,19,18,18 1,11 * Ruoff group, Science 000; outer shell of the multiwalled NT is what breaks. Table : omparison between molecular mechanics (MM) simulations and theoretical QFM predictions for the strength [GPa] of nanotubes with blunt cracks (adjacent vacancies of n atoms). n-atom defect MM* QFM *Belytschko (Xiao, Ruoff), zig zig tube But imagine 80 hexagons around the circumference for (80, 0) zig zag tube ~6.4 nm in diameter. 5 hexagons along this edge

0 4 1 3 50 hexagons along this edge 5 arm-chair tube zig zig tube We used a round hole for QFM But

Failure stresses (GPa) and strains of large NTs with holes calculated with the MTB-G potential and

QFM [100,0] [9,9] hole size index a) Tensional nergy-based Stress Strain at failure Stress Strain

12 hexagons along this edge 5 arm-chair tube zig zig tube We used a round hole for QFM But imagine 100 hexagons around the circumference for (100, 0) zig zag tube ~8 nm in diameter. Failure stresses (GPa) and strains of large NTs with holes calculated with the MTB-G potential and by QFM. QFM [100,0] [9,9] hole size index a) Tensional nergy-based Stress Strain at failure Stress Strain pristine

Rod Ruoff Northwestern University, 5 σ n σ 1+ ρ Q (1 + n ) 1, n>0 Nanotechnology 8 A.g., blunt cracks a = nq 1 0.

13 Rod Ruoff Northwestern University, 5 σ n σ 1+ ρ Q (1 + n ) 1, n>0 Nanotechnology 8 A.g., blunt cracks a = nq Quantized Strength Levels and Forbidden bands Northwestern University Rod Ruoff Nanotechnology

14 0(" %,)* Observed Strength/ Ideal Strength xperiments on β-si nanorods, α-si3n4 whiskers and MWNTs Quantized Levels Si3N4-59GPa Si3N4-75GPa Si-53GPa Si-68GPa MWNT-115GPa MWNT-104GPa n Thus, the strength is quantized as a consequence of the quantization of the defect size!

Fracture Behavior. Section

Fracture Behavior. Section Section 6 Fracture Behavior In January 1943 the one-day old Liberty Ship, SS Schenectady, had just completed successful sea trials and returned to harbor in calm cool weather when... "Without warning and