Tuesday, Part 1: Contact Mechanics - Northwestern...

” 2003 R.W. Carpick and D.S. Stone Tuesday 1

A Short Course on NanoscaleMechanical Characterization:

The Theory and Practice of ContactProbe Techniques

Tuesday, Part 1: Contact Mechanics

-Hertz & Boussinesq Problem-Adhesive elastic contact mechanics-Fracture approaches to the adhesive problem-Elastic-plastic problem-Viscoelastic problem-Rough surfaces


Hertz & Boussinesq Problem


• Assumptions:– contacting bodies are HILE

– contact radius is small compared to the size & curvature radius of the bodies

– frictionless surfaces

• Contact pressure, contact size, surface compression all depend on the load in anon-linear manner

• Consider two spheres in contact, radii d1/2=R1, d2/2=R2; moduli E1, E2; Poisson’sratios n1, n2. Applied load = P

Hertz Theory of Contact

question: what’swrong with thispicture? P P


Hertz Equations

†

surface compression d = d1 +d 2 =a2

R =9P2

16RE *Ê

Ë Á

ˆ

¯ ˜

1/3

†

contact radius: a =3PR4E *

Ê Ë

ˆ ¯

1/3

†

Define: Reff = 1 R1 +1 R2( )-1, E * = 1- n12( ) E1 + 1-n1

2( ) E2( )-1

†

max. normal pressure at interface: p0 = 32 pave =

6PE*2

p 3R2

Ê

Ë Á

ˆ

¯ ˜

1/ 3

where pave = P pa2

d1

sphere 1 shown only

†

normal stress: s z(r) = -p0a2 - r2

a

†

at the contact zone edge:s r = 0.13p0, sq = -0.13p0, s z = 0

which is a state of pure shear!


Hertz Theory of Contact• An analytic solution for all stress components at all points in space exists• Note: the largest shear stress tmax=0.31p0 occurs at z=0.48a below the

surface• The theory can also be applied to a sphere contacting a flat plane by letting

R1=Rsphere, R2=•

• Exercise: simplify the Hertz equations for this case, assuming the materialsare the same.

tmax=0.31p0


Boussinesq Problem: Rigid flat punch• The stress distribution is very different for a rigid circular punch:

†

surface compression: d =P

2aE*

†

a = constant

†

max. normal pressure: p0 = 12 pave =

P2pa2

†

normal stress: s z(r) = -p0a

a2 - r2


• Uniform pressure: stress is directly proportional to load, contact area is constant• Hertz pressure: stress depends nonlinearly on load, contact area is changing with

load

• Design considerations: bearing life is strongly influenced by contact stresses. Forexample, for ball bearings, fatigue life is related to p0

-9!! Thus, small reductions instress can substantially affect bearing performance.

Uniform pressure vs. Hertz pressure

r

stress

aa

sz

sr

sq

sz

sr

sq

sr

sq

sr

sqUniform Hertz

top view

sr

sq


Adhesive elastic problem


Rrecall: The JKR theory predicts the effect of adhesion on thecontact area

JKRHertz

R

L

aa A=π·a2

Hertz:

JKR:

A = p3R

4E* LÈ

Î Í

˘

˚ ˙

2 3

A = p3R

4E* L + 3pRg + 6pRgL + 3pRg( )2Ê Ë Á

ˆ ¯ ˜

È

Î Í

˘

˚ ˙ 2 3

A = contact area L = load

R = tip radius

E* = = reduced modulus

g = interfacial energy

1- n12

E1+

1 -n22

E2

Ê

Ë Á Á

ˆ

¯ ˜ ˜

-1


The area of contact depends upon the range of theattractive forces

l>5 (JKR) l<0.1 (DMT)

ac ac

Johnson-Kendall-Roberts (JKR): short-range forces, high adhesion,compliant materials

Derjaguin-Müller-Toporov (DMT): long-range forces, low adhesion, stiffmaterials

Intermediate cases: Maugis, Colloid Interface Sci. 1992, 150, 243.


Recall: Ac depends on the range of the attractive forces

ForceArea

distance

Actual

z0

ForceArea

distance

Hertz

ForceArea

distanceg

ForceArea

distance

DMT

g

JKR..

DMT=Derjaguin-Müller-Toporov modelDerjaguin, B. V.; Muller, V. M.; Toporov, Y. P. J. Colloid Interface Sci. 1975, 53, 314.


Recall: Ac depends on the range of the attractive forces

JKR.

