Basic Contact Mechanics Theories
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Transcript of Basic Contact Mechanics Theories
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).