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Shock and Vibration 10 (2003) 179–186 179IOS Press

High strain rate characterization of shockabsorbing materials for landmine protectionconcepts

Jennifer McArthura, Christopher Salisburya, Duane Cronina, Michael Worswicka and Kevin WilliamsbaUniversity of Waterloo, 200 University Ave W., Waterloo, ON, Canada N2L 3G1Tel.: +1 519 888 4567 x 2346; E-mail: [email protected] Research Establishment Valcartier, 2459 Pie XI Blvd. Nord, Val B elair, Quebec, Canada G3J 1X5

Received 3 December 2001

Revised 17 October 2002

Abstract. Numerical modelling of footwear to protect against anti-personnel landmines requires dynamic material properties inthe appropriate strain rate regime to accurately simulate material response. Several materials (foamed metals, honeycombs andpolymers) are used in existing protective boots, however published data at high strain rates is limited.Dynamic testing of several materials was performed using Split Hopkinson Pressure Bars (SHPB) of various sizes and materials.The data obtained from these tests has been incorporated into material models to predict the initial stresswave propagationthrough the materials. Recommendations for the numerical modeling of these materials have also been included.

1. Introduction

An estimated 110 million landmines are hiddenaround the world [1], interfering with agriculture aswell as industrial development. In 1997, the Ottawatreaty [2] was signed to ban landmines around theworld and many nations have responded to this prob-lem with extensive de-mining efforts to decontaminatemine-affected areas.

The design of footwear to protect against landminesis extremely complex and must take into account boththe shock waves from the blast and the rapid expansionof the explosive gases immediately below the foot. Ex-perimental testing is limited since it is extremelyexpen-sive and requires significant expertise. A method thatallows many concepts to be inexpensively comparedand ranked prior to the experimental trials is thus de-sirable. With such a method, many potential conceptsmay be evaluated and only those that show the mostpromise would then be tested experimentally. Numer-ical modelling allows for the evaluation of many con-cepts at a lower cost. To ensure that numerical mod-

els will be realistic, several simpler models have beenanalysed and validated with experimental results [3–5]. These models found that the explosive gases ex-pand at speeds as high as 1800 m/s. Since the footwearmust be thin enough for normal motion, the peak strainrates in the material will be very high and thus materialproperties obtained at these strain rates are desired.

This paper discusses the testing and modeling of var-ious materials that are used in existing protective bootsusing the Split Hopkinson Bar method. These mate-rials include closed-cell polyethylene foams, open-cellaluminum foams and honeycombs and polyurethanerubber.

2. Literature review

2.1. Material models – shock behaviour

The LS-DYNA hydrocode contains many materialmodels representing a variety of materials [6]. Ide-ally, both shock and deviatoric behaviour should be in-

ISSN 1070-9622/03/$8.00 2003 – IOS Press. All rights reserved

180 J. McArthur et al. / High strain rate characterization of shock absorbing materials for landmine protection concepts

Striker Incident Bar Transmitter Bar

Fig. 1. Set-up of compressive split Hopkinson bar (not to scale).

corporated into the material models. Shock may beaccounted for by the use of an equation of state andthere are few models that allow this equation of stateto be added. The simplest of these models include thenull material model which assumes that the material isa viscous fluid and neglects deviatoric properties, theelastic-plastic-hydrodynamic model that assumes thematerial follows a straight elastic curve to yielding andthen may have some strain hardening or softening in theplastic region. Neither accounts for extensive deforma-tion, however and a deviatoric model should be used todetermine the long-term deformation of any material.

The linear polynomial equation of state [6] may beused with the data tabulated in LASL shock hand-book [7] to describe the shock behaviour.

2.2. Material models – deviatoric behaviour

This paper contains stress-strain curves that may beincorporated into deviatoric material models. The fol-lowing models are recommended for the materials dis-cussed in this report.

Honeycombs and brittle foamsThe honeycomb model [6] uses the solid material

properties, densification strain and orthotropic loadcurves in compression to determine deviatoric be-haviour of the material. Isotropic materials such asbrittle foams may also use this material model and onlytwo load curves (uniaxial compression and shear) needto be defined in this case.

