Effect of Inertia on Energy Absorption

*Corresponding author.

International Journal of Impact Engineering 22 (1999) 955}979

Inertia e!ects in impact energy absorbing materialsand structures

J.J. Harrigan, S.R. Reid*, C. Peng

Department of Mechanical Engineering, UMIST, P.O. Box 88, Manchester M60 1QD, UK

Received 2 October 1998; received in revised form 29 June 1999

Abstract

Experimental data and numerical/computational models concerning the internal inversion of metal tubesand the dynamic crushing of aluminium honeycombs are presented and discussed as illustrations of impactenergy absorbers whose behaviour is strongly in#uenced by inertial e!ects. ( 1999 Elsevier Science Ltd.All rights reserved.

1. Introduction

Numerous materials and structural components have been investigated over the years for theirsuitability as impact energy absorbers. This work has generated an extensive literature some ofwhich has been reviewed [1,2]. Modelling the crushing of single structural components andassemblies of such components has resulted in the solution of a range of challenging large-de#ection, structural plasticity problems. As an example, recently Reid and Harrigan have de-scribed some results from a study of the behaviour of inversion tubes. Equally, there has beenconsiderable interest in understanding the behaviour of cellular materials as impact energyabsorbers, as exempli"ed by the recent work on wood by the authors [3,4]. The link between thesetwo studies is an attempt to clarify the role of inertia in the characteristics of the energy absorber,especially in terms of the magnitudes of the forces (stresses) generated during impact loading.

In this paper additional results are presented which highlight inertia e!ects. Early peak loadsoccur under impact loading of both inversion tubes and axially loaded honeycombs as a conse-quence of lateral inertia. The magnitude of these early peaks is governed by the properties of the

0734-743X/99/$ - see front matter ( 1999 Elsevier Science Ltd. All rights reserved.PII: S 0 7 3 4 - 7 4 3 X ( 9 9 ) 0 0 0 3 7 - 8

Nomenclature

A cross-sectional areaCP plastic wave speedE elastic modulusl0 original length¸D axial load at distal end of specimen¸SS quasi-static steady-state loadm, m

,, m

*masses of piston, knuckle region and fully inverted part of tube, respectively

p"

barrel pressurer die forming radiusS3

stress ratiov velocity<0 impact velocitye-

locking straine1-

plastic strainl Poisson ratiooL mass per unit lengtho0 original densityp0 initial peak crushing stressp1

plateau stressp#3

crushing strength of r-p-p-l materialpH stress inside shock front

material of the tube or honeycomb wall. Further details of the behaviour of inversion tubes arepresented which illustrate how the high inertia-generated forces can be moderated by changingthe geometry of the tube. The cellular structure of the honeycomb is responsible for furtherenhancements of the quasi-static crushing strength as the deformation continues. New resultsconcerning the dynamic crushing of aluminium honeycombs at high velocities are presented anddiscussed. The high initial peak load is moderated by pre-crushing the honeycomb specimens. Inboth cases the results are presented in summary form. More extensive treatments will be publishedelsewhere.

2. Internal inversion of tapered metal tubes

A recent in-depth study of internal inversion by Harrigan [5] has shown that there arecircumstances in which dynamic e!ects can lead to force peaks which exceed the steady-stateinversion forces and as such need to be explained and drawn to the attention of designers. Theinitial study of tubes with uniform thickness has been described in a recent paper by Reid andHarrigan [6]. This includes both experimental data and "nite element models generated using

956 J.J. Harrigan et al. / International Journal of Impact Engineering 22 (1999) 955}979

Fig. 1. Quasi-static loading arrangement and die geometries.

ABAQUS/Standard. A summary of some additional results is presented below in which particularemphasis is placed on the behaviour of tubes with tapered ends.

2.1. Quasi-static behaviour

Quasi-static inversion of tubes was achieved by compressing the tubes axially into dies withforming radii of 5 and 10 mm (Fig. 1). The material used was cold drawn steel (BS6323/4 1982 CFS3BK) which will be referred to throughout the text as `mild steela. For all tests the material wasused in the `as-supplieda condition without heat treatment. The specimens had an axial length of75 mm, a nominal outside diameter of 101.6 mm and were cut from cold-drawn, seamless metaltubing with a wall thickness in the range of 3}3.5 mm. Various wall thicknesses were produced bymachining the tubes.

