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Axial compression of foam-filled thin-walled circular tubes
Int. J. Impact Engng Vol. 7, No. 2, pp. 151 166, 1988 Printed in Great Britain
0734 743X/88 $3.00+0.00 © 1988 Pergamon Press plc
AXIAL COMPRESSION OF FOAM-FILLED CIRCULAR TUBES
THIN-WALLED
T. Y. REDDYand R. J. WALL Department of Mechanical Engineering, UMIST, P.O. Box 88, Manchester M60 IQD, U.K.
(Received 1 September 1987; and in revised form 27 January 1988) Summary--The effect of low density polyurethane foam on the axial crushing of thin-walled (D/t > 600) circular metal tubes is studied under quasi-static and dynamic loading conditions. The mode of deformation of the tube is found to change from irregular diamond crumpling to axisymmetric bellows folding due to the filler. Assuming the axisymmetric mode of crushing and using the model of foam recently developed (M. F. ASHBY, Metals Trans. 14A, 1755 1769, 1983), the behaviour is analysed. Theoretical predictions agree well with experiments. Numerical results show that there is an optimum foam density which produces a maximum specific energy absorption of the filled cylinders.
NOTATION
c,p D h 1o m f, mc
Mp
Pc Pt(O) Pmt(Oo)
Pm
Pf /~m t
T V
Vo O, 0 o 2e, £p Pf G GO, Gf
/;l
gr
constants in equation (13) diameter of the tube fold length of the concertina buckles, Fig. 10 initial length (height) of the tube mass of the foam and can, respectively fully plastic bending moment (per unit width) of the tube wall crushing load for diamond crumpling mode equilibrium load during concertina folding of an empty tube at any position 3, see Fig. 10 mean crushing load for a concertina fold up to a certain position defined by 0 0 Pmt at 00 = x/2 crushing load for foam only crushing load for foam and tube, equation (11) thickness of the tube wall time velocity of crushing velocity of impact compression of a fold, equation (2) angles defining intermediate and final positions of a concertina fold of tube half wavelengths of elastic and plastic buckles foam density stress yield stresses of metal and foam, respectively yield stress of foam at a given strain e, equation (21) strain locking strain for foam strain-rate rupture strain
Subscripts i,i-l,i-2
fold numbers during dynamic loading
Superscript d
dynamic
INTRODUCTION
In the design of metallic energy dissipating systems, see for example Johnson and Reid [1], thin-walled circular metal tubes under axial loading conditions have been identified as very 151
152
T.Y. RH)DY and R. J. WAL.L
efficient impact energy absorbing elements. The radius to thickness ratio of tubes extensively studied in this area is less than 50. Using various end fixtures, the tubes can be forced to (ij buckle progressively, (ii) invert externally or internally, or (iii) split into curls or flat strips. A brief review and discussion of these axial modes of deformation can be seen in a recent paper by Reid and Reddy [2]. The behaviour of a tube crushed axially between rigid platens depends on its geometrical parameters (L/D and D/t) as well as material parameters (%,+'r) and can range from axisymmetric bellows type of progressive buckling to an Euler column type of buckling failure, with progressive diamond crumpling modes also being observed. The first studies on the axial crushing of circular tubes into the plastic deformation range were due to Alexander [3] and Pugsley and Macaulay [4]. A survey of the behaviour of such tubes and literature published can be seen in Andrews et al. [5]. Recently Abramowicz and Jones [6, 7] have published further experimental data on the dynamic crushing of circular tubes and a summary of analyses produced to date by various authors to predict the mean crushing loads. If the wall thickness of a tube is very small, a non-axisymmetric buckling mode is observed in the elastic range, almost always resulting in several circumferential buckles merging together, causing instabilities leading to Euler buckling, or skewed deformations where the axis of the tube becomes zig-zagged, resulting in considerable loss of energy absorbing capacity. Such instabilities can be reduced by having a compressible filler in the tubes. Wirsching and Slater [-8], experimenting on beer cans, have found that even entrapped air can stabilize their crushing behaviour--particularly under dynamic loading conditions-and increase the energy absorbing capacity of such thin-walled tubes. Filling thin-walled tubes with rigid polyurethane foam is found to improve their stability and energy absorbing capacity. Thornton [9] has studied the behaviour of foam-filled tubular structures of various cross-sectional shapes under axial compression and has found that the mean crushing load of a tube was in excess of the sum of those of the individual components, i.e. the tube and the foam. He derived an empirical relation for the crushing load and observed that foam filling was not weight-effective and that using a stronger material or increasing the wall thickness of a given tube produced higher specific energy absorption, i.e. energy absorbed per unit mass of the component as it is crushed completely. Lampinen and Jeryan [10] have also studied the behaviour of foam-filled tubular struts of various cross-sections and have recognised the stabilising effect of the filler on the deformation of the structure. They have derived a regression model to obtain the mean crushing load of the foam-filled tubulars. Foams of higher densities were found to result in the Euler buckling mode of failure of a column which otherwise was crushed by progressive buckling. Recent studies on the crushing of thin-walled, square, sheet metal tubes ]-11] have shown that the large difference between the half wavelength, 2o, of the elastic buckles and plastic fold length, 2p, in empty tubes leads to a non-compact fold pattern, with relatively undeformed panel sections of length (2e - ;tp) in between adjacent plastic folds. A filler like polyurethane foam provides a foundation effect to the tube walls and reduces )++to the same order of magnitude as 2p which is less affected by the filler, thus causing crushing with the formation of contiguous, closely spaced, compact folds. It was found that thinner-walled tubes benefited more from foam filling in this respect. It may be noted here that the compact and non-compact crushing behaviour was observed earlier by Mahmood and Paluszny [12], respectively in thick and thin empty metal tubes. The mechanical properties of foam, i.e. the modulus of elasticity, El, and yield stress, at, of rigid polyurethane foam, depend on its density, pr. After yield, it behaves as an almost perfectly plastic material, hardening rapidly after a threshold (locking) strain, 4, which decreases with increasing density. Thus it can be considered as a perfectly plastic, rigid material [11], as shown in Fig. 1. Thornton [9] tested the foam he used and derived a simple empirical relation between yield stress under quasi-static compression and density. Hinkley and Yang [13] investigated the behaviour of polyurethane foams over a range of densities and strain rates, ~f, and derived empirical relations for modulus of elasticity as a function of density, pf, and for yield stress, ~f,
Compression of foam-filledtubes
153
o- t I
- - ACTUAL - - - IDEAIIBED
I I I I
STRAIN
CI
FIG. 1. Typicalstress-strain curve for polyurethanefoam. as a function of pf and 4--density and strain rate. More recently, Ashby and his associates published a series of papers on the quasi-static behaviour of cellular solids, a summary of which can be found in Refs [14, 15]. The mechanical properties of foam are related to the properties of cell wall materials and to the cell geometry. They describe the crushing of the cells as constituting the perfectly plastic part of the stress-strain curve and the beginning of densification of the crushed material as the threshold (locking) strain el. The yield stress derived from expressions in Refs [9, 13, 14] agree mutually as well as with the experiments [9, 11, 13, 14], for rigid polyurethane foam. In this paper, the behaviour of very thin-walled ( D / t ~ 600) cylindrical tubes--beer cans of several types--under axial compression, and the changes in the behaviour effected by filling them with rigid polyurethane foam, are studied under quasi-static and dynamic loading conditions. The behaviour of such unfilled tubes has also been studied in Refs [4, 6]. The behaviour of one type of tube is analysed in detail using the foam models for quasi-static [ 14] and dynamic [13] loading mentioned above. The filled tube behaviour is found to differ significantly from an unstable skewed deformation mode or a diamond fold mechanism with only a few lobes in an empty tube to a bellows mode in the filled cases. The predicted crushing loads agree well with the experiments. It is found that there is an optimum filler density for maximizing the specific energy absorption. The interaction and influence between the shell wall strength and foam strength are discussed. EXPERIMENTS A number of seamless aluminium alloy cans, of 440 ml capacity, formed by a deep drawing process, were used. These had dished bottoms and thicker sheet metal tops--see Fig. 2. There were four different types, based on the weight of the cans. These were filled with rigid polyurethane foam to obtain a range of densities, by mixing the liquid constituents (resin and hardener) of required quantities and casting the mixture into the cans, holding the mixture sealed to set inside the cans without allowing it to escape. The range of foam densities thus obtained ranged from 60 to 240 kg m-3. Material tests
Tension specimens were prepared from strips cut out of the cans, of type A, and were tested. The stress-strain curve thus obtained is shown in Fig. 3. A yield stress of ~ 550 MPa and a rupture strain of ~ 1 ~o were noted from these tests. Cylinders of foam of different densities were produced by casting foam in thin cylindrical tubes, as described above, and splitting open the metal sheet. Compression specimens of length equal to their diameter were prepared and compressed under quasi-static conditions to determine the stress-strain characteristics, which were similar to that shown in Fig. 1. The
T . Y . RH)I)Y and R..I. WAll
154
600
500
~400 w rr
II
I
300 200 t ¢
100
,/-- - %
I
o
1
I
0.5
I
]
1.o STRAIN, %
FIG. 2. Schematicof a can specimen.
