A Glimpse at Mythology■ Prometheus’ Legend

Hesiod’s Theogony (800 bc) – Aeschylus Trilogy (500 bc) Prometheus “κλέπτει” (steals/arranges) fire/knowledge 

from Olympus/Zeus via Athena’s helmet for miserable Humans

Survival

Humans survive but fight each other viciously/destruction Zeus sends Hermes to bring them consciousness/peace

Societies

Zeus sends Pandora (made by Hephaestus out of clay) Zeus sends Pandora (made by Hephaestus out of clay) Pandora’s jar of gifts (evils/pain/diseases + hope) Humans are left with “hope” striving to free themselves f h bl l

5

from their troubled + mortal nature

Civilization

■ Homer’s Automata Iliad

E 749 H h G i llE 749: Hera opens the Gates automatically

Σ 372: Hephaestus’ 20 golden self-moving tripods telecontrol?}

Σ 468-473: Hephaestus’ automated Lab modern casting unit?

Σ 410-420: Young Servant Girls (made out of gold withΣ 410 420: Young Servant Girls (made out of gold with mind/voice/movement) assisting “crippled” Hephaestus to walk human robots?

Odyssey

Θ 63 h i ’ Shi i “ i d f h i ”Θ 555-563: Phaeacians’ Ships possessing “mind of their own ” traveling at extremely high speeds at night and in clouds without fear to sink modern auto-pilots?

6

A Glimpse at Ancient Technologyp gy 800-700 bc: Empirical Techniques imported from the East

700-200 bc: Science/Mathematics 400 bc-100 ad: Technology

Teacher / Student Sequences

Th l A i d P th A h t Thales – Anaximander – Pythagoras – Archytas Socrates – Plato – Aristotle – Archimedes Euclid – Aristarchus – Hipparchus – Ptolemy Euclid Aristarchus Hipparchus Ptolemy Ctesibius – Philo - Heron

Geometry/Numbers/Forces/Compressed Air/Mirrors– Geodesy/Astronomy; Mining/Metallurgy; Statics/Optics/Engineering

B ildi /M t A d t /H b W /M i l I t t7

– Buildings/Monuments, Aqueducts/Harbors, War/Musical Instruments

■ The Great Alexandria Turning point in the History of Mankind

From a Top-Down to a Bottom-Up Approach

The search for the origin of the Universefrom first principles [e.g. the 4 elements – earth/water/fire/air]

h f f ldi th l f d lif b tisearch for unfolding the puzzle from everyday life observations measurement / fabrication / construction Modern Engineering

Great Thinkersare not Philosophers / Generals and Landlords / Merchants, BUT simple everyday men

Ctecibius (Aeroton, Hydraulic Pump, Hydraylis) - son of barber( , y p, y y ) Heron (Watt’s Steam Engine, Siphons, Lamps) – shoemaker

Th P f R i D Vi i G lil N8

The Precursor of Renaissance: Da Vinci, Galileo, Newton

150-100 bc: Antikythera Mechanism The 1st Computer/GPS

– Ancient Mechanical Computer designed to calculate the positions of sun/moon/stars, eclipses, calendar – 1st astronomical clock, p ,

– Remarkable miniaturization/precision and complexity: 3 Dials & over 30 gears with teeth (comparable to 18th Century clocks)D i d f Hi h ’ h f h M– Designed after Hipparchos’ theory of the Moon

The Antikythera Mechanism main fragment: 3216 10 cm

9

main fragment: 3216 10 cmThe Antikythera Mechanism’s Reconstruction:

Athens Archaeological Museum

A Glimpse at Ancient Nanotechnologyp gy

Ancient Egyptians/Greeks (~2000 bc) used Pd-S Nanoparticles(5 200 ) f h i l i(5-200 nm) for hair coloring

Ancient Thracians (~800 bc) used Au Nanoparticles (~70 nm) to Ancient Thracians ( 800 bc) used Au Nanoparticles ( 70 nm) to change Glass Cup Color with Light (Green Red)

Damascus (Alexander/Ottomans) used swords containing C-Nanotubes developed during forging + heat treatment

Roman Catholics used Au/Si Nanoparticles (~100 nm) in stained glass – Church Windows

10

Pd-S Nanoparticles (5-200 nm) for hair coloring

Damascus steel swords

Au Nanoparticles change Glass Cup Color with Light (Green Red)

11

A Glimpse at Current Nanotechnology

R. Feynmann (1918-88) Prophet of NanotechnologyC lt h’ L t D 29th 1959Caltech’s Lecture: Dec 29th 1959

“There’s Plenty of Room at the Bottom”Recent Examples of New Fields

K li h O L T h i

There s Plenty of Room at the Bottom

Kamerlingh Onnes: Low-Temperature physics

Percy Bridgman: High-Pressure physics

New Emerging Fieldg g Manipulating/controlling things on small scale

In 2000 we will wonder why it was not until 1960 that anybody began seriously moving in this direction

12

began seriously moving in this direction

Nanoscience / Nanotechnology

Why cannot we write the entire 24 volumes of the Encyclopedia Brittanica on the head of a pin?Brittanica on the head of a pin? All necessary to do is to reduce the letters by 25,000 times By Photoengraving to raise letters on a metal surface that areBy Photoengraving to raise letters on a metal surface that are

1/25,000 of their ordinary size Look through with an electron microscope and reverse the lenses

to read (demagnify/magnify)to read (demagnify/magnify)

24 million volumes Need 1 million pinheads i.e. 3 square yards or 35 pages of the Encyclopedia

This does not involve New Physicsyi.e. you can decrease size of things in a practical way, according to the laws of physics. I am not inventing anti-gravity, which is possible someday only if the laws are not what we thinksomeday only if the laws are not what we think. I am telling you what could be done if the laws are what we think; we are not doing it simply because we haven’t yet gotten around to it.

