4-th lecture

Physical Principles of Contact Mechanics

Basic Contact Mechanics Theories

Physical Principles of Contact Mechanics and Tribology on Micro- and NanoscalesSurface. Boundary atoms Heterogeneity of properties•Structure. Surface Layers. Surface coatings and composites•Different nature of Surface Forces•Multilevels Roughness Materials properties. Laws of deformations. Scale featuresZone of contactMicro- Nanolavels features of contact

Contents

Basic Contact Mechanics solutions

Hertz theory

Contact with adhesion effects (DKR, DMT)

Capillary forces

Viscoelastic contact

Multilayers contact

Models of Biological contact

1

∫∫−+−

−=

Sz dydx

yyxxyxp

Eyxu ''

)'()'()','(1),(

22

2

πν

1. Features of Contact Mechanics Problems

1

2

Physical factors in surfaces layers and in a gap is important

Physical Principles of Contact Mechanics

2

Adhesion forces

Cohesion forces

Elements of solids

2. Features of surfaces atoms

1 2

3

4

3. Structure of surface layers

3

1 2

3

4

Hµ , M Pa 340

300

260

220

180 0 10 20 30 40 50 h, um

2.3 10-16

3.8 10-16

1.5 10-16

Penetrability, m4/Ns

Metal surface structureCartilage materials structure

4. Comparison of biological and engineering structures

Substrate

Подложка Substrate

Substrate

Substrate

Cartilage composite

5. Types of surface composite materials

4

Е

η

σ

σ

2

σт

1

σ

σ

σ

σ

σ

σσ = ε Ε

•

==⋅

== εηdtdεη

dzdtdxη

dzdVησ

•

= εsignσσ т

•

+⋅=+= εηεEσσσ ηE

−−

= t

ηEexp1

Eσε

( )

−

−= τtηEexpεε τ

.

ησ

Еσεεε ηЕ +=+=

•

•••

−= t

ηEexpσσ

0

≥≥−=

<<=

.εεε-при εsingnσσ

;εεε- при Еεσ

тт

.

т

тт

6. Materials Deformations Lows

0

10– 9

10– 8

10– 7

10– 6

10– 5

m

Coulomb forces of repulsive

Molecular (Van-der-Waals Forces)

Capillary forces

Electrostaticforces

Damping of liquid layer

Probe body

Solid surface

7. Natures of Surface Forces

Magnitude of surface forces

5

x

y

z

Pictorial display of surface texture. (From Anonymous(1985), Surface Texture (Surface Roughness, Waviness and Lay), ANSI/ASME B46.1, ASME, New York.

Pre

cisi

on

cont

act

Waveness

10-1

Physical relief

10-410-3 10-2 10 -1 1 10 1 102

1

10-2

10-3

10-4

Mol

ecul

ar

cont

act

Mic

roco

ntac

t

Small scan AFM

Range of roughness

Microroughness

Height, um

Lateral size, um

STM

Opt

ical

pr

ofilo

met

ry

Styl

us

prof

ilom

etry

Large scan AFM

Atomic roughness

8. Levels of roughness

9. Micro-, nano-, atomic roughness

Multilevel (fractal) structure of roughness

Ceramic nanoroughness, AFM scan 1800x1700x21 nm

HPOG atomic roughness, STM scan 1.7x1.7x0.2 nm

6

10. Atomic level roughness

(a) Electronic charge density contours at a nickel (100)surface. (From Arlinghaus, F.J., Gay, J.G., and Smith, J.R. (1980), Phys. Rev.B , Vol. 21 (b) Charge transfer of the palladium (100) slab upon silver adsorption. (From Smith, J.R. and Ferrante, J. (1985), Mat. Sci. Forum , Vol. 4, pp. 21-38)

Fibre

1,4 nm 3–5 nm 20–100 nm

1–10 um

Microfibrille Collagen Fibrille

Glyc

opro

tein

Prot

eogl

ycan

λ 1

2h1

λ 3

2h3

λ 2

2h2

Col

lage

n st

ruct

ures

M

ultil

evel

rlie

f of

cart

ilag

e su

rfac

e

0.02 – 0.10.01 – 0.053Submicroroughness

0.5 0.1 – 0.22Micro roughness

30.0 – 50.01.0 – 2.01Macro roughness

Step, umAmplitude, um

LevelType of roughness

11. Role of collagenic structures at formation of a multilevel roughness

The surface of cartilage has a multilevel roughness

7

Real Contact Zone

Contact between roughness surfaces (Our computer simulation)

12. Real Contact Zone between roughness surfaces

∆Ari

Асi

Aa

Αа – nominal contact area;