-2 -1 0 1 2 3 4 50

1

2

3

4

5

6

Load

(Co

nta

ct A

rea

) /

π

Hertz

JKR

DMT

intermediate

DMT


Recall: How to tell if you are in the JKR or DMTregime (or in between)?

• compare the elastic deformation that adhesionforces cause with the range of the adhesion forcesthemselves

l =1.16 ⋅g 2 ⋅ R

E*2⋅ z0

3

Ê

Ë Á Á

ˆ

¯ ˜ ˜

1 3

g=adhesion energyR=tip radius

E*=elastic modulusz0=equilibrium spacing


Fracture mechanics approaches toadhesive contact and friction


Contact -- Crack• Tip/sample contact is analogous to an

external circular crack• System is otherwise elastic

crack tip


JKR Theory of ContactsJohnson, Kendall & Roberts,Proc. Roy. Soc. A (1971) 324, 301.

• Griffith’s concept of brittle fracture

• Total load:

• Mode I stress intensity factor KI, Irwin’s relationship:

• Final result:

†

P = PHertz + -Pa( ), PHertz =43 E * ⋅ a3

R

†

KI =12

⋅s a ⋅ 2pa =Pa

2a paG =

KI2

2E* = g

†

a3 =R

43 E * P + 3pgR + 6pgRP + 3pgR( )2Ê

Ë ˆ ¯

a = contact radiusP = load R = tip radiusg = adhesion energy E* = reduced Young’s modulus


Maugis’ Dugdale Model

Dugdale

D. Maugis, J. Coll. Interf. Sci., 1992, 150, 243.


Maugis’ Dugdale ModelD. Maugis, J. Coll. Interf. Sci., 1992, 150, 243.


Assumptions of Maugis’ model1. Linear elastic fracture mechanics applies2. Link between equilibrium spacing and cohesive

zone distance:

†

s0 = s theoretical ,LJ

s theoretical ,LJ =1.03 gz0

G = s0 ⋅d t , G = g

\ z0 = 1.03 ⋅d t


Shear strength measurements using the MD model are nowbeing made by several groups

• Lantz et al. (Phys. Rev. B 1997, 55, 10776): Si/NbSe2• Pietrement & Troyon (Langmuir 2001, 17, 6540): SiNx/carbon fiber, SiNx/mica,

SiNx/SiO, SiNx/epoxy• observations include: load-dependent shear strengths in some cases, more near-

ideal shear strengths, correlation between friction and adhesion

• E. Barthel (J. Colloid Interface Sci. 1998, 200, 7), has shown that the details of theinterfacial potential do not strongly affect the contact behavior

• K.L. Johnson (Proc.Roy.Soc.A, 1997, 453, 163) has considered mixed-modefracture approaches in frictional contacts.


Kim & Hurtado’s modelJ.A. Hurtado, K.-S. Kim, Proc. Roy. Soc. London A 1999, 455, 3363 and 3385

• why is the shear strength so large for AFM results,but typically much smaller for larger?

• use a dislocation mechanics approach• predicts a scale dependence of shear strength


Kim & Hurtado’s model


Kim & Hurtado’s model


Elastic-plastic problem


Elastic-plastic indentation

from Johnson, Contact Mechanics


Contact of Rough Surfaces


Mechanics Issues in MEMS: Friction and wear

• Currently, there are no commercial MEMS devicesthat involve contacting surfaces with extendedsliding

10 mm


10 x 10 µm scan, 100 nm scale

RMS roughness~11.5 nm 500 x 500 nm scan, 30 nm height

scale

AFM images of polycrystalline MEMS surface(E. Flater & R. Carpick)


A multi-scale understanding of friction could bedeveloped through experiment and modeling

•

micromachine interfaceprediction,

design,understanding

topographicstudies;

constitutive lawsfor friction & wear

mechanics ofmulti-asperity

contacts

Single asperity:Ff=tAc (?)

Multiple asperities:Ff=µL (?)