Nearly incompressible rubber materials (polyurethane)

The Mooney-Rivlin rubber model is a two-parametermodel for nearly incompressible rubber materials [6].The input data includes density, Poisson’s ratio and aload curve (either force-displacement or stress-strain)which it uses to calculate the two parameters A andB. This model, with an appropriate strain-rate curveat the expected strain rates, has been used for model-ing polyurethane rubber response to blast loading withsome success [3–5].

Polymer foamsThe simplest foam-specific model is the crushable

foam material model [6]. The inputs for this model arethe initial density of the foam and a stress-strain curveat a given strain rate.

2.3. Split hopkinson bars for testing low-impedancematerials

The Hopkinson Bar is the most commonly usedmethod for testing materials in the strain rate regime of102–104 s−1. Only compressive split Hopkinson Bar(CSHB) tests were used.

The CSHB consists of a striker bar, an incident barand a transmitter bar, shown in Fig. 1. The striker baris propelled from a gas gun at a desired velocity andimpacts the incident bar. A compressive stress waveis imparted into the incident bar and propagates un-interrupted until it reaches the sample. The wave ispartially transmitted through the sample into the trans-mitter bar and the rest is reflected into the incident bar.Strain gauges record the strain-time history of both theincident, reflected and transmitted waves.

Classical analysis of a split Hopkinson Bar requiresthe following assumptions [8]: (i) the bars must remainelastic throughout the test; (ii) no attenuation or disper-sion of the stress waves occur; (iii) the pulse is uniformover the cross-section of the bar and, (iv) the specimenremains in equilibrium throughout the test. These as-sumptions must be valid in order to apply the classicalHopkinson bar equations (Eqs (1)–(4)) to determine thetime histories of stress, strain and strain rate within thespecimen.

ε(t) :=2 · Cb

Ls· εr(t) (1)

σ(t) :=2 · Cb

Ls· εr(t) (2)

ε(t) :=2 · Cb

Ls·∫ t

0

εr(t)dτ (3)

Cb :=

√Eb

ρb(4)

3. Experimental methods

WhereCb is the speed of sound inside the bars,Eb

is the modulus of elasticity of the bars,ρb is the densityof the bars,Ab andAs are the cross-sectional areas of

J. McArthur et al. / High strain rate characterization of shock absorbing materials for landmine protection concepts 181

Table 1Sample sizes and maximum loading for low-rate material tests

Material Sample size (wx d x l) Aspect ratio (w:l) Strain rate

34 kg/m3 polyethylene foam (HL34) 10 cm × 10 cm × 10 cm 1:1 0.03/s34 kg/m3 polyethylene foam (LD24) 10 cm × 10 cm × 3.2 cm 1:3 0.03/sPolyurethane rubber 5 cm × 5 cm × 1 cm 1:5 0.01/sOpen-cell Al Foam (7% density, 10PPI) 3.8 cm × 3.6 m × 1.2 cm 1: 3.170.01/sOpen-cell Al Foam (7% density, 20PPI) 3.9 m× 3.9 m× 1.3 cm 1:3 0.01/sOpen-cell Al Foam (7% density, 40PPI) 3.9 cm × 3.75 cm × 1.3 cm 1:2.95 0.01/sOpen-cell Al Foam (11% density, 10PPI)2.5 cm × 2.2 cm × 1.8 cm 1:1.3 0.01/sAluminum Honeycomb (“ACG”) 10 cm × 10 cm × 1.6 mm 1:6.25 0.03/sAluminum Honeycomb (“CR-III”) 5 cm × 5 cm × 1.6 mm 1:3.125 0.03/s

0

0.5

1

1.5

2

2.5

3

0 0.2 0.4 0.6 0.8 1

Strain [mm/mm]

Str

ess

[Pa]

0.01/s

1500/s

Fig. 2. Stress-strain behaviour of HL34 foam.

the bars and specimen,Ls is the length of the specimenandεr andεt are reflected and transmitted strains.

