Plateau stress (MPa)
50 100
Figure 11.5 Energy absorbed per unit cost up to densification, plotted against plateau stress (which we take as the compressive strength at 25% strain) for currently available metal foams. Each foam is labeled with its density in Mg/m3
0.01
Figure 11.6 Selection of foam thickness, h, for decelerations of 20 g, 60 g and 200 g
from which
1 mv2
2 opi&DA
or using equation (11.3):
1 v2
with sD = 0.8 — 1.75(p/ps) (a best fit to recent data for metal foams). Manufacturers' data sheets for the foams give opl and eD, allowing h to be calculated for a given m, v and a*. Figure 11.6 shows aplot of equation (11.7). The use of equations (11.6) and (11.7) to design packaging is summarized in Table 11.3.
Table 11.3 Summary of the steps in initial scoping to select a foam for packaging
(1) Tabulate
• The limiting gfactor for the product (Table 11.1), a*
• The area of contact between the product and the package, A
• The design objective: minimum volume, or mass, or cost
(2) Calculate the foam crushstress (the plateau stress) which will just cause the limiting deceleration from equation (11.3).
(3) Plot this as a vertical line on Figures 11.3, 11.4 or 11.5, depending on the objective:
• Minimum volume: Figure 11.3
Only foams to the left of the line are candidates; those to the right have plateau stresses which will cause damaging decelerations. Select one or more foams that lie just to the left of the line and as high as possible on the energy scale. Note that the choice depends on the objective.
(4) Use these and the mass and velocity to calculate the required thickness of foam h to absorb all the kinetic energy without reaching the densification strain, using the plateau stress and densification strain which can be read from the data sheet of the chosen foam.
(5) Apply sensible safety factors throughout to allow for margins of error on mass, velocity and foam density.
11.3 Comparison of metal foams with tubular energy absorbers
Thinwalled metal tubes are efficient energy absorbers when crushed axially. By 'efficient' is meant that the energy absorbed per unit of volume or per unit weight is high. How do foams compare with tubes?
When a foam is compressed, its cell walls bend and buckle at almost constant stress until the cell faces impinge. The tube behaves in a different way: it buckles into a series of regular ringlike folds until, when the entire tube has buckled, the fold faces come into contact, as in Figures 11.7 and 11.8 (Andrews et al., 1983; Reid and Reddy, 1983; Wierzbicki and Abramowicz, 1983). The forcedisplacement curves for both have the approximate shape shown in Figure 11.1(a): a linearelastic loading line, a long plateau at the constant force, Fm, followed by a steeply rising section as the cell walls or tube folds meet. By dividing Fm by the nominal crosssection of the cylinder (nr2) and the displacement by the original length (€0) the forcedisplacement curve can be converted to an 'effective' stressstrain curve, as shown in Figure 11.1(b). The loading slope is now Young's modulus E (or 'effective' modulus for the tube), the plateau is now at the 'stress' api, and the plateau ends at the densification strain sD. The shaded area is the useful energy absorbed per unit volume of structure. Here we compare the energy absorbed by a metal foam with that absorbed by a tube of the same outer dimensions (Seitzberger et al., 1999; Santosa and Wierzbicki, 1998).