The specimen dimensions and compressive stress}strain curve are shown in Fig. 2. To obtain thematerial properties (Fig. 2(a)) rings with an initial height of 10 mm were cut from the mild steel tubeand compressed between the #at platens of a 3000 kN Amsler testing machine. It should be notedthat tubes with di!erent leading edge geometries were tested, three of them (square-edged,chamfered and short tapered) di!ering only at a detailed level, whilst the fourth had a taperedregion of approximately 20 mm in length. The quasi-static tests were carried out on either a 200 kNRDP or a 500 kN Instron (model 1345) testing machine, depending on the load capacity required.A cross-head speed of 10 mm/min was used. During quasi-static testing the specimens wereconstrained within a steel constraining tube (Fig. 1). A hardened steel pusher tube with an internaldiameter of 88 mm was used to compress the specimens into the dies. These quasi-static dies werecut from hardened tool steel (EN24T), with all contact surfaces polished. The polished surfacesof the die and constraining tube were coated with Shell Alvania Grease 2 and MolybdenumDisulphide Lubricant.

J.J. Harrigan et al. / International Journal of Impact Engineering 22 (1999) 955}979 957

Fig. 2. Stress}strain curve and dimensions of mild steel tubes.

For tube specimens with a 1 mm chamfer radius (Fig. 2), the e!ect that increasing the die radiushas on the load}displacement curves is illustrated in Fig. 3. As the forming radius r is increased thestage 1 load reduces as does the slope of the load}displacement curve at the beginning of stage 2.These e!ects have been discussed in detail with reference to "nite element modelling usingABAQUS by Reid and Harrigan [6], who used a coe$cient of friction of 0.15 to obtain thenumerical solutions shown in Fig. 3. A friction coe$cient of 0.15 may appear high for a welllubricated contact surface. However, this value has produced good correlation with experimentaldata for a range of wall thicknesses and die radii [6]. Furthermore the combination of pressure andsliding at the contact surface is such that the lubricant tends to be removed from this surface so thatthe tube and die appear dry after the test.


Fig. 3. Typical load}displacement curves and approximate shapes for quasi-static internal inversion.

Load}displacement results for the internal inversion of tubes with long and short tapers areplotted in Fig. 4. Some photographs of inverted specimens are provided in Fig. 5. The separatestages of the deformation processes are indicated in Fig. 4 for the 2 mm thick tubes with shorttapers at their leading edges.

Load}displacement results for inversion through the die with a 5 mm forming radius are plottedin Fig. 4(a). Specimens 1}4 are inverted to produce axisymmetric cylindrical regions free fromcircumferential buckle patterns (see Fig. 5). For all specimens inverted through the 5 mm radius die,the leading curled section struck the pusher tube at a stroke of approximately 45 mm, causing theload to rise sharply. For specimen 1 the three stages are clearly de"ned in the load}displacementcurve.

Further results for inversion through the die with a 10 mm forming radius, for tubes with taperedleading edges, are given in Fig. 4(b). The 2 mm thick tube with a short-taper leading end (specimen5) had a similar inversion characteristic when axially compressed into the 10 mm radius die toa specimen with a chamfer radius at its end (Fig. 3(b)). For these two tubes, second contact occurredat similar loads, unloading was evident to mark the second instability, and similar steady-stateinversion loads were reached. However, inversion of the 2 mm thick long-taper ended tube(specimen 6) resulted in a far smoother load}displacement relationship during stages 1 and 2 and


Fig. 4. Quasi-static load}displacement curves for inversion of tubes with long and short tapers.

no unloading was evident when this specimen was tested. Surprisingly, this tube also achievedsteady-state inversion at a noticeably lower load than the other tubes of equal wall thickness.Photographs of 2 mm thick tubes, inverted through the 10 mm radius die are shown in Fig. 5(b).Prominent circumferential buckle patterns can be seen at the leading rim of the inverted long-taperended specimen (6). Although barely visible, this circumferential ripple pattern runs axially alongthe full length of the `cylindricala inverted section of the specimen. Though less pronounced, butdetectable through touch, a similar pattern existed through the deformed portion of the short-taperended specimen (5). The chamfer-radius ended specimen had no buckle patterns in its invertedcylindrical section, although slight ripples can be seen at its leading edge.


Fig. 5. Quasi-statically inverted specimens.

It therefore appears that for tubes with thinner end sections, buckles that are "rst initiated inthese thin-walled regions can spread through the thicker regions of the tube as the specimen isinverting, resulting in a loss of axisymmetry in the inverted `cylindricala section. The circumferen-tial plastic buckling initiates circumferential plastic folding. This reduces the rate of dissipation ofenergy through circumferential compression and is the most likely cause of the lower than expectedsteady-state inversion load for specimen 6.