It should be noted that the real behaviour of the barsis more complex than assumed. Friction must be over-come before a specimen can expand or contract radi-ally. Davies [8] suggests certain criteria for specimendimensions to minimize this effect. Aspect ratios arevery important for ensuring that the friction effects atthe ends is minimal [9], and were kept at L/D= 1for the majority of the materials tested. In order tomeasure higher strain rates, the aspect ratio of the alu-minum foam and honeycomb specimens were as lowas L/D= 0.25. This was acceptable because these ma-terials exhibit very low Poisson’s ratios (approximately0.05 [10]) and thus do not deform significantly in theradial direction, so the frictional effects were much lesssignificant than for the other materials.

Polymeric bars require a different approach. Ba-con [11] developed an experimental method to correctfor the attenuation and dispersion of the wave as it prop-agates along the length of the bar. The method is basedon performing a free end test (where the end of thebar is not restricted) while measuring the incident andreflected wave. Since the bar is allowed to move freely

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.2 0.4 0.6 0.8 1

Strain [mm/mm]

Str

ess

[MP

a]

0.01/s

1600/s

Fig. 3. Stress-strain behaviour of LD24 foam.

at the interface end the entire wave must be reflected.By comparing the difference between the incident andreflected wave a measure of the attenuation and dis-persion is possible. This analysis leads to the applica-tion of a propagation coefficient (which is a functionof frequency) which is comprised of an attenuation co-efficient and a wave speed. The impractical nature ofperforming such calculations in the time domain, leadto the application of spectral methods for determiningthe propagation coefficients.

Once the free end test has been performed on boththe incident and transmitter bars, the application of thepropagation coefficient allows the velocity and forcetime histories at the end of the bars to be determined.The following fundamental relations can then deter-mine the strain rate, strain and stress within the sam-ple [12]:

ε(t) :=V1(t) − V2(t)

Ls(5)

ε(t) :=∫ t

0

ε(t)dt (6)

σs :=F2(t)As

(7)


0

2

4

6

8

10

12

14

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Strain [mm/mm]

Str

ess

[MP

a]

7% Density, 10PPI

7% Density, 20PPI

7% Density, 40PPI

11% Density, 10 PPI

Fig. 4. Low rate (0.01 s−1) stress-strain curves for open-celled aluminum foams.

Where,Ls andAs are the same as defined before,V1 andV2 are the velocities of the end of the incidentand transmitter bar at the sample interface andF2 is theforce in the transmitter bar.

Low strain rate testing of the materials was doneusing an Instron 4206 machine using steel platens andno lubricant at the ends of the samples. All sampleswere crushed beyond 90% engineering strain. Thesample sizes were determined according to the materialstiffness since the Instron could not accurately test atloads less than 4500 N. Table 1 shows the sample sizesused for each material.

It would have been preferable to test all materialswith samples with a 1:1 aspect ratio, however the lim-itations of minimum load and available material thick-ness prevented this.

High strain rate testing was done using compressivesplit Hopkinson bars. Three sets of bars were usedto avoid impedance mismatch with the materials. Thebars used in these tests included 25.4 mm diameteraluminum, 25.4 mm acrylic and 25.4 mm low-densitypolyethylene (LDPE). Both the incident and transmitterbars were 96” (2.4384 m) long for all materials. Thestriker bars used had various lengths; the aluminumstriker was 24” (605 mm), the acrylic striker was 28”(711 mm) and the LDPE striker was 6” (150 mm).Table 2 shows the specimen size and bar material(s) foreach sample material tested.

The choice of bar is critical to ensure that theimpedance is matched as closely as possible betweenthe specimen and the bars. When tested on the acrylic

bars, the impedance mismatch between the foam andthe bars resulted in a transmitted signal that was tooweak to record. With the LDPE bars, however, therewas a negligible mismatch since both materials werelow-density polyethylene and thus the transmitted sig-nal was much stronger.

4. Experimental results

4.1. Polyethylene foams

Two polyethylene foams were tested at a range ofstrain rates: HL34 with density34 kg/m3 and LD24with density24 kg/m3. Strain rates from 0.03 to 1500/swere achieved using the Instron 4026 and 25.4 mm di-ameter low-density polyethylene Split Hopkinson Barsrespectively. Data obtained from these tests is shownin Figs 2 and 3.

