Axisymmetric Contact Problems

Abstract

If the gap function $g_0(r)$ is axisymmetric, and if contact is assumed to occur only within a circle of some radius a, the problem of Eqs. (4.8)–(4.11) is reduced to the search for an axisymmetric harmonic function $\varphi (r, z)$ in cylindrical polar coordinates $(r,\theta , z)$.

Problems

1. A rigid conical punch of cone angle $\pi /2\!-\!\alpha$, ($\alpha \!\ll \!1$) is pressed into the surface of an elastic half-space, as shown in Fig. 5.4. Find the contact pressure p(r), the indentation depth $\varDelta$ and the applied force P, all as functions of the radius a of the contact area.

Fig. 5.4

The conical indenter

2. An elastic half-space is indented by an axisymmetric rigid punch with the power-law profile $g_0(r)\!=\!Cr^\lambda$, so the displacement in the contact area is

$$ u_z(r)=\varDelta -Cr^\lambda , $$

where $C,\lambda$ are constants. Show that the indentation force P, the contact radius a and the indentation depth $\varDelta$ are related by the equation

$$ P=\frac{2E \varDelta a}{(1-\nu ^2)}\left( \frac{\lambda }{\lambda +1}\right) . $$

3. A rigid flat punch has rounded edges, as shown in Fig. 5.5. The punch is pressed into an elastic half-space by a force P. Assuming that the contact is frictionless, find the relation between P, the indentation depth $\varDelta$ and the radius a of the contact area.

Fig. 5.5

Flat punch with rounded corners

4. The elastic half-space is loaded by a uniform pressure $p_0$ inside the circle $0\!\le \!r\!<\!a$, the rest of the surface being unloaded. Use Eq. (5.14) to determine the appropriate function h(t) and hence find the surface displacement $u_z(r)$ both inside and outside the loaded region.

5. The profile of a smooth axisymmetric frictionless rigid punch is described by the power law

$$ g_0(r)=A_nr^{2n}, $$

where n is an integer. The punch is pressed into an elastic half-space by a force P. Find the indentation $\varDelta$, the radius of the contact area a and the contact pressure distribution p(r). Check your results by comparison with the Hertz problem of Eq. (5.30) and give simplified expressions for the case of the fourth order punch

$$ g_0(r)=A_2r^4. $$

6. An elastic half-space is indented by a rigid cylindrical punch of radius a with a concave spherical end of radius $R\!\gg \! a$. Find the contact pressure distribution p(r) and hence determine the minimum force $P_0$ required to maintain contact over the entire punch surface. Do not attempt to solve the problem for $P\!<\!P_0$.

7.

• (i) By representing the function $\varphi$ in the form

  $$ \varphi =\mathfrak {I}\int _0^b\mathcal {F}(r,z, t)h(t)dt, $$

  determine the surface tractions [tensile and compressive] needed to establish the displacement field

  $$\begin{aligned} u_z(r)= & {} \varDelta \left( 1-\frac{r^2}{b^2}\right) ^2 \quad 0\le r<b \nonumber \\= & {} 0 \qquad \qquad \qquad r>b. \nonumber \end{aligned}$$

• (ii) The otherwise flat surface of a rigid punch of radius a contains a number of small widely spaced concave dimples of radius $b\!\ll \! a$ and depth $\varDelta$. Use your result from part (i) to find the minimum force that must be applied to the punch to ensure that contact is established throughout the surface.

8. A rigid axisymmetric punch has a Hertzian profile perturbed by a set of concentric sinusoidal waves, such that the initial gap function is defined as

$$ g_0(r)=\frac{r^2}{2R}+A\left[ 1-\cos (mr)\right] , $$

where $(mA)\ll 1$.

Assuming that the contact area comprises a single circle of radius a, determine the indentation depth $\varDelta$ and the normal force P as functions of a. Use these results to make a parametric plot of P as a function of $\varDelta$ and comment on the nature of this plot as A is increased. [This problem requires the use of Maple or Mathematica].

9. Find expressions analogous to (5.10), (5.11) for the surface values of the derivatives of the function $\varphi _2$ in Eq. (5.40). In particular, verify that condition (5.37) is satisfied for all functions $h_2(t)$.

10. Use the method of Sect. 5.4 to determine the contact pressure distribution under a flat-ended cylindrical punch of radius a loaded by a force P applied through the point (c, 0). Also, find the angle of tilt of the punch. Assume that the entire flat surface of the punch makes contact. What is the maximum value of c for which this assumption is correct?

11. A flat-ended rigid cylindrical punch of radius a is pressed into the curved surface of an elastic cylinder of radius $R\!\gg \! a$ by a force P. Use the method of Sect. 5.4 to find the contact pressure distribution for the case where P is sufficient to ensure that the entire flat surface of the punch makes contact.

12. Use the method of dimensionality reduction to find the relations between the indentation $\varDelta$, the radius of the contact area a and the applied force P for the power-law punch of Problem 5.