|
Preface |
6 |
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Contents |
8 |
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1 Kinematics of Contact |
19 |
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1.1 Reference Frame and the Initial Gap Function |
20 |
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1.2 Establishment of a Contact Region |
21 |
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1.2.1 Definition of Contact |
22 |
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1.2.2 The Boundary Value Problem |
22 |
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1.2.3 Signorini Problems |
23 |
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1.2.4 Asymptotic Arguments |
23 |
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1.2.5 The Discrete Problem |
25 |
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1.3 Nonlinear Kinematics |
26 |
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1.4 Almost Conformal Contact |
27 |
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|
2 Three-Dimensional Frictionless Elastic Problems |
30 |
|
|
2.1 The Half-Space Approximation |
30 |
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2.2 Normal Loading of the Half-Space |
31 |
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2.2.1 The Point Force Solution |
32 |
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2.2.2 Similarity, Equilibrium and Anisotropy |
33 |
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2.2.3 The Composite Elastic Modulus |
34 |
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2.3 Integral Equation Formulation |
35 |
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2.3.1 Field-Point Integration |
37 |
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2.3.2 Indentation by a Flat Elliptical Punch |
37 |
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2.4 Galin's Theorem |
40 |
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2.4.1 A Special Case |
41 |
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2.5 Interior Stress Fields |
42 |
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2.5.1 In-Plane Stress Components Near the Surface |
42 |
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3 Hertzian Contact |
45 |
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3.1 Transformation of Coordinates |
45 |
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3.1.1 Cylinders and Spheres |
47 |
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3.1.2 More General Cases |
48 |
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3.2 Hertzian Pressure Distribution |
49 |
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3.3 Strategy for Hertzian Contact Calculations |
50 |
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3.3.1 Eccentricity of the Contact Area |
50 |
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3.3.2 Dimensions of the Contact Area |
51 |
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3.3.3 Highly Elliptical Contacts |
54 |
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3.4 First Yield |
55 |
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4 More General Problems for the Half-Space |
58 |
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4.1 The Electrical--Mechanical Analogy |
59 |
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4.1.1 Other Mathematical Analogies |
61 |
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4.1.2 Boyer's Approximation |
63 |
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4.1.3 Fabrikant's Approximation |
64 |
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4.2 General Theorems for Frictionless Contact |
66 |
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4.3 Superposition by Differentiation |
70 |
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4.4 The Force--Displacement Relation |
72 |
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4.4.1 Non-conformal Contact Problems |