13 Nanolithography

Nanolithography Today• Chemical/electrochemical processes induced at predefined positions

+ AFM tip engraving on sample surface

Text written by using Dip-Pen Nanolithography (DPL) and imaged by using AFM

(Mirkin Group, Northwestern University)T. Schimmel et al, 2008

( p, y)

14

Nanomedicine Today/FutureNanomedicine Today/Future

Smart Bio-Nanotubes / Nanorobots

nanotube of lipid protein

spirals

lipidlipid layer

Bio nanotube: Selective Storage/Release of Drugs Bio nanorobot:Bio-nanotube: Selective Storage/Release of Drugs Nanotubes of lipid proteins encapsulated in a lipid

layer covered by protein spiralshtt // l t

Bio-nanorobot: Futuristic Targeted Drug Release

http://bionano.rutgers.edu/mru.htmlhttp://www.voyle.net

15

Nanostructured Li-batteries / NanosurgeonsCure of Alzheimer’s/Parkinson Diseases and Brain Damage/Paralysisg y

Mobile Cell RepairerY. Svidinenko, Nanotechnology News Network

Li-battery

Deep Brain StimulationDr. H. Mayberg, Univ. Toronto Mobile Artery Cleaner 16

New Electron Microscopes / Super Imaging

High-Resolution Transmission Electron Microscope (HRTEM/~0.8 Å)(C atoms imaged in diamond separated by only 0.89 Å/ Si at 0.78 Å)

Scanning Electron Microscope (SEM/~0.4 nm)

Scanning Tunneling Microscope (STM/Lateral ~0.1 nm, Depth ~0.1 Å)g g p ( , p )

Atomic Force Microscope (AFM/Lateral ~0.1 nm, Depth ~0.1 Å)

Most Recent Ultrahigh Resol tion TEM / 1 Å Most Recent Ultrahigh Resolution TEM /~1 ÅNew Technique overcoming the limit of diffraction

Reconstructed Image of a 9 nm diametergCdS quantum dotZuo et al, Univ. Illinois

17

Nanomanufacturing/Nanomechanics

Nanoholes with diameters of a few nm were drilled in a stainless steel foil using intense electron beams of 2.4 nm nominal probe size from a field emission electron gun in a HTREMsize from a field-emission electron gun in a HTREM

Nanoscrews of ZnO were manufactured with an average diameter of the tops ofwith an average diameter of the tops of about 250 nm, that of the roots of about 60 nm, and the average length in the order of several microns

Nanostamping Parts with characteristic dimensions below 1 nm

order of several microns SEM top view of the 18 sides of nanotips

Nanostamping Parts with characteristic dimensions below 1 nm and up to 100 nm are used for magnetic memory storage

Mi /N M Micro/Nano Motors

Outer rotor diameter: 10 mm Length from mount: 14 mm 18

Popular Nanomechanics TopicsNanotubes■ Nanotubes

Various Forms of CNTs5/7 Dislocation-like Defects in CNTs

Ropes of CNTsMultiwalled CNTs

■ Nanobiomembranes/M. sinense Cells

19

Nanopol cr stals: Obser ations/Metal Ph sics AspectsNANOMATERIALS & NANOMECHANICSNanopolycrystals: Observations/Metal Physics Aspects

Traditional Polycrystals 10 100 μm Nanopolycrystals 5 100 nm Grain Configuration at the Nanoscale

Traditional Polycrystals ……10 – 100 μm Nanopolycrystals…………….5 – 100 nm

G i dGrain d

1-50 nmCore r0

~1 nm

Grain size (d) of the same order as dislocation core (r0)10 nm grain size: 30% of atoms in the boundary

i ld

Nano Traditional Materials

Yield Strength

1 2kd

Amorphous

20Grain Size

Amorphous

Plasticity Mechanisms ? Inverse Hall-Petch Relation ?

Improved/Engineered Properties: ExamplesProperty Material Bulk NanoDensity (g/cc) Fe 7.5 6Modulus (GPa) Pd 123 88Fracture Stress (GPa) Fe 0.7 8

f S lf diff i ( ) CE for Self-diffusion (eV) Cu 2.0 0.64aE

In-situ TEM Deformation Testing/MTU Early Observ

In-situ TEM Deformation Testing/MTU Early Observ.

200 nm

21

Schematics 8 nm Au on Al: Nanovoid Coalescense

• Nanovoid Nucleation

8 nm Au on C: Nanocrack growth via nanopore formation

25 nm A on C: Periodic Crack profiles and bif rcation

22

25 nm Au on C: Periodic Crack profiles and bifurcation

Elementary Rosette Analysis

• Grain Rotation / Dislocation EmergenceElementary Rosette Analysis

Triangle angles (deg) Triangle lengths (nm)

Step α β γ a b c

Start 89 36 55 22.2 27.7 16.4

1 91 35 54 22.6 27.9 17.4

2 96 36 48 23.4 31.2 18.9

0160110011.005.0

St i T %20

3 102 33 45 21.7 32.0 18.0

24.000016.011.0

Strain Tensor %20eff

10 nm Au: 6-15 degrees grelative grain rotation

23100 nm Au film ~12 nm Ni nanopolycrystals

Diffusion in NanopolycrystalsOb i / Ph i l C fi i

• Grain Boundaries / Triple Junctions Observations / Physical Configuration

100

10-1

intercrystal grain boundarytriple junction

actio

n

10

10-2

Vol

ume

Fra

T i l J ti

10-3

100 101 102 103

1 nm

Grain Size (nm)

VGrain Boundary

Triple Junction

Mechanics of Diffusion

div c divj fTt

{ c }T f { }; 1 2 j

• Balance Eqs.

• Constitutive Eqs24

{ , , c } T f { , , }; 1,2 j • Constitutive Eqs.

