1. Suppose the asperity in Fig. 16.1 is replaced by a paraboloid whose radius at the apex is R . Using Bowden and Tabor’s assumptions, find the corresponding relation between indentation depth h , normal force P and volume of material removed V , and comment on the implications for the friction law at a single asperity.

2. A cylindrical column of diameter 50 mm and length 100 mm carries a compressive force of 5000 N. It is desired to measure small fluctuations in this force due to vibration by incorporating a load cell as shown in Fig. 16.31 .

All the contacting surfaces are ground to a surface finish of

\(0.5 \mu \) m RMS, bearing area results for which are given in Table

16.1 . You can assume that all the parts have Young’s modulus

\(E\!=\!200\) GPa, Poisson’s ratio

\(\nu \!=\!0.3\) and indentation hardness [flow pressure]

\(H\!=\!800\) MPa.

Fig. 16.31 All dimensions are in mm

Table 16.1 Bearing area data

Fit the bearing area data to a normal distribution and use Eq. (

16.1 ) to estimate the incremental stiffness of the column with this modification and compare it with the stiffness of a column of equal size without interfaces. The incremental stiffness is defined as

$$ k=\frac{dP}{d\varDelta } , $$

where

P is the applied compressive force and

\(\varDelta \) is the reduction in length of the column. Remember that there are several interfaces and each interface involves

two rough surfaces.

^{13} 3. At large values of

x , the complementary error function can be approximated using the asymptotic series

$$ \text {erfc}(x)\approx \frac{e^{-x^2}}{\sqrt{\pi }}\left( \frac{1}{x}-\frac{1}{2x^3}+\frac{1.3}{2^2x^5}-\frac{1.3.5}{2^3x^7}+\ldots \right) $$

[Carslaw and Jaeger 1959, Appendix II]. Assuming that the normal force can be approximated by the ‘bearing area’ Eq. (

16.16 ), use the above expression and Eq. (

16.18 ) to develop an approximation for the ratio

k /

P , where

P is the normal force and

k is the incremental stiffness of the contact. How much does this ratio change in the range

$$ 0.001<\frac{p_\mathrm{{nom}}}{H}<0.1 . $$

4. Show that if a surface can be represented by an exponential distribution of identical asperities, as in Sect.

16.3.1 , the incremental stiffness is given by

$$ \frac{dP}{dh}=-\frac{P}{\sigma } , $$

where

\(\sigma \) is defined in Eq. (

16.24 ). Use this same equation to find an expression for the bearing area for the surface comprised of an exponential distribution of asperities. Then use the bearing area data from Problem 2 to estimate

\(\sigma \) , and hence estimate the incremental stiffness

\(dP/d\varDelta \) for the problem of Fig.

16.31 .

5. If two large bodies of the same material make contact at a single circular area of radius a , the electrical resistance imposed by the interface is \(\rho /2a\) , where \(\rho \) is the resistivity of the material.

Suppose instead that one of the surfaces is plane and the other comprises an exponential distribution of asperities defined by Eq. (16.24 ). Find the contact resistance of the interface if the bodies are pressed together to the point where undeformed regions of the plane surface are at height \(h_0\) . Assume that the asperities are all spherical with radius R , so the Hertzian relations apply, including for example (16.29 ).

Show that the contact resistance is inversely proportional to the normal force P and find the constant of proportionality.

6. Find the contact resistance as in Problem 5, but for a Gaussian distribution of asperities defined by Eq. (16.31 ). Plot a suitably normalized graph of contact conductance [reciprocal of resistance] as a function of normal force and comment on the degree of deviation from linearity.

7. A particular surface comprises a single sine wave

$$ h(x, y)=h_0\cos (\omega _0 x+\phi ) , $$

so that the peaks and troughs are aligned with the

y -direction. Show that the profile PSD sampled in the

x -direction is

$$ P_P(k)=\frac{h_0^2\delta (k-\omega _0)}{4} . $$

A new surface is now constructed by superposition of sinusoidal surfaces of the same wavenumber

\(\omega \) and amplitude, but with different angular orientations

\(\theta \) relative to the

x -axis and arbitrary phases

\(\phi \) . Find the profile PSD for this surface assuming that all angles

\(\theta \) are equally represented [thus ensuring that the resulting surface is isotropic] and that the height variance is

\(\sigma ^2\) .

Then use this expression and Eq. (16.69 )\(_2\) to determine the corresponding surface PSD. Comment on your result.

8. Use results from Sects. 16.4.2 and 16.4.3 to express the two definitions of the plasticity index (16.36 ) in terms of the moments of the profile PSD (16.57 )–(16.61 ). What properties of the surface would maximize the difference between the two definitions? In such cases, comment on which definition might be most appropriate.

