Metal Foams: A Design Guide

Sandwich structures 129
with k ¾ D 0.58 (Allen, 1969). For a given core density, W is to be minimized
with respect to t and c subject to three strength-related constraints. These
constraints follow directly from the two expressions for the maximum stresses
and the wrinkling stress:
⊲face sheet yielding, f y � ⊳ ct � ⊲1/8⊳ℓ2⊲p/ f y ⊳
⊲face sheet wrinkling,
⊲10.24a⊳
f w � ⊳ ct � ⊲1/8⊳ℓ2 � p/k⊲EfE 2 c⊳1/3� ⊲10.24b⊳
⊲core shear yielding,
c
y � ⊳ c � ⊲1/2⊳ℓ⊲p/ c y⊳ ⊲10.24c⊳
From the above it is clear that failure is restricted to either face sheet yielding
(10.24a) or wrinkling (10.24b), depending on the smaller of f y and f w .Note
that, with (10.22a), f w D kEf˛2 2/3 ⊲ c/ps⊳ 4/3 . Consequently, for an aluminum
alloy (Table 10.2) and a core with a quality factor ˛2 D 1, the wrinkling stress
exceeds the yield strength of the face sheets when c/ s > 0.036. We proceed
by considering only core materials with relative densities larger than 0.036,
with the consequence that face sheet wrinkling (10.24b) is eliminated as a
constraint. Were the core density below 0.036, face sheet wrinkling would
supersede face sheet yielding.
The deflection υ at the centre of the panel is given by (Allen, 1969; Gibson
and Ashby, 1997) (see 10.1).
υ D 2pℓ4 pℓ2
C 2
B1Eftc B2Gcc
⊲10.25⊳
where the first term is due to face sheet stretching and the second is the
contribution due to core shear. For the present case, the coefficients are B1 D
384/5 andB2 D 8. More results for a selection of loadings and boundary
conditions are given in Table 10.3. The maximum deflection will be required
to be no greater than υ.
Based on the dimensionless load and weight indices,
5 D p/ f y , D W/⊲ fbℓ 2 ⊳ ⊲10.26⊳
the two operative constraints in equation (10.24) can be re-expressed as
⊲face sheet yielding⊳ ⊲c/ℓ⊳⊲t/ℓ⊳ � ⊲1/8⊳5 ⊲10.27a⊳
⊲core shear yielding⊳ ⊲c/ℓ⊳ � ⊲1/2⊳⊲ f y / c y⊳5 ⊲10.27b⊳
The corresponding deflection constraint is
� �
fy
5 � � �� �
/Ef 2 ℓ ℓ
C
⊲c/ℓ⊳ B1 t c
Ef
�
�
B2Gc
υ
ℓ
⊲10.27c⊳