Αсi -contour area of contact;

∆Αri – real contact area

Physical contact

13. Structure of contact area of multilevels roughness surfaces

8

14. GENERAL FACTORS IN MICRO- AND NANOTRIBOLOGY

Surface interactions dominate as machine scale is reduceddegradation phenomena (friction, stiction and wear)

Micro- and nano roughness

Adsorption layers

Elastic deformation and micro elastic properties

Van-der-Waals, electrostatic forces

Capillary forces

10-4

10-3

10-2

10-1

1

0 5 10 15 20 25 30 35σ, nm

metalls

polymers

elastomers

Аr/Aa

Effect of roughness

10–3

10–2

10–1

100

101

102

Ra,um

∇ 6

1 2

∇ 8

∇ 13

Met

all-E

lasto

mer

Elas

tom

e r-E

lasto

mer

Met

all

Poly

mer

Poly

mer

-

Poly

mer

M

etal

l-m

etal

l

Effect of Surface Forces

15. Basic factors estimations

9

103

10-6

10-9

mNano

Micro

Macro

Cohesion

Adhesion

Contact mechanics

Mechanics of discrete contactNanomechanics

Mec

hani

c al

p rop

ert ie

s

Topo

grap

hy

Surf

ace

forc

es

Scale Model Factors Friction Components

16. Scale Levels of Contact Mechanics and Tribology

Ffr=Fad + Fc

Basic Contact Mechanics Theories and Methods

10

Johnson K.L. (1985), Contact Mechanics, Cambridge UniversityPress, Cambridge

Fischer-Cripps A. C. (2000), Introduction to Contact Mechanics, Springer, Berlin

Sviridenok A.I., Chizhik S.A., Petrockvetc M.I. (1990) Mechanics of Discrete Friction Contact, Nauka I Technika, Minsk (in russian )

References

Handbook of micro- and nanotribology. Ed.by B. Bhushan (1999) CRC Press, London, New York

Modern Tribology Handbook V.1.Principles of TribologyEd.by B. Bhushan (2001) CRC Press, London, New York

Contact of Spheres

23

21

*34

∆= REP

Ra ⋅∆=

17. Hertz theory

Assumptions

1. Homogeneous of materials 2. Low deformations 3. R >>a

11

Elastic contact of spheres;subsurface stresses along the axis of symmetry

Contours of maximum shear stress normalized by Hertz stress p0

18. Subsurface stresses in Hertz Contact

Contours of maximum shear stress normalized by Hertz stress p0

19. Subsurface stresses in sliding Hertz Contact

Microslip

for spheres

for cylinders

12

DMT Only compressive elastic forces inregion of contact

No forces outside region of contact

20. Adhesion Contact of Elastic Bodies

∗=EPRa

433

0PPP adh +=

))3(63( 23 γπγπγπ RPRRPKRa +++=

10-2

10-1

100

10-1 101 103 K, GPa

∆γ, N/m

1 2 3 4 5

JKR

DMT

21. Adhesion contact parameters analysis

1 – 1 – R= 10-8 m; 2 –10-7; 3 –10-6; 3 –10-6; 4 –10-5;5 –10-4

31

3

222232

∆=

εγπ

πµ RK

13

22. Comparison of adhesion contact modelsAssumptions

Hertz No surface forces

DMT Long-ranged surface forces act only outside contact area.

JKR Short-ranged surface forces act only insidecontact area.Contact geometry allowed to deform.

Limitations

Hertz Not appropriate for low loads if surface forces present

DMT May underestimate contact area due to restricted geometry. Applies to low systems only

JKR May underestimate loading due to surface forces. Applies to high d systems only

23. Model of viscoelastic adhesion contact

Maxwell’s model is used to describe the viscoelastic behavior of materials.Adhesion is describe by DMT model and the applied load is changed according to the law :

γπ RtctPtP 2)sin()( 0 +=

14

)sin()( 0 tctPtP =

∫ −=t

dttPdtdttRta

0

3 ')'('

