Metal Foams: A Design Guide

Sandwich structures 135
To construct such plots, the stiffness S and weight W are first defined. With
υ as the deflection and P as the transverse load amplitude, the compliance
per unit width of panel can be obtained directly from equation (10.25) as
(Allen, 1969):
1
S
b 2ℓ3 ℓ
D C
P/υ 2
B1Eftc B2cGc
⊲10.29⊳
The first term is the contribution from face sheet stretching and the second
from core shear. The result applies to essentially any transverse loading case
with the appropriate choice for B1 and B2 (Table 10.3). The weight index is:
D W
fℓ 2 2t
D C
b ℓ
c c
f ℓ
These basic results are used in all subsequent derivations.
The global minimum
⊲10.30⊳
In the search for the global optimum, the free variables are t, c, andc, with
due recognition that Gc in equation (10.29) depends on c. If the core density
c is taken to be prescribed so that Gc is fixed, the optimization proceeds
by minimizing with respect to t and c for specified stiffness. Inspection
of equation (10.29) reveals that the most straightforward way to carry out
this process is to express t in terms of c, allowing equation (10.29) to be
re-expressed with c as the only variable. For this problem, the expressions
are sufficiently simple that the minimization can be carried out analytically:
the results are given below. In other cases, the minimization may not lead to
closed-form expressions. Then, the most effective way to proceed is to create
a computer program to evaluate (or W itself) in terms of c and to plot this
dependence for specified values of all the other parameters over the range
encompassing the minimum. Gibson and Ashby (1997) have emphasized the
value of this graphical approach which can be extended to consider variations
in core density simply by plotting a series of curves for different c, analogous
to what was done in Figure 10.12.
If the global minimum is sought, the dependence of the shear modulus of
the core must be specified in terms of its density. In the following examples,
the material comprising the face sheets is assumed to be the same as the parent
material for the core (Ef D Ec D Es, when c D s D f). The dependence
of Young’s modulus of the core on c is again expressed by equation (10.22a).
Taking the Poisson’s ratio of the cellular core material to be 1 (Gibson and
3
Ashby, 1997) then Gc/Ef D ⊲ 3
8⊳⊲ c/ s⊳2 .
Although it seems paradoxical, the search for the global optimum gives
rise to simpler expressions than when the core density is fixed. The result of