Metal Foams: A Design Guide

136 Metal Foams: A Design Guide
minimizing with respect to t, c and c with S prescribed is:
�
c 18˛2B2S
D 2
ℓ B 2 1Ef �1/5 t
� B1 c �3 c
D D 8
ℓ 96˛2B2 ℓ
s
t
⊲10.31⊳
c
Note that t, c and c are explicitly defined at the global minimum. It
is readily verified that the globally optimized beam has the following two
characteristics: (1) the compliance, S 1 D 6ℓ3 /⊲B1Eftc2⊳, has exactly twice
the contribution from the core as from the face sheets (the second term in
equation (10.29) is twice the first term); (2) perhaps more surprisingly, the
weight of the core is exactly four times that of the combined weight of the
two face sheets.
At the minimum, equation (10.31) enables the weight index, Y, to be
expressed in terms of the stiffness index X,
Y D 5
16⊲48X⊳3/5 where
�
8B1
Y D
3˛2B2
� 1/2
and X D
B 1/2
1
⊲3˛2B2/8⊳ 3/2
S
Ef
⊲10.32a⊳
⊲10.32b⊳
These are the two non-dimensional quantities plotted as the global minimum
in Figure 10.13. They contain all the information needed to characterize the
support and load conditions encompassed by Table 10.3, inclusive of the coefficient
determining the stiffness of the cellular metal, ˛2. The core density at
the global minimum can also be expressed as a function of X:
c/ s D ⊲48X⊳ 2/5 /8 ⊲10.33⊳
Fixed core density
Return now to the minimization of weight with c fixed. This is done for the
same class of sandwich beams: parent core material the same as the face sheet
material. Minimization of weight at prescribed stiffness now relates c and t to
a free parameter such that,
c
ℓ D
� 3˛2B2⊲ c/ s⊳ 3
4B1
� 1/2
and
t
c D ⊲1 2 ⊳⊲ c/ s⊳
4
⊲10.34⊳
Each value of generates a minimum weight beam for the fixed core density,
with the stiffness specified by the index X, defined in equation (10.11), now
given by:
X D 16p 2⊲ c/ s⊳ 5/2 3/2
1 4 2 ⊲10.35⊳