Metal Foams: A Design Guide

118 Metal Foams: A Design Guide
considering equilibrium of the face sheet and yield of the face sheet and core.
We assume that the bending moment in the face sheets attains a maximum
local value of Mp at the edge of the indenter and also a value of Mp at a
distance from the edge of the indenter. It is further assumed that the foam
core yields with a compressive yield strength c y and exerts this level of stress
on the face sheet, as shown in Figure 10.3. Then, force equilibrium on the
segment of a face sheet of length ⊲2 C a⊳ gives
FI D ⊲2 C a⊳b c y ⊲10.15⊳
and moment equilibrium provides
Mp
1
4 bt2 f y
D 1
4 b 2 c y ⊲10.16⊳
Relations (10.15) and (10.16) can be rearranged to the form of (10.13) and
(10.14), demonstrating that the lower and upper bound solutions coincide. We
conclude that, for a rigid-perfectly plastic material response, these bounds give
exact values for the collapse load and for the span length between plastic
hinges. The presence of two indenters on the top face of the sandwich beam in
four-point bending results in a collapse load twice that for three-point bending,
but with the same wavelength .
Core shear
When a sandwich panel is subjected to a transverse shear force the shear force
is carried mainly by the core, and plastic collapse by core shear can result. Two
competing collapse mechanisms can be identified, as shown in Figure 10.4 for
the case of a beam in three-point bending. Mode A comprises plastic hinge
formation at mid-span of the sandwich panel, with shear yielding of the core.
Mode B consists of plastic hinge formation both at mid-span and at the outer
supports.
Consider first collapse mode A (see Figure 10.4). A simple work balance
gives the collapse load FA, assuming that the face sheets on the right half of
the sandwich panel rotate through an angle , and that those on the left half
rotate through an angle . Consequently, the foam core shears by an angle
. On equating the external work done Fℓ /2 to the internal work dissipated
within the core of length ⊲ℓ C 2H⊳ and at the two plastic hinges in the face
sheets, we obtain
FA D 2bt2
ℓ
f
y C 2bc c y
�
1 C 2H
ℓ
�
⊲10.17⊳
where c y is the shear yield strength for the foam core. Typically, the shear
strength of a foam is about two-thirds of the uniaxial strength, c y D 2 c y /3.