Chapter 4 Response of Materials to Surface Traction
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Response of Materials
1. Deformation of the surface and
subsurface
2. Fracture of Solids
2
Point Force Applied to a Semi-Infinite Elastic Solid
F
θ
D
r
φ
R
m
z
σrr
σθθ
σ
zz
σrz
Point force P is applied at the origin of the
cylindrical coordinate system(r, θ, z).
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Point Force Applied to a Semi-Infinite Elastic Solid (Boussinesq solution)
4
Point Force Applied to a Semi-Infinite Elastic Solid -- Resultant stress on face m
3P z2
3P cos φ 3P 1
σ =σ +σ =
=
=
2
2 2
2
2 2
2
2π (r + z ) 2π (r + z ) 2π D
where
R
2
zz
2
rz
2
D is the diameter of the sphere passing throu
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Point Force Applied to a Semi-Infinite Elastic Solid
6
Trajectory of Principal stresses due to a Point Force
Diagram removed for copyright reasons.
See Figure 4.2 in [Suh 1986]: Suh, N. P. Tribophysics. Englewood Cliffs NJ: Prentice-Hall, 1986. ISBN: 0139309837.
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Contour of Principal Normal Stresses (ν=0.25)
Diagram removed for copyright reasons.
See Figure 4.3 in [Suh 1986]: Suh, N. P. Tribophysics. Englewood Cliffs NJ: Prentice-Hall, 1986. ISBN: 0139309837.
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Angular Variation of Principal Stress Components in Boussinesq Field (ν=0.25)
Diagram removed for copyright reasons.
See Figure 4.4 in [Suh 1986]: Suh, N. P. Tribophysics. Englewood Cliffs NJ: Prentice-Hall, 1986. ISBN: 0139309837.
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Hertzian Contact
z1
y
R1
r
x
R2
z2
Figure 4.5 Two spherical bodies in contact. At zero load the contact occurs at a point x = y = z1 = z2 = 0.
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Hertzian Stress due to Spherical Indenters
(Normal Stress Distribution) 11
Hertzian Stress due to Spherical Indenters
(Normal Stress Distribution) 12
Location of the Max. Radial Stress
1.8
1.4
e
pl
m
Co
te s
l
ip t
y
or
e
h
K=0.4
No-slip theory
rmax/a
1.6
1.2
0.3
0.2
0.1
1.0
0.2
0.4
Coefficient of friction, µ
13
Figure 4.6 Position of the maximum radial tensile stress
Stress Field due to Line Load
F
P
Q
x
θ
z
Figure 4.7 Semi-infinite elastic solid loaded by a concentrated
load
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Stress Field due to Line Load
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Semi-infinite elastic solid loaded by a elliptical distributed load
p0
-a
q0
a
ξ ||
dξ
Figure 4.8 Semi-infinite elastic solid loaded by a elliptical
distributed load. The maximum normal and tangential stresses
are p0 and q0, respectively.
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Semi-infinite elastic solid loaded by a elliptical distributed load
17
The maximum normal and tangential stresses are p0 and q0, respectively.
Semi-infinite elastic solid loaded by a elliptical distributed load
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The maximum normal and tangential stresses are p0 and q0, respectively.
Semi-infinite elastic solid loaded by a elliptical distributed load
19
The maximum normal and tangential stresses are p0 and q0, respectively.
Semi-infinite elastic solid loaded by a elliptical distributed load
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The maximum normal and tangential stresses are p0 and q0, respectively.
Semi-infinite elastic solid loaded by a elliptical distributed load
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The maximum normal and tangential stresses are p0 and q0, respectively.
Contour and Variation of Stress
Graphs removed for copyright reasons.
See Figures 4.9 through 4.15 in [Suh 1986].
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Crack Growth of Hertzian Crack
(soda-lime glass in air)
(a)
(b)
(c)
(d)
Figure 4.16
23
Surface Traces of Hertzian Cracks on Surface of Silicon
Photos removed for copyright reasons.
See Figure 4.18 in [Suh 1986].
24
Crack Patterns on Sod-lime Glass Produced by Sliding Tungsten Carbide Sphere
Photos removed for copyright reasons.
See Figure 4.17 in [Suh 1986].
25
“Star” Crack in Soda-Lime Glass produced by Conical Indenter
Photos removed for copyright reasons.
See Figure 4.19 in [Suh 1986].
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Sequence of Crack Formation and Growth Events during Loading and Unloading
Photos removed for copyright reasons.
See Figure 4.20 in [Suh 1986].
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