00.20.40.60.811.2-60

-40

-20

0

20

40

60

00.20.40.60.811.2

-60

-40

-20

0

20

40

60 d

Contact mechanics analysis:the Greenwood-Williamson Theory

rigid plane, lowered

rough elastic surfacedistribution ofsurface heights

distribution ofsummit heights

Greenwood & Williamson, Proc. Roy. Soc. A 295 300 (1966)


GW Theory: Assumptions• linear, elastic, homogeneous, isotropic• distribution of peak heights is Gaussian• radius of peaks is constant (or narrowly distributed

about a mean)• Hertzian contact (no adhesion, no tangential

stresses)• key parameters:

– number of peaks/area– peak curvature– std. dev. of peak height distribution– elastic modulus


The GW Theory: Derivation

• Greenwood & Williamson, Proc. Roy. Soc. A 295 300 (1966)

†

Li = KR1/2d i3/ 2

†

Ai = pRd i

z

d

zi

di

rigid surface

ith summit out of N total summits

†

di = zi - d

Hertz:

†

Li = KR1/2(zi - d)3/ 2

†

Ai = pR(zi - d)

†

di = zi - d


The GW Theory


†

Li = KR1/2(zi - d)3/ 2

†

Ai = pR(zi - d)

†

Let f(z) = asperity height distributioni.e. there are Nf(z)dz asperities between z and z + dz

†

toatal contact area A = NpR (z - d) ⋅f(z)dzd

•

Ú

†

total load L = NKR1/2 (z - d)3/ 2 ⋅f(z)dzd

•

Ú


The GW Theory


†

toatal contact area A = NpR (z - d) ⋅f(z)dzd

•

Ú

†

total load L = NKR1/2 (z - d)3/ 2 ⋅f(z)dzd

•

Ú

†

Assume Gaussian distr. of summit heights: f(z) = 1s 2p e-z 2 2s 2

†

then AL =

pR1/2

Ks1/2 ⋅(x - a)e- x 2 dx

a

•

Ú(x - a)3/ 2e-x 2 dx

a

•

Ú=

pR1/2

Ks 1/ 2 ⋅ I


Area is directly proportional to load!!!•

10-5 10-4 10-3 0.01 0.1 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

I, in

tegr

al ra

tio

LNKR 1/2 s3/2

The effect of increasing the load is not just to increase thearea of contact between asperities already in contact, butalso to bring new asperities into contactNote: real contact area may only be a fraction of a percent of theapparent contact area!


An explanation for macroscopic friction?

•

†

then AL =

pR1/2

Ks1/2 ⋅(x - a)e- x 2 dx

a

•

Ú(x - a)3/ 2e-x 2 dx

a

•

Ú=

pR1/2

Ks 1/ 2 ⋅ I

†

Ff = tA = tpR1/ 2

Ks1/ 2 L = mL

m = tpR1/ 2

Ks1/ 2

Assumes only elastic deformation!


Mistakes in the analysis1. “Peaks” found on a line are not true“summits” within a given area

• How to correct for this?• The answer comes from oceanography!

– developed by Longuet-Higgins, Phil. Trans. R. Soc. Lond. (1957)– adapted to contact problems by Nayak, J. Lubr. Technol. (1971)– see work by J. I. McCool

2. Roughness can’t be representedby a single length scale

summit

line peak


Rough surfaces with adhesion• Fuller & Tabor: JKR case• The effect of surface roughness on the adhesion of elastic solids

• Proceedings of the Royal Society of London A. 1975; 345(1642): 327-42

• Maugis: DMT case• On the contact and adhesion of rough surfaces

• Journal of Adhesion Science and Technology. 1996; 10(2): 161-75

1© 2003 R.W. Carpick and D.S. Stone

Scale Effects in Plasticity

39


Hall-Petch Effect

2/10

−+= kdσσd = grain size

Yield stress increases with decreasing grain size Hardness of molybdenum thin films

Yoder et al, 2003N.J. Petch, Fracture, Proceedings of Swampscott Conference 1959, p. 54.

40


Indentation Size Effect & Reverse Indentation Size Effect

41



Ma and Clark J Mater. Res. 10(4)

41a



Tambwe et al, J Mat. Res. 1999

41b

42” 2003 R.W. Carpick and D.S. Stone Tuesday

Surfaces

• Surface crystallography

• Relaxation and reconstruction

• Surface states

• Surface energy

• Characterization methods


Surface crystallography

• A 2-D lattice is referred to as a net• The area unit is referred to as a mesh• 5 Bravais nets in 2-D

– Square– Hexagonal– Rectangular– Centered rectangular– Oblique

• Mesh defined by mesh basis vectors c1, c2

• For a reconstructed surface, (c1, c2)≠(a1,a2)


Wood’s notation:-used for reconstructed surfaces or overlayer (adsorbate) structures

• General relation between c’s and a’s:

• Wood’s notation:

