1. Use the field-point integration method of Sect.

2.3.1 to determine the tangential surface displacements

\(u_x(r), u_y(r)\) due to the traction distribution

$$ q_x(r)=\frac{2f{{E}^{*}}\sqrt{a^2-r^2}}{\pi R}\;\;\;\;\;0\le r<a $$

in the region

\(r\!>\!a\) ,

outside the loaded circle.

Hence find the direction of the slip displacements in

\(b\!<\!r\!<\!a\) due to the Cattaneo–Mindlin distribution

$$ q_x(r)=\frac{2f{{E}^{*}}}{\pi R}\left( \sqrt{a^2-r^2}-\sqrt{b^2-r^2}\right) , $$

where the square roots are to be interpreted as zero in regions where their respective arguments are negative. Find the angle between the resultant slip displacement and the local tangential traction as a function of

r and comment on the results.

2. A rigid punch in the form of a half-cylinder of radius

R is pressed into an elastic half-plane such that the plane side of the punch remains vertical, as shown in Fig.

9.7 . Find the relationship between the indenting force

P and the width

a of the contact area.

Fig. 9.7 Indentation by a semi-cylindrical punch

A tangential force \(Q_x\) is now applied to the punch, with P remaining constant. Use the Ciavarella–Jäger theorem to find the extent of the slip zone as a function of \(Q_x\) .

3. A rigid cylinder of radius R is pressed into an elastic half-plane by a normal force P leading to the contact pressure distribution of Eq. ( 6.25 ). This force is then held constant whilst a tangential force \(Q_x\) oscillates between zero and fP / 2, where f is the coefficient of friction. Find the maximum extent of the slip zones in the steady state.

4. The flat and rounded punch of Fig.

9.8 is loaded by a constant normal force

\(P_0\) and a periodic tangential force

\(Q_x\) which alternates between

\(\pm Q_0\) , where

\(Q_0\!<\!fP_0\) . Use the Ciavarella–Jäger theorem to show that the flat region

\(-b\!<\!x\!<\!b\) never slips.

Fig. 9.8 A flat and rounded punch subjected to periodic tangential forces

Microslip in the region \(b\!<\!|x|\!<\!a\) can be expected to cause wear which will modify the profile of the punch and hence the contact pressure distribution. Show that whatever the form of the worn profile, the flat region will never slip and hence will also never wear. Discuss the implications for fretting fatigue problems.

5. Use the field-point integration method of Sect. 2.3.1 to determine the tangential displacement in the stick zone due to the traction distribution (9.4 ). Notice that the origin lies in the stick zone, so it is sufficient to set \(x\!=\!y\!=\!0\) in \(C_0(\theta ), C_1(\theta )\) of Eq. ( 2.26 ), which considerably simplifies the integration.

Use this result to establish the relation between the tangential force \(Q_x\) and the tangential surface displacement \(u_x\) during periodic loading, and hence determine the energy loss in friction per cycle as a function of the amplitude \(Q_0\) in periodic loading between \(Q_0\) and \(-Q_0\) .

6. A cylinder of radius R is pressed into an elastic half-plane by a normal force P and a tangential force \(Q_x\) that vary with time according to Eq. (9.23 ) with \(Q_0\!=\!0\) and \(P_1\!=\!Q_1\!=\!0.2P_0\) . If the coefficient of friction is 0.4, find the extent of the permanent stick zone.

7. A symmetrical elastic body is pressed into an elastic half-plane by a normal force

P , whilst the half-plane is subjected to a bulk tension

\(\sigma \) . Use Eq. (

9.31 ) as the basis for an incremental formulation analogous to that in Sect.

6.3.1 to show that the entire instantaneous contact area will stick as long as

$$ \frac{d P}{d t}>0 \quad \text{ and } \quad \left| \frac{d \sigma }{d P}\right| <\frac{4f}{\pi a}, $$

where

t is time and

a is the semi-width of the instantaneous contact area.

8. An uncoupled two-dimensional Hertzian contact is loaded initially by a purely normal force

\(P_0\) , but is then subjected to a combination of normal loading and bulk stress increasing in time

t according to the equations

$$ P=P_0+\frac{P_1t}{t_0};\;\;\;\sigma =\frac{\sigma _1 t}{t_0}\;\;\;\;\;\;\;\;\;0<t<t_0, $$

where

\(P_1\!>\!0\) , but the ratio

\(\sigma _1/P_1\) is sufficiently large to ensure that full stick is impossible throughout this second phase.

Write down the boundary conditions defining the difference between the traction and displacement states at \(t\!=\!0\) and \(t\!=\!t_0\) and show that this differential problem has a similar form to that in Sect. 9.4.1 . Hence obtain an equation whose solution defines the extent of the stick zone at \(t\!=\!t_0\) .

9. The inclined rigid punch of Fig. 6.3 is loaded by a normal force \(P_0\) which is then held constant whilst bulk stresses \(\sigma _1,\sigma _2\) and a tangential force \(Q_0\) are applied in proportion. If the problem is uncoupled and \(P_0\) is insufficient to ensure full contact, find the extent of the stick zone under the assumption of ‘moderate bulk stress’ and hence determine the condition which must be satisfied for this assumption to be appropriate.

10. An uncoupled two-dimensional Hertzian contact is loaded by a normal force and bulk tension [in body 1 only] that increase in proportion up to their maximum values \(P_0,\sigma _0\) . Assume that \(\sigma _0\) is sufficiently small to ensure that this process involves no slip. Use an incremental formulation to find the resulting distribution of tangential traction.

These loads are then maintained constant whilst an increasing tangential force is applied, with maximum value \(Q_0\) . Describe in words, what slip zones you expect to see developed and set up the equations which must be satisfied by the corrective solution — i.e. the difference between the final state and that before \(Q_0\) was applied.

11. Use the Goodman approximation to formulate the problem of a two-dimensional flat rigid punch loaded by normal and tangential forces that increase in proportion — i.e.

\(P(t)\!=\!C_Nt\) ,

\(Q_x(t)\!=\!C_Tt\) , where

t is time and

\(C_T\!<\!fC_N\) . Find two equations for the coordinates

b ,

c defining the extent of the stick region

\(-b\!<\!x\!<\!c\) from (i) the consistency condition in the equation analogous to (

9.58 ) and (ii) the equilibrium condition

$$ Q_x=\int _{-a}^a q_x(x)dx. $$

Do not attempt to solve these equations.

12. Use the potential function representation of Appendix A, Sects. A.1 and A.2 to formulate the coupled problem of a rigid cylindrical flat punch of radius a pressed into an elastic half-space by a force P , assuming a finite coefficient of friction f . Satisfy the boundary conditions outside the contact area identically by choosing appropriate functions related to that in Eq. ( 5.4 ) to represent the potential functions. Then use the remaining boundary conditions to set up integral equations for the unknown functions [e.g. h (t ) in Eq. ( 5.4 )]. Do not attempt to solve these equations.

13. The flat punch of Sect. 9.5.1 is first loaded by a normal force P , after which a compressive bulk stress \(\sigma _1\!=\!-s\) is applied to the half-plane. Set up the Cauchy integral equations defining the resulting traction distributions and then use the Goodman approximation to estimate the extent of the slip zones, if the coefficient of friction is f .

What do you think would happen if instead a tensile bulk stress were applied?