9. Use Eq. (

16.69 ) to show that the even moments of the profile PSD can be expressed in terms of the surface PSD as

$$ m_{2n}=\frac{2\pi (2n-1)!!}{(2n)!!}\int _0^\infty \omega ^{2n+1}P_S(\omega )d\omega . $$

Use this result to evaluate the bandwidth parameter

\(\alpha \) for a surface for which

$$\begin{aligned} P_S(\omega )= & {} \frac{A}{\omega ^m} \quad \omega _0<\omega<\lambda \omega _0 \nonumber \\= & {} 0 \quad \omega <\omega _0\qquad \text {and}\qquad \omega >\lambda \omega _0 \nonumber , \end{aligned}$$

where

\(A, m,\lambda \) are constants and

\(\lambda \!>\!1\) . Hence show that the minimum value of

\(\alpha \) is 3 / 2 and comment on likely values for practical surfaces.

10. Use Eq. (16.55 ) to find the PSD for a profile defined by Archard’s exponential autocorrelation function (16.48 ). Hence show that this profile is fractal at large \(\omega \) with a fractal dimension \(D\!=\!1.5\) .

11. Suppose that a random surface is defined by the surface PSD

$$ P_S(\omega )=A\omega ^{2D-8} \quad \omega _1<\omega <\omega _2, $$

where

A is a constant. Outside this range

\(P_S(\omega )\!=\!0\) . Evaluate the moments

\(m_0, m_2, m_4\) using the integral expression given in Problem 9, and hence determine the bandwidth parameter

\(\alpha \) as a function of

\(\lambda \!\equiv \!\omega _2/\omega _1\) and the fractal dimension

D . Plot a graph of

\(\alpha \) as a function of

\(\lambda \) for

\(D\!=\!2.1\) and 2.5.

12. If a rigid fractal surface is pressed against an elastic half space, we anticipate that the resulting contact area will also be a fractal. Consider the special case where the contact area at any scale comprises a number N of circular contact areas, all of the same radius a . Find the dependence of N on a , assuming that the total contact area has fractal dimension \(D_A\) . Use this result to determine the total perimeter S of the contact area, and hence show that the fractal dimension of the length S is the same as that of the area A .

13. Find the autocorrelation function C (z ) for the Weierstrass profile of Eq. (16.83 ). Hence prove the result (16.87 ).

14. Use Persson’s theory [Eq. (16.122 )] to estimate the limiting fractal dimension of the total actual contact area A at a given nominal pressure, for a surface whose PSD is \(P_S(\omega )\!=\!C\omega ^{-2-H}\) , where H is the Hurst exponent.

15. The nominal contact pressure

\(p_\mathrm{{nom}}\) can be expressed in terms of the probability distribution

\(\varPhi (p, V)\) as

$$ p_\mathrm{{nom}}=\int _0^\infty \varPhi (p,V)\,p\, dp. $$

Use Persson’s differential equation (

16.118 ) and the boundary condition

\(\varPhi (0,V)\!=\!0\) to show that

\(p_\mathrm{{nom}}\) is independent of

V and hence that Persson’s theory defines the evolution of

\(\varPhi (p)\) as roughness is added at constant

\(p_\mathrm{{nom}}\) , for any initial condition

\(\varPhi (p, 0)\) .

16. From Sect. 12.2.4 and Fig.

12.3 , we expect that a spherical asperity would jump into contact at

\(\varDelta \!=\!0\) , but not pull out of contact until

\(\varDelta \!=\!\varDelta _A\) , where

$$ \hat{\varDelta }_A\equiv \frac{\beta ^2 \varDelta _A}{R}=-\frac{3\pi ^{2/3}}{4} , $$

from Eq. (

12.39 ). This implies that a surface defined by the Gaussian distribution of asperities of Eq. (

16.31 ) will exhibit different force–displacement relations during loading and unloading. Find expressions for these relations using the approximation (

16.132 ) and make appropriate normalized plots for the cases where the minimum approach is

\(h_0\!=\!\sigma \) and

\(h_0\!=\!2\sigma \) respectively. [Note: If you cannot devise a universal normalization for these plots, just use representative engineering values for the remaining parameters.]

17. Use the Maugis-Dugdale force law of Sect. 12.4.3 to approximate the force–displacement relation for the rigid sinusoidal surface of Sect. 16.7.2 contacting a rigid plane. Compare your result with Fig. 16.21 .

Suppose we now wish to use the same force law, but include the effects of elastic deformation. Two cases can be distinguished, depending on whether regions near the troughs of the sinusoid lose adhesion (i) before or (ii) after the contact region shrinks to zero as the bodies are pulled apart. Use a suitable superposition with the Westergaard solution to solve the problem for case (ii), when the nominal pressure \({\bar{p}}\) is sufficient to ensure that the contact region is non-null.

18. Suppose that an elastic half space is indented by a rigid body that contains two spherical asperities of radius R and the same height, whose peaks are separated by some distance d . Find the relation between total force P and indentation depth \(\varDelta \) , assuming that the displacement at one asperity due to the force at the other can be approximated by the point force solution, as in Sect. 4.1.2 .

Hence show that the incremental stiffness \(dP/d\varDelta \) is an increasing function of d . Can you extend this argument to show that the incremental stiffness is always overestimated if interaction between asperities is neglected?