73 |
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5 Axisymmetric Contact Problems |
77 |
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5.1 Green and Collins Solution |
77 |
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5.1.1 The Flat Punch Solution |
79 |
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5.2 Non-conformal Contact Problems |
80 |
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|
5.3 Annular Contact Regions |
82 |
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5.4 The Non-axisymmetric Cylindrical Punch |
83 |
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5.5 The Method of Dimensionality Reduction (MDR) |
84 |
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6 Two-Dimensional Frictionless Contact Problems |
90 |
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6.1 The Line Force Solution |
91 |
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6.2 Integral Equation Formulation |
93 |
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6.2.1 Edge Conditions |
94 |
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6.3 Incremental Solution of Non-conformal Contact Problems |
98 |
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6.3.1 Symmetric Problems |
98 |
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6.3.2 Bounded-Singular Problems |
99 |
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6.4 Solution by Fourier Series |
99 |
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6.4.1 Rigid-Body Rotation |
100 |
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6.4.2 Galin's Theorem, Chebyshev Polynomials and Recurrence Relations |
102 |
|
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6.5 Periodic Contact Problems |
104 |
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6.5.1 Sinusoidal Contact Pressure |
104 |
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6.5.2 Fourier Series Methods |
105 |
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6.5.3 The Periodic Green's Function |
106 |
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6.5.4 The Cotangent Transform |
106 |
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6.5.5 Manners' Solution |
107 |
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6.5.6 Westergaard's Problem |
109 |
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6.6 The Smirnov--Sobolev Transform |
110 |
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6.6.1 Inversion of the Transform |
111 |
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6.6.2 Example: Uniform Loading Over the Circle |
111 |
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6.6.3 Anisotropic Problems |
112 |
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6.7 Displacements in Two-Dimensional Problems |
113 |
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6.7.1 Kalker's Line Contact Theory |
115 |
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7 Tangential Loading |
121 |
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7.1 Kinematics |
121 |
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7.1.1 Gross Slip and Microslip |
122 |
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7.2 Green's Functions for Tangential Forces and Displacements |
123 |
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7.2.1 Three-Dimensional [point] Loading |
123 |
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7.2.2 Two-Dimensional [line] Loading |
125 |
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7.2.3 Normal-Tangential Coupling |
126 |
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7.3 Two-Dimensional Flat Rigid Punch with No Slip |
127 |
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7.3.1 Uncoupled Problem |
129 |
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7.3.2 Oscillatory Singularities |
129 |
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7.4 Axisymmetric Flat Rigid Punch with No Slip |
