Use results from Sect. 17.1 to determine the normal surface displacement of a traction-free half space due to a steady-state heat source *Q* per unit length around the circle \(r\!=\!a\), the rest of the surface being unheated.

Hence show that the contact pressure distribution in a steady-state axisymmetric thermoelastic contact problem is unaffected by the thermal boundary conditions [e.g. convection or radiation to the environment] outside the circular contact area.

2. A traction-free half-space is subjected to surface heating. Show that the maximum outward thermoelastic displacement must occur in a heated area and the maximum *inward* displacement in a cooled area.

3. The layer

\(0\!<\!z\!<\!h\) rests on a frictionless rigid foundation at

\(z\!=\!0\) and the surface

\(z\!=\!h\) is traction-free. The foundation is a thermal insulator and the free surface is subjected to the steady-state heat input

$$ q_z(x)=q_0\cos (mx). $$

Use Dundurs’ theorem to show that the layer will not separate from the foundation and find the amplitude of the sinusoidal perturbation in the free surface due to thermoelastic distortion.

4. Suppose that the thermal resistance for the system in Fig.

17.5 is defined by Duvaut’s law

$$\begin{aligned} R(p)= & {} \frac{C}{p}\;\;\;\;\;p>0\nonumber \\= & {} \infty \;\;\;\;\;g>0. \nonumber \end{aligned}$$

If the temperature difference

\((T_B\!-\!T_A)\) is specified, find the minimum value of the constant

*C* if the steady-state solution is to be unique for all values of

\(g_0\).

5. Assuming that the two rods in Fig. 17.8 are in a steady state with the gap *g* positive, find the equation determining *g* if the thermal resistance is defined by a known function *R*(*g*). Find a function *f*(*g*) in the range \(0\!<\!f(g)\!<\!1\) that allows you to express this equation in a form similar to Eq. (17.53) and its solution by an intersection as in Fig. 17.6. What must be the definition of \(\tilde{g}\) if this result is to apply also to the case of contact at pressure *p*?

6. An assembly like that in Fig. 17.13, but with only two rods, is sliding at speed *V* and transmitting a constant force *P*. Initially both rods are in contact and they have identical temperature and stress fields. By postulating the occurrence of exponentially growing perturbations in these fields, determine the condition for this state to be stable if wear is governed by Archard’s wear law \(\dot{w}(t)\!=\!\gamma f|V| p(t)\) where *w* is the depth of material removed, *p* is the contact pressure, *f* is the coefficient of friction and \(\gamma \) is a constant. The rods are made of a material with properties \(\alpha , E, K, k\) and they each have cross-sectional area *A* and length *L*, which you can assume to be sufficiently large for perturbations in temperature at the non-contact end to be negligible.

7. An elastic half-plane is pressed against a rigid plane by a pressure

\(p_0\) and slides at speed

*V* in the

*y*-direction. Wear occurs, governed by Archard’s wear law

\(\dot{w}(x,y,t)\!=\!\gamma f|V| p(x,y, t)\) where

*w* is the depth of material removed,

*p* is the contact pressure,

*f* is the coefficient of friction and

\(\gamma \) is a constant. Find the critical speed above which an out-of-plane perturbation of the form

$$ T(x,y,z, t)=\varTheta (z)e^{bt}\cos (mx), $$

is unstable.

8. Find the function

\(\varTheta (z)\) such that the temperature perturbation

$$ T(x,z, t)=\varTheta (z)e^{bt}\cos (mx) $$

satisfies the transient heat conduction equation

$$ \nabla ^2T=\frac{1}{k}\frac{\partial T}{\partial t}. $$

Hence find the relation between growth rate

*b* [assumed real] and wavenumber

*m* for the out-of-plane sliding problem of Sect.

17.5.3 and verify that the maximum growth rate corresponds to the wavenumber

\(m_0\) of Eq. (

17.88).