Specimens 7 and 8 (t0"1.5 mm, r"10 mm, ST and LT, respectively) inverted at similar

steady-state loads (Fig. 4(b)). Circumferential buckles were again more evident in the specimen withthe longer taper (Fig. 5(c)) and this specimen's steady-state inversion load is again slightly lowerthan that of the short-taper ended tube. The leading edge of specimen 8 curled to a greater extentthan any other specimen inverted through the 10 mm radius and testing this specimen producedthe smoothest load}displacement relationship obtained for inversion.

2.2. Dynamic response

A series of dynamic tests was carried out using mild steel tube specimens that were of the samematerial and had identical dimensions to tubes that had been tested quasi-statically, in order todetermine the correlation between the quasi-static load}displacement characteristic and the


Fig. 6. Set-up and piston dimensions for dynamic inversion.

equivalent dynamic behaviour. Dynamic inversion was achieved using shaped projectiles (pistons)of mass m (approximately 1.1}1.3 kg) which were "red axially at stationary tubes using the facilityshown in Fig. 6. These pistons had the same internal forming radii as the dies that were used inthe quasi-static tests and the impact took place within the barrel of a pneumatic gun. Theinstrumentation allowed for the force pulse transmitted to the support to be measured (see [6]for details).

Typical load traces are plotted in Fig. 7 for impact velocities of approximately 100 m/s. Themaximum loads recorded during each test are plotted in Fig. 8(a) and plots of energy absorbedagainst "nal stroke are given in Fig. 8(b). Photographs of deformed specimens are providedin Fig. 9.

Stage 3 loads recorded at the distal end of the tubes during dynamic inversion were lessthan those measured during equivalent quasi-static tests. This phenomenon is a result of tubeinertia and has been explained by means of a simple theoretical model for stage 3 of dynamicinternal inversion by Reid and Harrigan [6]. The dynamic stage 3 load for forced inversion isgiven by

¸D"P

"A#(¸

SS!P

"A#2o

Lv2)

(m#m,#2m

*)

(m#m,#4m

*)!2o

Lv2, (1)


Fig. 7. Experimental results and ABAQUS solutions for dynamic inversion (mild steel, tO"2 mm).

where ¸D

is the load at the distal end of the specimen, p"

is the pressure behind the piston(see Fig. 6), A is the cross-sectional area of the piston, ¸

SSis the quasi-static steady-state load, o

Lis the mass per unit length of the tube, v is the instantaneous velocity of the piston, and m, m

,and m

*are the masses of the piston, the knuckle region and the fully inverted portion of the

tube, respectively.


Fig. 8. Maximum loads recorded and energy absorbed during dynamic inversion.

The load trace plotted in Fig. 7(a) illustrates the early peak load that was recorded duringdynamic inversion by the 5 mm piston for all 2 mm thick specimens which had a short-taperend-shape. Tests on specimens with the same thickness but long-taper ends produced load traceswith a much reduced early maximum, which was generally exceeded by loads recorded during stage3 of the load pulse as shown in Fig. 7(b). Comparing Fig. 9(a) with the photographs of specimensinverted quasi-statically through the 5 mm die (Fig. 5(a)) it can be seen that for this forming radiusthe dynamically and quasi-statically inverted tubes are very similar in shape, i.e. no buckling has


Fig. 9. Dynamically inverted specimens.

occurred during any of the tests and all tubes have inverted well to form approximately co-axialcylinders of smaller diameter than the original.

For tests performed using pistons with a 10 mm forming radius, typical photographs of testedspecimens are shown in Fig. 9(b). Again the maximum load recorded for short-tapered specimensoccurred early on in the crushing period although peak loads were less than those recorded duringtests on similar tubes with the 5 mm piston (Fig. 8(a)). Noticeable again was a reduction in peakloads for tests conducted on long-taper ended tubes (see Figs. 7(c) and 8(a)(ii)).

Comparing the photographs of dynamically inverted tubes (Fig. 9(b)) with the quasi-staticequivalents (Fig. 5(b)) it can be seen that the leading edge curled to a far less degree during dynamictesting. Furthermore, the 10 mm piston never created a co-axial cylinder from the test specimen,but rather a circumferentially buckled conical section, with the leading section apparently reducingin circumference during the `steady phasea.

Obviously, reductions in tube wall thickness increases the tendency of tubes to buckle undercompressive loading. This showed itself in the appearance of circumferential buckle patterns incertain tubes.