131 |
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7.5 The `Goodman' Approximation |
133 |
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7.6 Uniform Tangential Displacement in a Prescribed Area |
135 |
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7.6.1 Tangential Loading over a Circular Area |
135 |
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7.6.2 Tangential Loading over an Elliptical Area |
136 |
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7.6.3 Two Conjectures |
138 |
|
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7.7 Non-conformal Contact Problems with No Slip |
139 |
|
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7.7.1 Uncoupled Hertzian Contact with Tangential Loading |
140 |
|
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7.7.2 The Coupled Axisymmetric Problem under Purely Normal Loading |
141 |
|
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7.7.3 The Coupled Two-Dimensional Problem |
142 |
|
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7.7.4 Relaxation Damping |
144 |
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8 Friction Laws |
149 |
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8.1 Amontons' Law |
149 |
|
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8.1.1 Continuum Problems |
150 |
|
|
8.1.2 Two-Dimensional Problems |
151 |
|
|
8.1.3 Existence and Uniqueness Theorems |
151 |
|
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8.2 The Klarbring Model |
152 |
|
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8.2.1 General Loading Scenarios |
154 |
|
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8.2.2 The Critical Coefficient of Friction |
154 |
|
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8.2.3 Wedging |
155 |
|
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8.3 Multinode Systems |
156 |
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8.3.1 The Evolution and Rate Problems |
157 |
|
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8.3.2 Algorithms for Two-Dimensional Problems with Time-Varying Forces |
157 |
|
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8.3.3 History-Dependence and Memory |
158 |
|
|
8.3.4 Klarbring's P-Matrix Criterion |
159 |
|
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8.4 Periodic Loading |
160 |
|
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8.4.1 A Uniqueness Proof for Uncoupled Systems |
161 |
|
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8.4.2 Shakedown |
163 |
|
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8.4.3 Coupled Systems |
163 |
|
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8.4.4 Asymptotic Approach to a Steady State |
163 |
|
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8.5 A Simple Continuum Frictional System |
164 |
|
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8.5.1 Unloading |
167 |
|
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8.5.2 Periodic Loading |
168 |
|
|
8.5.3 Discrete Model of the Strip Problem |
169 |
|
|
8.5.4 The Inverse Problem |
169 |
|
|
8.6 More Complex Friction Laws |
170 |
|
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8.6.1 Instabilities During Steady Sliding |
171 |
|
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8.6.2 Velocity-Dependent Friction Coefficient |
171 |
|
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8.6.3 Stick-Slip Vibrations |
173 |
|
|
8.6.4 Slip-Weakening Laws |
174 |
|
|
8.6.5 Rate-State Laws |
175 |
|
|
9 Frictional Problems Involving Half-Spaces |
181 |
|
|
9.1 Cattaneo's Problem |
181 |
|
|
9.2 The Ciavarella--Jäger Theorem |
184 |
|
|
9.2.1 Three-Dimensional Problems |
186 |
|
|
9.3 More General Loading Scenarios |
187 |
|
|
9.3.1 Constant Normal Force |
187 |
|
|
9.3.2 Variable Normal Force |
188 |
|
|
9.3.3 Memory and `Advancing Stick' |
190 |
|
|
9.4 The Effect of Bulk Stress |
191 |
|
|
9.4.1 Hertz Problem with Superposed Bulk Stress |