( 1) [ ]T 1 f j

The Simplest Model1 1 2 2; ; c ( 1) [ ]

T 1 f j

21 1 2

21 21 1 2 22 1 1 2 2= D , =( ) ( )D

t t

● Solution

t t

12

Dt t

t

1 21 1 2

1 22

1 1 1 2= + [ ]dD D

h (x,D t) h ( , ) h ( , )t t

D te e e x x

2h ;= h 1/ 21 21 2 2 1 1 2

1 21 2 1 2 1 2

2, , ( ) ( )D D D t D t

D D D D D D

● Uncoupling / Higher-order Diffusion Eq.2

22

2 4ED D

2 tt t

11 2 1 2 2 1 1 2 1 2( ) , ( ) , ( ) , ED D D D D D D D

25

2 2 11 2

1 2 1 2

; efft D D D D Dt

1 2(1 )f D f D

64 in PolycrystallineCu Cu 67 i N t lliC C Experiments / Diffusion Penetration Profiles

in Polycrystalline Cu Cu

b. u

nits

)

67 in Nanocrystalline Cu Cu

rb. u

nits

)

ntra

tion

(arb

activ

ity (a

r

Con

cen

x6/5 (105 m6/5) Spec

ific

x2 (104 nm2)

59 in compacted Fe n Pd 182in nano O ZrO

C

26x [μm]

Initial Simple – Minded Models

“Bulk” Phase, Hall – Petch relation

- Volume Fraction3

• Model: 2–Phase Material / Rule of Mixtures

“B d ” Ph C t t

11G d

3

3( )df

d

“Boundary” Phase, Constant strength related to amorphous material or grain

- Rule of Mixtures(1 )G GBf f

boundary sliding ( )GB

•Continuum Model predicts behavior of N C t lli M t i lNanoCrystalline Materials

•Continuum Model can sort out conflicting Materials Science data

•Continuum elasticity Model has also been developed, which shows the importance of gradients in

27

the importance of gradients in elasticity of nanophase materials

Improved Inverse Hall-Petch Relation

1G GBH H f H f GBG HdddHddH 33333

1 20 ,G G GH H k d 1 2

0 ,GB GB GBH H k d

0

0

lnln

rdrdkk

cGGB

3 330 1 2

0 3 30

lnlnG G

c

d rd d dH H k d

d rd d

0c

28(a) & (b): nanocrystalline metals; (c) & (d): intermetallics

Elasticity of NanopolycrystalsG di El i i / G d l

● “Bulk” phase and “boundary” phase“Bulk” Phase

Gradient Elasticity / Gradela

● Bulk phase and boundary phaseoccupy the same material point andinteract via an internal body force “Boundary”

● Equilibrium

Phase

,div div 1 2fσ σ f .......for each phase0,div 1 2σ σ σ σ .......total

● Elasticity for Each Phase

Assume that each phase obeys Hooke’s Law and that the interaction force is p y fproportional to the difference of the individual displacements

, 1,2; )(k k k k 1 2f u uσ L u

29

, 1,2;ˆ

)ˆ;

(

; Tk k

k

div

k k k

k

1 2f u uσ L u

L G G I

Uncoupling

2 22 ( ( ))graddiv c graddiv u uu u 0

● Gradient Elasticity – Gradela The above implies the following gradient-elasticity relation

2) ) 2( (2 ctr tr σ ε I ε Iε ε

i.e.

) ) 2( (2 ctr tr σ ε I ε Iε ε

elasticity of nanopolycrystals depends on higher – order gradients in strain

● Ru-Aifantis Theorem

30

20cu uu

● Gradela: 2 0 2 01 ; 1( ) ( )

Gradela Dislocation Nanomechanics● Gradela:

● Screw Dislocation

2 0 2 01 ; 1ij ij ij ij( ) ( )c c

;zb y y K r c - Stress / Strain :

12 ; ...4

;

zxz yz

z

y y K rr r

b y y K

c

r

c

c

12 ; ...4xz yzKr c

rr

c

r 0 K r 0cc

- Self – energy : 2

ln ... 0.577; Euler constant4E Ez

sb RW

, 1 xz yzr 0 K r 0r

c

f gy ;4 2s c

0r 0 no need for ad hoc dislocation core rz

c/x0yz

yzx

z

31

0yz

yz

yz

c/y

y

● Dislocation Dipoles [insight to nucleation / annihilation]

d 2 c d 10 c

b x / c x / c

M d III C k [ ti di t ib ti f di l ti ( )]

… characteristic distance of “strong” interactiond 10 c

● Mode III Crack [continuous distribution of dislocations n(x)]

y yz 10

.O

. . . . x n(x) 0

10z0

-10-5 5

x

/

c/u

32

c/x

Barenblatt’s “smooth closure” condition

● Comparison with MD Simulations (Stilliger – Weber Potential)2

2

0 1 2

2

b R R R 2W ln 2K 2 K4 1 R R2

b R 1

c cc c c

b R 1R W ln

4 1 22 c 2o

c 0.4 A

b-edge-b

Wcore / atomistic simulations

c 0.4 A

b-edge-a• AtomisticGradelaElasticity

}

2oc 2.0 A

2o

Elasticity

b-edge-c

Wcore / atomistic simulations

oc 0.2 AWcore / atomistic simulations

o

oc 0.2 2.2 A

33

core 0W r 0.33 0.00c 8 eV A Invariant Relations: o

gW (b) b 0.3 0.00c 8 eV A

● X-ray Line Profile AnalysisGradela Soltn for of edge bb eAccording to Gradela (e.g. ECA 2003) the εxx component of the strain tensor corresponding to an edge dislocation with Burgers vector isb xb e

- Gradela Soltn for of edge b xb exx

p g g g x

2 2

2 2 2 2xx 4 1 2

1 2 r 2xb b y y r 3x y4 1 r 2 1

where 1 24 231 21 1 2K r , K r

r rrcc c

c

2 2 2, r x y

2

- The first results for calculating 2L

L

34log L

- X-ray line profile for deformed Cu single crystal

The measured (count intensity I) line profile of the (111) reflection of a deformed singlety

I of the (111) reflection of a deformed single crystal Cu sample: the intensity is plotted as a function of K - K0, where and K is the K value at the exact Bragg

K 2sin Inte

nsit

and K0 is the K value at the exact Bragg position. The intensity scale is logarithmic

K - K0

- for deformed Cu single crystal 2L

2L The mean square strain as a function of

log L, determined experimentally for 2L

g p ydeformed Cu single crystal by FT. It is noted that obtained this way is not singular, but it tends to a finite value for L 0