In terms of energy-absorbed against "nal stroke, the dynamic results agree well with the quasi-static energy-displacement relationships for inversion through a 5 mm forming radius (Fig. 8(b)).Considering the strain-rate sensitivity of mild steel, this level of agreement is somewhat surprising.When a 10 mm forming radius was used to promote inversion, the quasi-static energy}displace-ment curves did not predict the dynamic energy absorption with such accuracy (Fig. 8(b)(ii)). There


is also a more noticeable di!erence between the shapes of dynamically and quasi-statically invertedspecimens (Figs. 5 and 9).

2.3. Modelling

ABAQUS/Standard was employed in the "nite element modelling of the internal inversionproblems and details of the approach used are given in Ref. [6]. The material was assumed to beelastic (E"200 GN/m2, l"0.3), linear strain-hardening (p

0"570#211 e

1-MN/m2) with no

strain-rate dependence. A density of 7850 kg/m3 was assumed. The tube was made up of eight-noded axisymmetric solid-continuum elements. Axisymmetric rigid surface elements monitoredcontact between the rigid surfaces and the tube. The nodes at the distal end of the tube were axiallyconstrained while the piston rigid-surface was given a mass and an initial velocity. A coe$cient offriction of 0.15 was used for the cases considered.

As indicated in Figs. 3 and 7, there is often excellent agreement between the ABAQUSsimulation and the experimental results despite the fact that the "nite element analysis isrestricted to axisymmetric deformation modes. After an initial set of tests the ABAQUS modelswere used to select geometries for which the high initial peak loads were removed. When theCowper-Symonds equation was used to model the strain-rate e!ects the dynamic load predictionsgreatly over-estimated those that were recorded experimentally. This is not surprising as theCowper-Symonds equation only applies to small strains and not the large plastic strains whichoccur during inversion. More details of the "nite element results will be provided in a futurepublication.

3. Dynamic crushing strength of aluminium honeycombs

Aluminium honeycombs are low density structures with relatively high speci"c strength,sti!ness and energy absorption characteristics. The quasi-static behaviour of aluminium honey-combs (i.e. under low loading rates) has been investigated by several authors, however there is littlepublished information on their dynamic response apart from the very recent paper by Wu andJiang [10].

The cell geometry of man-made honeycombs is similar to that seen in natural cellular compositematerials such as wood, see Reid et al. [2]. The similarities are not only con"ned to the geometry ofthe cellular assemblies but also extend to the shapes of their load}compression curves and even totheir local deformation mechanisms. It is the similarity in cellular structure and quasi-staticload}compression curves that enables the same treatment to be applied to the two materials as thelocal deformations are not considered in the analysis. For wood there is a signi"cant enhancementin the crushing strength with increasing impact velocity and this has been attributed by Reid et al.[2,3] to inertial e!ects in the cellular array including microinertia which a!ects the initial crushingstrength of wood loaded along the grain in the same way that lateral inertia of the cell wallsincreases the initial impact load for honeycomb. Additionally the shock formation which elevatesthe plateau crushing strength of woods is mirrored in the results for honeycomb. A shock model forwood has been described in [3] and similar models have been formulated to predict the dynamicresponse of man-made foams were developed by Jahsman [7] and Zaretsky et al. [8].


Goldsmith and Sackman [9] provided experimental data on the uniaxial dynamic crushing ofaluminium honeycombs at impact velocities up to 35 m/s using slightly rounded projectile as partof a study of the impact response of sandwich panel with honeycomb cores. They noted an increaseof about 20}50% in the plateau crush stress over the range of loading. Recently, Wu and Jiang [10]investigated the static and impact crushing of various types of aluminium honeycombs. Theirimpact tests were performed at velocities up to 27 m/s. They observed a signi"cant increase ofabout 74% in the average dynamic crush strength when the specimens were loaded dynamicallycompared with the quasi-static crush strength.

This section summarises the results of an investigation of the in#uence of dynamic e!ects uponthe initial peak stress, the plateau crushing strength and the energy absorption characteristics ofaluminium honeycomb samples under out-of-plane loading. Experimental results obtained fromimpact tests on short cylindrical aluminium honeycomb specimens over a range of impactvelocities up to 300 m/s are compared with the theoretical results from the simple shock modelpreviously developed by Reid and Peng [3].

