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IMAC-U course 6th semester
Tribology 3. Contact of solid bodies
Assoc. Prof. Takeshi YAMAGUCHI [email protected]
Plane contact Line contact Point contact
Contact types
Elastic contact (Hertzian contact) Plastic contact
Hertzian contact problems
Line contact Point contact
The following assumptions are made in determining the solutions of Hertzian contact problems: 1. ( ). 2. The surfaces are continuous and non-conforming
(implying that the area of contact is much smaller than the characteristic dimensions of the contacting bodies).
3. Each body can be considered an elastic half-space. 4. The surfaces are frictionless.
Contact between two cylinders
Contact area
Contact pressure distribution
p(x,y) =pmax 1−x
a
2−
y
a
2
pmax=3W
2πa2=
3
2pmean
1E′
=12
1−ν12
E1+
1−ν22
E2
(3.1)
(3.2)
(3.3)
(3.4)
(3.5)
Contact pressure distribution
Radius of contact circle
Hertzian maximum contact pressure
Reduced Young’s modulus
Reduced radius of curvature
1R=
1R1
+1R2
E1, n1
E2, n2
Contact between two spheres Contact area
Contact pressure distribution
p=pmax 1−x
b
2
pmax=2
π
W
bL=
4
πpmean
1E′
=12
1−ν12
E1+
1−ν22
E2
(3.6)
(3.7)
(3.8)
(3.9)
(3.10)
Contact pressure distribution
Half width of contact rectangle
Hertzian maximum contact pressure
Reduced Young’s modulus
Reduced radius of curvature
1R=
1R1
+1R2
E1, n1
E2, n2
Plastic deformation due to indentation of a sphere
x
z Starting point of plastic deformation
a -a
W1
0.47a
W3
Plastic region
Elastic region
W2
Plastic region
Elastic region
Elastic region
( ) ( ) ( )
pmean < 1.1 s0 1.1 s0<pmean<3s0 pmean≒3s0
(H) Yield strength
Indentation hardness
Ball-flat contact with normal load and tangential force
tmax contour
T = 0.5mP
J. ホーリング編, トライボロジ，近代科学社（１９８４）
7
1. Basic concept of hardness
(1) Touch with a finger Lower contact pressure means softer.
e.g.) Rubber is soft and iron is hard.
(2) Rubbing each other Scratched surface is softer.
e.g.) Glasses can scratch steels.
2. Types of hardness test
(1) ( ) ⇒use hard indenter instead of your finger
(2) Dynamic hardness testing
(3) Scratch hardness testing
Hardness?
Indentation hardness testing methods
HV = 2 W sin (q/2)
=
L2
Vickers hardness
q
Diamond
pyramid indenter
W
L
Hardness value ( ) means contact pressure necessary for a plastic deformation of surface (plastic
contact).
Vickers hardness of engineering materials
10000
5000 4000 3000
2000
1000
500 400 300
200
100
50 40 30
20
10
5 4 3
2
1
Diamond
Cubic boron nitride (cBN) Boron nitride (BN) Titanium carbide (TiC)
Vic
kers
har
dn
ess
, kgf
/mm
2
Tungsten carbide (WC-Co) Hard chromium plate Martensite
Cementite
Fully hardened tool steel Glass
Titanium Gray iron Annealed mild steel Brass
Nickel Pure iron
Zinc, gold, magnesium Silver Polystyrene Pure aluminum
Vinyl resin Graphite
Tin Lead
Contact between rough surfaces
Normal load
Aa = a×b Apparent contact area
Apparent contact area
Real contact area is ( )
of apparent contact area.
Ar= 𝑨𝒓𝒊𝒊=𝒏𝒊=𝟏
(2.19)
(2.20)
Real contact area
Real contact area
Ar= Ai
n
i=1
= Wi
p0
n
i=1
=W
p0 (3.11)
Plastic flow pressure
Real contact area is proportional to normal load and inverse proportional to plastic flow pressure (≒3s0 or H) .
Plasticity index Y
Ψ =E′2H
σβ
Ψ >
Ψ =
Ψ < Elastic contact
Plastic contact
Plastic and elastic contact
Because of the height distribution of surface asperities, not all asperities contact with plastic deformation. The plasticity index express average contact state of asperities (elastic, elastic and plastic , and plastic)
E’: reduced Young’s modulus s: reduced roughness ([(Rq1)2+(Rq2)2]1/2)
b: radius of surface asperity tip H: hardness
Plastic? Elastic?
(3.12)
Contact stiffness
W
W
W
W
Deflection Sd
No
rma
l lo
ad
P
Contact stiffness
Asperities on the surfaces decrease stiffness when contact with each other.
Stiffness decreases as the number of contact area increases.
Contact stiffness
Asperities on the surfaces
Without asperities on the surfaces
大西清著、JISにもとづく標準製図法、第13全訂版、理工学社（2010）
Indication of surface roughness and chamfer
Contact between elastic flat surface and rigid body without chamfer
Stress concentration
W Contact geometry Contact pressure distribution
Necessity of chamfer
W
R R
Contact between elastic flat surface and rigid body with chamfer