Figure 11.9 shows the loaddeflection curves for the compression of a foam, a tube and a foamfilled tube. The tubes show regular wavelike oscillations of load, each wave corresponding to the formation of a new fold. A circular tube of length I, outer radius r and wall thickness t(t — r) and yield strength fm, S Fm, S
i"
i"
Tube
Figure 11.7 A foam cylinder and a tubular energy absorber
Foam
Foam
Tube
Figure 11.7 A foam cylinder and a tubular energy absorber 
 Figure 11.8 Sections through tubes with and without foam fillings, after partial crushing. (Figure courtesy of Seitzberger et al., 1999)
Axial compression (mm)
Figure 11.9 Loaddeflection curves for a foam, a tube and a foamfilled tube. The fourth curve is the sum of those for the foam and the tube. The foamfilled tube has a higher collapse load, and can have a higher energy absorption, than those of the sum. (Figure courtesy of Seitzberger et al., 1999)
Axial compression (mm)
Figure 11.9 Loaddeflection curves for a foam, a tube and a foamfilled tube. The fourth curve is the sum of those for the foam and the tube. The foamfilled tube has a higher collapse load, and can have a higher energy absorption, than those of the sum. (Figure courtesy of Seitzberger et al., 1999)
oys crushes axially at the load
The load remains roughly constant until the folds of the tube lock up at a compaction strain s'Dube, giving an axial displacement
The energy absorbed per unit volume of the tube is then jirTube FmU a ( 5 ^ „Tube
The quantity 2t/r is the effective 'relative density' of the tube, p/ps, giving wTube = 2l/3 ' a^£Tube (n 1Q)
The foam absorbs an energy per unit volume of
(using equations (11.4a) and (4.2)) with C1 ^ 0.3. The densification in both the tube and the foam involves the folding of tube or cell walls until they touch and lockup; to a first approximations the strains sTTDlbe and gFoam are equal at the same relative density. Thus the tube is more efficient than the foam by the approximate factor
WFoam \psJ
For all realistic values of p/ps the tube absorber is more efficient than the foam, on an energy/volume basis, by a factor of about 3.
The equivalent results for energy absorbed per unit weight are yyTube _ _ ^ / ^ ays Tube w ~ 2nrtlps ~ \r) ps D or, replacing 2t/r by p/ps,
ps ps
That for the foam is wFoam = Ci£_\/ ^y±eFoam (11.14)
\Pj Ps giving the same ratio as before  equation (11.12)  and with the same conclusions.
More detailed calculations and measurements bear out the conclusions reached here. Figure 11.10 shows the calculated energy per unit mass absorbed by tubes plotted against the upperbound collapse stress (the plateau stress) compared with measured values for foams. Axially compressed tubes outperform foams by a small but significant margin. Foams retain the advantage that they are isotropic, absorbing energy equally well for any direction of impact. Tubes hit obliquely are less good.
Figure 11.10 The energy absorbed per unit mass by tubes (full line) and by metal foams, plotted against plateau stress, apl. The data for tubes derive from an upper bound calculation of the collapse stress. Each foam is labeled with its density in Mg/m3
Figure 11.10 The energy absorbed per unit mass by tubes (full line) and by metal foams, plotted against plateau stress, apl. The data for tubes derive from an upper bound calculation of the collapse stress. Each foam is labeled with its density in Mg/m3
Foamfilled sections
A gain in efficiency is made possible by filling tubes with metal foam. The effect is demonstrated in Figure 11.9 in which the sum of the individual loads carried by a tube and a foam, at a given displacement, is compared with the measured result when the foam is inserted in the tube. This synergistic enhancement is described by
where the additional energy absorbed, W1nt', arises from the interaction between the tube and the foam. This is because the foam provides internal support for the tube wall, shortening the wavelength of the buckles and thus creating more plastic folds per unit length (Abramowicz and Wierzbicki, 1988; Hanssen et al., 1999) A similar gain in energyabsorbing efficiency is found in the bending of filled tubes (Santosa et al., 1999).
The presence of the foam within the tube reduces the stroke S before the folds in the tube lock up, but, provided the density of the foam is properly chosen, the increase in the collapse load, Fm, is such that the energy FmS increases by up to 30% (Seitzberger et al., 1999).
11.4 Effect of strain rate on plateau stress
Impact velocities above about 1 m/s (3.6km/h) lead to strain rates which can be large: a 10 m/s impact on a 100 mm absorber gives a nominal strain rate of 100/s. It is then important to ask whether the foam properties shown here and in Figures 4.64.11 of Section 4, based on measurements made at low strain rates (typically 10~2/s), are still relevant.