3.1. Experimental testing programme

3.1.1. Material specixcation

Material: Aluminium alloy 5050Core designation: 2.3 1/4 10 5052Nominal core density: o

0"37 kg/m3

Measured core density: o0"42 kg/m3

Nominal cell size: s"6.35 mmNominal cell side length: l"s/J3"3.67 mmNominal cell wall thickness: t"0.254 mmMeasured cell wall thickness: t"0.049 mm (single wall)Honeycomb specimen length: 30 mm (undeformed)

25 mm (pre-crushed)Skins: Aluminium alloy 2024 of 1 mm thicknessSolid material density: o

4"2700 kg/m3

Measured tensile yield stress of cell wall material: p:"135 N/mm2

Aluminium Young's modulus: E4"68.5 kN/mm2

Solid material Poisson's ratio: l4"0.3

Each cell has six sides, four of which are single-thickness and two sides are double-thick-ness.

The honeycomb specimens tested formed two sets, uncrushed single-skin samples and pre-crushed single-skin samples, respectively. The skin referred to is a 1 mm thick aluminium alloy2024 disc glued to one of the end surfaces of the honeycomb specimen. The impact surface of thespecimens tested is the end surface with no aluminium skin. The overall testing programme consistsof uniaxial quasi-static and uniaxial impact tests on aluminium honeycomb cylindrical specimensof approximately 45 mm diameter and 30 mm length.


The static tests were performed using a Universal testing machine, Instron 4507 at a loading rateof 1 mm/min under displacement control. The aim of these static tests was to produce thequasi-static load versus de#ection characteristics for comparison purposes.

In the impact testing only uncrushed samples are tested under both laterally free (uniaxial stress)and laterally constrained (uniaxial strain) conditions, whereas the pre-crushed samples were testedonly under laterally constrained conditions. Each group of tests was divided into two sets ofsamples: samples without and with an additional aluminium backing disc of 45 mm diameter and12 mm thickness (i.e. 50 gm mass) attached to their rear skins. The added mass provided moreenergy to crush the specimens. The results for the backed, initially uncrushed specimens will be theprincipal ones discussed in this paper, although some data for the unbacked and pre-crushedspecimens will be included where appropriate. For each set of tests, the samples were tested overa range of "ve impact velocities from 20 up to 300 m/s approximately. The impact testing rig usedwas the same as that used in the study of wood and is described in detail in Ref. [3]. It consists ofa pneumatic launcher, a Hopkinson pressure bar (HPB) load cell for measuring the dynamic forcepulse and a thick-walled chamber for lateral constraint tests. The HPD was 50 mm diameter, 2 mlong and was made of EN24T steel.

3.2. Quasi-static test results

3.2.1. Deformation mechanismsThe mechanics of deformation of thin-walled aluminium honeycomb samples are characterised

by localised and progressive buckling (i.e. diamond type axial buckling) seen as a crush band whosewidth is almost linearly proportional to the axial displacement. The onset of this microbucklingoccurs at the weakest zones which can be anywhere in the specimen. Both initially uncrushed andpre-crushed specimens have the same modes of deformation. Some of the tested samples weresectioned and photographed immediately after being tested with a view to observing theirdeformation pro"les. Fig. 10 shows longitudinal cross-sections of these samples, revealing themicrobuckling deformation. This diamond type of deformation is very similar to that observed insingle hexagonal tubes and in woods of low relative density when subjected to axial compression(i.e. compression along the grain for woods), see Fig. 10(c).

3.2.2. Static load levels and energy absorption characteristicsThe load versus de#ection curves of the honeycomb samples are typical of cellular materials

generally and are characterised by an initial peak force (i.e. which separates the elastic frominelastic behaviour) followed by a small load drop and then a slightly increasing plateau upto the locking displacement where a sudden and steep load increase occurs due to materialdensi"cation. The small load drop described above did not occur for the pre-crushed samples,see Fig. 11. For pre-crushed samples the initial crush stress, p

0, was typically 0.56 N/mm2

compared with 0.93 N/mm2 for initially uncrushed samples. Pre-crushing the samples initiatedthe inelastic deformation, pre-disposing the specimens to crush at or close to the plateau stresslevels. The values of plateau stresses, p

P, and locking strain, e

-, were 0.53 N/mm2 and 0.65 for

all samples.The energy of deformation was determined from the area under the load versus displacement

curves. The locking energy is the energy used to deform the specimens up to their locking


Fig. 10. Honeycomb specimens subjected to uniaxial quasi-static compression: (a) Development of the crushing bandfrom the end surface; (b) Initiation and growth of a crushing band from the central parts of the specimen; (c) Similarcrushing bands in Pine wood specimen.

displacements. All the samples tested absorb approximately the same levels of locking energy ofabout 18.3 J implying a speci"c energy absorption capacity of 383.5 kJ/m3 or 9.1 kJ/kg.