F1G. 3. Nominal stress-engineeringstrain curve for the sheet metal of can, type A.
yield stress and locking strain were obtained from the stress-strain curves. The locking strain was obtained as e = el at a stress tr; = 2trf from the experimental stress-strain curves, as in Ref. [11]. The variations of trf and el with foam density Pr are shown in Fig. 4 where the predictions from formulae due to Hinkley and Yang [ 13] and Ashby [ 14, 15] are also shown. Physically, sections of 5 - 1 0 m m depth were observed to deform progressively.
Quasi-static tests Empty and filled specimens were crushed in an Instron universal testing machine at a cross-head speed of 20mm min-1. The empty specimens exhibited several axial and circumferential buckles in the elastic stage. After plastic collapse, around the mid-region of the specimens, several adjoining circumferential buckles merged to produce a large dent on one side with only three to five other lobes around the sections. This always led to either an Euler buckling type failure or a skewed deformation where the top and bottom faces of the specimen were parallel but not co-axial. In all the filled specimens, plastic deformation started with the collapse of the toroidal region at the lower end (of the specimen) and continued with crushing of the cylindrical regions, about l0 mm away from the ends, with a larger number of circumferential buckles than in empty tubes. Typically, the circumferential and axial half wavelengths were about 12 mm and l0 mm, respectively, in a filled specimen with pf = 90 kg m - 3. All the tubes with pr greater than 100 kg m - 3 were crushed, with the formation of axisymmetric folds. In a few specimens with foam of higher densities, one or two axial cracks were observed at the outer edges of the folds. After certain progressive buckling from the lower end, crushing started from the top end. Tubes having higher density foams were crushed partially at top and bottom and then column type buckling resulted. In such cases the tube wall invariably ruptured on the tension side of the buckling 'column'. Typical load-compression characteristics for specimens of type A are shown in Fig. 5. There is an initial peak in the load-deflection curve; this was associated with the crushing of the toroidal base. Crushing progresses at an almost constant (or mildly increasing) load with little fluctuation. After a certain level of compression, the load starts to increase rapidly, possibly indicating the starting of foam densification. While the initial peak load and
Compression of foam-filledtubes
155
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o n
:%
,' /
,'/i ,'/
u~ rr"
,S/ / /.~/ ,,#
EXPT o • ---.....
REF (11) PRESENT EQUATION(6) EQUATION(15) (£f= 0.II sec)
o,',~" 100
i "
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"~Z~\" " ~ . ' ~ ..UNCONSTRAINED "-\ ~ ~---~---_A.~-
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/,Tz~-~.~
/~ 20 /•--
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~" ~ I
t
IOO 200 FOAM DENSITY,kgm-3
EQUATION(TO) CONSTRAINED ]
300
FIG. 4. Variation of yield stress and locking strain with foam density.
constant crushing load increase with the density of foam, the compression at rapid hardening decreases. For higher density foams, mildly hardening load-compression behaviour is observed before more severe hardening. A summary of the tests carried out and the results are given in Table 1, where the mean crushing load of the specimen, Ptm, is taken as the load at the beginning of compression. The percentage compression is calculated from a deflection where the load started to increase steeply or where the tube wall ruptured. To examine the effect of the relatively thick top and bottom rings on the deformation process, these were cut off in two filled specimens, which were then tested as before. Apart from the absence of the initial peak collapse load, the characteristics were broadly similar. The load-compression curves for specimens with and without rings having nearly the same foam density are shown in Fig. 6. Typical specimens crushed under quasi-static loading are shown in Fig. 7. It may be observed from sectioned specimens that foam is trapped in the diamond lobed folds, while in axisymmetric buckles the foam remains inside as a core, acting as a mandrel to the tube.
Dynamic tests All the dynamic tests were carried out on a drop weight apparatus with a tup mass of 52 kg impacting on the specimen from the same nominal height of ~ 1.75 m. The velocity was measured in each case, and was less than the calculated value by a maximum of 12 %. In all the tests the load transmitted through the specimen to the base was measured using a piezo-
156
T . Y . REt)D'~ and R. J. WALl.
/
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Pf= 232.5 kgm-3
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SPECIMEN HEIGHT.. 148mm DIA ........ 65ram WALLTHICKNESS...0II5mn
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COMPRESSION,ram
FIG. 5. Typical load-compression curves for empty and filled specimens.
TABLE
Specimen
1 . SUMMARY
OF
QUASI-STATIC
Foam density
Initial peak load
Initial crush load
(kN)
(kN)
~°~oCompression
S.E.A. (kJ k g - i )
2.0 5.3 5.2 7.2
88 71 68 65 76 62 63 40
1.54 4.31 4.30 4.9 4.9 5.66 6.0 5.32
88 75 73 71 65 60
1.9 3.82 5.7 5.78 5.8 5.6
Type
No.
(kg m - 3)
A me = 38g
1 2 3 4 5 6 7 8
0 61.6 63.2 86.8 90.1 139.1 141.2 232.2
11.4
0.45 2.0 2.0 3.3 3.0 5.5 6 11
1 2 3 4 5 6
0 39.7 49.6 61.1 102.7 204.4
1.3 2.0 3.5 3.5 4.5 10
0.25 1.1 2.2 2.6 4.0 9.5
D
mc = 20.5g
TESTS
7.7
Remarks
No end rings
No end rings
electric (Kistler, type 4091) load-washer. The load-time trace was stored in an oscilloscope and plotted on an x - y plotter. Empty cans deformed in a diamond folding progressive buckling mode in a more consistent manner than under quasi-static loading. The folds were more even and regular. N o n e of the empty (air-filled) cans 'bottomed out', while those with foams of densities less than 80 kg m - 3 did bottom out, experiencing larger compression than the empty specimens. Some specimens slid along the base plate upon impact of the drop weight, thus being subjected to 'off-axis' loading. A few specimens buckled at the centre after folding
157
Compression of foam-filled tubes
15
14
(to=137.mm)
/////
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25
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75 100 COMPRESSION,rnm
FIG. 6. Load-compression characteristics for specimens with and without end rings.
AI
a
A2
A6
A8 FIG. 7. Photographs of some quasi-statically crushed specimens, type and number as in Table 1.
I
158
T. Y. REI)DY and R. J. WAiL
A9
AI2
ii
AIO
All AIr~
09
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DIO
DII
FIG. 8. Photographs of some dynamically crushed specimens, type and number as in Table 2.
progressively at each end. One specimen, not of the highest density tested, ruptured after a certain deformation. Some typical specimens are shown in Fig. 8. As in quasi-statically deformed specimens, the axisymmetric deformations occurred with no foam trapped in the folds and acting as a crushable mandrel. Typical load-time traces are shown in Fig. 9. A summary of the tests carried out, along with the mean crush loads at which deformation begins, the mean load over the period of deformation and the period of deformation as measured from the load-time traces, is given in Table 2. The compressed length of the specimens was measured to obtain the compression, also shown in Table 2. In all the specimens, deformations were predominantly axisymmetric buckles, although a few
Compression of foam-filled tubes
/
159
~ . . t Pi =61-1 kgrfi 3
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FIG. 9. Typical dynamic load-time traces for specimens of type A. TABLE 2. SUMMARY OF DYNAMIC TESTS
Specimen
load (kN)
Shortening (ram)
Time for deformation (msec)
Crush
Mean
load (kN)
Type
No.
Foam density (kg m - 3}
A mc = 38g
9 10 11 12 13 14 15
0 81.1 94.3 115.5 148.2 176.1 183.3
2.0 2.25 5.5 6.5 9 11 12
6.5 -7.5 8.75 11 8.5 12.5
108 125 117 101 76 105 61
22.5 31.0 32.0 32.0 30.0 29.0 26.0
B mc = 39.2g
1 2 3 4 5
0 76.1 88.9 91.0 96.6
2.0 4 4.5 5.25 6.25
6.25 --8.4 9
118 115 108 95 93
22.5 31.5 31.5 31.5 31.0
C mc = 27g
1 2 3 4
62 93 104 115
3.75 5.25 6.3 8.3
-8.5 9.25 10
116 93 91 88
29.5 31.0 31.0 31.5
D mc = 20.5g
7 8 9 10 11
0 67.5 79.4 96.7 106.8
1.75 3 4 2.5 6.25
6.5 --
115 117 118
-9.0
93
22.5 28 29 26 32
1 2 3 4 5
0 116 135.7 152 163.6
2 8.5 8.75 10 11.25
7 9.2 10.5 11.5 11.5
113 97 82 75 58
22.5 31 30 30 25
E mc = 40.6g
Remarks
Bottomed out
Can ruptured
Crushed & bent Skewed and bottomed out Bottomed out
Bottomed out Bottomed out Skewed and bottomed out
diamond folds were also observed. Generally, the degree of compression reduced with increasing density of foam in the tubes• Sectioned specimens showed that there was no adhesion between the tube and the foam. A very thin and flimsy layer of foam not attached either to the tube wall or the foam core was seen adjacent to the wall of the tube.