Tests on aluminumbased foams show that the dependence of plateau stress on strain rate is not strong (Kenny, 1996; Lankford and Danneman, 1998; Deshpande and Fleck, 2000). Data are shown in Figures 11.11 and 11.12 for an Alporas closedcell foam and an ERG Duocel (Al6101T6) opencell foam. They suggests that the plateau stress, apl, increases with strain rate £ by, at most, 30%, over the range
Tests on magnesiumbased foams (Mukai et al., 1999) show a stronger effect. For these, the plateau stress is found to increase by roughly a factor of 2 over the same range of strain rate.
It is important to separate the effect of strain rate and impact velocity on the dynamic response of a metallic foam. The negligible effect of strain rate is associated with the fact that aluminum displays only a minor strainrate sensitivity. In contrast, material inertia leads to enhanced stresses at high impact velocities. At the simplest level, the effects can be understood in terms of a onedimensional shock wave analysis, elaborated below.
16 14
CO CO CD
Figure 11.11 Stressstrain curves for Alulight foam with a relative density of 0.18 at two strain rates: 3.6 x 10 ~3/s and 3.6 x 10 C3/s (Deshpande and Fleck, 2000)
16 14
Figure 11.11 Stressstrain curves for Alulight foam with a relative density of 0.18 at two strain rates: 3.6 x 10 ~3/s and 3.6 x 10 C3/s (Deshpande and Fleck, 2000)
Velocity m/s 102
100 Strain rate/s
Velocity m/s 102

 1G
100 Strain rate/s
Figure 11.12 The plateau stress corrected for relative density, plotted against strain rate. It is essentially independent of strain rate up to 3.6 x 10C3/s (Deshpande and Fleck, 2000)
Figure 11.12 The plateau stress corrected for relative density, plotted against strain rate. It is essentially independent of strain rate up to 3.6 x 10C3/s (Deshpande and Fleck, 2000)
11.5 Propagation of shock waves in metal foams
When a metal foam is impacted at a sufficiently high velocity, made precise below, a plastic shock wave passes through it and the plateau stress rises. Consider the idealized nominal compressive stressstrain curve for a metallic foam, shown in Figure 11.13. It has an initial elastic modulus, E, and a plateau stress, apl, before compaction occurs at a nominal densification strain, sD. When such a foam is impacted an elastic wave propagates through it; and if the stress rises above apl, this is followed by a plastic shock wave. In the simplified onedimensional case sketched in Figure 11.14(a) it is imagined that the bar is initially stationary and stressfree. At a time t = 0, the lefthand end of the bar is subjected to a constant velocity, V. In response, an elastic wave travels
pl e
Figure 11.13 A schematic compressive stressstrain curve for a metal foam; the stress jumps from the plateau level apl at U to the value aD at D
Plastic wave front
Stress = aD 
Stress = Jpl 
Velocity = VD 
Velocity = 0 
Density = p /(1  eD) 
Density = p 
Plastic wave front


Mass M 
Stress = JD Velocity = VD Density = p /(1  eD) 
Stress = Jpl Velocity = O Density = p 
Figure 11.14 (a) The stress, velocity on either side of the plastic shockwave front for an impacted foam (b) The stress, velocity on either side of the plastic shock wave front for an impacted foam carrying a buffer plate of mass M
quickly along the bar at an elastic wave speed c,A = ~jEjp and brings the bar to a uniform stress of api and to a negligibly small velocity of v = api/(pcei). Trailing behind this elastic wave is the more major disturbance of the plastic shock wave, travelling at a wave speed cp1. Upstream of the plastic shock front the stress is api, and the velocity is vv & 0. Downstream of the shock the stress and strain state is given by the point D on the stressstrain curve: the compressive stress is aD, the strain equals the densification strain, sd, and the foam density has increased to pD = p/(1 — sd).
