Metal Foams: A Design Guide

74 Metal Foams: A Design Guide
(b) Metal foams
The shear moduli of open-cell foams scales as ⊲ / s⊳ 2 and that of closedcell
foams has an additional linear term (Table 4.2). When seeking torsional
stiffness at low weight, the material index characterizing performance (see
Appendix) is G/ or G 1/2 / (solid and hollow shafts). Used as shafts, foams
have, at best, the same index value as the material of which they are made;
usually it is less. Nothing is gained by using foams as torsion members.
6.7 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, c), tensile fracture (caused by the tensile stress, t) or
yielding (caused by the shear stress s). 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 c, t and s.
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 1
4 . 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, 1 and E2, 2) are pressed together by a force F.
If the shear stress s exceeds the shear yield strength y/2, a plastic zone
appears beneath the centre of the contact at a depth of about a/2 andspreads
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:
H ³ 3 y
(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 c:
H ³ c