160
T.Y. REDDYand R. J. WALL ANALYSIS
General expressions for crushing forces and deformations are derived in this section. The crushing strength of a foam-filled tube is due to the plastic deformations of the shell and crushing of foam. The interaction between the filler and container is considered to modify the mode of deformation of the container, from an irregular diamond mode folding to an axisymmetric (or nearly axisymmetric) bellows mode deformation. The degree of compression of a given fold of the tube is also limited and governed by the strain at which the crushing of foam is completed and its densification begins. The limited degree of deformation, in turn, increases the mean crushing load of the shell. The foam strength is assumed not to be affected by the small lateral compression imposed on it by the shell. Numerical calculations have been carried out for tubes of type A whose material properties were obtained from tension tests mentioned above.
Quasi-static compression The axial crushing of a thin empty tube deforming in a diamond crumpling mode has been studied by Pugsley and Macaulay [4]. Pugsley [16] has suggested an expression for the crushing load Pc, which can be written as Pc = 228.5 mp,
(1)
where Mp = aotZ/4 is the fully plastic bending moment per unit width of the tube sheet with yield stress a 0 and thickness t. This equation produces a value of 0.4 kN, to be compared with an experimental value of 0.45 k N - - F i g . 5, Table 1. For axisymmetric concertina buckling observed in filled tubes, Alexander's deformation model [3] is used with modifications to account for the presence of foam. It is assumed that the tube wall moves only outwards from its initial position and that its deformation takes place due to rotation at the three hinges, as shown in Fig. 10. Equating the rate of work for plastic deformation due to bending at the plastic hinges and stretching of the regions in between to the rate of external work done at any position defined by the angle 0, i.e. at a compression 6 = 2h(1 - cos 0),
/
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,, / /
f
/
l
l
g ,/
_l_ t
D
i
(2)
i
.,r
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FIG. 10. Idealizedmodel of deformationfor concertina mode of buckling of an axiallycompressed cylindrical shell.
Compression of foam-filledtubes
161
the equilibrium load for the tube can be shown to be 2rtMp [-D h 1 ~- + sin 0 + -cost 0 j ,
Pt(O) = ~ [
(3)
where D is the diameter of the tube and h is the optimum value of the fold length (see Fig. 10) to produce a minimum value for the mean load when the fold is completely formed. It can be shown that h ,,~ x/(Dt). The mean load up to a given compression defined by angle 0 o can be obtained by rearranging and integrating equation (3) between limits 0 = 0 and 0 = 0 o, as
[DOo+2sinOo Pmt(0o)=2nMp ~
1--cos0 o
] t-1 .
(4)
When 0 o = rt/2, one gets Pm= 2rOMp
+ 2) + 1 ,
(5)
which is applicable to empty tubes. The presence of foam slightly changes the plastic fold length, 2h, from that of the empty tube [11]. This consideration is beyond the scope of the present investigation, and in the present case h ~- ,f(Dt) is assumed to be unaffected by pf. Foam is crushed by a deforming fold and causes the deformation of the fold to cease at 0o < ~/2. The value of 0o will be found in the following by assuming that the foam becomes rigid at a locking strain which is taken as the strain at the beginning of its densification. The plateau stress, trt, for polyurethane foam of density Of is shown [14] to be adequately represented by the expression
trf = 0.3try(pf/ps) 1"5,
(6)
where try 127 MPa and Ps = 1200 kg cm-3 are respectively the yield stress and density of the cell walls of rigid polyurethane foam [ 14]. As the foam is compressed further, the yielding of foam ceases and densification commences at a 'locking strain', el, which is given as [15] =
el = 1 - 3pr/ps.
(7a)
It may be mentioned here that in Ref. [14], a different expression for el, given by
el = 1 - 2pr/ps.
(7b)
is suggested. Equation (7a) is used in the following and the choice is discussed below. Now, 0 o can be determined from 131.A cylinder of foam having an initial height 2h is compressed to 2h cos 0 o, causing a nominal strain in foam given by /~f = (1 - - COS 0 0 ) .
(8)
Equating ef to el, equation (7a), the value of 0 o can be seen, after substituting for ps, to be 0 o = cos- 1(3pf/p~).
(9)
Thus the mean crushing load of the tube can be obtained from equation (4) as a function of foam density. The crushing strength of foam itself can be written as Pf = trf~zD2/4.
(10)
Hence the crushing force of the foam-filled tube is given by P'm = Pmt(O0) + Pt.
(11)
The energy absorbed by a specimen of an initial length Io undergoing a maximum deformation e,lo is E.A. = ptmeflo and the specific energy absorption is S.E.A. = E.A./(mf + m~),
(12)
162
T. Y, REDDY and R. J. WALL
_~15 //~
< LIA tt3 Z
Pmox
Z//
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13..
//'
10
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"/~
,Y./,"
ExPT
./". /
..0
Pint 1
o
t
100
200
of, kgr63
I
300
FIG. 11. Variation of initial crush load, crush load at locking strain and specificenergy absorption against foam density for specimens of type A crushed under quasi-static conditions. where mr = nD2lopf/4 is the mass of foam in the can and mc is the mass of the can. The variation of Ptm and S.E.A. against pf is shown in Fig. 11 along with experimental values. The variation of mean crushing load of tube only (which depends on 0o and hence on pf) is also shown in this figure.
Dynamic loading The modes of deformation of specimens under quasi-static and dynamic loadings were identical. Hence the inertia effects can be assumed to be negligible and only the strain-rate effects need to be considered. The strain-rate dependence of the tube wall can be described by the C o w p e r - S y m o n d s [17] relation = GO
t Y i" l+iri\j
,
(13)
where a~ is the yield stress at a strain rate ~ and c = 6500 s - 1 and p = 4 are constants for aluminium alloys. Thus for a given, ith, fold with a mean strain-rate o f ~ , the mean dynamic crushing load of the tube wall will be
s m. =
pmt[ 1 ~,i o.25 ,_ + ( 6 5 - ~ ) 1 "
(14)
The strain-rate dependence of polyurethane foams is described by the empirical equation due to Hinkley and Yang [11] for the dynamic yield stress of foam: af~ = (552 + 19.5 In ef)pr1"64.
(15)
Strain-rate varies from fold to fold as well as varying during a fold. For simplicity, velocity is assumed to remain constant during the deformation of a fold. The mean hoop strain in the metal sheet is eh = h sin Oo/D and the strain in the foam is el during the ith fold, when the velocity of compression is Vi_ 1. The above strains are caused in a time T, -
2h(1 - cos 0o)
Vi-i
(16)
Compression of foam-filledtubes
I0 Z
163
-,ooI \/, -
50
//
~// /,.~///
--EXPT --- THEORY
I
0
I00
I
200
pf, kgm-3
FIG. 12. Variation of mean dynamic load and compressionagainst foam density for specimens of type A. (A typical predicted load-time trace is also shown.) Hence the strain rates in the tube wall and foam, for the ith fold, can be shown to be
and
o e~i =
V~_ 1 sin 00 2D(1 - cos 0o)
o
Vi_ 1
~i =
(17) (18)
2h
Equations (17) and ( 18 ) can be used with equations (14) and ( 15 ) to obtain the dynamic crush load for the ith fold as d xD2 PCmi = pdmti + O'fl
~
•
(19)
The velocity of crushing, Vi- ~, for the ith fold can be obtained from an energy balance before and after the ( i - 1)th fold: 1 2 ~MoVi_ 2 = ½MoVi 2_ , + Pat i_ ,2h(1 - cos 0o),
(20)
where Mo is the mass of the drop weight and V~_2 is the velocity at the beginning of the ( i - 1)th fold. Pati_ 1 is the crushing force for the ( i - 1)th fold. Using equations (14)-(20), the dynamic load-time trace has been computed, and is shown along with the experimental trace in Fig. 9. The mean loads and corresponding compressions calculated from the above equations are plotted in Fig. 12 against Pr. The relevant experimental values (Table 2) are also shown in this figure. DISCUSSION
Bucklino modes
The number of circumferential lobes and axial fold lengths in axially compressed thinwalled tubes in the plastic deformation range are unpredictable because of the merging of several adjoining elastic buckles. This is due to the considerably smaller resistance to bending than to stretching of the tube wall. Hence the deformations are inextensional. The edges of an inward elastic buckle transform to travelling plastic hinges after collapse. These travel outwards, feeding material from adjacent buckles into the inward buckle, causing it to grow into a large dent. Several such mechanisms greatly reduce the number of lobes during plastic deformation. The extent of growth of these dents will be limited by the number of such
164
T.Y.REDDY R. WALL and
J.
mechanisms around the circumference. A sharp corner at the travelling hinge or meeting of two plastic hinges travelling in opposite directions stops the growth of the dent; folding will continue. Unstable bending failure results if a dent becomes too large. The presence of crushable material like foam, despite having little crush resistance in itself, acts as a foundation and resists the otherwise almost free inward movement of the tube wall, thus increasing the number of lobes in diamond crumpling, leading to axisymmetric bellows "crinkling" mode of plastic deformation. Both the strength of foam and change of mode of deformation of the tube increase the crushing force.