191 |
|
|
9.4.2 Combined Bulk Stress and Tangential Force |
193 |
|
|
9.5 Coupled Problems |
196 |
|
|
9.5.1 Indentation by a Two-Dimensional Flat Rigid Punch |
196 |
|
|
9.5.2 Normal Loading for More General Geometries |
199 |
|
|
9.5.3 Combined Normal and Tangential Loading |
201 |
|
|
9.5.4 Unloading |
201 |
|
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9.5.5 Periodic Loading |
202 |
|
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10 Asymptotic Methods |
206 |
|
|
10.1 Indentation by a Frictionless Rigid Punch |
206 |
|
|
10.1.1 Eigenfunction Series |
208 |
|
|
10.1.2 More General Frictionless Indentation Problems |
209 |
|
|
10.1.3 Non-conformal Problems |
210 |
|
|
10.1.4 Both Materials Deformable |
211 |
|
|
10.2 No-Slip Conditions |
212 |
|
|
10.3 Frictional Slip |
213 |
|
|
10.3.1 Slip-Separation Transition |
214 |
|
|
10.3.2 Slip--Stick Transition |
215 |
|
|
10.4 Indentation by an Elastic Wedge |
216 |
|
|
10.4.1 Right-Angle Wedge of the Same Material |
217 |
|
|
10.4.2 A Slipping Interface |
218 |
|
|
10.5 Local Fields |
219 |
|
|
10.5.1 The Flat and Rounded Indenter |
220 |
|
|
10.5.2 Fretting in Non-conformal Contact |
222 |
|
|
10.5.3 Edge Slip Zones with a Rigid Punch |
223 |
|
|
10.5.4 Slip Zones in Conformal Contact |
225 |
|
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11 Receding Contact |
232 |
|
|
11.1 Characteristics of Receding Contact |
233 |
|
|
11.1.1 Examples of Receding Contact |
234 |
|
|
11.2 Frictional Problems |
237 |
|
|
11.2.1 Frictional Unloading |
237 |
|
|
11.3 Thermoelastic Problems |
239 |
|
|
11.4 Almost Conformal Contact Problems |
240 |
|
|
12 Adhesive Forces |
244 |
|
|
12.1 Adhesion Between Rigid Bodies |
247 |
|
|
12.2 The JKR Theory |
248 |
|
|
12.2.1 Axisymmetric Problems |
249 |
|
|
12.2.2 Indentation by a Sphere |
250 |
|
|
12.2.3 Energetic Considerations and Stability |
252 |
|
|
12.2.4 Hysteretic Energy Dissipation |
254 |
|
|
12.2.5 JKR Solution for More General Axisymmetric Bodies |
254 |
|
|
12.2.6 Guduru's Problem |
256 |
|
|
12.3 The Tabor Parameter |
257 |
|
|
12.3.1 An Adhesive Length Scale |
259 |
|
|
12.3.2 Limitations on the JKR Solution |
260 |
|
|
12.4 Solutions for Finite Tabor Parameter |
261 |
|
|
12.4.1 Jump-In at Large Tabor Parameter |
262 |
|
|
12.4.2 Simplified Force Laws |
263 |
|
|
12.4.3 Maugis' Solution |
264 |
|
|
12.4.4 The `double-Hertz' Approximation |
267 |
|
|
12.4.5 More General Axisymmetric Geometries |
269 |
|
|
12.5 Other Geometries |
269 |
|
|
12.5.1 Two-Dimensional Problems |
269 |
|
|
12.5.2 Elliptical Contact Area |
270 |
|
|
12.5.3 General Three-Dimensional Geometries |
271 |
|
|
13 Beams, Plates, Membranes and Shells |
274 |
|
|
13.1 Contact of Beams |
274 |
|
|
13.1.1 A Heavy Beam Lifted from the Ground |
276 |
|
|
13.1.2 Adhesive Forces |
277 |
|
|
13.1.3 Piston Ring in a Cylinder |
278 |
|
|
13.1.4 Two and Three-Dimensional Effects |
281 |
|
|
13.1.5 Matched Asymptotic Expansions |
282 |
|
|
13.2 Contact of Plates |
285 |
|
|
13.2.1 Displacement Due to a Concentrated Point Force |
286 |
|
|
13.2.2 Indentation by a Rigid Sphere |
286 |
|
|
13.3 Membrane Effects |
288 |
|
|
13.3.1 `Membrane Only' Solutions |
289 |
|
|
13.4 Contact of Shells |
292 |
|
|
13.5 Implications for Finite Element Solutions |
296 |
|
|
14 Layered Bodies |
300 |
|
|
14.1 Esll El: Plate on an Elastic Foundation |
301 |
|
|
14.1.1 Choice of Foundation Modulus |
302 |
|
|
14.1.2 Two-Dimensional Problems |
302 |
|
|
14.1.3 Three-Dimensional Problems |
305 |
|
|
14.2 Esgg El: Layer on a Rigid Foundation |
306 |
|
|
14.2.1 Frictionless Unbonded Layer |
307 |
|
|
14.2.2 Bonded Compressible Layer |
309 |
|
|
14.2.3 Bonded Incompressible Layer |
309 |
|
|
14.2.4 Flat Punch Problems |
314 |
|
|
14.2.5 Frictional Problems |
315 |
|
|
14.2.6 Effect of Adhesive Forces |
315 |
|
|
14.3 Winkler Layer on an Elastic Foundation |
318 |
|
|
14.3.1 Nonlinear Layers |
319 |
|
|
14.4 Fourier Transform Methods |