2L

35

but it tends to a finite value for L 0

log L

● Image Force – Inverse Hall Petch Behavior2Gb R R - Self -energy:2

E0

Gb R RW ln Kc c2 2

Gb 1 1 d - Image Stress: 1Gb 1 1 dK2 d 2 2c c

derived by differentiation and evaluation at the center of a grain of diameter d

- stress to move a dislocation situated at the center of a grain of diameter dstress to move a dislocation situated at the center of a grain of diameter d

0.001

R d 20.0006

0.0008

Gd* 9 nm

d*

0.0002

0.0004

36d b0 5 10 15 20

0

■ Gradela Crack Nanomechanics (Mode III) ● Gradela: Mode III Cracking

- Gradela: ij ij ij ij1 & 1 ; λ tr 2μc c 0 0 0 0 0σ σ ε ε σ ε 1 ε

Target: Non-Singular Stresses/Strain Estimation at the crack tipTarget: Non Singular Stresses/Strain Estimation at the crack tip

- Boundary Conditionsy

Far field coincidence of stresses: ij ijlim

0

rσ σ

Vanishing of stresses at the origin: ij0lim 0

r

σ

Zero tractions on crack surfaces: zyσ (x,0 ) 0 ; x a

37

● Nonsingular stress distribution in Mode III

IIIK θ K θ IIIxz

K θσ sin 1 exp rr

c22π

IIIyz

K θσ cos 1 exp r22 r

cπ

G di t St i l G di S i lτ

Gradient Stress non-singular Gradient Stress non-singular

2a

τ. .. .Classical Stress singularClassical Stress singular

Note: max at cr1 e r r c1.25

38 (Stress Fracture Criterion)

max max III IIIyz xz 4 4

K K0.2c

544c

IIIK a

Plasticity of Nanopolycrystals Gradient Plasticity: A Scale Invariance Argument

• Momentum Balance for Dislocated StateMomentum Balance for Dislocated Stateˆdiv ; ; div 0 D D LT f T S T S

ˆ

• Yield Condition

ˆ...dislocation stress; ...dislocastion-lattice interaction forceDT f

L Lˆ ˆˆ ˆj ; Lf T M• Yield Condition j ; f T M

s= , = ,

M n n

n s

p p p p Dm n, , t t

D M W T M Np 1 p 1max tr ; tr 0, tr 1 2 ; J tr

22 J

L p 2 p L L LT D M M D T T T

pJ

39

J

Structure of Kinematic Hardening Plasticity

• Inhomogeneous Back Stress: D inhT Tg- = homogeneous back stress … as before

pˆ inhT g n p p p p

, ,

T g n

n n

2 p pdiv inh 2T n grad n

-

- Integrate over all possible orientations of

2 p p,ij i j i jdiv n inhT

( , )n

2 pdiv inhT

• 2p pc . . . Gradient – dependent flow rule (YC)

40

Thermodynamics applied to gradient theories : The theories of Aifantis and Fleck & Hutchinson and their generalizationThe theories of Aifantis and Fleck & Hutchinson and their generalization

[J. Mech. Phys. Sol. 57, 405-421 (2009)]

M.E. Gurtin/Carnegie-Mellon & L. Anand/MIT

Abstract : We discuss the physical nature of flow rules for rate-independent (gradient) plasticity laid down by Aifantis and Fleck and Hutchinson. As central results we show that:

Aifantis Fleck and Hutchinsonthat: the flow rule of Fleck and Hutchinson is incompatible with thermodynamics

unless its nonlocal term is dropped. Fleck and Hutchinson incompatible

If the underlying theory is augmented by a general defect energy dependent on γp

and γp , then compatibility with thermodynamics requires that its flow rule reduce to that of Aifantisreduce to that of Aifantis

compatibility with thermodynamics requires that its flow rulereduce to that of Aifantis.

Refs• E C Aifantis On the microstructural origin of certain inelastic models Trans ASME J

reduce to that of Aifantis

• E.C. Aifantis, On the microstructural origin of certain inelastic models, Trans. ASME, J. Engng. Mat. Tech. 106, 326-330 (1984).

• E.C. Aifantis, The physics of plastic deformation, Int. J. Plasticity 3, 211-247 (1987). • N A Fleck and J W H tchinson A reform lation of strain gradient plasticit J M h Ph

41

• N.A. Fleck and J.W. Hutchinson, A reformulation of strain gradient plasticity, J. Mech. Phys. Solids 49, 2245-2271 (2001).

S lf C i A i i A Note on the Origin of Gradients● Self - Consistent Approximation

- Simple Shear

( ) , 12121 2 , S

1 ( ) , V

dVV

x r 343

V R

1 (2) 2 1 2 11( ) ( ) ...; 02!

n n

V

dV x r x r r r r

22R 2

10,

R / 2R d

2R 7 5 2( )

10( ) ;

R h 7 5

15(1 )/

h d d

42

2

( )10

Rc h 2c Cd

- Various Models for α

• Lin 1954

K (1958) / B di k W (1962)

12121/(1 )S

1• Kroner (1958) / Budiansky – Wu (1962)

• Berveiller Zaoui 1979

1

1 H • Berveiller – Zaoui 1979(Secant Model)

,1 ( / 2 )

HH

(7 5 )h • Hill (1965) / Hutchinson (1970)

(Tangent Model) 1212

(7 5 )6 (4 5 ) 15 (1 ) (1 2 )

hh S

(1 ) ;2 (1 )

h dhh d

43

Shear Bands

● Constitutive Eq.n

1 2 ;S Dp n

w

2c 12 , ;2

S S

D D

● Linear Stability / SB Orientation

2 D D

iqz+ t ;v L x ve max

00 & &

04

crcr

cr

hq

● Nonlinear Solution / SB Thickness

●

zzc

0 4w c

44

● ~ 0.4w c

C i t t Bulk Nanostructured Fe-10%Cu Polycrystals

- Compression testsd ~1370 nm, σy ~750 Mpa d ~540 nm, σy ~960 MPa

angle ~ 490 angle ~ 490

(MPa

)

(MPa

)

