Metal Foams: A Design Guide

Design for creep with metal foams 111
The shear deflection rate is calculated from the creep of the core. The
power-law creep of the foam core under uniaxial stress is given by:
� �nc Pε DPε0
⊲9.16⊳
0
where Pε0, 0 and nc are the creep parameters of the core material. The core
is subjected to both normal and shear loading; in general, for metallic foam
cores, both are significant. The creep shear strain rate is calculated using
equations (9.6) and (9.7):
and
Pε12 D f⊲ O ⊳
O 2 D
Noting that
e D
∂ O
∂ 12
⊲9.17⊳
1
1 C ⊲˛/3⊳ 2 [ 2 e C ˛2 2 m ] ⊲9.18⊳
� 211 C 3
2
2 3
12 C 2
and that m D 11/3 gives
O D
�
2
21
2
11 C
3
2
1 C ⊲˛/3⊳ 2 ⊲ 2 12 C 2 21⊳ �1/2 Taking the partial derivative with respect to 12 gives:
∂ O
∂ 12
D
3
2 12
1 C ⊲˛/3⊳ 2
�
211
C
3
1 C ⊲˛/3⊳ 2
� 1/2
2
12
Using equation (9.9) for f⊲ O ⊳ gives:
� �nc O 3 12
P D 2 Pε12 DPε0
1 C ⊲˛/3⊳ 2
�
211 3
C
1 C ⊲˛/3⊳ 2
0
� 1/2
2
12
⊲9.19⊳
⊲9.20⊳
⊲9.21⊳
Noting that the normal stress varies through the depth of the beam, in
general the shear strain rate has to be integrated over the depth of the beam.
The derivative of the shear deflection with respect to x, along the length of
the beam, is given by (Allen, 1969):
d Pυs
dx DPc
d
⊲9.22⊳