320 |
|
|
14.4.1 Elastic Layer Bonded to a Rigid Foundation |
320 |
|
|
14.4.2 Multilayered Bodies |
324 |
|
|
14.5 Functionally Graded Materials |
324 |
|
|
14.5.1 Exponential Variation of Modulus |
325 |
|
|
14.5.2 Power-Law Grading |
326 |
|
|
14.5.3 Linear Variation of Modulus |
329 |
|
|
15 Indentation Problems |
333 |
|
|
15.1 The Hardness Test |
333 |
|
|
15.2 Power-Law Material |
334 |
|
|
15.2.1 Graded Materials |
336 |
|
|
15.3 Other Constitutive Laws |
337 |
|
|
16 Contact of Rough Surfaces |
339 |
|
|
16.1 Bowden and Tabor's Theory of Friction |
339 |
|
|
16.1.1 The Ploughing Force |
340 |
|
|
16.1.2 Plastic Deformation at an Actual Contact |
341 |
|
|
16.1.3 The Effect of Surface Films |
342 |
|
|
16.2 Profilometry |
343 |
|
|
16.2.1 The Bearing Area Curve |
344 |
|
|
16.2.2 The Contact Problem |
346 |
|
|
16.3 Asperity Model Theories |
347 |
|
|
16.3.1 The Exponential Distribution |
349 |
|
|
16.3.2 The Gaussian Distribution |
350 |
|
|
16.3.3 The Plasticity Index |
352 |
|
|
16.4 Statistical Models of Surfaces |
353 |
|
|
16.4.1 Discrete Models |
353 |
|
|
16.4.2 Random Process Models |
355 |
|
|
16.4.3 Determining Asperity Parameters |
361 |
|
|
16.5 Fractal Surfaces |
362 |
|
|
16.5.1 Archard's Model |
362 |
|
|
16.5.2 Self-affine Fractals and the Fractal Dimension |
362 |
|
|
16.5.3 The Weierstrass Function |
364 |
|
|
16.5.4 Generating Realizations of Fractal Profiles and Surfaces |
366 |
|
|
16.6 Contact of Fractal Surfaces |
369 |
|
|
16.6.1 Majumdar and Bhushan's Theory |
369 |
|
|
16.6.2 Elastic Contact for a Fractal Surface |
370 |
|
|
16.6.3 The Weierstrass Profile |
372 |
|
|
16.6.4 Persson's Theory |
374 |
|
|
16.6.5 Implications for Coulomb's Law of Friction |
378 |
|
|
16.7 Adhesive Forces |
379 |
|
|
16.7.1 Asperity Model Predictions |
380 |
|
|
16.7.2 The Sinusoidal Profile |
381 |
|
|
16.7.3 Adhesion of Random Rough Surfaces |
384 |
|
|
16.8 Incremental Stiffness and Contact Resistance |
385 |
|
|
16.8.1 Asperity Model Predictions |
386 |
|
|
16.8.2 Clustering of Actual Contacts |
387 |
|
|
16.8.3 Bounds on Incremental Stiffness |
388 |
|
|
16.8.4 Persson's Theory of Incremental Stiffness |
390 |
|
|
16.8.5 Gaps and Fluid Leakage |
391 |
|
|
16.9 Finite-Size Effects |
392 |
|
|
16.9.1 Integral Equation Formulation |
393 |
|
|
16.9.2 Unit Cells and the Constriction Alleviation Factor |
396 |
|
|
16.9.3 Contact of Rough Spheres |
397 |
|
|
17 Thermoelastic Contact |
405 |
|
|
17.1 Thermoelastic Deformation |
406 |
|
|
17.1.1 Fourier Transform Solutions |
406 |
|
|
17.1.2 Steady-State Temperature |
407 |
|
|
17.1.3 Thermoelastic Distortion Due to a Point Heat Source |
408 |
|
|
17.1.4 Dundurs' Theorem |
409 |
|
|
17.1.5 Moving Heat Sources |
410 |
|
|
17.2 The Axisymmetric Thermoelastic Hertz Problem |
411 |
|
|
17.2.1 The Heat Conduction Problem |
412 |
|
|
17.2.2 Thermoelastic Distortion |
413 |
|
|
17.2.3 Solution of the Contact Problem |
413 |
|
|
17.3 Existence and Uniqueness |
415 |
|
|
17.3.1 A One-Dimensional Model |
416 |
|
|
17.3.2 Effect of a Thermal Interface Resistance |
417 |
|
|
17.3.3 Imperfect Thermal Contact |
419 |
|
|
17.3.4 The Hertz Problem Revisited |
420 |
|
|
17.3.5 Stability |
420 |
|
|
17.3.6 Contact of Dissimilar Materials |
423 |
|
|
17.3.7 Two-Dimensional Stability Problems |
423 |
|
|
17.4 Solidification Problems |
425 |
|
|
17.5 Frictional Heating |
427 |
|
|
17.5.1 The Rod Model |
429 |
|
|
17.5.2 Burton's Stability Analysis |
430 |
|
|
17.5.3 Out-of-Plane Sliding |
431 |
|
|
17.5.4 In-Plane Sliding |
433 |
|
|
17.5.5 Limiting Configurations |
435 |
|
|
17.5.6 Effect of Geometry |
437 |
|
|
17.5.7 Numerical Solutions |
439 |
|
|
18 Rolling and Sliding Contact |
443 |
|
|
18.1 Rigid-Body Kinematics |
443 |
|
|
18.1.1 Three-Dimensional Motions |
445 |
|
|
18.2 Johnson's Belt Drive Problem |
448 |
|
|
18.3 Tractive Rolling of Elastic Cylinders |
451 |
|
|
18.3.1 Dissimilar Materials |
455 |
|
|
