1. The profiles of two half-planes are defined by the symmetric gap function \(g_0(x)\!=\!Cx^4\) , where C is a constant. Use any of the various methods in this chapter to determine the contact pressure distribution p (x ) and the applied force P as functions of the semi-width of the contact area a .

2. A rigid punch has the wedge-shaped profile shown in Fig.

6.12 , where the angle

\(\alpha \!\ll \!1\) . Find the contact semi-width

a and the contact pressure

p (

x ) as functions of the applied force

P . Discuss the nature of the pressure distribution near

\(x\!=\!0\) and

\(x\!=\!a\) .

Fig. 6.12 The wedge-shaped indenter

3. A rigid punch has the form of a truncated wedge as shown in Fig. 6.13 . Find the relation between the applied normal force P and the half-width of the contact area a .

4. 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.

6.14 . Find the relationship between the indenting force

P and the width

a of the contact area, and hence determine the contact pressure distribution

p (

x ) using the incremental method of Sect.

6.3 .

Fig. 6.13 The truncated wedge-shaped indenter

Fig. 6.14 Indentation by a half-cylinder

5. Use Eq. (6.27 ) and the incremental method of Sect. 6.3 to determine the contact pressure distribution p (x ) for the flat and rounded punch of Fig. 6.4 . Plot the resulting expression for representative values of b / a .

We know that the contact pressure for the flat punch is singular at the edges, so we must anticipate a local maximum of contact pressure as \(a\!\rightarrow \! b\) . Find the value of this maximum and make a log-log dimensionless plot as a function of \(k\!=\!\sqrt{1-b^2/a^2}\) . Comment on the shape of this plot as \(k\!\rightarrow \!0\) .

6. Use the Fourier series method of Sect. 6.4 to find the contact pressure distribution p (x ) for Problem 4.

7. Find the contact pressure distribution for the problem of Fig. 6.5 a if the line of action of the force P passes through the point \(x\!=\!-c\) and the contact area is defined by \(-a\!<\!x\!<\!a\) . Hence determine the value \(c_0\) such that the entire punch face remains in contact if and only if \(|c|\!<\!c_0\) .

Now suppose that \(c_0\!<\!c\!<\!a\) . Find the new contact pressure distribution p (x ), the angle of tilt \(\alpha \) and the extent of the contact area.

8. Use the potentials (6.66 ) to find the complete stress and displacement field in the half-plane due to the contact pressure distribution (6.65 ), and hence verify Eq. (6.68 ).

9. Solve Westergaard’s problem [Fig. 6.6 ] by substituting (6.76 ) in (6.75 ) and solving the resulting Cauchy singular integral equation as in Sect. 6.2 .

10. By writing \(p_0\!=\!P/\pi a^2\) and proceeding to the limit where \(a\!\rightarrow \!0\) , show that Eq. (6.107 ) reduces to the point force solution ( 2.7 ).

11. Use the method of Sect. 5.1 to determine the surface displacements due to the axisymmetric pressure distribution \(p(r)\!=\!p_0H(a\!-\!r)\) and hence verify that they are given by Eq. (6.107 ).

12. A rectangular block of height

h and width

w rests on two rigid cylinders, each of radius

R as shown in Fig.

6.15 . The block is loaded only by its own weight. Use arguments similar to those in Sect.

6.7 to estimate the vertical displacement of the centre of gravity of the block due to elastic deformation. The material has density

\(\rho \) , Young’s modulus

E and Poisson’s ratio

\(\nu \) .

Fig. 6.15 A rectangular block resting on two cylinders

Do you expect the spacing between the two supports to influence the result, and if so, in what range of the parameters w , h , R ?

13. A rigid flat punch with a rectangular cross section \(20a\!\times \!2a\) is pressed into an elastic half-space by a force P . Use Kalker’s line contact theory to estimate the distribution of contact stress. How does your estimate for the indentation depth d compare with that for an elliptical flat punch of semi-axes 10a and a ?

Hint: Notice that \(g_0\!=\!0\) throughout the contact area, so the integral term in Eq. (6.119 ) is also zero. You will need to write a simple numerical code to solve the resulting Fredholm equation [e.g. by assuming that \(F(\eta )\) is piecewise constant and using collocation at the mid-point of each element].

14. If the annular punch in Fig.

5.2 is ‘thin’, meaning

\(2c\equiv (a\!-\!b)\!\ll \! a\) , the contact pressure will be locally approximately two dimensional and given by

$$ p(x)\approx \frac{F}{\pi \sqrt{c^2-x^2}}\qquad \text{ where }\qquad x=r-R, $$

and

\(F\!=\!P/2\pi R\) is the applied force per unit circumference around the mean line

\(R\!=\!(a\!+\!b)/2\) .

Estimate the rigid-body indentation

\(\varDelta \) of the punch by (i) using Eq. (

6.64 ) to calculate the indentation of a two-dimensional flat punch relative to the points

\(x\!=\!\pm L\) , and then (ii) adding the average of the displacements at the points

\(r\!=\!R\!-\!L\) and

\(r\!=\!R+L\) , equidistant on the two sides of a ring force of

F per unit circumference, for which

$$\begin{aligned} u_z(r)= & {} \frac{4F}{\pi {{E}^{*}}}K\left( \frac{r}{R}\right) \;\;\;\;\;0\le r<R \nonumber \\= & {} \frac{4FR}{\pi {{E}^{*}}r}K\left( \frac{R}{r}\right) \;\;\,\;\;\;r>R.\nonumber \end{aligned}$$

The resulting relation between

P and

\(\varDelta \) will depend on your choice of

L which must lie in the range

\(c\!<\!L\!<\!R\) . How sensitive is the result to this choice and what do you think is the most appropriate value?