Momentum conservation for the plastic shock wave dictates that the stress jump (aD — apl) across the shock is related to the velocity jump, vD, by
and material continuity implies that the velocity jump, vD, is related to the strain jump, eD, by vD = CpiSD (11.17)
Elimination of vD from the above two relations gives the plastic wave speed, cpi:
p where the tangent modulus, Et, is the slope of the dotted line joining the downstream state to the upstream state (see Figure 11.13), defined by
The location of the point D on the stressstrain curve depends upon the problem in hand. For the case considered above, the downstream velocity, vD, is held fixed at the impact velocity, V; then, the wave speed cpi is cpi = vD/sD = V/sd, and the downstream stress aD is constant at aD = api + pcpivD = api + pV2/sD. This equation reveals that the downstream stress, aD, is the sum of the plastic strength of the foam, api, and the hydrodynamic term pV2/sD. A simple criterion for the onset of inertial loading effects in foams is derived by defining a transition speed Vt for which the hydrodynamic contribution to strength is 10% of the static contribution, giving
Vt D
Olcygp P
Recall that the plateau strength for metal foams is approximated by (Table 4.2)
where oys is the yield strength of the solid of which the foam is made and C1 is a constant with a value often between 0.2 and 0.3. The densification strain scales with relative density according to eD = a — fi(p/ps), where a & 0.8 and fi & 1.75. Hence, the transition speed, Vt, depends upon foam density according to
Examination of this relation suggests that Vt shows a maximum at p/ps = a/3fi = 0.15. Now insert some typical values. On taking C1 = 0.3, p/ps = 0.15, oys = 200MPa and ps = 2700kgm—3, we find Vt = 21.5ms—1 (77 km h—1). For most practical applications in ground transportation, the anticipated impact speeds are much less than this value, and we conclude that the quasistatic strength suffices at the conceptual design stage.
Kinetic energy absorber
Insight into the optimal design of a foam energy absorber is gained by considering the onedimensional problem of endon impact of a long bar of foam of crosssectional area A by a body of mass M with an impact velocity V0, as sketched in Figure 11.14(b). After impact, a plastic shock wave moves from the impact end of the bar at a wave speed cp1. Consider the state of stress in the foam after the plastic wave has travelled a distance I from the impacted end. Upstream of the shock, the foam is stationary (except for a small speed due to elastic wave effects) and is subjected to the plateau stress op1. Downstream, the foam has compacted to a strain of sD, is subjected to a stress oD and moves at a velocity vD equal to that of the mass M. An energy balance gives
Using the relation vD = cplsD and the momentum balance oD = opl + pcplvD, the downstream compressive stress, oD, exerted on the impacting mass is p MV2q  2apiAleD/(\  eD)
Thus, the compressive stress decelerating the mass M decreases with the length of foam compacted, 1/(1 — bd). In the limit I = 0 the peak compressive stress on the mass is
in agreement with the findings of the shock wave analysis above. The length of foam 1/(1 — bd) required to arrest the mass is determined by putting vD = 0 in the above equation, giving
I MV0
1  Sd 2apISDA
On noting that aplsD equals the energy, W, absorbed by the foam per unit volume, we see that the minimum length of foam required for energy absorption is obtained by selecting a foam with a maximum value of W, consistent with a value of upstream stress apl which does not overload the structure to which the foam is attached. The plots of W versus apl for metallic foams (Figures 11.3, 11.4 and 11.5) are useful for this selection process.
