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