Metal Foams: A Design Guide

72 Metal Foams: A Design Guide
A thin-walled elastic tube will buckle inwards under an external pressure
p 0 , given in the last box. Here I refers to the second moment of area of a
section of the tube wall cut parallel to the tube axis.
(b) Metal foams
The moduli of open-cell metal foams scale as ⊲ / s⊳ 2 , that of closed-cell
foams has an additional linear term (Table 4.2). When seeking elastic-buckling
resistance at low weight, the material index characterizing performance (see
Appendix) is E 1/2 / (beams) or E 1/3 / (panels). As beam-columns, foams
have the same index value as the material of which they are made; as
panels, they have a higher one, meaning that the foam panel is potentially
lighter for the same buckling resistance. Sandwich structures with foam cores
(Chapter 10) are better still. Clamping metal foams requires special attention:
see Section 6.7.
6.6 Torsion of shafts
(a) Isotropic solids
A torque, T, applied to the ends of an isotropic bar of uniform section, and
acting in the plane normal to the axis of the bar, produces an angle of twist
. The twist is related to the torque by the first equation below, in which
G is the shear modulus. For round bars and tubes of circular section, the
factor K is equal to J, the polar moment of inertia of the section, defined in
Section 6.2. For any other section shape K is less than J. ValuesofK are
given in Section 6.2.
If the bar ceases to deform elastically, it is said to have failed. This will
happen if the maximum surface stress exceeds either the yield strength, y, of
the material or the stress at which it fractures. For circular sections, the shear
stress at any point a distance r from the axis of rotation is
D Tr
K D G r
ℓ
The maximum shear stress, max, and the maximum tensile stress, max, are
at the surface and have the values
max D max D Td0
2K
D G d0
2ℓ
If max exceeds y/2 (using a Tresca yield criterion), or if max exceeds the
MOR, the bar fails, as shown in Figure 6.5. The maximum surface stress
for the solid ellipsoidal, square, rectangular and triangular sections is at the