Foam behaviour The model of foam behaviour of Ashby [14, 15] is seen to provide agreeable results for quasi-static cases, as does the strain-rate sensitive model of Hinkley and Young [13]. The locking strain of foam has a linear dependence on its initial density. The two expressions suggested by Ashby [equation (7)] are plotted in Fig. 4. The experimental values plotted in this figure are derived as strains at twice the yield stress, from tests on constrained [11] and unconstrained foam compression tests with which the correspondence of equations (7a&b) is clear. It may be mentioned that the yield stress of foam does not show such dependence on the type of testing; see Fig. 4. The locking strain e,~ has an effect on the S.E.A. and this is discussed below. A few diamond crumpling folds were observed in specimens with low density foams, changing to axisymmetric folds as the deformation progressed. This is possibly because of the weaker foundation stiffness the low density foams initially provide to the tube wall. As compression progresses, the density of foam just below the folds increases, resulting in an increase in foundation stiffness. This is possibly due to the non-adhesion between the tube and foam, or the fact that 2h (or its equivalent in diamond crumpling mode) is smaller than the unit crushing height of a foam column, (observed as ~ 10 mm), as in the present study. The load-compression curves, Fig. 5, show hardening to a certain degree, depending upon pf, before the rapid hardening of the locking type. This can be attributed to the similar hardening stress-strain behaviour of foam. In Ref. [15], an expression for the strain dependent flow stress a~ of foam at a strain e is derived as
Ii
1--(Pf~l/3 -~ _
(
\P,/
P~\l-e/J
_I
_J
where af is given by equation (6). Using this equation, the crushing load for foam at locking strain is computed, added to the mean load for the tube and plotted in Fig. 11 as Pmax.The experimental values observed (at the experimental locking strains) also shown in this figure agree well with the predictions.
Eneroy absorbing capacity of foam-ill.led tubes The enhancement in the energy absorbing capacity due to filling is not only because of the crush strength of foam but also because of its interaction with the container resulting in a change of the mode of deformation from an irregular diamond crumpling mode to concertina mode. In addition, the degree of compression is changed because of the filler. Increased foam density decreases ef and thus 00, leading to an increase in the crush load while decreasing the stroke. The overall effect of density on specific energy absorption is shown in Fig. 11. For the tube considered, we find, theoretically, that the S.E.A. is highest when p f - 150 kg m-3. The experimental values also shown in the graph broadly agree with the theoretical prediction. A reason for the considerable overestimation of the analysis should be the use of a mean crushing strength for foam, calculated as an average of the initial crushing strength and that at the locking strain. A perfectly plastic material model for foam, with a crush load given by equation (10), improves the agreement between calculated and measured values.
Compression of foam-filledtubes
165
As Thornton [9] has observed, it is possible to obtain the increase in the crushing load by using stronger material (increased try) or by increasing the thickness of the tube. He argues that in either case similar or better weight effectiveness (S.E.A.) can be achieved. However, stability of deformation will not be improved by increasing try, while a large increase in thickness (to make D / t < 100) will be required to obtain stable compression behaviour of an empty tube. An axisymmetric bellows m o d e is assumed throughout in the analysis; this is found to provide sufficiently accurate prediction of the loads and energy absorbing capabilities of the tubes tested. The assumption made in the analysis of dynamic loading, i.e. that a fold forms at a constant velocity, equal to that at the beginning, is validated by the results; see Fig. 12. This need not be true in cases where only a few folds form to dissipate all the energy. The stepwise decrease of velocity assumed above is a good approximation only when the number of steps, i.e. the number of folds, is large. In the present case, both the dynamic crushing loads and periods of compression as computed agree well with the experiment. Dynamic behaviour o f empty (air-filled) tubes
Better stability of empty cans under dynamic loading can be attributed to two reasons. First is the near adiabatic compression, which provides a foundation effect. Second is that, under dynamic loading, travelling hinges cannot form and move at a speed required to create dents and instabilities. Further analysis is required to strengthen these arguments. However, it may be mentioned that the air-filled cans did not bottom out as did the specimens with low density foam. There is also a larger difference between the initial kinetic energy of the tup and the energy absorbed through plastic deformation (mean load crush length) in the air-filled cans than in the foam-filled cans of all the categories, which can only be due to the energy dissipated during the compression of air.
CONCLUSIONS The axial crushing behaviour of very thin (D/t > 600) empty and foam filled circular tubes under quasi-static and dynamic loading conditions is studied experimentally and theoretically. It is seen that the stability of crushing is improved by the presence of a filler such as polyurethane foam. The analytical models of such a foam described by Ashby [14, 15] are used to predict the effect of foam on the crush-strength of the tube. The increase in the crushing load is not only because of the crush strength of foam but also because of its effect in changing (i) the mode of deformation and (ii) the compressibility of the tube wall, both of which increase the mean crush strength of the tube. For a given set of tube parameters (tr 0, t) there is an optimum density of foam which provides a m a x i m u m specific energy absorption. Acknowledgements--The authors would like to thank Prof. S. R. Reid for his helpful comments on the manuscript and Mrs M. Mellor and Miss C. Tyler for typing it.
REFERENCES 1. W. JOHNSONand S. R. REID,Metallic energy dissipating systems. Appl. Mech. Rev., 31,277-288 (1978). See also Applied Mechanics Update (Edited by STEELEC. R. and SPRINGERG. S.), pp. 303-320. ASME (1986). 2. S. R. REIDand T. Y. REDDY,Axiallyloaded metal tubes as impact energy absorbers. In Proc. IUTAM Syrup. Inelastic Behaviour Plates and Shells. Springer, Bedin (1986). 3. J. M. ALEXANDER,An approximate analysis of the collapse of thin cylindrical shells under axial loading. Q. J. Mech. appl. Math. 13, 10-15 (1960). 4. A. PUGSLEYand M. A. MACAULAY,The large scale crumpling of thin cylindrical columns. Q. J. Mech. appl. Math. 13, 1-9 (1960). 5. K.R.F. ANDREWS,G. L. ENGLANDand E. GrlANI,Classificationof the axial collapseofcylindrical tubes under quasi-static loading. Int. J. Mech. Sci. 25, 687-696 (1983). 6. W. AaRAMOWlCZand N. JONES,Dynamic axial crushing of circular tubes. Int. J. Impact Engng 2, 263-281 (1984).
166
T.Y. REDDY and R. J. WALL
7. W. ABRAMOWICZand N. JONES, Dynamic progressive buckling of circular and square tubes. Int. J. Impact Engn,q 4, 243--270 (1986). 8. P.H. WIRSCH~NGand R. C. SLATER,The beer can as a shock absorber. J. Engn9 Matls Tech. Trans. ASME 95, 224 226 (1973). 9. P. H. THORNTON, Energy absorption by foam filled structures. SAE paper 800081 (1980). 10. B. E. LAMPINEN and R. A. JERYAN, Effectiveness of polyurethane foam in energy absorbing structures. SAE paper 820494 (1982). 11. S. R. RE1D,T. Y. REDDYand M. D. GRAY,Static and dynamic axial crushing of foam-filled sheet metal tubes. Int. J. Mech. Sci. 28, 295-322 (1986). 12. H. F. MAHMOOD and A. PALNSZNY, Design of thin-walled columns for crash energy management--their strength and mode of collapse. Proc. 4th Int. Conf. Vehicle Struct. Mech. SAE paper 811302, 7-18 (1981). 13. W.M. HINKLEYand J. C. S. YANG,Analysis of rigid polyurethane foam as a shock mitigator. Expl Mech., Proc. SESA 177 183 (1975). 14. M. F. ASHBY, The mechanical properties of cellular solids. Metals Trans. 14A, 1755-1769 (1983). 15. S.K. MAITI, L. J. GIBSON and M. F. ASHBY,Deformation and energy absorption diagrams for cellular solids. Acta Metall. 32, 1963-1975 (1984). 16. A. PUGSLEY,On crumpling of thin tubular struts. Q. J. Mech. appl. Math. 32, 1-7 (1979). 17. G. R. COWPERand P. S. SYMONDS,Strain hardening and strain-rate effects in the impact loading of cantilever beams. Brown University, Tech. Rep. 28 (1957).