)'(83)( φ

Contact radius- time curves with different values of relaxation time

24. Results of viscoelastic adhesion contact study

Contact radius- time curves with different values of surface energy

25. Capillary Forces

+=∆

21

11rr

p γ

2

211

11 xrr

pAFc

+=∆= πγ

( )21 coscos2 θθπγ += rFc

021 ≈= θθ

RFc πγ4=

for hidrophylic surfaces

15

( )[ ] 2121 12 Rah +=ρ

( )Raha 212 +−= ρρ

γπ= aF 21

( )haaFFF +γπ≈+= 1221

( )12

11

22

−− ρ−ργπ= aF( )hhRF o−= 14 γπ

( )[ ]{ }RaRaahsep −+= 2121

( )hhF 014 +πβγ≈

0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

0 50 100 150 200 250 300 h, nm

Fi,×10-7 N

123

approach

moving off

26. Capillary forces for deformed spherical contact

Spherical or cylindrical indenter in contact with a layered half-space; (b)normal contact pressure profiles beneath a frictionless rigid spherical indenter for an elastic layered half-space with different values of E1E2,a0 is the Hertzian semi-dimension when E1 = E2; (c) indentation depth vs. applied load, P0 is the normal load for the case E1 = E2 and a0 = h. (Adapted from O’Sullivan, T.C. and King, R.B. (1988), Sliding contact stress field due to a spherical indenter on a layered elastic half-space, Trans. ASME J. Trib., 110, 2345-240)

27. Contact of layered bodies

16

Epoxysilane SAMs

28. Analytic approximation for layered contact

2

1

Hertz10 nm

41

2

34

8.01

8.0

+

+=

hertz

hertzhertz

at

atJ

aa

( )

−=

1

21

413E

PRahertzν

21

22

2

1

11

νν

−−

=EEJ

RKa 2

=δ

1

2

T,nm E1,MPa11.5 4.520.3 0.631.1 0.3850.0 0.75

δ

P

δ,m

P, N

29. Finite element methods analysis

17

M. Doblareґ *, E. Cueto, B. Calvo, M.A. Martıґnez, J.M.Garcia, J. Cegon˜ino On the employ of meshless methods in biomechanics. Comput. Methods Appl. Mech. Engrg. 194 (2005) 801–821

30. FEM and NEM method of Bio Contact simmulations

( ) imax

ii khEP δ=δ

Z

zi

Si=∆xi∆yi

H

zmax

li

h

контртело

шероховатая поверхность

312

38 −−∆= ii lEkhdz εγ

( ) ( )∑ −∆∆−=i

izhyxKhP ( ) ( )∑−

−∆∆γε∆=i

is zhyxhF32

38 '

31. “Winkler layer” computer 3-D simulation of contact

18

32. Computer simulations of adhesion contact

Load

0

0.2

0.4

0.6

0.8

0 5000 10000 15000 20000 R, nm

h

E=108 N/m2

E=109 N/m2

E=1010 N/m2

0

1000

2000

3000

4000

2 4 6 8 P, ×10-7N

P, N

A, nm2

Hertz JKR

CS, ∆γ=0,5 2

CS, ∆γ=0 2

33. Computer Simulation (CS).

Comparison with Hertz and JKR theories

19

34. Molecular dynamic model

Typical setup for a molecular dynamics simulation(from Nancy Burnham and Adrzej A. Kulik Surface Forces and Adhesion. Handbook of micro- and nanotribology. Ed.by B. Bhushan.

Molecular dynamics (MD) was first used in thermodynamics and physical chemistry to calculate the collective or average thermochemical properties of various physical systems including gases, liquids, and solids. It has been recently applied to simulate the instantaneous atomic behavior of amaterial system. There are two basic assumptions made in standard molecular dynamics simulations:

(1) Molecules or atoms are described as a system of interacting material points, whose motion is describeddynamically with a vector of instantaneous positions and velocities.The atomic interaction has a strong dependence on the spatial orientation and distances between separate atoms. This model is often referred to as the soft sphere model, where the softness is analogous to the electron clouds of atoms.

(2) No mass changes in the system.Equivalently, the number of atoms in the system remains the same.

35. Molecular Dynamic equations

Pair-wise potentials and the interatomic forces: (a) Lennard–Jones, (b) Morse.

The dependence of the potential function Uon the separation between atoms and molecules and their mutual orientation can in principle be obtained from quantum

mechanical (QM) calculations.

20

Friction behavior of ahydrocarbon system [109]: (a)simulation model of friction between two molecular surfaces; (b) comparison

of the friction coefficients forhydrogen-terminated carbon surfaces and clean carbon surfaces.

Indentation pattern in agolden substrate: MDsimulation. Actual imprint size can be tens-to-hundreds of nanometers.