4%

St i Strain

Stre

ss (

Stre

ss (

2 mm 2 mm

~ 4%p~ 4%p

- Shear band width analysis

Strain Strain

2R Model2( ) c

~ 0.4w c e (n

m) o

ModelExperiment

2

~ ( )10

c R h

7 5 Gra

in S

ize

45

7 515(1 )

G

w

G

Shear Band Width (microns)

Statistical / Random Aspects

• Microscopic vs. Macroscopic Constitutive Eq. ; vs ; vs. ,

, ... random microscopic fieldsi fi ld

T l i + A i

, ... average macroscopic fields

• Taylor expansion + Averaging

2 2

2 2c c

2 2x x

1 12 2( ) ( )r rh h

2 2

0 0

( ) ( ); ;r r

c c h hr r

46

... correlation function ; ... hardening coefficientr h

• Compression Tests – Shear Band Patterns Bulk Nanostructured Fe-10%Cu Polycrystals

p

cor

d = 150nm85μm cor

d = 300nm386μm cor

d = 830nm1143μm

• Correlation Function and Corresponding Correlation Length- Moving Average Process

cor

2

2

( ) : stationary random process1 ( ) ,: window of observation

x L

Lx L

ξ xξ ξ x dx

LL

- Variance/Correlation Function + Correlation Length2 2 22( ) , ( ) (1 ) ( ) ; : variance

L

Lrf L g g f L Λ r dr g

L L

0L L

121( ) 1 1 , 2

2

m m m

cor cor

r rmΛ r m

47

cor cor lim ( )cor L

L f L

- Shear band width analysis1

2( ) ,k γ γτ c 1

22

20

( ) ,corr

Λ rc h hr

w α c

- Calibration for 300nm grain size

0

12 2 2

0( ) ( ) 0.85corr

w α h Λ r r α h α h

- Modeling of experimental data2R M d l

e (n

m) o

ModelExperiment

Self-consistent Probabilistic

Gra

in S

ize Probabilistic

48

G

wShear Band Width (microns)

More on Nano Shear Bands: n-Fe (Ma et al)

Stress strain behavior and development of shear bands Compression test

49

Stress-strain behavior and development of shear bands. Compression testof a Fe sample with an average grain size of 268 nm with loading,unloading, and reloading at various strain levels (~0.3%, 3.7%, and 7.8%).

Additional Benchmark Problems■ Gradient Solid / Solid Interface

20y(m) ci = (0.3 N, 0.5 N)

i = (3 μm, 3.5 μm) Gi = (30 GPa,40 GPa)

10

0

x

yFilm (material 1)

0.024 0.026 0.028 0.030 0.032 0.034

10

20

xSubstrate (material 2)

• Elastic Bimaterial / Elastic Interface: κi=Giγ ; τI=GIγI

γi

2ic( )

Elastic Bimaterial / Elastic Interface: κi Giγ ; τI GIγI

- Aifantis (1984) 1 2 1 2 1 1 2 2I

G G G c G c G

f ( )

Fleck H (1994)

I1 2 y 0 1 2 2 2 1 1

GG G c G G c

1 2 1 2 1 1 2 2G G G c G cG

50

- Fleck-H (1994) I

1 1 2 2 y 0 1 2 2 2 1 1

GG G c G G c

● Elastic Bimaterial / Inelastic Interface: κi=Giγ ; τI=GIγI

- Scaled adhesive energy (Rose et al.): * * *0E E E (1 )exp gy ( )

M ll R l

0 ( ) p

*I

* * * *( ) d 0

0.8

1.00.8

- Maxwell Rule: *

* * * *( ) d 0

0 1 2 3 4

0.0

0.2

0.4

0.6

0 0 *0 1 2 3 4

0.0 γ*0 1 2

GaN 2 0y(nm)

GaN 2.0

1.0 Ga

γ0.1 0.2 0.3 0.4

-1.0

-2.0

Si

γ

51

Plastic Boundary Layers● Fleck/Van Der Giessen/Needleman (2000)

τx2

Discrete Dislocations (DD) Fleck-Hutchinson (F-H)U( ) 0 τ

τ

2

x1

u1 = U(t), u2 = 0

u1 = 0, u2 = 0 av.( )

● Aifantis (1984) / Gurtin (2000)0 2

02 2 cosh(x / )G 1

G G h(H / )G

0

/ 2

0

G G cosh(H / )

1 2 H(x )dx 1 tanh

52

02 2

/ 2

(x )dx 1 tanhG G 2

● Plastic Strain Profiles / Size Effects 2x

HAifantis/G ti

0.1 mF-H

2xH

Γ=0.0070.02

GurtinDD

0

/ 5H 10

0

/ 20H 30

0

H/d4080

53

240

Effective Moduli of NanopolycrystalsId li d U i C ll● Idealized Unit Cell

y τh

dGg Ggb

y gb

g gb

0 , y=0

Bc's y d / 2

xh

dGg

y g y gb

g

Bc s , y d / 2

/ G , y h d / 2

τg

2ii c

● Average Strain/Effective Modulus

d / 2 h d / 2

d / 2 h d / 2

gb g eff0 0

1 dy dy , G /h d / 2

54

● Size Dependence / Experiments

0.9

1.0

Gg

0.7

0.8

d=0.5 nm d=1.0 nm

Gef

f / G

0 20 40 60 80 100

0.6 Nanocrystalline Fe

Grain Size d (nm)

3 )

0.076

A-3)

● Observations

Ni 0.09

Den

sity

(Å-3

0.074Fe

Den

sity

(A

0.06

Ato

mic

D

0.072

Ato

mic

D

55

-150 0 150Distance from Interface Plane (Å)

-10 -5 0 5 100.070A

Distance from Interface Plane (A)

Size Effects in Microtorsion

M

600

700

M 2 = 12 m 2 = 15 m

300

400

5003

(MPa)2 = 20 m 2 = 30 m 2 = 170 m

0

100

200

l 5 m

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

s

2( ) )(k ( ) Nk k 12( ) ) ,(o k c ( ) , Nok k 1( ) Nc c

23NM N l 3

32 1 ,3 3 1

No s

oM Nk

Nl

N

/ ol c k

56

Size Effects in Microbending

10

12

14

24bhM

h = 12.5 m

Aifantis (l 11.4 – 15.6 m) Fleck (lc 6.2 m)