0734 743X/88 $3.00+0.00 © 1988 Pergamon Press plc
AXIAL COMPRESSION OF FOAM-FILLED CIRCULAR TUBES
THIN-WALLED
T. Y. REDDYand R. J. WALL Department of Mechanical Engineering, UMIST, P.O. Box 88, Manchester M60 IQD, U.K.
(Received 1 September 1987; and in revised form 27 January 1988) Summary--The effect of low density polyurethane foam on the axial crushing of thin-walled (D/t > 600) circular metal tubes is studied under quasi-static and dynamic loading conditions. The mode of deformation of the tube is found to change from irregular diamond crumpling to axisymmetric bellows folding due to the filler. Assuming the axisymmetric mode of crushing and using the model of foam recently developed (M. F. ASHBY, Metals Trans. 14A, 1755 1769, 1983), the behaviour is analysed. Theoretical predictions agree well with experiments. Numerical results show that there is an optimum foam density which produces a maximum specific energy absorption of the filled cylinders.
NOTATION
c,p D h 1o m f, mc
Mp
Pc Pt(O) Pmt(Oo)
Pm
Pf /~m t
T V
Vo O, 0 o 2e, £p Pf G GO, Gf
/;l
gr
constants in equation (13) diameter of the tube fold length of the concertina buckles, Fig. 10 initial length (height) of the tube mass of the foam and can, respectively fully plastic bending moment (per unit width) of the tube wall crushing load for diamond crumpling mode equilibrium load during concertina folding of an empty tube at any position 3, see Fig. 10 mean crushing load for a concertina fold up to a certain position defined by 0 0 Pmt at 00 = x/2 crushing load for foam only crushing load for foam and tube, equation (11) thickness of the tube wall time velocity of crushing velocity of impact compression of a fold, equation (2) angles defining intermediate and final positions of a concertina fold of tube half wavelengths of elastic and plastic buckles foam density stress yield stresses of metal and foam, respectively yield stress of foam at a given strain e, equation (21) strain locking strain for foam strain-rate rupture strain
Subscripts i,i-l,i-2
fold numbers during dynamic loading
Superscript d
dynamic
INTRODUCTION
In the design of metallic energy dissipating systems, see for example Johnson and Reid [1], thin-walled circular metal tubes under axial loading conditions have been identified as very 151
152
T.Y. RH)DY and R. J. WAL.L
efficient impact energy absorbing elements. The radius to thickness ratio of tubes extensively studied in this area is less than 50. Using various end fixtures, the tubes can be forced to (ij buckle progressively, (ii) invert externally or internally, or (iii) split into curls or flat strips. A brief review and discussion of these axial modes of deformation can be seen in a recent paper by Reid and Reddy [2]. The behaviour of a tube crushed axially between rigid platens depends on its geometrical parameters (L/D and D/t) as well as material parameters (%,+'r) and can range from axisymmetric bellows type of progressive buckling to an Euler column type of buckling failure, with progressive diamond crumpling modes also being observed. The first studies on the axial crushing of circular tubes into the plastic deformation range were due to Alexander [3] and Pugsley and Macaulay [4]. A survey of the behaviour of such tubes and literature published can be seen in Andrews et al. [5]. Recently Abramowicz and Jones [6, 7] have published further experimental data on the dynamic crushing of circular tubes and a summary of analyses produced to date by various authors to predict the mean crushing loads. If the wall thickness of a tube is very small, a non-axisymmetric buckling mode is observed in the elastic range, almost always resulting in several circumferential buckles merging together, causing instabilities leading to Euler buckling, or skewed deformations where the axis of the tube becomes zig-zagged, resulting in considerable loss of energy absorbing capacity. Such instabilities can be reduced by having a compressible filler in the tubes. Wirsching and Slater [-8], experimenting on beer cans, have found that even entrapped air can stabilize their crushing behaviour--particularly under dynamic loading conditions-and increase the energy absorbing capacity of such thin-walled tubes. Filling thin-walled tubes with rigid polyurethane foam is found to improve their stability and energy absorbing capacity. Thornton [9] has studied the behaviour of foam-filled tubular structures of various cross-sectional shapes under axial compression and has found that the mean crushing load of a tube was in excess of the sum of those of the individual components, i.e. the tube and the foam. He derived an empirical relation for the crushing load and observed that foam filling was not weight-effective and that using a stronger material or increasing the wall thickness of a given tube produced higher specific energy absorption, i.e. energy absorbed per unit mass of the component as it is crushed completely. Lampinen and Jeryan [10] have also studied the behaviour of foam-filled tubular struts of various cross-sections and have recognised the stabilising effect of the filler on the deformation of the structure. They have derived a regression model to obtain the mean crushing load of the foam-filled tubulars. Foams of higher densities were found to result in the Euler buckling mode of failure of a column which otherwise was crushed by progressive buckling. Recent studies on the crushing of thin-walled, square, sheet metal tubes ]-11] have shown that the large difference between the half wavelength, 2o, of the elastic buckles and plastic fold length, 2p, in empty tubes leads to a non-compact fold pattern, with relatively undeformed panel sections of length (2e - ;tp) in between adjacent plastic folds. A filler like polyurethane foam provides a foundation effect to the tube walls and reduces )++to the same order of magnitude as 2p which is less affected by the filler, thus causing crushing with the formation of contiguous, closely spaced, compact folds. It was found that thinner-walled tubes benefited more from foam filling in this respect. It may be noted here that the compact and non-compact crushing behaviour was observed earlier by Mahmood and Paluszny [12], respectively in thick and thin empty metal tubes. The mechanical properties of foam, i.e. the modulus of elasticity, El, and yield stress, at, of rigid polyurethane foam, depend on its density, pr. After yield, it behaves as an almost perfectly plastic material, hardening rapidly after a threshold (locking) strain, 4, which decreases with increasing density. Thus it can be considered as a perfectly plastic, rigid material [11], as shown in Fig. 1. Thornton [9] tested the foam he used and derived a simple empirical relation between yield stress under quasi-static compression and density. Hinkley and Yang [13] investigated the behaviour of polyurethane foams over a range of densities and strain rates, ~f, and derived empirical relations for modulus of elasticity as a function of density, pf, and for yield stress, ~f,
Compression of foam-filledtubes
153
o- t I
- - ACTUAL - - - IDEAIIBED
I I I I
STRAIN
CI
FIG. 1. Typicalstress-strain curve for polyurethanefoam. as a function of pf and 4--density and strain rate. More recently, Ashby and his associates published a series of papers on the quasi-static behaviour of cellular solids, a summary of which can be found in Refs [14, 15]. The mechanical properties of foam are related to the properties of cell wall materials and to the cell geometry. They describe the crushing of the cells as constituting the perfectly plastic part of the stress-strain curve and the beginning of densification of the crushed material as the threshold (locking) strain el. The yield stress derived from expressions in Refs [9, 13, 14] agree mutually as well as with the experiments [9, 11, 13, 14], for rigid polyurethane foam. In this paper, the behaviour of very thin-walled ( D / t ~ 600) cylindrical tubes--beer cans of several types--under axial compression, and the changes in the behaviour effected by filling them with rigid polyurethane foam, are studied under quasi-static and dynamic loading conditions. The behaviour of such unfilled tubes has also been studied in Refs [4, 6]. The behaviour of one type of tube is analysed in detail using the foam models for quasi-static [ 14] and dynamic [13] loading mentioned above. The filled tube behaviour is found to differ significantly from an unstable skewed deformation mode or a diamond fold mechanism with only a few lobes in an empty tube to a bellows mode in the filled cases. The predicted crushing loads agree well with the experiments. It is found that there is an optimum filler density for maximizing the specific energy absorption. The interaction and influence between the shell wall strength and foam strength are discussed. EXPERIMENTS A number of seamless aluminium alloy cans, of 440 ml capacity, formed by a deep drawing process, were used. These had dished bottoms and thicker sheet metal tops--see Fig. 2. There were four different types, based on the weight of the cans. These were filled with rigid polyurethane foam to obtain a range of densities, by mixing the liquid constituents (resin and hardener) of required quantities and casting the mixture into the cans, holding the mixture sealed to set inside the cans without allowing it to escape. The range of foam densities thus obtained ranged from 60 to 240 kg m-3. Material tests
Tension specimens were prepared from strips cut out of the cans, of type A, and were tested. The stress-strain curve thus obtained is shown in Fig. 3. A yield stress of ~ 550 MPa and a rupture strain of ~ 1 ~o were noted from these tests. Cylinders of foam of different densities were produced by casting foam in thin cylindrical tubes, as described above, and splitting open the metal sheet. Compression specimens of length equal to their diameter were prepared and compressed under quasi-static conditions to determine the stress-strain characteristics, which were similar to that shown in Fig. 1. The
T . Y . RH)I)Y and R..I. WAll
154
600
500
~400 w rr
II
I
300 200 t ¢
100
,/-- - %
I
o
1
I
0.5
I
]
1.o STRAIN, %
FIG. 2. Schematicof a can specimen.
