1. Use Boyer’s method to estimate the shape function

S for an equilateral triangle of side

a . You will need to represent the triangle as the sum of four smaller triangles, as shown in Fig.

4.5 . Notice that the force on the central triangle will differ from those on the other three.

Fig. 4.5 An equilateral triangle of side 2a

2. A rigid flat punch with a rectangular cross section of dimensions \(a\!\times \! 5a\) is pressed into the surface of an elastic half-space by a force P . Find an approximate expression for the indentation depth \(\varDelta \) by considering five separate punches [see Fig. 4.6 ], each of \(a\!\times \!a\) square cross section and each indented to the same depth. Use Nakamura’s result for the diagonal elements in the resulting stiffness matrix and Eq. (4.33 ) for the off-diagonal elements. Compare your result with that for an elliptical punch of semi-axes \(5a/2,\; a/2\) using Eq. ( 2.34 ).

3. Use Eq. ( 2.17 ) to determine the surface displacement at the point (c , 0) due to a uniform contact pressure \(p_0\) acting over the square \(-a/2\!<\!x\!<\!a/2,\; -a/2\!<\!y\!<\!a/2\) , where \(c\!>\!a/2\) . Hence, determine whether Eq. (4.33 ) overestimates or underestimates the off-diagonal elements of the stiffness matrix in Problem 2.

4. Show that the boundary of the ellipse in Fig.

2.7 can be expressed in polar coordinates as

$$ a(\theta )=\frac{1}{\sqrt{C_2(\theta )}}, $$

where

\(C_2(\theta )\) is defined by Eq. (

2.27 ). Then use Eqs. (

4.36 ), (

4.39 ) to verify that Fabrikant’s solution gives the exact result for the pressure distribution under an elliptical rigid flat punch with indentation

\(\varDelta \) .

5. Figure

4.7 shows a rigid punch with two parallel plane faces

\(A_1, A_2\) , pressed into an elastic half-space by a force

P sufficient to cause all points in both of these areas to be in contact. Show that the work done by

P during loading is least when the two areas have the same height [i.e. when they lie in the same plane]. Hence, or otherwise, show that of all punches of given planform

A [convex or concave], the work done loading to a given force

P [sufficient to establish full contact] is least when the punch is flat.

Fig. 4.6 Planform of a rectangular punch considered as five adjacent squares

Fig. 4.7 A rigid punch with two plane faces

6. Show that if the gap function

\(g_0(x, y)\) is convex, meaning

$$ \frac{\partial ^2 g_0}{\partial {x}^2}+\frac{\partial ^2 g_0}{\partial {y}^2}>0\qquad \text{ all } \text{(x, } \text{ y), }\qquad $$

the contact area

\(\mathcal {A}\) in a frictionless contact problem for the half-space must be simply connected for all applied forces.

7. An axisymmetric rigid punch is defined by the piecewise-linear gap function \(g_0(r)\) , where the slope \(g_0^\prime (r)\) is a non-decreasing function of r . Show that the resulting force–displacement relation \(P(\varDelta )\) is continuous up to the first derivative.

8. Use Eq. (

2.14 ) to find an expression for the surface curvature

$$ \nabla _2(u_z)\equiv \frac{\partial ^2 u_z}{\partial {x}^2}+\frac{\partial ^2 u_z}{\partial {y}^2} $$

when the surface is loaded by a concentrated compressive force

P .

Use an integral formulation [as in Sect. 2.3 ] to generalize this expression to a distribution of compressive normal tractions p (x , y ), and use your result to prove that a local maximum value of \(u_z(x, y)\) can occur only in a loaded region.

9. Use Eqs. ( 2.43 ), ( 2.46 ) with \(a\!=\!b\) to obtain the normal displacement in the contact area \(0\!\le \! r\!<\!a\) for the pressure distribution \(p(r)\!=\!C(a^2\!-\!r^2)^{3/2}\) . Show that an appropriate derivative of this distribution, in combination with lower order axisymmetric fields, can be used to solve the problem of a cylindrical flat-ended rigid punch of radius a indenting the curved surface of an elastic cylinder of radius \(R\!\gg \! a\) if the indenting force is sufficiently large to ensure full contact. Comment on possible methods for solving this problem at lower values of the indenting force.

10. Use the method of Sect. 4.4 to determine the force–displacement relation for Problem 9 in the range where the entire punch face makes contact with the cylindrical surface.