1. Use Eq. (

7.5 ) to find the

surface dilatation $$ e_S(x, y)\equiv \frac{\partial u_x}{\partial x}+\frac{\partial u_y}{\partial y}\quad \text{ at }\quad z=0, $$

due to a concentrated tangential force

\(\varvec{Q}\) .

Hence or otherwise show that for any uncoupled problem [

\(\beta \!=\!0\) ] with no slip, the tangential contact tractions

\(\varvec{q}=\{q_x, q_y\}\) must satisfy the equation

$$ \int \!\!\int _{\mathcal {A}}\frac{(\varvec{r}-\varvec{r}_0)\varvec{\cdot q}\, d\mathcal {A}}{|(\varvec{r}-\varvec{r}_0)|^3}=0, $$

where

\(\varvec{r}_0\) is any point in

\(\mathcal {A}\) . Comment on the interpretation of the apparent singularity in this integral as

\(\varvec{r}\!\rightarrow \!\varvec{r}_0\) .

2. Table

7.1 shows representative values of the elastic constants for a selection of materials. Estimate Dundurs’ bimaterial constant

\(\beta \) for the following material combinations: carbon steel on rubber, stainless steel on glass, nylon on glass, rubber on ice, compact bone on stainless steel. Within this set of materials, which combination gives the greatest value of

\(\beta \) and what is that value?

Table 7.1 Elastic properties of selected materials

3. A two-dimensional flat rigid punch is pressed into an elastic half-space by a normal force P , such that the contact area is defined by \(-a\!<\!x\!<\!a\) . A tangential force \(Q_y\) is now applied in the y -direction [antiplane]. Assuming no-slip conditions, use Eq. (7.18 ) to determine the additional contact tractions due to the application of \(Q_y\) .

4. A two-dimensional flat rigid punch is pressed into an elastic half-space by a normal force P , such that the contact area is defined by \(-a\!<\!x\!<\!a\) . The line of action of the force passes through the point \(x\!=\!c\) , so some rigid-body rotation is to be anticipated, as in Fig. 6.5 a. Use the method of Sect. 7.3 to determine the resulting contact tractions under the assumption of full contact and no slip.

5. If the Goodman approximation is applied to the problem of a normally loaded axisymmetric flat punch, the first equation in (

7.43 ) is simplified to

$$ \frac{\partial \varphi }{\partial z}=-\frac{E^{{*}}\varDelta }{2} $$

and the solution for

\(h_1(t)\) is given in Sect.

5.1.1 . Use this result and other results from Sect.

5.1 to define an integral equation for the function

\(h_2(t)\) , and hence determine the tangential tractions

\(\sigma _{zr}(r)\) in the contact area.

6. Show that if the contact area \(\mathcal {A}\) has three-fold symmetry [for example, like an equilateral triangle], then the tensor \(\varvec{C}\) in Eq. (7.69 ) must be isotropic — i.e. \(\varvec{C}\!=\!C\varvec{I}\) , where \(\varvec{I}\) is the identity matrix.

Two identical half-spaces are spot-welded together such that the welded area comprises three circles of radius

a whose centres are located at the vertices of an equilateral triangle of side

b , as shown in Fig.

7.11 . The circles do not overlap, so

\(b\!>\!2a\) .

Fig. 7.11 Welded area between two half spaces

Estimate the tangential compliance C by using Eq. (7.57 ) for the displacement at a circle due to a force distributed over the same circle, and approximating the effect of forces at the other circles by replacing them by equal point forces at their respective centres. Comment on the possible relevance of your results to the second conjecture in Sect. 7.6.3 .

7. Due to surface roughness, two half-spaces of the same material make contact at a large number N of small elliptical areas each of semi-major axis a and eccentricity e . These ellipses are of random orientations and are sufficiently widely separated for their elastic fields to be independent of each other. Use Eqs. (7.68 , 7.70 ) to estimate the normal and tangential compliances of the system and comment on the effect of \(\nu \) and e on the ratio between them.

8. The uncoupled Hertzian contact of Sect.

7.7.1 is subject to the monotonically increasing forces

$$ P=Ct;\;\;\;Q_x=0.2Ct;\;\;\;Q_y=\frac{0.075Ct^2}{t_0},\quad \text{ where }\quad 0< t<t_0. $$

Find the contact tractions

\(p, q_x, q_y\) under no-slip conditions as functions of

x ,

y and time

t , and hence determine the minimum coefficient of friction

f if this is to be a reasonable assumption.

9. Use the Goodman approximation to estimate the tangential tractions developed when a rigid cylinder of radius R is pressed into an elastic half-plane by a normal force P . The normal tractions and the extent of the contact area will then be given by the two-dimensional Hertzian solution, but the incremental tangential tractions include the effect of normal-tangential coupling and are given by Eq. (7.55 ) [with \(Q_x\!=\!0\) ].

10. An elastic sphere of radius

R is pressed into an elastic half-space and then subjected to the periodic loading

$$ P=P_0+P_1\cos (\omega t);\;\;\;Q_x=Q_1\sin (\omega t);\;\;\;Q_y=0, $$

where

\(P_0\!>\!P_1\!>\!0\) . Find the energy dissipation per cycle if both materials have Young’s modulus

E and Poisson’s ratio

\(\nu \) , and the friction coefficient is sufficiently large to prevent all slip.

11. For a general uncoupled three-dimensional contact problem, the incremental tangential compliance

\(\varvec{C}(P)\) under no-slip conditions is a Cartesian tensor function of the normal force

P which we write in the form

$$ \frac{\partial \varvec{U}}{\partial \varvec{Q}}=\varvec{C}(P)=\varvec{\varLambda }(P)C_N(P), $$

where

\(C_N(P)\) is the corresponding normal compliance. Show that Eq. (

7.104 ) for the energy dissipation per cycle is then generalized to

$$ W=-\frac{1}{2}\int _{P_{\min }}^{P_{\max }}(\varvec{Q}_A-\varvec{Q}_B)\varvec{\cdot }\left[ \varvec{\varLambda }(P)(\varvec{Q}_A-\varvec{Q}_B)\right] C_N^\prime (P)dP. $$

If we assume that

\(\varvec{\varLambda }(P)\) is bounded between the in-plane and antiplane limits, as conjectured in Sect.

7.6.3 , can we place corresponding bounds on the error associated with the approximation

\(\varvec{\varLambda }(P)\!=\!\mathcal {R}_T\varvec{I}\) , where

\(\varvec{I}\) is the identity matrix?