18.3.2 Antiplane Loading |
456 |
|
|
18.3.3 Rolling of Misaligned Cylinders |
456 |
|
|
18.3.4 Three-Dimensional Rolling Contact Problems |
457 |
|
|
18.3.5 Kalker's Strip Theory |
458 |
|
|
18.3.6 The Incipient Sliding Solution |
460 |
|
|
18.3.7 Transient Problems |
460 |
|
|
18.3.8 Rail Corrugations |
461 |
|
|
18.4 Steady Sliding |
462 |
|
|
18.4.1 Two-Dimensional Problems |
462 |
|
|
18.4.2 Three-Dimensional Problems |
464 |
|
|
18.5 Wear |
465 |
|
|
18.5.1 Archard's Wear Law |
465 |
|
|
18.5.2 Long-Time Solution |
466 |
|
|
18.5.3 Transient Problems |
467 |
|
|
18.5.4 Galin's Eigenfunction Method |
469 |
|
|
18.5.5 Non-conformal Contact Problems |
471 |
|
|
18.6 Sliding of Rough Surfaces |
472 |
|
|
18.6.1 Flash Temperatures |
473 |
|
|
18.6.2 Bulk Temperatures |
478 |
|
|
18.6.3 Transient Asperity Interactions |
479 |
|
|
19 Elastodynamic Contact Problems |
484 |
|
|
19.1 Wave Speeds |
485 |
|
|
19.1.1 Rayleigh Waves |
486 |
|
|
19.2 Moving Contact Problems |
487 |
|
|
19.2.1 The Moving Line Force |
487 |
|
|
19.2.2 Integral Equation Formulation |
488 |
|
|
19.2.3 The Subsonic Problem |
489 |
|
|
19.2.4 The Speed Range cR |
490 |
|
|
19.2.5 The Solution of Slepyan and Brun |
491 |
|
|
19.2.6 The Transonic Solution c2 |
493 |
|
|
19.2.7 The Superseismic Solution V>c1 |
494 |
|
|
19.2.8 Three-Dimensional Problems |
496 |
|
|
19.3 Interaction of a Bulk Wave with an Interface |
499 |
|
|
19.3.1 SH-Waves Transmitted Across a Frictional Interface |
499 |
|
|
19.3.2 In-Plane Waves |
505 |
|
|
19.4 Interface Waves |
507 |
|
|
19.4.1 Slip Waves |
508 |
|
|
19.4.2 Slip Waves at a Sliding Interface |
509 |
|
|
19.4.3 Slip--Stick Waves |
510 |
|
|
19.5 Stability of Frictional Sliding |
512 |
|
|
19.6 Transient Elastodynamic Contact Problems |
513 |
|
|
19.6.1 Impulsive Line Force |
513 |
|
|
19.6.2 A Uniform Pressure Suddenly Applied |
513 |
|
|
19.6.3 Integral Equation Formulation of the Transient Contact Problem |
514 |
|
|
19.6.4 Normal Indentation by a Rigid Body |
515 |
|
|
19.6.5 Superseismic Indentation |
516 |
|
|
19.6.6 Self-Similar Indentation Problems |
517 |
|
|
19.6.7 Three-Dimensional Transient Problems |
518 |
|
|
20 Impact |
522 |
|
|
20.1 Hertz' Theory of Impact |
523 |
|
|
20.1.1 Duration of the Impact |
524 |
|
|
20.1.2 Homogeneous Sphere |
526 |
|
|
20.1.3 Range of Validity of the Theory |
526 |
|
|
20.1.4 The Superseismic Phase |
527 |
|
|
20.2 Impact of a Cylinder |
528 |
|
|
20.3 Oblique Impact |
530 |
|
|
20.3.1 The Equation of Motion |
531 |
|
|
20.3.2 The Tangential Contact Problem |
532 |
|
|
20.3.3 Complete Stick |
532 |
|
|
20.3.4 Gross Slip |
535 |
|
|
20.3.5 Partial Slip |
535 |
|
|
20.3.6 The Complete Trajectory |
536 |
|
|
20.3.7 Rebound Conditions |
537 |
|
|
20.4 One-Dimensional Bar Problems |
538 |
|
|
20.4.1 The Semi-infinite Bar |
539 |
|
|
20.4.2 The Infinite Bar |
540 |
|
|
20.4.3 Reflections |
541 |
|
|
20.4.4 The Impact Problem |
542 |
|
|
20.4.5 A Rigid Mass Impacting an Elastic Bar |
542 |
|
|
20.4.6 Frictional Problems |
545 |
|
|
20.4.7 Continuous Frictional Supports |
547 |
|
|
Appendix A Potential Function Solutions for Elasticity Problems |
551 |
|
|
A.1 Frictionless Problems |
551 |
|
|
A.2 Problems with Tangential Tractions |
552 |
|
|
A.3 Two-Dimensional Problems |
554 |
|
|
Appendix B Integrals over Elliptical Domains |
555 |
|
|
B.1 Mathematical Preliminaries |
556 |
|
|
B.1.1 The Singular Field n=0 |
557 |
|
|
B.1.2 The Hertzian Field n=1 |
557 |
|
|
B.2 Applications |
558 |
|
|
B.2.1 Normal Loading of an Isotropic Half-Space |
558 |
|
|
B.2.2 The Anisotropic Half-Space |
559 |
|
|
B.2.3 Tangential Loading of an Isotropic Half-Space |
559 |
|
|
B.3 Evaluation of Integrals |
561 |
|
|
Appendix C Cauchy Singular Integral Equations |
562 |
|
|
C.1 Integral Equations of the First Kind |
562 |
|
|
C.2 Integral Equations of the Second Kind |
564 |
|
|
Appendix D Dundurs' Bimaterial Constants |
566 |
|
|
References |
568 |
|
|
Index |
588 |
|