– a omitted if it is 0°.– p or c is included at the beginning to indicate primitive or

centered, e.g. c(4x2), p(2x2)• The p is optional

– The surface is listed first, e.g. Si(111) (7x7)– If an adsorbate structure is being described, it is included

afterward, e.g. Si(111) (1x1)H

†

c1

c2

Ê

Ë Á

ˆ

¯ ˜ = P

a1

a2

Ê

Ë Á

ˆ

¯ ˜ =

P11 P12

P21 P22

Ê

Ë Á

ˆ

¯ ˜

a1

a2

Ê

Ë Á

ˆ

¯ ˜

†

c1

a1

¥c2

a2

Ê

Ë Á

ˆ

¯ ˜ Ra, e.g. 3 ¥ 3( )R30°, (2 ¥1)



Examples:• See: http://www.chem.qmw.ac.uk/surfaces/scc/ (Prof. R. Nix, U. London)

Substrate : fcc(100)c( 2 x 2 )(√2 x √2)R45

Substrate : fcc(110)c( 2 x 2 )

Substrate : fcc(111)(√3 x √3)R30


Surface Relaxation and Reconstruction

• Surfaces are defects: bonds are broken (all of them!),creating dangling bonds

• Atoms will re-arrange to make up for this lack ofcoordination

• Surface relaxations: motion of atoms inward oroutward from surface plane– More frequently observed with metallic and ionic materials

(much weaker in ionic materials)– See table– Compensation for “electron overspill” (sketch)

• Surface reconstruction: rebonding of surface and sub-surface atoms– Surface lattice is not in registry with the bulk– More frequently observed with covalent materials– Can be removed by terminating the surface with other atoms– e.g. Diamond(111)


Surface relaxations

back


Relaxation

• Relaxation: motion perpendicular to surface– change in the interlayer spacing.

– Usually affects the first 1-3 layers, with the strongest effectat the surface


Example: Bulatov et al, JOM-e, 49 (4) (1997)

Figure 1. The ions in a 552 ion cluster of LaF3 areallowed to move, subject to the forces from all other ionsin the nanocluster, until the target temperature of 10 K isreached.

Figure 2. Cross-section view of the ionmovement depicted in Figure 1.

http://www.tms.org/pubs/journals/JOM/9704/Bulatov/Bulatov-9704.html


Relaxation: NaCl(001)

A sodium chloride (NaCl) surface. There is a slight rearrangement of the surface: sodium ions moveinto the surface and chloride ions relax away from the surface.

http://www.molecularuniverse.com/Bound/bound1.htm


Surface Reconstruction

• Re-bonding by re-arrangement ofsurface and sub-surface atoms(parallel and perpendicular to thesurface plane)

• Si(100)

• Si(111)


Diamond (111), with andwithout H

diamond(111)-(1x1)H

Bulk termination

diamond(111)-(2x1)

Reconstructed


Surface energy

• All surfaces are energetically unfavorable becausethey have a positive free energy of formation withrespect to the bulk crystal– Bonds must be broken to form the surface

• Surface energy is minimized by:– Reducing the amount of exposed surface area– Exposing surface planes with low surface tension– Altering local atomic geometry (relaxation and

reconstruction)

• Low surface energy=high surface stability


Rules of thumb for lowsurface energies (in vaccum)• High surface atom density

• High coordination number

fcc(100) fcc(110) fcc(111)

Arrange these surfaces from lowest to highest stability.


Surface tension

• Total energy: U=TS-PV+µN+gA• g = surface tension

= the reversible work done in forming a unit area of surface

≠ surface energy, but lots of people call it that!

units: energy/area or force/distance (e.g. soap film)

†

g =∂U∂A T ,V ,N

†

Ss = -A∂g∂T e

†

s ij = gdij +∂g∂eij T

†

Ss = surface entropyT = temperatures ij = surface stress (force/distance)eij = surface strain

We can derive the following relations (Zangwill):


Surface tension is anisotropic in a crystal• To minimize the excess surface free energy,

†

gÚ dA = minimum. But g = g(hkl)

\ gÚ (hkl)dA(hkl) = minimum.

http://www.fysik.dtu.dk/CAMP/hot0009_img3.html

PALLADIUM NANOCRYSTALS ON AL2O3

Wulff construction: easy method forfinding the low-temperatureequilibrium shape

Pb


Surface tensions of variousmaterials

Material Surface energy (mJ/m2)

W 4400Cu 2000Ag 1500Al 1100

Water 73Ethanol 22.8Teflon 18.3

from Israelachvili, “Intermolecular and Surface Forces”


Summary

Tuesday, Part 1: Contact Mechanics - Northwestern...

Documents

Transcript of Tuesday, Part 1: Contact Mechanics - Northwestern...