3.3. Impact test results

The impact tests were performed by "ring cylindrical aluminium honeycomb specimens of45 mm diameter and 30 mm length from a pneumatic launcher on to a Hopkinson pressure bar(HPB) load cell of 50 mm diameter and 2.4 m length (see Ref. [3]). A range of impact velocities wascovered by varying the air pressure in the reservoir or the position of the specimen in the gunbarrel. These velocities were measured using pairs of photodiodes and light sources close to the


Fig. 11. Quasi-static uniaxial load}displacement curves for aluminium honeycomb specimens.

impact area. The pulses detected by using full-bridge strain gauge stations attached to the HPBload cell were converted into force}time pulses.

The aluminium cylindrical backing disc attached to some of the honeycomb specimens hada mass signi"cantly greater than that of the honeycomb specimen. The mass of the specimen wasapproximately 2 gm without the skin, 6.3 gm with the skin and 56.3 gm with the backing disc. Forbacked specimens it was assumed that the measured pulse was applied to the mass and byintegrating the equation of motion of the mass (neglecting the mass of the honeycomb) anapproximate dynamic force versus displacement relationship was derived and the correspondingenergy absorption capacity of the sample deduced.

3.3.1. Deformation mechanismsThe deformation of the honeycomb cylinders under uniaxial impact loading conditions pro-

gresses through initiation and rapid development of strain localisation which result from local cellinstabilities and dynamic microbuckling of the cell walls. This process is similar to that observed inthe quasi-static loading conditions. However, dynamically, localisation of deformation shows itselfin the form of a crushing wave front moving usually from the impact (proximal) end surfacetowards the distal surface of the specimens, see Fig. 12. In terms of deformation modes, there are nosigni"cant di!erences between the uncrushed and pre-crushed samples and samples with andwithout backing discs. Samples impacted at high energy often disintegrated fully. This wasa limiting factor when attempting to test samples at high impact velocities, particularly the sampleswith aluminium backing discs.

3.3.2. Force pulses and energy absorption characteristicsThe force pulses from the impact tests on uncrushed and pre-crushed honeycomb specimens are

shown in Figs. 13 and 14, respectively. In all of these impact tests, the impact face of the honeycomb


Fig. 12. Deformation pro"les of honeycomb specimens subjected to uniaxial impact loading conditions: (a) Specimenwithout a backing disc impacted at <

0"49 m/s; (b) Specimen with a backing disc impacted at <

0"27 m/s.

specimen is the face without the aluminium skin. As can be seen in these "gures, the force}timeplots of impacted samples are similar in shape to the load versus de#ection curves of staticallytested samples despite the di!erence in the independent variable. The e!ects of impact velocity canbe clearly seen in the levels of enhancement of the initial peak crushing and plateau force, both ofwhich tend to increase with increase in the impact velocity. These increases can be expressed interms of stress ratio, S

3. For the initial peak loads the stress ratio is de"ned as the initial peak load

in the dynamic load pulse divided by the corresponding quasi-static peak. Similarly the stress ratioin the plateau region is de"ned as the dynamic load plateau divided by the static value. Typicalstress ratio versus impact velocity curves are shown in Fig. 15 for both the initial peaks andsubsequent plateau forces recorded.

As can be seen in Figs. 15(a) and (b), the level of stress enhancement appears to be insensitive tolaterally free and fully constrained loading conditions. However, for the same impact velocity, thestress ratios of specimens with backing discs are slightly greater than those of specimens withoutbacking discs due possibly to inertial e!ects. Some of the force pulses measured were integratednumerically assuming the force to be applied to the backing disc and neglecting the core mass of thehoneycomb to obtain its displacement-time history. In this way an approximate load}de#ectionrelationship was obtained and subsequently the energy absorption characteristics were deduced.These force}de#ection curves and their corresponding energy versus de#ection plots are shown inFigs. 16 and 17. In contrast to the quasi-static behaviour, the impact load}de#ection curves ofpre-crushed samples show a load drop immediately following the initial peak force, see Fig. 17(a).These results indicate clearly that the pre-crushing of the samples does not eliminate the initialforce spike when the samples are loaded dynamically, thus suggesting that it is likely to be ofinertial nature. For the same impact velocity, the initial peak crushing forces of uncrushed samples


Fig. 13. Force pulses of uncrushed honeycomb specimens: (a) Fully constrained specimens without backing discs;(b) Fully constrained specimens with backing discs.

are slightly greater than those of pre-crushed samples. However, the plateau forces and energyabsorbing characteristics of both uncrushed and pre-crushed samples are almost equal. Thesigni"cant enhancement of energy absorption capability of the honeycombs under impact loadingcompared to the static energy resulted from the increase in plateau crushing forces with increasingimpact velocity, see Figs. 16(b) and 17(b). The source of this stress enhancement is brie#y discussedin the next section.


