Metal Foams: A Design Guide

Design analysis for material selection 59
Using equations (5.2) and (5.3) to eliminate t in equation (5.1) gives the
performance equation for the performance metric, m:
m ½
� �
Ł 2
1/3
12S b
B1
�
2
ℓ
E1/3 �
⊲5.4⊳
This equation for the performance metric, m, is the objective function – it is
the quantity we wish to minimize.
All the quantities in equation (5.4) are specified by the design except the
group of material properties in the last bracket, /E1/3 . This is the material
index for the problem. The values of the performance metric for competing
materials scale with this term. Taking material M0 as the reference (the incumbent
in an established design, or a convenient standard in a new one), the
performance metric of a competing material M1 differs from that of M0 by
the factor
m1
D
m0
⊲ 1/E 1/3
1 ⊳
⊲ 0/E 1/3
0 ⊳
⊲5.5⊳
where the subscript ‘0’ refers to M0 and the ‘1’ to M1.
Panel of specified strength and minimum mass
If, for the panel of Figure 5.1, the constraint were that of bending strength
rather than stiffness, the constraining equation becomes that for failure load,
Ff, per unit width, meaning the onset of yielding:
Ff D B2 yI
btℓ ½ FŁ f ⊲5.6⊳
where B2, like B1, is a constant that depends only on the distribution of the
load; it is tabulated in Chapter 6, Section 6.4. The performance metric, again,
is the mass, m:
� �
m �
�
Ł
6Ffb 2
�1/2 ℓ
B2
3/2
1/2
y
⊲5.7⊳
where y the yield strength of the material of which the panel is made and F Ł f b
is the desired minimum failure load. Here the material index is / 1/2
y .Taking
material M0 as the reference again, the performance metric of a competing
material M1 differs from that of M0 by the factor
m1
m0
D ⊲ 1/ 1/2
y,1 ⊳
⊲ 0/ 1/2
y,0 ⊳
⊲5.8⊳