6

8

10bho h = 25 m h = 50 m

2

4

;

0.02 0.04 0.06 0.08 0.100

εb h/2

;

21 2( ) ( )mc ck 3 3( )

2 4o pk E

, m = 0.34;

24o

Mbh

124

22 1

2 213 3

mp m

b bmo

mlh

E

1/ 21 / m

pc El = 1 +

57

Size Effects in Micro/Nano indentation

,PH 2tan h

● Definitions,

pA D2

22

DA h 22 (tan )pA h

● Gradient Theory (Symmetric Stress) and Tabor’s Rule1/ 2( ) ;c 2~ tanh

D

● Gradient Theory (Symmetric Stress) and Tabor s Rule

22 2 tan (tan )~D D h

03 3 3 ~ 1 ;H H H H lh

tan3 3l cH

58

h oH

● Couple Stress Theory (Asymmetric Stress)

4 04.5 (111) single crystal Cu (Nix et al. 1988)

2 53.03.54.0

H2 (GP 2)

1 01.52.02.5(GPa2)

Symm Stress Theory

0 1 2 3 4 5 6 70.00.51.0

1

Symm. Stress TheoryAsymm. Stess Theory Experiment

Gradient Theory Ho 0.35 GPa, l 4.6 m (c/G)2 6.73 105 m

1/h (m-1)

Gradient Theory Ho 0.35 GPa, l 4.6 m (c/G) 6.73 10 m

Couple Stress Theory H 0 581 GPa l 1 6 m

59

Couple Stress Theory Ho 0.581 GPa, l 1.6 m

Johnson’s Spherical Cavity Model Revisited● Core Incompressibility

2 2 22 ( ) tana du a a dh a da

c

a 2 ( ) tana du a a dh a da

● Geometric Similarityda a

ep epdr r

● Constitutive Assumptions

– Rigid Perfect Plasticity , 0pYd d g y , 0Yd d

– Gradient Yield Condition: 2Y c

60

● Displacements / Stresses / Tractions3 2

2 22

2 4(1 )( )

(1,

)3e

p

e

e

p pY rr l

r cu r lr E E

5 52

2 2

2( ) 2 lo 4 (1 ) 9 1915 4 (1 )

g( )3

e epYY

p

p

e

Y rlr

t rr l rr

● Johnson’s Modified Relations

int ( ) 2 2log( )3

ep

Y Y

rp t

5.00

int

Y

p

Hardness H ~ pintdependence on

H52

2 2

4 ( 1 ) 9 191

5 1 )

4 (

e

e

p

p

rlr l

4.00

Y

intclassicp

dependence onplastic zone size

12 2 2

2

3

2 2

(3 20 (tan2 (1 )

1 ))3( 4 (1 ))

ep p epe r rra l

r E l

0.00 2.00 4.00 6.00 8.00

3.00

int

Y

p

r

61

2 (1 ) 3( 4 (1 ))Y epra l epr

P di i f H/● Prediction of H/σY vs rep

6.5

5.5

6

ness

Equation (26)McElhaney et al

4

4.5

5du

ced

Har

dn

3

3.5Red

0 2000 4000 6000 8000 10000 120002.5

Plastic zone size (nm)

E = 129 8 GPa v = 0 34 σ = 0 36 GPaE = 129.8 GPa, v = 0.34, σY = 0.36 GPa

The fit determines the internal length ℓ ≈ 15.77 nm

62

g

More Recent Results for n-Polycrystals

Activation Volume (υ)● ln3 kT

● Rule of Mixtures

3 kT

1 1 11f f g gb

3 3 ;f d d 0 1 21 1g g gk d

me

vatio

n Vo

lum

Act

iv

630 3 3 31000 , 30 , 2 , 0.3 g gb gb b nm k nm b Grain Size (nm) Grain Size (nm)

Pressure – Sensitivity Parameter (α)

● 0J p Mohr-Coulomb Yield Condition used for the prediction of shear

● Rule of Mixtures 1f f

Mohr Coulomb Yield Condition used for the prediction of shear band Angle in Fe-10%Cu Nanopolycrystals

● Rule of Mixtures 1g gbf f

3 33 0 1 2 31g g gbd d k d d d

ffici

ent

ffici

ent

oulo

mb

Coe

f

oulo

mb

Coe

f

Moh

r-Co

Moh

r-Co

640 0.02, 0.16, 2 , 0.7 g gb gnm k nm Grain Size (nm) Grain Size (nm)

Stress-Strain Curves for Nanopolycrystals

● H-P Type Behavior

t hph

tanhf s f

s f

0 1 2 1 2k d h h k d 0 1 2 1 2, , , 0,s f s f s f hk d h h k d

0 0230 92 5 70k MP kP MP65

0 0230 92.5 70s s fk MPa, kPa m , MPa,

0104 , 827 86f hk h k kPa m MPa, kPa m

● Inverse H-P Type Behavior

3

1 5

2

2.5

ss (G

Pa)

0 5

1

1.5

True

Stre

0.02 0.04 0.06 0.080.00

0.5

Plastic Strain

00.5 4 140 GPa, GPa, kPa mf s sk

66

● Strain Rate & Temperature Effects

0

1 lns c

1m

ref melt refT T T T 0

s (M

Pa)

(MPa

)

890 15ln s

rue

Stre

ss

ue S

tress

383

MPa

MPa

s

Tr Tru

True Strain True Strain

383

95296

MPa

GPaK

f

hT

(MPa

)

(MPa

)296

1356

K

Kref

melt

T

T

Stre

ss (

Stre

ss ( 2.6m

67True Plastic Strain Plastic Strain

● Simultaneous Grain Size & Strain Rate Dependence

MPa

)

Pa)

Stre

ss (M

tress

(MP

S

Strain Strain

Pa)

Pa)

Stre

ss (M

P

Stre

ss (M

P

S S

Plastic Strain Plastic Strain

68

p0f f f 0

p

1 2k k 1 m ln( )dd

p

p0 3 4k k 1 m ln( )

s s s 0p

1 m ln( )dd

70MPa 265MPa 3GPa

0f

0s 0h

70MPa 265MPa 3GPa

1k kPa m 2k kPa m fm 3k kPa m 4k kPa m sm 5k kPa m 6k kPa m

386 1634 0.045 437 1207 0.016 60851 216646

69

A Note on Size Effects in Tension (Gradients or Not?)● Experiments

b) Ag micro-wiresa) Steel cylindrical macro-bars

D=1.5 mmD=15 mm

Experiments

σeng

Probe B D=15 mmIRS 4IRS 5 D=1.5 mmIRS 6

Data from Ngan 2011

NO macroscopic gradients; Size effect modeling?Data from ECA et al 1998 εeng

NO macroscopic gradients; Size effect modeling?