36. Examples of MD method applications to Contact Mechanics

W.K. Liu *,1, E.G. Karpov, S. Zhang, H.S. Park , An introduction to computational nanomechanics

and materials. Comput. Methods Appl. Mech. Engrg. 193 (2004) 1529–1578

4-th lecture Conclusions

•The contact mechanics described a complex of mechanics and physics phenomena on the surfaces

•It is necessary to consider properties of a surfaces on micro- and nanolevels

•The major are following factors and properties: roughness, mechanical properties and low of deformations, structure of surface layers

•The tendency to consider the maximal number of factors is observed at the description of contact

21

4-th lecture Physical Principles of Contact Mechanics Basic Contact Mechanics Theories This lecture is dedicated to statement and main approaches to solution of contact mechanics problems. It consists of two parts. In first part, we will consider physical aspects of surface layer structure of engineering surfaces and their role in formation of actual contact. In second part, main problems of contact mechanics that are used for estimation of bearing capacity and friction of materials will be examined. General statement of problem in contact mechanics provides for examination of deformations and stresses only near zone of the surface contact, in thin material layers adjacent to this zone (Slide 1). Main integral equation describes equilibrium conditions in the contact zone. That’s why adequate description of contact demands considering of physical features of material surface layer structure. Atoms of the material laying on the surface, in contrast to that ones being inside the bulk, experience no balancing interaction from neighboring atoms in all directions. From environment (or other phase) side they have unbalanced force fields (Slide 2). Surface of engineering materials are structurally heterogeneous in layers that is conditioned by their machining and interaction with environment (Slide 3). For comparison, biological structure of cartilage also has layered heterogeneity near its surface. And, for example, distribution of permeability for cartilage and microhardness for engineering surfaces are similar (Slide 4). Besides, in the engineering, working surfaces usually have coatings. Slide 5 shows optional character of composite coatings. To describe contact, we need to know regularities of material surface deformation. Shown are examples of different deformation laws: elastic, plastic, elastoplastic, viscoelastic (Slide 6). Heterogeneity of surface force field effecting on trial body approached to the surface is conditioned by its dependence on the separation and difference in physical nature of surface forces (Slide 7). Discussed are general provisions of multilevel character of engineering surface geometry and compliance of measurement techniques to the evaluated levels (Slide 8). Presented are examples of micro- and nanoroughness (Slide 9), as well as nature of atomic-level roughness of surfaces (Slide 10). For comparison, nature of multilevel roughness of cartilage surface originated from collagen fiber structure is shown (Slide 11). Demonstrated is role of roughness in formation of actual discrete contact (Slide 12). Multilinked structure of actual contact area is conditioned by multilayer organization of surface roughness (Slide 13). Shown is a scheme of complex effect of different heterogeneity factors at formation of physical contact (Slide 14).Presented are criteria of complex joint effect of factors of elasticity, surface forces and roughness. Precision contact (smooth and/or elastic) demands to take into account all factors (Slide 15). Slide 16 shows scheme of scale levels in contact mechanics and tribology that makes clear importance of the factors in

22

23

contact mechanics and tribology dependingly on the level of their consideration: macro-, micro-, nano-. Further, considered are classical and non-classical problems of a contact for the most spread contact geometries: sphere-sphere, sphere-plane. Hertzian solution is basic here. Shown are main assumptions and main formulas (Slide 17). Discussed is character of stress distribution around the contact area (Slide 18) . including case of sliding (Slide 19). Considered is problem definition with account of effect of surface forces in models of JKR and DMT (Slide 20). Shown are solutions of the problems in comparison with Hertzian solution. Effect of surface forces is estimated. Discussed is criterium of application of either JKR or DMT theories (Slide 21). Compared are main parameters of adhesive contact in different theories. Slide 22 gives description of advantages and limitations of each of them (Slide 22). Considered is problem of adhesive contact at viscoelastic deformation of material (accounting of viscoelasticity according to Maxwell model) (Slide 23). Discussed are dependence of contact area on time for various adhesion force and time of preloading (Slide 24). Discussed is problem of effect of capillary forces. Described is an approach to the problem definition and solution (Slides 25, 26). Shown are peculiarities of contact formation at presence of coating and its dependence on the material elastic properties and load (Slide 27). The solution can be applied for definition of mechanical properties of thin coatings by indentation technique (Slide 28). Slide 29 shows characteristics of stress distribution within contact area at the layer deformation. Discussed is a possibility of contact problem solution on example of knee joint using finite element method (Slide 30). Shown are possibilities of approximate solution of contact problem at modelling of a deformed layer by Winkler layer (Slide 31). Adequacy of the received solution is shown on the example of adhesive contact (Slide 32). Conducted is quantitative comparison of the solutions with that ones received at more precise problem definition. They demonstrated efficiency of Winkler model application (Slide 33). Discussed is approach to contact modeling on nanometer level for material composed of atoms or nanometer-sized particles. Molecular dynamics approach was used here. General principles of the approach are presented (Slides 34 – 36).