Metal Foams: A Design Guide

where
D
�
2⊲1
Eft
2 f ⊳GcR
Sandwich structures 141
⊲10.45b⊳
Here, R is the shell radius, t the face sheet thickness, c the core thickness.
The parameter measures the relative shear compliance of the core. It must
be less than 1/ p 2 to avoid localized shear kinking of the shell wall. This
result (10.45) applies if the length of the shell ℓ is at least several times the
axial buckle wavelength. It also assumes that sufficiently strong end support
conditions are in effect. End conditions modify (10.45) slightly, but not as
much as imperfections, to be discussed later.
The condition for face yielding of the perfect shell prior to (or simultaneous
with) buckling is:
N D 2t f y
⊲10.46⊳
Yielding of the core does not directly affect the load-carrying capacity since
it supports no significant load. However, it will affect the ability of the core to
suppress face sheet wrinkling and to maintain the shear stiffness necessary for
post-buckling load-carrying capacity. Yielding of the core prior to buckling
is avoided if the axial strain in the unbuckled shell (ε D N/⊲2tEf⊳) does not
exceed the uniaxial compressive yield strain of the core. In other words, core
yielding is excluded if its yield strain is larger than that of the face sheets.
If neither the core nor the face sheets yield, face sheet wrinkling may occur.
It is governed by the condition introduced earlier (equation (10.24b)). In terms
of the load per unit circumferential length, the onset of face sheet wrinkling
in the perfect shell occurs when
N D 2tk⊲EfE 2 c ⊳1/3
⊲10.47⊳
For shells with core parent material identical to the face sheet material
( s D f and Ef D Es), with core modulus and density still related by
equation (10.22a), the weight index is
D
W
2 R 2 ℓ f
D 2 t c c
C
R s R
⊲10.48⊳
When the shell buckles elastically, uninfluenced by yielding or wrinkling,
global optimization of the shell could be carried out analytically to obtain the
values of t, c and c which minimize the weight of the shell for prescribed
N. This elastic optimization implies that the core weight is twice that of the
two face sheets. However, it is of little practical value because face sheet
yielding or wrinkling invariably intervene at the levels of load index wherein
the sandwich shell is weight competitive.