21 ● Modeling

21; , qd D M

i e σ depends on ε and an internal variable α which evolves

70

i.e. σ depends on ε and an internal variable α which evolves inhomogeneously through a diffusive transport term2

Adiabatic elimination of α: reaches steady states much faster than εy

Radial symmetry 0 r

c D M

Bc’s α finite

0 0qr AK r c BI r c

c D MM

0 0r A q

c

r Rr c c

… zero flux of α at r = 0

Assume *0 0ˆ ˆ;n mY k k … Ludwig type

m

* 20 0 1 2 ( )

mqmnY k k e

( ) Models the si e effects in tension of pre io s slide

Note: Grain size dependence can be introduced in according to H-P f l i d t t i t i i (d) d t i i (D) i

( ) Models the size effects in tension of previous slide

71

P, for example, in order to capture intrinsic (d) and extrinsic (D) size effects simultaneously

0k 40 MPa; m n 0.6; q 1; 3.5 μm - Size effects on tensile strength of Mg microwires

2.2μ108

σ [Pa] D=20 μm D=30 μmD=40 μm

D=50 μm

0k 40 MPa; m n 0.6; q 1; 3.5 μm

1.8μ108

2.μ108

1.4μ108

1.6μ108

Y 165 MPaY 150 MPa*

0

d 3.5μm

k 260 MPa

1.6μ108

0.05 0.10 0.15 0.20 0.25ε [ ]σ [Pa] D=20 μm

D=30 μmD=40 μm

1.2μ108

1.4μ108

Y 80 MP

D=30 μm

D=50 μm

6 μ107

8.μ107

1.μ108

d 40.6μm

Y 80 MPaY 70 MPaY 50 MPa

(Chen & Ngan, 2011)

0.02 0.04 0.06 0.08 0.10 0.12 0.14

6.μ10 *0k 400 MPa Y 25 MPa

72ε [ ]

- Modified Hall-Petch Relation: Application on Ag nanowires*

Y Y0

k kY Y dd

Y

*Y Y

D d 3: k 300 MPa m (H-P)

D d 3: k 100 MPa m; k 210 N m (Modified H-P)

according to the modified H-P relation

di t H P l tiY [Pa]

1.5μ108according to H-P relation

D d 31.0μ108

D d 3D d 3

0 1 0 2 0 3 0 4 0 5 0 6

5.0μ107

73

0.1 0.2 0.3 0.4 0.5 0.6d-1/2 [μm-1/2]

[Data by Chen & Ngan, 2011]

- Size effects in yield

mn *Υ k ln 1 ε k ln 1 ε 1 4 1 ε

1 0μ109

mn * ε 2 ε 2

Υ k ln 1 ε k ln 1 ε 1 4 1 εDσ Υ kε k ε 1 2βe ; β e σ

R 1 ε

1.0μ10

5.0μ108

3.0μ108

7.0μ108

*

ε 0.01;Υ k 30 MPa;n 0.2;

k 280 MPa;m 1.6; 1μm

Y [Pa]

1.0μ108

2.0μ108

1.5μ108

[ ]

1μ10-6 2μ10-6 5μ10-6 1μ10-5 2μ10-5

(Dimiduk et al, 2005) D [m]

8

1μ 109

2μ 109*

ε 0.1;Υ 40 MPa;k 20 MPa;n 0.25;

k 240 MPa;m 1.8; 0,3μm

1μ 108

2μ 108

5μ 108

Y [Pa]

1.0000000 μ 10-65.0000000 μ10-7 5.0000000 μ10-62.0000000 μ10-63.0000000 μ 10-7 3.0000000 μ 10-61.5000000 μ10-67.0000000 μ10-7 7.0000000 μ 10-6

D [m](Greer et al, 2005)

- Size effects in yieldm 2 m 2m 2m *

0 0 m 2εσ σ k (ε) ε ; ε ; (ε) σ σ kD ε D

5.0μ108

2.0μ108

3.0μ108

*

0

k 353ΜPa;σ 20 MPa;m 1.4;

1μm

Y [Pa]

1.0μ108

1.5μ1081μm[ ]

1μ10-6 2μ10-6 5μ10-6 1μ10-5 2μ10-5

(Dimiduk et al, 2005) D [m]*

5μ 108

1μ 109

2μ 109

Y [Pa]

*

0

k 270 ΜPa;σ 40 MPa;m 1.6;

0.3μm

1μ 108

2μ 108

5μ 10Y [Pa] 0.3μm

1.0000000 μ 10-65.0000000 μ10-7 5.0000000 μ10-62.0000000 μ 10-63.0000000 μ 10-7 3.0000000 μ10-61.5000000 μ10-67.0000000 μ 10-7 7.0000000 μ 10-6

D [m](Greer et al, 2005)

A Note on Entropy & Power Laws • Boltzmann-Gibbs Entropy

-23B BS k P( I )ln P( I ) ; k 1.38065 10 J KB B

i( ) ( ) ;

• Tsallis Entropy

1 1 11

q

qI

S ( P ) P( I ) ; qq

: entropic index

0λ tdξ λ ξ ξ λ

• Exponential Relaxation

L t11 0λ tdξ λ ξ ξ e ; λ

dt : Lyapunov exponent

- Presence of fractality qdξ λ ξ1

- Presence of fractality qλ ξdt 1

11 1

qq( q ) t

AI (i d f l d )

74

1 11 1 ( q )p I

B( q )I(instead of as commonly done) p I ~ I

• Acoustic emission during creep of ice single crystals

Λ= -1.6

(Miguel et al, 2001)

2 4 6 8 10

2

4)E(plog10 I ≡ E (AE Energy)

2 4 6 8 10

6

4

2 )E(log10

60000 25

AB

---

10

80 251 65

B .q .