F1G. 3. Nominal stress-engineeringstrain curve for the sheet metal of can, type A.
yield stress and locking strain were obtained from the stress-strain curves. The locking strain was obtained as e = el at a stress tr; = 2trf from the experimental stress-strain curves, as in Ref. [11]. The variations of trf and el with foam density Pr are shown in Fig. 4 where the predictions from formulae due to Hinkley and Yang [ 13] and Ashby [ 14, 15] are also shown. Physically, sections of 5 - 1 0 m m depth were observed to deform progressively.
Quasi-static tests Empty and filled specimens were crushed in an Instron universal testing machine at a cross-head speed of 20mm min-1. The empty specimens exhibited several axial and circumferential buckles in the elastic stage. After plastic collapse, around the mid-region of the specimens, several adjoining circumferential buckles merged to produce a large dent on one side with only three to five other lobes around the sections. This always led to either an Euler buckling type failure or a skewed deformation where the top and bottom faces of the specimen were parallel but not co-axial. In all the filled specimens, plastic deformation started with the collapse of the toroidal region at the lower end (of the specimen) and continued with crushing of the cylindrical regions, about l0 mm away from the ends, with a larger number of circumferential buckles than in empty tubes. Typically, the circumferential and axial half wavelengths were about 12 mm and l0 mm, respectively, in a filled specimen with pf = 90 kg m - 3. All the tubes with pr greater than 100 kg m - 3 were crushed, with the formation of axisymmetric folds. In a few specimens with foam of higher densities, one or two axial cracks were observed at the outer edges of the folds. After certain progressive buckling from the lower end, crushing started from the top end. Tubes having higher density foams were crushed partially at top and bottom and then column type buckling resulted. In such cases the tube wall invariably ruptured on the tension side of the buckling 'column'. Typical load-compression characteristics for specimens of type A are shown in Fig. 5. There is an initial peak in the load-deflection curve; this was associated with the crushing of the toroidal base. Crushing progresses at an almost constant (or mildly increasing) load with little fluctuation. After a certain level of compression, the load starts to increase rapidly, possibly indicating the starting of foam densification. While the initial peak load and
Compression of foam-filledtubes
155
/
o n
:%
,' /
,'/i ,'/
u~ rr"
,S/ / /.~/ ,,#
EXPT o • ---.....
REF (11) PRESENT EQUATION(6) EQUATION(15) (£f= 0.II sec)
o,',~" 100
i "
/
"~Z~\" " ~ . ' ~ ..UNCONSTRAINED "-\ ~ ~---~---_A.~-
~.
/,Tz~-~.~
/~ 20 /•--
/ - ~ EQUATION(7b)
~" ~ I
t
IOO 200 FOAM DENSITY,kgm-3
EQUATION(TO) CONSTRAINED ]
300
FIG. 4. Variation of yield stress and locking strain with foam density.
constant crushing load increase with the density of foam, the compression at rapid hardening decreases. For higher density foams, mildly hardening load-compression behaviour is observed before more severe hardening. A summary of the tests carried out and the results are given in Table 1, where the mean crushing load of the specimen, Ptm, is taken as the load at the beginning of compression. The percentage compression is calculated from a deflection where the load started to increase steeply or where the tube wall ruptured. To examine the effect of the relatively thick top and bottom rings on the deformation process, these were cut off in two filled specimens, which were then tested as before. Apart from the absence of the initial peak collapse load, the characteristics were broadly similar. The load-compression curves for specimens with and without rings having nearly the same foam density are shown in Fig. 6. Typical specimens crushed under quasi-static loading are shown in Fig. 7. It may be observed from sectioned specimens that foam is trapped in the diamond lobed folds, while in axisymmetric buckles the foam remains inside as a core, acting as a mandrel to the tube.
Dynamic tests All the dynamic tests were carried out on a drop weight apparatus with a tup mass of 52 kg impacting on the specimen from the same nominal height of ~ 1.75 m. The velocity was measured in each case, and was less than the calculated value by a maximum of 12 %. In all the tests the load transmitted through the specimen to the base was measured using a piezo-
156
T . Y . REt)D'~ and R. J. WALl.
/
/ 15
i
Pf= 232.5 kgm-3
J J
Z
t..
SPECIMEN HEIGHT.. 148mm DIA ........ 65ram WALLTHICKNESS...0II5mn
/'
c~ oU..
10
/ 0f=139"1 kOjn-31.~.j'~ ' ./"
I
j.
,,.. ,,./--~-v,-,"-"'-"
\,j-J
ef=e6.8,
.............. I
~
" .......... - - ~'-"
" ~ " "
~ " ~ F
-/ .....
k,
3
/
_ ,---'"
/
/
pf=61.6 kgm-3 ~ . / "
~ ' ~ "- J ~ - ' ' % - - - - "
°~
- ""--'~-
- ~"""
/
EMPTY J I
I
7s
5o
,6o
I
125
COMPRESSION,ram
FIG. 5. Typical load-compression curves for empty and filled specimens.
TABLE
Specimen
1 . SUMMARY
OF
QUASI-STATIC
Foam density
Initial peak load
Initial crush load
(kN)
(kN)
~°~oCompression
S.E.A. (kJ k g - i )
2.0 5.3 5.2 7.2
88 71 68 65 76 62 63 40
1.54 4.31 4.30 4.9 4.9 5.66 6.0 5.32
88 75 73 71 65 60
1.9 3.82 5.7 5.78 5.8 5.6
Type
No.
(kg m - 3)
A me = 38g
1 2 3 4 5 6 7 8
0 61.6 63.2 86.8 90.1 139.1 141.2 232.2
11.4
0.45 2.0 2.0 3.3 3.0 5.5 6 11
1 2 3 4 5 6
0 39.7 49.6 61.1 102.7 204.4
1.3 2.0 3.5 3.5 4.5 10
0.25 1.1 2.2 2.6 4.0 9.5
D
mc = 20.5g
TESTS
7.7
Remarks
No end rings
No end rings
electric (Kistler, type 4091) load-washer. The load-time trace was stored in an oscilloscope and plotted on an x - y plotter. Empty cans deformed in a diamond folding progressive buckling mode in a more consistent manner than under quasi-static loading. The folds were more even and regular. N o n e of the empty (air-filled) cans 'bottomed out', while those with foams of densities less than 80 kg m - 3 did bottom out, experiencing larger compression than the empty specimens. Some specimens slid along the base plate upon impact of the drop weight, thus being subjected to 'off-axis' loading. A few specimens buckled at the centre after folding
157
Compression of foam-filled tubes
15
14
(to=137.mm)
/////
z I0 o" <
8
A ~
"
I
I
25
50
I
I
75 100 COMPRESSION,rnm
FIG. 6. Load-compression characteristics for specimens with and without end rings.
AI
a
A2
A6
A8 FIG. 7. Photographs of some quasi-statically crushed specimens, type and number as in Table 1.
I
158
T. Y. REI)DY and R. J. WAiL
A9
AI2
ii
AIO
All AIr~
09
D8 A
DIO
DII
FIG. 8. Photographs of some dynamically crushed specimens, type and number as in Table 2.
progressively at each end. One specimen, not of the highest density tested, ruptured after a certain deformation. Some typical specimens are shown in Fig. 8. As in quasi-statically deformed specimens, the axisymmetric deformations occurred with no foam trapped in the folds and acting as a crushable mandrel. Typical load-time traces are shown in Fig. 9. A summary of the tests carried out, along with the mean crush loads at which deformation begins, the mean load over the period of deformation and the period of deformation as measured from the load-time traces, is given in Table 2. The compressed length of the specimens was measured to obtain the compression, also shown in Table 2. In all the specimens, deformations were predominantly axisymmetric buckles, although a few
Compression of foam-filled tubes
/
159
~ . . t Pi =61-1 kgrfi 3
/
15
, ~;
z
11i
c5 l0 <
S
.
,'-,/...)/zl;
,. ;,
,~ ;,
,, ,,,
t
,
f',,
iI,
, " :,., ;., ';,,., '
.
p.'l
10
x
',
11 \
t/'. ~.b./I ./,../~-
ts,<;<.,X,->"" 5
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,.
,,,' • a~.k~,~,
/'W,s s
-4D'~'/'-.--Y--
/
'\
,, I/1,THEORY
",-----,'V (~:,,s.s) \ EpTV
~
~
15 20 25 30 T i M E A F T E R IMPACT, r n s e c
FIG. 9. Typical dynamic load-time traces for specimens of type A. TABLE 2. SUMMARY OF DYNAMIC TESTS
Specimen
load (kN)
Shortening (ram)
Time for deformation (msec)
Crush
Mean
load (kN)
Type
No.