11.6 Blast and projectile protection
Explosives create a pressure wave of approximately triangular profile, known as a 'blast' (Smith and Hetherington, 1994). Protection against blast involves new features. The blast imparts an impulse, /¡, per unit area of a structure, equal to the integral of the pressure over time:
The blast wave is reflected by a rigid structure, and the details of the pressuretime transient depend on the orientation of the structure with respect to the pressure wave. In design, it is conservative to assume that the structure is at normal incidence and fully reflects the blast. Figure 11.15 shows the peak pressure, p0, and the resulting impulse, /¡, caused by the detonation of a charge of TNT, at a radial distance, R, from the charge. Curves are shown for a reflected blast in air, and in water. The impulse and the distance are normalized by the cube root of the mass, M, of the charge in kg. As an a e p
1G6 1G5 1G4 1GGG
1G6 1G5 1G4 1GGG


Impul 
se, water st 
Pressure, water Impulse, air 
Pressure^ air 








radius/(mass of TNT)1/3
radius/(mass of TNT)1/3
Figure 11.15 Peak pressure and impulse as a function of distance R from an explosion of a mass M of TNT
example, a charge of 1 kg of TNT in water produces a peak pressure of about 100 MPa and an impulse of 104Ns/m2 at a distance of 1m. The curves for water blast are approximated by the formulae po D 10S
1.13
MPa and
Ji d 1.1S5 x 104M1/3
0.S6
Ns m
where the mass, M, of TNT is given in kilograms and the distance from the explosion, R, is given in metres. The energy content of other common chemical explosives is similar to that of TNT, as shown in Table 11.4. In order to estimate the blast from other explosives the simplest method is to scale the mass of the explosive by its energy content relative to that of TNT: this scale factor is included in the table.
Blast protection (and protection from projectile impact, which is treated in a similar way) is achieved by attaching a heavy buffer plate, mounted on an energy absorber, to the face of the object to be protected. The impulse accelerates the buffer plate; its kinetic energy is dissipated in a benign way by the energy absorber. Let the buffer plate have a thickness b and density pb. Then
Table 11.4 Energy density and TNT equivalents of explosives
Explosive Mass specific TNT equivalent energy (Qx/Qtnt)
Amatol 80/20 (80% ammonium nitrate,
20% TNT) 
2650 
0.586 
Compound B (60% RDX, 40% TNT) 
5190 
1.148 
RDX (Cyclonite) 
5360 
1.185 
HMX 
5680 
1.256 
Lead azide 
1540 
0.340 
Mercury fulminate 
1790 
0.395 
Nitroglycerine (liquid) 
6700 
1.481 
PETN 
5800 
1.282 
Pentolite 50/50 (50% PETN, 50% TNT) 
5110 
1.129 
Tetryl 
4520 
1.000 
TNT 
4520 
1.000 
Torpex (42% RDX, 40% TNT, 18% 


aluminum) 
7540 
1.667 
Blasting gelatin (91% nitroglycerine, 7.9% 


nitrocellulose, 0.9% antacid, 0.2% water) 
4520 
1.000 
60% Nitroglycerine dynamite 
2710 
0.600 
the impulse, /¡, imparts a momentum, Mt, to a unit area of the face plate where
and thereby accelerates the plate to a velocity v. At this point it has a kinetic energy
2 2pbb and it is this that the energy absorber must dissipate. Note that the thicker and heavier the buffer plate, the lower is the kinetic energy that the absorber must dissipate.
The selection of a metal foam as an energy absorber follows the method of Section 11.2. It is necessary to absorb Ui per unit area at a plateau stress apl which will not damage the protected object. Let the energy absorbed per unit volume up to densification by a foam with a plateau stress apl be Wvol. Then the thickness hbiast of foam required to absorb the blast is J2
2pbbWVoi
The efficiency of absorption is maximized by using a heavy buffer plate (pbb) and choosing the foam with the greatest Wvoi for a given api.
An alternative strategy might be to minimize the combined mass per unit area, mt of the buffer plate and the foam. The mass per unit area of the buffer plate is mb = pbb, and the mass per unit area of the foam is mf = phbiast. Substitution for hbiast from equation (11.33) into the expression for mf gives pJ2
2pbbWvoi and minimization of mt with respect to the buffer plate mass mb = pbb gives mb = mf, and


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