introduction of nano science and tech -...
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ME 498 © 2006-09 Nick Fang, University of Illinois. All rights reserved. 1
Basics of Solid Mechanics in Nanostructures
Nick FangCourse Website: nanoHUB.org
Compass.illinois.edu
Introduction of Nano
Science and Tech
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ME 498 © 2006-09 Nick Fang, University of Illinois. All rights reserved. 2
First Midterm
•
Friday, Sept 25 1-2PM•
Coverage: –
Scaling
–
Quantum Effects–
Molecular Dynamics of Transport
–
Nanoscale
Solid Mechanics
•
A Review Lecture on Monday, Sept 21
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ME 498 © 2006-09 Nick Fang, University of Illinois. All rights reserved. 3
Mechanics at Nanoscale
•
Why?–
Promising material behaviors (reduced defects and faster recovery)
–
Coupling and quantum effect on mechanical response (this lecture)
(E.G. mechanical thermometer)
E.G. artists’
view of space elevators using CNTs
Cleland and Roukes, Nature, 1998
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•
Look back at the potential function for two atoms.
•
Derivative is related to strength.
•
Approximate strength curve with a sinusoid:
–
o
= theoretical cohesive strength
Theoretical Cohesive Strength
ar
o sin
radrd)r(
StrengthoroEnergy
(r)
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ME 498 © 2006-09 Nick Fang, University of Illinois. All rights reserved. 5
How Close is Theory?
•
Theory states:
•
Way too high for common materials (100-1000x too high)
•
Look at “whiskers”–
Small, “defect-free”
fibers
–
Agreement is a little better.
E
o
From: Hertzberg, p.76.
~8oEOR
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Defects in Solids•
To this point we have assumed perfect order in crystals–
Defects always exist in real materials
–
Sometimes we add “defects”
-
alloying
•
Classifications of defects–
Usually referring to geometry or dimension of defect
–
Point : 1-2 atomic positions (10-10
m)-
e.g. vacancies, interstitials
–
Line: 1-Dimensional (10-9
to 10-5
m)-
e.g. dislocations–
Interfacial: 2-Dimensional (10-8
– 10-2
m) -
e.g. grain boundaries
–
Volume: 3-Dimensional (10-4
– 10-2 m) -
e.g. pores, cracks
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Bulk Dislocation Movement•
Shear must act in direction of Burgers vector
•
Edge–
Positive & negative•
Screw–
Right-hand & left-hand•
Analogies for motion–
Caterpillar crawling
From: Callister, p.156
b
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Stress Fields•
Extra half plane of atoms cause lattice distortions
•
Result in tensile, compressive, and shear strains in neighboring atoms–
Magnitude decreases with distance
–
Pure compression and tension directly above and below slip line
–
Over most of the effected region combination of stresses
•
Screw dislocation–
Pure shear
Adapted from Fig. 7.4, Callister 7e.
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E
F
F
Microscopic View of Strain-Stress
(MPa
)
(%)
E
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Flaws are Stress Concentrators!
t
Results from crack propagation•
Griffith Crack
where t = radius of curvatureo = applied stressm = stress at crack tip
m 2o
at
1/ 2
Kto
For a crack, typically havea = 10-3m
= 10-9m 2000*local applied
a stress concentration factor
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thickness, t
Griffith Approach
Minimum criterion for stable crack growth:
Strain energy goes into surface energy
applied
applied
2w
l2a
a2a
aaU
ta4a
ta2E2a s
22
2 sEa
a
V
UT
UE
acrit
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So When Does Crack Propagate?
Crack propagates rapidly if above critical stress
where–
E = modulus of elasticity–
Gc = specific energy release rate–
a = one half length of internal crack
1/ 2c
cEG
a
i.e., m >
c
2C s pG Ductile:2C sG Brittle:
C ca EG = constant!!
Fracture Toughness, Kc
Measurable (fixed) materials properties
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Quantum Fracture MechanicsApplying Griffith’s approach to atomic lattices, e.g. graphenes:
a
V
UT
UE
acrit
aaU
U
a a
From N. Pugno
and R. S. Ruoff, Quantized fracture mechanics, Philosophical Magazine 84 (2004), 2829-2845
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“Quantized” Critical Strength!
From N. Pugno
and R. S. Ruoff, Quantized fracture mechanics, Philosophical Magazine 84 (2004), 2829-2845
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Space Elevator: Out of Order?
Nicola Pugno, Nano
Today, 2007(2)44-47
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Mechanical coupling at Nanoscale
•
E.G. Piezo- electricity (i.e.
electric potential in response to applied stress)
S = strain (relative length change ∆L/L, dimensionless) L0= ceramic length [m]E = electric field strength [V/m]dij= piezoelectric coefficient of the material [m/V] See: www.physikinstrument.com
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Principle of Piezo-electric EffectU+
U-
Asymmetry in crystal leads to separation of charges under stress
i ij j ji jD E d Strain effectElectric polarization
i ij j ij jC e d E Hooke’s law piezo
effect
Inversely, polarization in crystal induces stress
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Piezoelectric Sensing/Actuation
Quartz Tuning Forks, typically f=32KHz
Stack of piezo-tubes
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Magneto-restrictive Effect
Magnetostriction is the strain of a material in response to change of magnetization.
“giant”
magnetorestriction found in nanostructured
materialsApplication: Flat panel speakers (e.g. sound bugs) http://www.feonic.com/#commInfo
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Shape Memory Effect
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Memory Alloys Example
A nitinol
wire reverts back to its original shape upon heating
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Mechanical Nanoresonators
Putting Mechanics intoQuantum Mechanics, Keith C. Schwab and Michael L. Roukes, Physics Today, 2005, 36-42)
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Thermal Noise in ResonatorsCantilever total energy:
Each are subject to thermal noise 1/2kT
From We get
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ME 498 © 2006-09 Nick Fang, University of Illinois. All rights reserved. 24
Additional Reading
•
Callister, Chapter 7&8, in Materials Science and Engineering, 7th Edition, John Wiley, 2007
•
NanoHUB
resource: “Synthesis & Mechanics of Nanostructures & Nanocomposites”
by
Rod Ruoff
•
Cleland and Roukes, “Noise processes in nanomechanical
resonators”, JAP,
92(2002)2758