1. Define new variables \(s\!=\!a\!-\!r\) in Eq. ( 5.21 )\(_1\) and \(t\!=\!r\!-\!a\) in Eq. ( 5.22 ) and then expand these equations for small values of s , t respectively. Show that the leading terms in these expansions have the form of Eq. (10.15 ), and find the corresponding value of the constant \(B_1\) .

2. Extract the square-root singular term in Eq. ( 5.14 ) by (i) integrating by parts and then (ii) performing the differentiation with respect to r . Hence show that the constant \(B_1\) in Eq. (10.15 )\(_1\) is proportional to h (a ). Then show that Eq. ( 5.17 ) defines a square-root bounded expression for displacement of the form (10.15 )\(_2\) with the same multiplier \(B_1\) .

3. Two similar electrically conducting bodies

\(0\!<\!\theta \!<\!\pi \) and

\(-\pi \!<\!\theta \!<\!0\) make perfect electrical contact over the half-line

\(\theta \!=\!0\) and are separated [and hence also insulated] over

\(\theta \!=\!\pm \pi \) . Show that the local potential fields can be expressed as an eigenfunction series of the form

$$ V(r,\theta )=\sum _{n=1}^\infty B_nr^{\lambda _n}f_n(\theta ) $$

and find the eigenvalues

\(\lambda \) and eigenfunctions

\(f_n(\theta )\) .

Hint : The governing equations for electrical conduction are (

4.13 ,

4.14 ) and in polar coordinates

$$ \varvec{\nabla }=\left\{ \frac{\partial }{\partial r},\frac{1}{r}\frac{\partial }{\partial \theta }\right\} ;\;\;\;\nabla ^2=\frac{\partial ^2 }{\partial {r}^2}+\frac{1}{r}\frac{\partial }{\partial r}+\frac{1}{r^2}\frac{\partial ^2 }{\partial {\theta }^2}. $$

4. In Problem 3, suppose that in the contact region, there is a contact resistance

R such that the local current density

$$ i_\theta (r, 0)=\frac{(V_2(r, 0)-V_1(r, 0))}{R}, $$

where

\(V_1(r,\theta ), V_2(r,\theta )\) are the electrical potentials in bodies

\(0\!<\!\theta \!<\!\pi \) and

\(-\pi \!<\!\theta \!<\!0\) respectively. Show that a parameter

L with the dimensions of length can be constructed from

R and the resistivity

\(\rho \) , and hence that the asymptotic problem is not now self-similar. What form would you expect the fields to take in the region where

\(r\!\ll \!L\) ?

5. Two similar thermally conducting bodies at different uniform temperatures are brought into contact at time

\(t\!=\!0\) over an area

\(\mathcal {A}\) . By focussing on a region very close to the boundary of

\(\mathcal {A}\) , and using symmetry, we can describe the local temperature field

\(T(r,\theta , t)\) in one of the bodies by the conditions

$$ T(r,\theta , 0)=0;\;\;\;T(r, 0,t)=T_0;\;\;\;\frac{\partial T}{\partial \theta }(r,\pi , t)=0, $$

where

\(T_0\) is a constant and

\(r,\theta \) are defined as in Fig.

10.3 . The temperature field must also satisfy the transient heat conduction equation

$$ \frac{\partial ^2 T}{\partial {r}^2}+\frac{1}{r}\frac{\partial T}{\partial r}+\frac{1}{r^2}\frac{\partial ^2 T}{\partial {\theta }^2}=\frac{1}{k}\frac{\partial T}{\partial t}, $$

where

k is the thermal diffusivity of the material. Show (i) that the solution for

T depends only on the dimensionless coordinates

\(r/\sqrt{kt}, \theta \) , and (ii) that the heat flux

\(q_\theta (r, 0,t)\) in the contact region tends to a constant as

\(r/\sqrt{kt}\!\rightarrow \!\infty \) . Find the value of this constant and describe in words the evolution of the heat flux

\(q_\theta (r, 0,t)\) with time.

6. Find the first eigenvalue for the problem of Fig. 10.5 a with \(\alpha \!=\!60^{\circ }\) , if the materials of the two bodies are the same and no slip occurs at the interface. Hence find the minimum coefficient of friction for the no-slip assumption to be reasonable.

7. Using the incremental formulation of Sect.

6.3.1 , show that for symmetric non-conformal problems with contact semi-width

a , the constant

B in Eq. (

10.50 ) is given by

$$ B=\frac{F(a)}{\pi }\sqrt{\frac{2}{a}}, $$

where

F (

a ) is defined in Eq. (

6.29 ). Hence show that under ‘Cattaneo’ loading, as defined by Fig.

9.1 , the length of each slip zone for sufficiently small ratios

Q /

P is given by

8. A frictionless rigid punch has the form of a truncated wedge, as shown in Fig.

10.11 .

Fig. 10.11 The truncated wedge-shaped indenter

Define a coordinate system with origin at the point

\(x\!=\!b\) , and use an asymptotic method to find the dominant singular term in the contact tractions at this point due to the discontinuity of slope. Notice that the displacement boundary conditions will then take the inhomogeneous form

$$ \frac{\partial u_\theta }{\partial r}(r, 0)=0\quad \text{ and }\quad \frac{\partial u_\theta }{\partial r}(r,\pi )=\alpha , $$

so this is not an eigenvalue problem. Show that the multiplier on the singular term depends only on

\(\alpha \) and is independent of the force

P .

9. Use the method of Sect. 10.5.1 to estimate the relation between P and a in the axisymmetric Problem 5.3, assuming \(R\ll b\) .

10. A flat rigid frictionless elliptical punch of semi-axes a , b is pressed into an elastic half-space by a normal force P . The edges of the indenter are slightly rounded, with uniform radius \(R\!\ll \! a, b\) . Use the solution of Sect. 2.3.2 and the asymptotic method of Sect. 10.5.1 to estimate the magnitude and location of the maximum contact pressure.

11. A two-dimensional Hertzian contact between similar materials is loaded by a normal force P which is then held constant whilst a bulk stress difference \(\sigma \) and a tangential force \(Q_x\) are applied in proportion. Assuming full-stick conditions, determine the resulting mode II stress-intensity factors at the two edges and hence estimate the sizes of the two slip zones using an asymptotic method. Check your results against Eq. ( 9.37 ) for the case where the tangential force \(Q_x\!=\!0\) and \(k\!\ll \!1\) .