Indentation Problems

Abstract

If an elastic–plastic material is indented by a rigid quadratic indenter such as a sphere, the contact problem will exhibit an elastic range, in which the stress field and the force–displacement relation are defined by the Hertzian analysis. The maximum shear stress associated with the axisymmetric Hertzian problem occurs at a depth of 0.48a, where a is the radius of the contact circle and this point reaches both the Tresca and von Mises yield conditions when $P=P_Y=\frac{21.2S_{Y}^{3}R^{2}}{E^{*}\,^2},$ where $S_Y$ is the yield stress in uniaxial tension (Johnson in Contact Mechanics, Cambridge University Press, Cambridge, 1985, Johnson 1985).

Problems

1. Show that for indentation of a homogeneous half-space by a rigid conical indenter, the contact pressure p(r) is a function of the ratio r / a only, where a is the radius of the contact area, and does not otherwise depend upon the applied force P.

2. Table 15.1 shows values of applied force P and indentation depth $\varDelta$ from a series of microindentation measurements using a pyramidal indenter that can be assumed rigid. Assuming the material to be linear elastic, use an appropriate logarithmic plot to comment on the likely form of the variation of modulus with depth.

Table 15.1 Force P and indentation $\varDelta$ in an indentation test

3. A spherical indenter is pressed into an elastic half-space by a normal force that cycles between a maximum value of $P_0$ and a minimum value of $\rho P_0$, where $0<\!\rho \!<1$. Frictional microslip at the interface leads to a dissipation of energy W per cycle. Can we argue that the dissipation has a power-law dependence on $P_0$ [i.e. $W\!\sim \! P_0^\lambda$] for a given value of $\rho$ and if so, what is the value of $\lambda$?

4. A half-plane is indented by a two-dimensional power-law rigid indenter defined by the gap function $g_0\!=\!C|x|^\alpha$. Assuming the constitutive law is defined by Eq. (15.4), find the expected power-law form of the relation between the indenting force P and the contact semi-width a.

5. Brinell hardness tests conducted on the same material with the same maximum force P, but different indenter diameters D show different values for the hardness H calculated as $P/\pi a^2$, where a is the radius of the residual indentation.

If the results are a good fit to a relation

$$ \frac{H}{H_0}=\left( \frac{D}{D_0}\right) ^\lambda , $$

where $H_0, D_0,\lambda$ are constants, show that they are consistent with a power law constitutive relation (15.4) and find the appropriate value of $\beta$ as a function of $\lambda$.