Similarly, for a double-hat section
[6.72]
and
[6.73]
Test points fall within the bounds given by the two formulae, for each type of hat section.
The interaction between the in-filled foam and tube walls can be under- stood as follows. The foam provides constraint when a tube wall buckles inwardly – similar to a compression-only spring. In the case of weak or non- bonding between the foam and tube, the tube wall can bend freely outward without any action from the foam. This constraint of foam has two con- sequences. First, depending upon the level of plateau stress, the foam may change the tube collapse mode: for circular tubes, from the diamond mode of an empty tube to the ring mode, and for square tubes, from non-compact to compact mode, see Fig. 6.22(b). Second, even if the collapse mode may appear the same, ring or compact mode, the plastic fold length decreases with the presence of foam; so does the proportion of inward bending – a very strong foam would completely prohibit any inward bending.
When foams are compressed, densification occurs after the plateau stage and the stress increases rapidly with the strain. This corresponding strain, the locking strain, is the limit of a compression stroke achievable by the tube and this reduces the effective length of each fold, leading to a higher average force.
Previous theoretical models for empty tubes can be modified to account for the effect of in-filled foams. For circular tubes with an axisymmetric col- lapse mode, we assume that the tube wall moves only outwards and hence the model in Fig. 6.4 is applicable. We note that the collapse of one fold
Displacement (mm) 0
10 20 30 40 50 60
Force (kN)
Density = 35 kg/m3 Density = 140 kg/m3 Density = 60 kg/m3
0 20 40 60 80 100 120 140
6.21 Force–displacement curves for a circular aluminium tube filled with polyurethane foam with three densities (Guillow et al., 2001) (reproduced with kind permission of Elsevier).
stops when the overall axial strain reaches the locking strain of the in-filled foam,el. The nominal axial strain of a tube is
[6.74]
Hence
[6.75]
qo=cos-1(1-el) el = -1 cosqo
Displacement (a)
rf = 110kgm–3 Pm = 17.5kN
Empty, Pm = 7.5kN 40
35 30 25 20 15 10 5 0
Load (kN)
(b)
6.22 Comparison of an empty and a foam-filled square tube:
(a) force–displacement curves; (b) crushing modes (c=75 mm, h=0.76 mm) (Reid et al., 1986).
This defines the complete position of a collapsing fold. The locking strain of foam is related to its relative density,r*/rs, where r* is the density of foam and rsis the density of the solid from which the foam walls are made.
If we take the strain as corresponding to a stress three times that of its plateau stress, 3sp,sp being the plateau stress, then the locking strain is approximated as
[6.76]
Substituting this equation into Eq. [6.75], we have
[6.77]
In Eqs [6.1] and [6.2], using q0instead of p/2, the modified average force for the tube alone is
[6.78]
The presence of foam reduces the fold length 2Hslightly. But, if we assume that it is the same as an empty tube, or (Eq. [6.5]), we obtain
[6.79]
By ignoring any possible increase in the cross-sectional area and from Eq.
[10.35] of the relationship between the plateau stress and the relative density, the crushing force for the foam is
[6.80]
whereYsis the yield stress of the solid cell wall of the foam. Hence, the average force for a foam-filled circular tube is
[6.81]
This analysis is due to Reddy and Wall (1988), and Eq. [6.81] agrees fairly well with their experiments (Fig. 6.23). Note that the force for this tube alone, Pmt, is also plotted and is shown to increase slightly with r*. This reflects the interaction between the foam and tube walls discussed earlier.
The increase of Pmtwith r* would be larger if the foam is assumed to lock earlier, say, when the stress is 2sp. The choice of this lock strain is a little arbitrary here.
Similar theoretical analysis can be performed for square or rectangular tubes (Reid et al., 1986), based on a modification of the analysis presented before for an empty square tube. Two empirical equations are worth
Pm =Pmt+Pf
P D
Y D
f =spp = s( s)
r r p
2 1 5 2
4 0 3
. * . 4
P M D
mt o h
s s
s
= ( )+
(
( ))
- +
È ÎÍ
˘
˚˙
- -
2 3 2 3
1 3 1
1 1
p r r r r
r r
cos * sin cos *
* Hª Dh
P M D
mt o o H
o o
o
q p q q
( )= ( + q )
( - ) +
È ÎÍ
˘
2 2 ˚˙
1 sin 1
cos qo=cos-1(3r* rs)
el = -1 3 *r rs
mentioning. Based on a numerical analysis, it is found that (Santosa and Wierzbicki, 1998b)
[6.82]
where cis the side length of a square tube and Pmhere is the average force for an empty tube. To incorporate the strain-hardening effect, the equiva- lent flow stress is taken
[6.83]
for a material having stress–strain relation
[6.84]
In Eq. [6.82], if the coefficient of the second term being 2, instead of 1, this indicates the enhancement of force as a result of foam–tube wall interac- tion. It appears that this interaction is more significant than is the case with circular tubes (Fig. 6.23).
Equation [6.84] is verified for very low strength foams (sp<1.48 MPa).
For tubes filled with aluminium foams with sp in the range 1–12 MPa, another empirical formula was proposed by Hanssen et al. (1999):
s s e
= Êe Ë
ˆ
u ¯
u n
s s
w n u
n
n n
h
= c
+ +
Ê Ë
ˆ
¯ Ê Ë
ˆ 2 23 ¯
1 2
2
2 3
4 9
.
P P c
Yc h c
m mt p
p
= +
= +
2
14 2
2 1 3
5
3 2
s s
Experiment Theory
0 5 10 15
100 200 300
Pmt Pm (kN)
r* (kgm–3)
6.23 Average compression force versus foam density for a circular section (Reddy and Wall, 1988).
[6.85]
This equation correlates very well with a large number of experiments con- ducted. The crushable length (maximum stroke) is reduced by the foam and a new maximum stroke,dmax, is given by
[6.86]
whereLis the tube length.The effect of strain-rate on Pmtcan be taken into account in the same way as in Section 6.1.2. Similarly, the enhancement of the plateau stress of foam due to the strain-rate effect can be considered.
To estimate the value of the strain-rate, assume that the foam is compressed uniformly over a fold of length 2H. For an impact velocity Vo, the initial strain-rate is the Vo/2H. The average strain-rate is then
[6.87]
For Vo=10 m/s and 2H=2.5 ~3.5 mm, the calculated strain-rate is between 100 and 200 s-1. This could lead to an increase of 50 % in plateau stress for polyurethane foams.