At low strain rates, both polymers had similar be-haviour though the LD24 had a lower yield strengthand modulus (as expected for a foam of lower den-sity). After the initial elastic loading, there was a sig-nificant amount of crushing at nearly constant stressuntil the stress increased dramatically at densification.The HL34 foam had a higher yield stress and exhibitedmore energy dissipation than the LD24 foam due to itshigher density.

These foams show significant strain-rate depen-dence, in line with the observations of other re-searchers [13,14]. Gibson and Ashby [13] determined


Table 2Sample sizes, Hopkinson bars used, and achieved strain rates for high-rate material tests

Material Bar material Sample sizes Strain rates achieved

34 kg/m3 polyethylene foam (HL34) LDPE 12.7 mm diameter× 12.7 mm tall 6 mmdiameter× 6 mm tall

1500/s

34 kg/m3 polyethylene foam (LD24) LDPE 12.7 mm diameter× 12.7 mm tall 6 mmdiameter× 6 mm tall

1600/s

Polyurethane rubber Acrylic and aluminum 12.7 mm diameter× 12.7 mm tall 6 mmdiameter× 6 mm tall

3000/s

Open-cell Al Foam (7% density, 10PPI) Acrylic 12.7 mm diameter× 12.7 mm tall 12.7 mmdiameter× 6 mm tall

1800/s

Open-cell Al Foam (7% density, 20PPI) Acrylic 12.7 mm diameter× 12.7 mm tall 1200/sOpen-cell Al Foam (7% density, 40PPI) Acrylic 12.7 mm diameter× 12.7 mm tall 2200/sOpen-cell Al Foam (11% density, 10PPI) Acrylic 12.7 mm diameter× 12.7 mm tall 12.7 mm

diameter× 6 mm tall 12.7 mm diameter×3 mm tall

1500/s

Aluminum Honeycomb (“ACG”) Acrylic 7-cell cluster× 1.6 mm, 8 mm tall 2400/sAluminum honeycomb (“CR-III”) Acrylic 7-cell cluster× 1.8 mm, 9 mm tall 1400/s

0

1

2

3

4

5

6

7

8

9

10

0 0.1 0.2 0.3 0.4 0.5 0.6

Strain [mm/mm]

Str

ess

[MP

a]

0.01/s

1500/s

Fig. 5. Behaviour of 11%, 10PPI Duocel foam.

that the compression of the fluid within the cells (air,in this case) is very rate-dependent and that this is thegoverning behaviour for foams of this density.

4.2. Open-celled foamed aluminum

Foamed aluminum is available in two forms: open-celled, with lamellae rather than solid cell walls, andclosed-cell foams. Open-celled foams are widelyused in crumple zones for cars [15] and other energy-absorption applications due to their crush strength andnegligible strain-rate sensitivity at moderate strain rates(< 1200 s−1).

Several varieties of Duocel open-celled aluminumfoams were tested: 7% relative density with 10, 20and 40 pores per inch, and 11% relative density with10 pores per inch. Low-rate tests were conducted at astrain rate of approximately 0.01/s. Figure 4 comparesthese results.

0

5

10

15

20

25

30

35

40

45

0.0 0.2 0.4 0.6 0.8 1.0

Strain [mm/mm]

Str

ess

[MP

a]

500-1000/s 1300/s

1800-2200/s

0.01/s

Fig. 6. Behaviour of 7%, 40PPI Duocel foam.

Note that, at this rate, the relative density has a muchlarger effect on the foam behaviour than the size ofthe pores. Since there is much more metal in a higherdensity foam, it logically follows that the area of metalin a given cross-section will be larger (regardless ofpore size) and thus the material will be stronger asdensity increases. Other researchers have found thatthis is generally true for aluminum foams [13], andspecifically for Duocel open-celled foams [15].