Foam density (kg m - 3}
A mc = 38g
9 10 11 12 13 14 15
0 81.1 94.3 115.5 148.2 176.1 183.3
2.0 2.25 5.5 6.5 9 11 12
6.5 -7.5 8.75 11 8.5 12.5
108 125 117 101 76 105 61
22.5 31.0 32.0 32.0 30.0 29.0 26.0
B mc = 39.2g
1 2 3 4 5
0 76.1 88.9 91.0 96.6
2.0 4 4.5 5.25 6.25
6.25 --8.4 9
118 115 108 95 93
22.5 31.5 31.5 31.5 31.0
C mc = 27g
1 2 3 4
62 93 104 115
3.75 5.25 6.3 8.3
-8.5 9.25 10
116 93 91 88
29.5 31.0 31.0 31.5
D mc = 20.5g
7 8 9 10 11
0 67.5 79.4 96.7 106.8
1.75 3 4 2.5 6.25
6.5 --
115 117 118
-9.0
93
22.5 28 29 26 32
1 2 3 4 5
0 116 135.7 152 163.6
2 8.5 8.75 10 11.25
7 9.2 10.5 11.5 11.5
113 97 82 75 58
22.5 31 30 30 25
E mc = 40.6g
Remarks
Bottomed out
Can ruptured
Crushed & bent Skewed and bottomed out Bottomed out
Bottomed out Bottomed out Skewed and bottomed out
diamond folds were also observed. Generally, the degree of compression reduced with increasing density of foam in the tubes• Sectioned specimens showed that there was no adhesion between the tube and the foam. A very thin and flimsy layer of foam not attached either to the tube wall or the foam core was seen adjacent to the wall of the tube.
160
T.Y. REDDYand R. J. WALL ANALYSIS
General expressions for crushing forces and deformations are derived in this section. The crushing strength of a foam-filled tube is due to the plastic deformations of the shell and crushing of foam. The interaction between the filler and container is considered to modify the mode of deformation of the container, from an irregular diamond mode folding to an axisymmetric (or nearly axisymmetric) bellows mode deformation. The degree of compression of a given fold of the tube is also limited and governed by the strain at which the crushing of foam is completed and its densification begins. The limited degree of deformation, in turn, increases the mean crushing load of the shell. The foam strength is assumed not to be affected by the small lateral compression imposed on it by the shell. Numerical calculations have been carried out for tubes of type A whose material properties were obtained from tension tests mentioned above.
Quasi-static compression The axial crushing of a thin empty tube deforming in a diamond crumpling mode has been studied by Pugsley and Macaulay [4]. Pugsley [16] has suggested an expression for the crushing load Pc, which can be written as Pc = 228.5 mp,
(1)
where Mp = aotZ/4 is the fully plastic bending moment per unit width of the tube sheet with yield stress a 0 and thickness t. This equation produces a value of 0.4 kN, to be compared with an experimental value of 0.45 k N - - F i g . 5, Table 1. For axisymmetric concertina buckling observed in filled tubes, Alexander's deformation model [3] is used with modifications to account for the presence of foam. It is assumed that the tube wall moves only outwards from its initial position and that its deformation takes place due to rotation at the three hinges, as shown in Fig. 10. Equating the rate of work for plastic deformation due to bending at the plastic hinges and stretching of the regions in between to the rate of external work done at any position defined by the angle 0, i.e. at a compression 6 = 2h(1 - cos 0),
/
/
I
i
,/
,, / /
f
/
l
l
g ,/
_l_ t
D
i
(2)
i
.,r
~-
f
i
FIG. 10. Idealizedmodel of deformationfor concertina mode of buckling of an axiallycompressed cylindrical shell.
Compression of foam-filledtubes
161
the equilibrium load for the tube can be shown to be 2rtMp [-D h 1 ~- + sin 0 + -cost 0 j ,
Pt(O) = ~ [
(3)
where D is the diameter of the tube and h is the optimum value of the fold length (see Fig. 10) to produce a minimum value for the mean load when the fold is completely formed. It can be shown that h ,,~ x/(Dt). The mean load up to a given compression defined by angle 0 o can be obtained by rearranging and integrating equation (3) between limits 0 = 0 and 0 = 0 o, as
[DOo+2sinOo Pmt(0o)=2nMp ~
1--cos0 o
] t-1 .
(4)
When 0 o = rt/2, one gets Pm= 2rOMp
+ 2) + 1 ,
(5)
which is applicable to empty tubes. The presence of foam slightly changes the plastic fold length, 2h, from that of the empty tube [11]. This consideration is beyond the scope of the present investigation, and in the present case h ~- ,f(Dt) is assumed to be unaffected by pf. Foam is crushed by a deforming fold and causes the deformation of the fold to cease at 0o < ~/2. The value of 0o will be found in the following by assuming that the foam becomes rigid at a locking strain which is taken as the strain at the beginning of its densification. The plateau stress, trt, for polyurethane foam of density Of is shown [14] to be adequately represented by the expression
trf = 0.3try(pf/ps) 1"5,
(6)
where try 127 MPa and Ps = 1200 kg cm-3 are respectively the yield stress and density of the cell walls of rigid polyurethane foam [ 14]. As the foam is compressed further, the yielding of foam ceases and densification commences at a 'locking strain', el, which is given as [15] =
el = 1 - 3pr/ps.
(7a)
It may be mentioned here that in Ref. [14], a different expression for el, given by
el = 1 - 2pr/ps.
(7b)
is suggested. Equation (7a) is used in the following and the choice is discussed below. Now, 0 o can be determined from 131.A cylinder of foam having an initial height 2h is compressed to 2h cos 0 o, causing a nominal strain in foam given by /~f = (1 - - COS 0 0 ) .
(8)
Equating ef to el, equation (7a), the value of 0 o can be seen, after substituting for ps, to be 0 o = cos- 1(3pf/p~).
(9)
Thus the mean crushing load of the tube can be obtained from equation (4) as a function of foam density. The crushing strength of foam itself can be written as Pf = trf~zD2/4.
(10)
Hence the crushing force of the foam-filled tube is given by P'm = Pmt(O0) + Pt.
(11)
The energy absorbed by a specimen of an initial length Io undergoing a maximum deformation e,lo is E.A. = ptmeflo and the specific energy absorption is S.E.A. = E.A./(mf + m~),
(12)
162
T. Y, REDDY and R. J. WALL
_~15 //~
< LIA tt3 Z
Pmox
Z//
G
13..
//'
10
///,' ,' /
"/~
,Y./,"
ExPT
./". /
..0
Pint 1
o
t
100
200
of, kgr63
I
300
FIG. 11. Variation of initial crush load, crush load at locking strain and specificenergy absorption against foam density for specimens of type A crushed under quasi-static conditions. where mr = nD2lopf/4 is the mass of foam in the can and mc is the mass of the can. The variation of Ptm and S.E.A. against pf is shown in Fig. 11 along with experimental values. The variation of mean crushing load of tube only (which depends on 0o and hence on pf) is also shown in this figure.
Dynamic loading The modes of deformation of specimens under quasi-static and dynamic loadings were identical. Hence the inertia effects can be assumed to be negligible and only the strain-rate effects need to be considered. The strain-rate dependence of the tube wall can be described by the C o w p e r - S y m o n d s [17] relation = GO
t Y i" l+iri\j
,
(13)
where a~ is the yield stress at a strain rate ~ and c = 6500 s - 1 and p = 4 are constants for aluminium alloys. Thus for a given, ith, fold with a mean strain-rate o f ~ , the mean dynamic crushing load of the tube wall will be
s m. =
pmt[ 1 ~,i o.25 ,_ + ( 6 5 - ~ ) 1 "
(14)
The strain-rate dependence of polyurethane foams is described by the empirical equation due to Hinkley and Yang [11] for the dynamic yield stress of foam: af~ = (552 + 19.5 In ef)pr1"64.
