Chapter 4 Response of Materials to
Surface Traction
1
Response of Materials
1. Deformation of the surface and subsurface
2. Fracture of Solids
2
Point Force Applied to a Semi-Infinite Elastic Solid
m
r D
R
z
F
φ
θ
σθθ σrr
σrz σzz
Point force P is applied at the origin of the cylindrical coordinate system(r, θ, z). 3
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
R 2 3P cos2φ 3P 1σ =σ zz +σ2 rz =
3P z22 2 = = 2 22π (r2 + z ) 2π (r2 + z ) 2πD2
where D is the diameter of the sphere passing throu
5
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.
7
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.
8
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.
9
z1
z2
R1
R2
y
r
x
Hertzian Contact
Figure 4.5 Two spherical bodies in contact. At zero load the contact occurs at a point x = y = z1 = z2 = 0. 10
Hertzian Stress due to Spherical Indenters(Normal Stress Distribution)
11
Hertzian Stress due to Spherical Indenters(Normal Stress Distribution)
12
r max
/a K=0.4
1.0
Location of the Max. Radial Stress
1.8
Complet
e slip th
eory
No-
slip
theo
ry
0.2 0.4 Coefficient of friction, µ
1.6
1.4
1.2 0.2
0.1
Figure 4.6 Position of the maximum radial tensile stress13
0.3
Stress Field due to Line Load
F
P
Q
z
x
θ
Figure 4.7 Semi-infinite elastic solid loaded by a concentrated load
14
Stress Field due to Line Load
15
Semi-infinite elastic solid loaded by a elliptical distributed load
p0
q0-a 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.
16
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
18
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
20
The maximum normal and tangential stresses are p0 and q0, respectively.
Semi-infinite elastic solid loaded by a elliptical distributed load
21
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].
22
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].
26
Sequence of Crack Formation and Growth Events during Loading and Unloading
Photos removed for copyright reasons. See Figure 4.20 in [Suh 1986].
27
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