In order to properly represent the material, samplesizes were chosen such that each dimension includedat least 10 cells. An exception was made for the 10PPIspecimens, which had only 5 cells in each directiondue to the maximum allowable specimen size for thebars. Due to these limitations, strain rates were lim-ited to 1500/s for the 10PPI foams and 2200/s for the40PPI foams. The results suggest that this was an ac-ceptable specimen size since the stress-strain curvesobtained were consistent with the trends observed by


0

0.5

1

1.5

2

2.5

3

3.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Strain (mm/mm)

Str

ess

(MP

a)

1

2

3

Fig. 7. Low rate out-of-plane stress-strain curves for ACG Aluminumhoneycomb.

0

5

10

15

20

25

30

35

40

45

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9Strain [mm/mm]

Str

ess

[MP

a]

1

2

3

Fig. 8. Low rate out-of-plane stress-strain curves for CR-III Alu-minum honeycomb.

1 32 4

Fig. 9. Honeycomb specimens for dynamic testing ACG before (1)and after (2), CR-III before (3) after (4).

Danneman [16] for 40PPI Duocel aluminum foam andDeshpande [17] for 20PPI Duocel aluminum foam.

Dynamic tests of these materials show very similarresults at strain rates lower than1500 s−1 in 11% rela-tive density, 10PPI (Fig. 5) and less than1300 s−1 for7% relative density, 40PPI foam (Fig. 6).

Danneman [16] suggested that strain rate effects arenegligible up to 1200/s strain rate for the 40PPI foamand this is confirmed by the data obtained. At strainrates higher than1300 s−1, a strain rate dependence isseen at strains above 50%, though this may be relatedto the specimen size. Further high-strain rate testsshould be done with these materials using bars capableof higher firing pressures so that the same specimensizes may be used for all tests. A definite strain rate

0

1

2

3

4

5

6

7

8

9

10

0 0.2 0.4 0.6 0.8 1Strain (mm/mm)

Str

ess

(MP

a)

0.06/s

1200/s

1800/s

2400/s

Fig. 10. Out-of-plane stress-strain curves for ACG Aluminum Hon-eycomb.

0

5

10

15

20

25

30

35

40

45

50

0 0.2 0.4 0.6 0.8 1

Strain [mm/mm]

Str

ess

[MP

a] 0.03/s

850/s

1400/s

Fig. 11. Out-of-plane stress-strain curves for CR-III Aluminum Hon-eycomb.

effect is seen at strain rates exceeding1500 s−1 forthe 40PPI foam. Deshpande [17] suggested that therewas little strain rate sensitivity until 5000/s, but thatconclusion was based on a 20PPI foam. It is possiblethat the reduced pore size in the 40PPI foam increasesthe strain rate effect because the cell walls are muchcloser together, thus the foam behaves more like solidaluminum in this regime.

4.3. Honeycombed aluminum

Honeycombed materials are used in a variety of ap-plications, most often as an energy absorbing mate-rial. Its strength is much higher out-of-plane and itcrushes in three modes of deformation [13]: linear elas-tic loading (1), plastic buckling (2), and crushing (3), asshown in Figs 7 and 8. Two types of honeycomb weretested: ACG (ACG-1/4-4.8) and CR-III (CRIII-1/8-5052-.006N-2.21-STD),both manufactured by Hexcel.These graphs clearly show the three modes of defor-mation listed above.


0

10

20

30

40

50

60

70

80

90

100

0.0 0.2 0.4 0.6 0.8

Strain [mm/mm]

Str

ess

[MP

a]

3000/s

2500/s

2150/s

1600/s

1300/s

1000/s

0.01/s

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0.000 0.002 0.004 0.006 0.008 0.010

Strain [mm/mm]

Stre

ss [M

Pa]

Fig. 12. Stress-strain curves for polyurethane rubber.

Gibson and Ashby [13] calculate the plastic bucklingstress (σpl) of honeycombs out-of-plane from the yieldstress of the solid material (Sys), wall thickness (t) andcell wall length (l). Equation (8) applies to honeycombwith single thickness cell walls (such as ACG) andEq. (9) for honeycombs where two of the six walls aredouble-thickness (CR-III):

σpli := 6.6 ·(

tili

) 53

· Sys (8)

σpli := 5.6 ·(

tili

) 53

· Sys (8)

The plastic buckling stresses predicted using theseequation are 31 MPa for the CR-III (Eq. (9)) and2.4 MPa for ACG (Eq. (8)) The experimentally ob-served values, 35 MPa and 2.3 MPa for CR-III andACG respectively, compare reasonably well with thesepredictions.