(15)
Strain-rate varies from fold to fold as well as varying during a fold. For simplicity, velocity is assumed to remain constant during the deformation of a fold. The mean hoop strain in the metal sheet is eh = h sin Oo/D and the strain in the foam is el during the ith fold, when the velocity of compression is Vi_ 1. The above strains are caused in a time T, -
2h(1 - cos 0o)
Vi-i
(16)
Compression of foam-filledtubes
I0 Z
163
-,ooI \/, -
50
//
~// /,.~///
--EXPT --- THEORY
I
0
I00
I
200
pf, kgm-3
FIG. 12. Variation of mean dynamic load and compressionagainst foam density for specimens of type A. (A typical predicted load-time trace is also shown.) Hence the strain rates in the tube wall and foam, for the ith fold, can be shown to be
and
o e~i =
V~_ 1 sin 00 2D(1 - cos 0o)
o
Vi_ 1
~i =
(17) (18)
2h
Equations (17) and ( 18 ) can be used with equations (14) and ( 15 ) to obtain the dynamic crush load for the ith fold as d xD2 PCmi = pdmti + O'fl
~
•
(19)
The velocity of crushing, Vi- ~, for the ith fold can be obtained from an energy balance before and after the ( i - 1)th fold: 1 2 ~MoVi_ 2 = ½MoVi 2_ , + Pat i_ ,2h(1 - cos 0o),
(20)
where Mo is the mass of the drop weight and V~_2 is the velocity at the beginning of the ( i - 1)th fold. Pati_ 1 is the crushing force for the ( i - 1)th fold. Using equations (14)-(20), the dynamic load-time trace has been computed, and is shown along with the experimental trace in Fig. 9. The mean loads and corresponding compressions calculated from the above equations are plotted in Fig. 12 against Pr. The relevant experimental values (Table 2) are also shown in this figure. DISCUSSION
Bucklino modes
The number of circumferential lobes and axial fold lengths in axially compressed thinwalled tubes in the plastic deformation range are unpredictable because of the merging of several adjoining elastic buckles. This is due to the considerably smaller resistance to bending than to stretching of the tube wall. Hence the deformations are inextensional. The edges of an inward elastic buckle transform to travelling plastic hinges after collapse. These travel outwards, feeding material from adjacent buckles into the inward buckle, causing it to grow into a large dent. Several such mechanisms greatly reduce the number of lobes during plastic deformation. The extent of growth of these dents will be limited by the number of such
164
T.Y.REDDY R. WALL and
J.
mechanisms around the circumference. A sharp corner at the travelling hinge or meeting of two plastic hinges travelling in opposite directions stops the growth of the dent; folding will continue. Unstable bending failure results if a dent becomes too large. The presence of crushable material like foam, despite having little crush resistance in itself, acts as a foundation and resists the otherwise almost free inward movement of the tube wall, thus increasing the number of lobes in diamond crumpling, leading to axisymmetric bellows "crinkling" mode of plastic deformation. Both the strength of foam and change of mode of deformation of the tube increase the crushing force.
Foam behaviour The model of foam behaviour of Ashby [14, 15] is seen to provide agreeable results for quasi-static cases, as does the strain-rate sensitive model of Hinkley and Young [13]. The locking strain of foam has a linear dependence on its initial density. The two expressions suggested by Ashby [equation (7)] are plotted in Fig. 4. The experimental values plotted in this figure are derived as strains at twice the yield stress, from tests on constrained [11] and unconstrained foam compression tests with which the correspondence of equations (7a&b) is clear. It may be mentioned that the yield stress of foam does not show such dependence on the type of testing; see Fig. 4. The locking strain e,~ has an effect on the S.E.A. and this is discussed below. A few diamond crumpling folds were observed in specimens with low density foams, changing to axisymmetric folds as the deformation progressed. This is possibly because of the weaker foundation stiffness the low density foams initially provide to the tube wall. As compression progresses, the density of foam just below the folds increases, resulting in an increase in foundation stiffness. This is possibly due to the non-adhesion between the tube and foam, or the fact that 2h (or its equivalent in diamond crumpling mode) is smaller than the unit crushing height of a foam column, (observed as ~ 10 mm), as in the present study. The load-compression curves, Fig. 5, show hardening to a certain degree, depending upon pf, before the rapid hardening of the locking type. This can be attributed to the similar hardening stress-strain behaviour of foam. In Ref. [15], an expression for the strain dependent flow stress a~ of foam at a strain e is derived as
Ii
1--(Pf~l/3 -~ _
(
\P,/
P~\l-e/J
_I
_J
where af is given by equation (6). Using this equation, the crushing load for foam at locking strain is computed, added to the mean load for the tube and plotted in Fig. 11 as Pmax.The experimental values observed (at the experimental locking strains) also shown in this figure agree well with the predictions.
Eneroy absorbing capacity of foam-ill.led tubes The enhancement in the energy absorbing capacity due to filling is not only because of the crush strength of foam but also because of its interaction with the container resulting in a change of the mode of deformation from an irregular diamond crumpling mode to concertina mode. In addition, the degree of compression is changed because of the filler. Increased foam density decreases ef and thus 00, leading to an increase in the crush load while decreasing the stroke. The overall effect of density on specific energy absorption is shown in Fig. 11. For the tube considered, we find, theoretically, that the S.E.A. is highest when p f - 150 kg m-3. The experimental values also shown in the graph broadly agree with the theoretical prediction. A reason for the considerable overestimation of the analysis should be the use of a mean crushing strength for foam, calculated as an average of the initial crushing strength and that at the locking strain. A perfectly plastic material model for foam, with a crush load given by equation (10), improves the agreement between calculated and measured values.
Compression of foam-filledtubes
165
As Thornton [9] has observed, it is possible to obtain the increase in the crushing load by using stronger material (increased try) or by increasing the thickness of the tube. He argues that in either case similar or better weight effectiveness (S.E.A.) can be achieved. However, stability of deformation will not be improved by increasing try, while a large increase in thickness (to make D / t < 100) will be required to obtain stable compression behaviour of an empty tube. An axisymmetric bellows m o d e is assumed throughout in the analysis; this is found to provide sufficiently accurate prediction of the loads and energy absorbing capabilities of the tubes tested. The assumption made in the analysis of dynamic loading, i.e. that a fold forms at a constant velocity, equal to that at the beginning, is validated by the results; see Fig. 12. This need not be true in cases where only a few folds form to dissipate all the energy. The stepwise decrease of velocity assumed above is a good approximation only when the number of steps, i.e. the number of folds, is large. In the present case, both the dynamic crushing loads and periods of compression as computed agree well with the experiment. Dynamic behaviour o f empty (air-filled) tubes
Better stability of empty cans under dynamic loading can be attributed to two reasons. First is the near adiabatic compression, which provides a foundation effect. Second is that, under dynamic loading, travelling hinges cannot form and move at a speed required to create dents and instabilities. Further analysis is required to strengthen these arguments. However, it may be mentioned that the air-filled cans did not bottom out as did the specimens with low density foam. There is also a larger difference between the initial kinetic energy of the tup and the energy absorbed through plastic deformation (mean load crush length) in the air-filled cans than in the foam-filled cans of all the categories, which can only be due to the energy dissipated during the compression of air.
CONCLUSIONS The axial crushing behaviour of very thin (D/t > 600) empty and foam filled circular tubes under quasi-static and dynamic loading conditions is studied experimentally and theoretically. It is seen that the stability of crushing is improved by the presence of a filler such as polyurethane foam. The analytical models of such a foam described by Ashby [14, 15] are used to predict the effect of foam on the crush-strength of the tube. The increase in the crushing load is not only because of the crush strength of foam but also because of its effect in changing (i) the mode of deformation and (ii) the compressibility of the tube wall, both of which increase the mean crush strength of the tube. For a given set of tube parameters (tr 0, t) there is an optimum density of foam which provides a m a x i m u m specific energy absorption. Acknowledgements--The authors would like to thank Prof. S. R. Reid for his helpful comments on the manuscript and Mrs M. Mellor and Miss C. Tyler for typing it.
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T.Y. REDDY and R. J. WALL
7. W. ABRAMOWICZand N. JONES, Dynamic progressive buckling of circular and square tubes. Int. J. Impact Engn,q 4, 243--270 (1986). 8. P.H. WIRSCH~NGand R. C. SLATER,The beer can as a shock absorber. J. Engn9 Matls Tech. Trans. ASME 95, 224 226 (1973). 9. P. H. THORNTON, Energy absorption by foam filled structures. SAE paper 800081 (1980). 10. B. E. LAMPINEN and R. A. JERYAN, Effectiveness of polyurethane foam in energy absorbing structures. SAE paper 820494 (1982). 11. S. R. RE1D,T. Y. REDDYand M. D. GRAY,Static and dynamic axial crushing of foam-filled sheet metal tubes. Int. J. Mech. Sci. 28, 295-322 (1986). 12. H. F. MAHMOOD and A. PALNSZNY, Design of thin-walled columns for crash energy management--their strength and mode of collapse. Proc. 4th Int. Conf. Vehicle Struct. Mech. SAE paper 811302, 7-18 (1981). 13. W.M. HINKLEYand J. C. S. YANG,Analysis of rigid polyurethane foam as a shock mitigator. Expl Mech., Proc. SESA 177 183 (1975). 14. M. F. ASHBY, The mechanical properties of cellular solids. Metals Trans. 14A, 1755-1769 (1983). 15. S.K. MAITI, L. J. GIBSON and M. F. ASHBY,Deformation and energy absorption diagrams for cellular solids. Acta Metall. 32, 1963-1975 (1984). 16. A. PUGSLEY,On crumpling of thin tubular struts. Q. J. Mech. appl. Math. 32, 1-7 (1979). 17. G. R. COWPERand P. S. SYMONDS,Strain hardening and strain-rate effects in the impact loading of cantilever beams. Brown University, Tech. Rep. 28 (1957).