Fig. 14. Force pulses of pre-crushed honeycomb specimens: (a) Laterally free specimens without backing discs;(b) Laterally free specimens with backing discs.

3.3.3. Shock behaviour of cellular solid materialsThe localised nature of deformation and the load-de#ection characteristics of aluminium

honeycombs under uniaxial quasi-static compression loading are typical features of cellularmaterials. Previous work by Reid et al. [2,3] showed that a structural shock wave propagationtheory could be used to predict reasonably well the dynamic response of one-dimensional metalring systems and woods, particularly when loaded across the grain. The same shock theory isapplied to assess the dynamic crushing strength of aluminium honeycombs under uniaxial impactloading since their quasi-static load}de#ection characteristics are monotically increasing and


Fig. 15. Comparisons between the shock theory and the experiments: (a) Initial peak stress ratios of uncrushedhoneycombs; (b) Plateau stress ratios of uncrushed honeycombs.

steepening and convex curve towards the de#ection axis apart from the initial peak. The simpleshock model based upon a "rst-order approximation of the behaviour of woods was derived in [3].It is based upon the assumption that the stress}strain curve of the material is rigid-perfectly-plasticlocking (r-p-p-l) curve which retains the two key features of the inelastic behaviour of woods,namely the crushing strength p

#3and a locking strain e

-as shown in Fig. 18.

As can be seen in Fig. 18, a mass carrying a specimen of initial length l0, cross-sectional area

A and density o0is impacting on to a rigid target normally with an impact velocity <

0. As soon as

the impact is initiated, a shock wave is produced which moves towards the opposite end of the


Fig. 16. Impact of uncrushed honeycombs with backing discs: (a) Static and dynamic load}displacement curves;(b) Static and dynamic energy}displacement curves.

specimen. Because of the rigid behaviour (r-p-p-l), at the moment of contact the stress in all thematerial ahead of the shock is instantaneously raised to p

#3. The material behind the compaction

front is locked to the strain e-, its particle velocity becomes zero, its density o

0is raised up to the

locking density o1and the stress is raised to pH. This stress inside the shock front is de"ned in terms

of the material constants p#3

and e-and the velocity change at the shock front. The equations of

propagation of the shock are derived from algebraic equations of conservation of mass andmomentum which hold across the discontinuity front. These equations were derived in [3]. Herethe key equation for the enhancement of the crushing stress is applied to both the initial peak stress


Fig. 17. Impact of pre-crushed honeycombs with backing discs: (a) Static ad dynamic load}displacement curves;(b) Static and dynamic energy}displacement curves.

and the plateau stress. This is done by setting the crushing stress (r#3

of Fig. 18) equal to the initialpeak stress p

#3"p

0in Fig. 15(a) and then equal to the plateau stress p

#3"p

Pin Fig. 15(b).

The initial dynamic peak stress, p*0, is given by

p*0"p

0#

o0<2

0e-

. (2)

and the dynamic plateau crush stress, p*P, by

p*P"p

P#

o0<2

0e-

. (3)


Fig. 18. Shock propagation model for r-p-p-l material.

The stress ratio is de"ned as

S3"

p*0

p0

, or"p*P

pP

. (4)

3.4. Comparison between shock theory and experimental data

The theoretical predictions are plotted with the experimental stress ratio versus impact velocityfor initially uncrushed honeycomb specimens in Fig. 15. As can be seen, the model tends tounderestimate the initial peak stress ratios but it gives better prediction for the plateau stress ratios.Similar results are obtained for the pre-crushed specimens. As can be seen in Figs. 14(b) and 15(a),there are no signi"cant di!erences between the results from laterally free and fully constrainedimpact tests. Fig. 15(a) shows that the model tends to underestimate more signi"cantly theexperimental results from tests on specimens with backing discs. But in Fig. 15(b), for the plateaustress ratios, the agreement between the shock model and the experimental data is good. Fig. 15(b)includes also some low velocity impact data obtained from Wu and Jiang [10]. Their experimentaldata agree reasonably well with our data and shock theory. Despite its limitations, the shocktheory provides good estimates for the enhancement of the dynamic plateau crush stress foraluminium honeycombs.