Dynamic testing was done using a 25.4 mm diameteraluminum split Hopkinson Bar. The specimens usedfor this testing consisted of a 7-cell repeated group ofthe honeycomb, shown in Fig. 9. This specimen shapedoes not account for the extra energy required to breakthe adjoining cell walls, thus the data collected fromthese specimens underestimates the actual strength ofthe material.

The stress-strain data obtained is shown in Figs 10and 11.

The CR-III honeycomb shows a reduced strain rateeffect, most likely because it tends to fail by shearingat the glued interfaces between the layers, which is notstrain rate dependent. It is recommended that largerspecimens of the CR-III be tested dynamically to de-termine whether the mode of failure changes with thenumber of cells tested.

4.4. Polyurethane (Elastomer)

Polyurethane was tested using the 25.4 mm diameteracrylic and 25.4 mm aluminum Hopkinson bars. Theagreement between both bars was excellent, howeveronly data from the acrylic bar is included in the graphssince the recorded signal was much less noisy. Thelikely reason for this is that the signal in the aluminumtransmitter bar was much weaker than that in the acrylictransmitter bar, thus the noise was much larger withrespect to the signal. Figure 12 shows the strain-ratesensitivity at all strain rates tested, with the behaviourat low strains enlarged.

A clear dependence on strain rate is seen at lowstrains, similar to the results obtained by Yang et al [18].At higher strains, however, this dependence is not seenas clearly and the curves for strain rates of 1000–3000s-1 do not show a direct dependence on strain rate at highstrains.

5. Conclusions

The material testing presented here included highand low strain rate testing of polymer foams, foamedaluminum, aluminum honeycomb, and polyurethanerubber. The polymer foams were found to be verystrain-rate dependent, comparable with results fromGibson and Ashby and Zhao [13,14]. The foamed alu-minum tests showed two things: first, that the densityhas a much more significant effect than cell density atall strain rates, and second, that there is little strain ratesensitivity in the10−2 to 103 strain rate regime. Thisobservation has been noted by other researchers [16,17]. The aluminum honeycomb was found to be strain-rate dependent, though there was little published data


to compare these results with. The polyurethane rubbershowed a clear strain-rate dependence at low strains,however at higher strains, the strain rate effect in the1000–3000s-1 regime was not clear.

The deviatoric models discussed in the literature re-view should be used with the high strain rate data pro-vided in this paper to model the long-term deviatoricbehaviour of the materials.

Acknowledgements

Financial support for this research from the DefenceResearch and Development Valcartier (DRDV) andSuffield (DRDS), Materials and Manufacturing Ontario(MMO), the Canadian Centre for Mine Action Tech-nologies (CCMAT), National Science and EngineeringResearch Council (NSERC) and Med-Eng Systems,Inc. is gratefully acknowledged.

References

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[10] G. Gioux, T.M. McCormack, L.J. Gibson, Failure of alu-minum foams under multiaxial loads,International Journal ofMechanical Sciences 42 (2000), 1097–1117.

[11] C. Bacon, An experimental method for considering dispersionand attenuation in a viscoelastic Hopkinson bar,ExperimentalMechanics 38 (1998), 242–249.

[12] C. Salisbury, Spectral Analysis of Wave Propagation througha Polymer Hopkinson Bar, M.A.Sc. Thesis, University of Wa-terloo, 2001.

[13] Gibson, J. Lorna and Ashby, F. Michael,Cellular Solids Struc-ture and Properties, Pergamon Press, 1988.

[14] H. Zhao, Testing of polymeric foams at high and mediumstrain rates,Polymer Testing 15 (1997), 507–516.

[15] E. Andrews, W. Sanders and L.J. Gibson, Compressive andtensile behaviour of aluminum foams,Materials Science andEngineering A270 (1999), 113–124.

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