--

0.1

• Acoustic emission during paper fractureI ≡ E (AE Energy)

10 5

0.001(Salminen et al, 2002)

Λ= -1.3

P(E)

0 2A .

I ≡ E (AE Energy)

-

10 7

10 5

0 0121 65

B .q .

-

100 104 106

75AE Energy (arb. units)

• Strain bursts in Mo micropillars under compression

0.1

1(Zaiser et al, 2010) p(s) I ≡ s (strain burst)

Λ= -1.5

0.001

0.011 515

A .B

1.00.5 5.00.1 10.0 50.0100.0

10 4 1 65q .

Burst si e s [nm]Burst size s [nm]

• Pop-ins in Zr52.5Al10Ni10Cu15Be12.5 glasses under indentation2.0 I ≡ Δh/h (normalized pop in)

1 0

1.5

2.0

obab

ility)

1.0

1.5

babi

lity)

250A

(Ngan et al, 2007) I ≡ Δh/h (normalized pop-in)

Λ= -2.7

0.0

0.5

1.0

log

(pro

1 mN/min, k = 3.00.3, R2 = 0.96710 mN/min k = 2 80 1 R2 = 0 985

0.0

0.5

log

(pro

b 2503501 25

ABq .

76-2.4 -2.0 -1.6 -1.2

-0.5

log (h/h)

10 mN/min, k = 2.80.1, R = 0.985 100 mN/min,k = 2.70.2, R2 = 0.990 - 2.4 - 2.2 - 2.0 - 1.8 - 1.6 - 1.4 - 1.2

log (Δh/h)

q

• Slip AvalanchesSlip Avalanches

r = 1 and q > 1r qr q r

dξ μ ξ ( λ μ )ξ ;dt

dt

111

q( )

b

111

1 11

qq q ( q ) te

0.01

0.1

(Zaiser & Aifantis, 2003) p(s)I ≡ s (avalanche size)

0.0001

0.001

1

1 8

0 0022 05

q .

.q

Λ= 1

1 10 100 1000

1. ΄ 10- 6

0.00001

Avalanche size s

2 050 3

q .b .

Λ= -1

77

Avalanche size s

A Note on Mindlin’s Strain Gradient Elasticity Theory

• Strain Energy Density1 ii jj ij ij ij, j ik,k ii,k jk, j ii,k jj,k

ij,k ij,k ij,k jk,

3

5 i

1 2

4

w2

E E Eij ij ij ij ij ijE

ij ij

w w 1 νσ λε δ 2με ; ε σ σ δε σ 2μ 2μ(1 ν)

ij ijε σ 2μ 2μ(1 ν)

w 1ε δ ε δ ε δ ε δ 2ετ α α δ

i , jk j , ik ,i jk , j ik k , ij

,k ij ij

ijk

,k ik

1 2ij,k

3 , j jk4 ,5 i

ε δ ε δ ε δ ε δ 2ετ α αε

α α

δ2

2 ε δ 2 ε α ε ε

,k ij ij,k ik3 , j jk4 ,5 i

E E Eij ij ji... elastic like stress ; ... 6 components

78

ij ijk jik... dipolar like stress ; ... 18 components

● Equilibrium● Equilibrium

Ej ij k ijk 0

j j j

1 5 2j ij ij i , j j , j ,ij k ,k

2 2kk ij i3 j4

2

2 0

i.e. formidable to solve in general kk ij i3 j4

● Special Solutions

(i) Feynman 1962 Linear Theory of Gravity- (i) Feynman 1962 … Linear Theory of Gravity

Eij, j5 ijk, jk0 ; 0 ; 0

metric : stra in t ensor4D gradient theory

gravitation : metrical elasticity of spacetime

79

2 2 2 22 0 2 2 2 2i , j , ji ,i j1 2 3, ji ,i ij j4 ,2 0

2 2j , j i i , i2 4 2 3 ,1 1 2 2 0

1 2 3 42 c ; c is identity choose

ijk,k ik jmn n,kmij

1 inc2

j , j ,ij

2 2ij k , k ij kk,ij ik,kj jk,ki kk ij

2 c

i.e.3D linear Einstein tensor used in the gauge theory of dislocations (Malysev/Lazar)

80

(Malysev/Lazar)

- (ii) ECA 1992 … Linear theory of Gradela

1 2 5 3 40 ; 2,c c

E E Eij ij ij,k ij,k ij,k ij,kE

ij,k ij,k

1 W WW ; ,c c2

c2

Eijk ij,k ,k ij ij,kc 2c

Eij ij ij, jijk,kLet ; 0

2ctr 2 tr 2 1 1

81

■ A Note on Electromagnetism

1 2 Tk T

● MacCullagh’s (~1850) Eqs of the Rotationally Elastic Aether

; 1 2 Tk T u u 2 2

2 2div curlcurl 0k

u uT u2 2t t

Letting curl &k a a

t

uu E B

curl =0 ; div 0t

BE B

... electric field ; ... magnetic fluxE B

82

● By also noting the identities

divcurl 0 curl curl 0&t t

uu u

● By also noting the identities

t t 1div 0 curl 0&

t

EE B0 t

0where , k 0

i

curl 0 ; div 0

B E B

i.e.

1 curl 0 ; div 0

t

E B E

83

0t

A SENSE OF SCALE

From: 10-34 – 1024 m

84

From: 10-32 – 1028 m

85

■ Below Newton’s Apple

10‐15

10‐11

10‐17

10‐33

86

i i■ Interesting (?) Analogies

Hubble : 1928

Hot Big Bang Spacetime Foam

Penzias & Wilson : 1965

87COBE : 1992 WMAP : 2003