It is noted that the initial peak stress increases in an approximately linear fashion as shown inFig. 15(a). It is conjectured that this behaviour is controlled by uniaxial plastic wave e!ects which


precede the onset of the progressive collapse mechanism. Employing a bilinear model for the cellwall material, the initial peak impact stress p* can be estimated using uni-directional stress}wavetheory to give

pH">#o0C

P(<

0!<

Y), (5)

where > is the yield stress, CP

is the plastic wave speed and <Y

is the impact velocity required toinitiate a plastic wave. The stress ratio is calculated by normalising the predicted impact stresspH with respect to the quasi-static initial peak stress p

0. The initial peak stress ratio obtained using

the plastic wave theory is plotted in Fig. 15(a) for a plastic wave speed CP

of 1200 m/s. This valuehas not been determined experimentally. Further work is being developed to improve the presenttheory and this will be the subject of future papers by the authors.

The results from the uniaxial impact crushing of honeycombs indicate a signi"cant enhancementof the dynamic crush stresses with increasing impact velocity. When expressed in terms of stressratio, the level of this enhancement could be as high by an order of magnitude at impact velocitiesof about 300 m/s. It has been observed in both initial peak crush stress and in the dynamic plateaucrush stress. This increase in the dynamic plateau stresses leads to an overall increase in the energyabsorption capacity of the honeycomb specimen.

4. Conclusions

The two examples above illustrate the signi"cant e!ects that inertia can have on the performanceof energy absorbing materials and structures. In the case of the inversion tube, the principal e!ect fora tube of uniform thickness is to generate an initial peak force in excess of the steady-state stage3 force due to the phenomenon of `second impacta. This is of considerable importance to thedesigner. In addition the reduction in the steady-state force compared with its quasi-static value hasbeen con"rmed both experimentally and by both "nite element analysis and a simpli"ed theoreticalmodel. The study of the dynamic crushing of aluminium honeycombs has once more con"rmed thesensitivity of the crushing stress of a cellular material to impact velocity. The in#uence of inertia at thecell wall level modi"es the crushing mechanisms leading to enhancements in both the initial crushingstress and the plateau stress. The latter is well-predicted by the simple rigid/perfectly plastic/lockingshock model, whilst the former appears to be governed by uniaxial plastic wave e!ects.

Acknowledgements

The work outlined in this paper was supported "nancially by the DRA, Fort Halstead (projectSMCFU/9) and by the Engineering and Physical Sciences Research Council (EPSRC) under grantGR/K18832. The authors would like to express their gratitude for this support.

References

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[2] Reid SR, Reddy TY, Peng C. Dynamic compression of cellular structures and materials. In: Jones N, Wierzbicki T,editors. Structural Crashworthiness and failure. London: Elsevier, 1993. p. 295}340.

[3] Reid SR, Peng C. Dynamic uniaxial crushing of wood. Int J Impact Engng 1997;19:531}70.[4] Harrigan JJ, Reid SR, Reddy TY. Inertial e!ects on the crushing strength of wood loaded along the grain. In:

Allison IM, editor. Experimental mechanics. Rotterdam: Balkema, 1998. p. 193}8.[5] Harrigan JJ. Internal inversion and nosing of laterally constrained metal tubes. PhD Thesis, UMIST, UK, 1995.[6] Reid SR, Harrigan JJ. Transient e!ects in the quasi-static and dynamic plastic internal inversion and nosing of

metal tubes. Int J Mech Sci 1998;40:263}80.[7] Jahsman WE. Static and dynamic material behaviour of syntactic foam. In: Lindholm US, editor. Mechanical

behaviour of materials under dynamic loads. Beriln: Springer, 1968. p. 361}87.[8] Zaretsky E, Ben-dor G. Compressive stress}strain relations and shock Hugoniot curves of #exible foams. ASME

J Engng Material Technol 1995;117:278}87.[9] Goldsmith W, Sackman JL. An experimental study of energy absorption in impact on sandwich plates. Int J Impact

Engng 1992;12:241}62.[10] Wu E, Jiang W-S. Axial crush of metallic honeycombs. Int J Impact Engng 1997;12:439}56.


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Effect of Inertia on Energy Absorption