## 67 Contact stresses

(a) Isotropic solids

When surfaces are placed in contact they touch at one or a few discrete points. If the surfaces are loaded, the contacts flatten elastically and the contact areas grow until failure of some sort occurs: failure by crushing (caused by the compressive stress, ac), tensile fracture (caused by the tensile stress, at) or yielding (caused by the shear stress as). The boxes in Figure 6.6 summarize the important results for the radius, a, of the contact zone, the centre-to-centre displacement u and the peak values of oc, at and as.

The first box in the figure shows results for a sphere on a flat, when both have the same moduli and Poisson's ratio has the value Results for the more general problem (the 'Hertzian Indentation' problem) are shown in the second box: two elastic spheres (radii R1 and R2, moduli and Poisson's ratios E1, v1 and E2, v2) are pressed together by a force F.

If the shear stress as exceeds the shear yield strength ay/2, a plastic zone appears beneath the centre of the contact at a depth of about a/2 and spreads to form the fully plastic field shown in the second figure from the bottom of Figure 6.6. When this state is reached, the contact pressure (the 'indentation hardness') is approximately three times the yield stress, as shown in the bottom box:

(b) Metal foams

Foams densify when compressed. The plastic constraint associated with indentation of dense solids is lost, and the distribution of displacements beneath the indent changes (bottom figure in Figure 6.6). The consequence: the indentation hardness of low-density foams is approximately equal to its compressive yield strength oc:

Ri F  Plastic indent: dense solid Plastic indent: metal foam Figure 6.6 Contact stress (°c)max = 3F 2 n a2 (ss)max = F 2 n a2 (st)max = Radii of spheres (m) Modulii of spheres (N/m2) Poission's ratios Load (N) Radius of contact (m) Dadius of contact (m) Stresses (N/m2) Yield stress (N/m2) Figure 6.6 Contact stress u If the foam is not of low density, the indentation hardness is better approximated by This means that foams are more vulnerable to contact loads than dense solids, and that care must be taken in when clamping metal foams or joining them to other structural members: a clamping pressure exceeding ac will cause damage.
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