1. A rigid conical punch of cone angle

\(\pi /2\!-\!\alpha \) , (

\(\alpha \!\ll \!1\) ) is pressed into the surface of an elastic half-space, as shown in Fig.

5.4 . Find the contact pressure

p (

r ), the indentation depth

\(\varDelta \) and the applied force

P , all as functions of the radius

a of the contact area.

Fig. 5.4 The conical indenter

2. An elastic half-space is indented by an axisymmetric rigid punch with the power-law profile

\(g_0(r)\!=\!Cr^\lambda \) , so the displacement in the contact area is

$$ u_z(r)=\varDelta -Cr^\lambda , $$

where

\(C,\lambda \) are constants. Show that the indentation force

P , the contact radius

a and the indentation depth

\(\varDelta \) are related by the equation

$$ P=\frac{2E \varDelta a}{(1-\nu ^2)}\left( \frac{\lambda }{\lambda +1}\right) . $$

3. A rigid flat punch has rounded edges, as shown in Fig.

5.5 . The punch is pressed into an elastic half-space by a force

P . Assuming that the contact is frictionless, find the relation between

P , the indentation depth

\(\varDelta \) and the radius

a of the contact area.

Fig. 5.5 Flat punch with rounded corners

4. The elastic half-space is loaded by a uniform pressure \(p_0\) inside the circle \(0\!\le \!r\!<\!a\) , the rest of the surface being unloaded. Use Eq. (5.14 ) to determine the appropriate function h (t ) and hence find the surface displacement \(u_z(r)\) both inside and outside the loaded region.

5. The profile of a smooth axisymmetric frictionless rigid punch is described by the power law

where

n is an integer. The punch is pressed into an elastic half-space by a force

P . Find the indentation

\(\varDelta \) , the radius of the contact area

a and the contact pressure distribution

p (

r ). Check your results by comparison with the Hertz problem of Eq. (

5.30 ) and give simplified expressions for the case of the fourth order punch

6. An elastic half-space is indented by a rigid cylindrical punch of radius

a with a

concave spherical end of radius

\(R\!\gg \! a\) . Find the contact pressure distribution

p (

r ) and hence determine the minimum force

\(P_0\) required to maintain contact over the entire punch surface. Do not attempt to solve the problem for

\(P\!<\!P_0\) .

7.

(i) By representing the function

\(\varphi \) in the form

$$ \varphi =\mathfrak {I}\int _0^b\mathcal {F}(r,z, t)h(t)dt, $$

determine the surface tractions [tensile and compressive] needed to establish the displacement field

$$\begin{aligned} u_z(r)= & {} \varDelta \left( 1-\frac{r^2}{b^2}\right) ^2 \quad 0\le r<b \nonumber \\= & {} 0 \qquad \qquad \qquad r>b. \nonumber \end{aligned}$$

(ii) The otherwise flat surface of a rigid punch of radius a contains a number of small widely spaced concave dimples of radius \(b\!\ll \! a\) and depth \(\varDelta \) . Use your result from part (i) to find the minimum force that must be applied to the punch to ensure that contact is established throughout the surface.

8. A rigid axisymmetric punch has a Hertzian profile perturbed by a set of concentric sinusoidal waves, such that the initial gap function is defined as

$$ g_0(r)=\frac{r^2}{2R}+A\left[ 1-\cos (mr)\right] , $$

where

\((mA)\ll 1\) .

Assuming that the contact area comprises a single circle of radius a , determine the indentation depth \(\varDelta \) and the normal force P as functions of a . Use these results to make a parametric plot of P as a function of \(\varDelta \) and comment on the nature of this plot as A is increased. [This problem requires the use of Maple or Mathematica].

9. Find expressions analogous to (5.10 ), (5.11 ) for the surface values of the derivatives of the function \(\varphi _2\) in Eq. (5.40 ). In particular, verify that condition (5.37 ) is satisfied for all functions \(h_2(t)\) .

10. Use the method of Sect. 5.4 to determine the contact pressure distribution under a flat-ended cylindrical punch of radius a loaded by a force P applied through the point (c , 0). Also, find the angle of tilt of the punch. Assume that the entire flat surface of the punch makes contact. What is the maximum value of c for which this assumption is correct?

11. A flat-ended rigid cylindrical punch of radius a is pressed into the curved surface of an elastic cylinder of radius \(R\!\gg \! a\) by a force P . Use the method of Sect. 5.4 to find the contact pressure distribution for the case where P is sufficient to ensure that the entire flat surface of the punch makes contact.

12. Use the method of dimensionality reduction to find the relations between the indentation \(\varDelta \) , the radius of the contact area a and the applied force P for the power-law punch of Problem 5.