Energy Absorption of Monolithic and Fibre Reinforced Aluminium Cylinders

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Energy Absorption

of

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Energy Absorption

of

Monolithic and Fibre Reinforced

Aluminium Cylinders

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. dr. ir. J.T. Fokkema, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op maandag 13 februari 2006 om 15.30 uur

door

Johannes Ludovicus Carola Gerardus DE KANTER

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Prof. dr. Z. G¨urdal

Prof. dr. ir. S. van der Zwaag

Samenstelling promotiecommissie:

Rector Magnificus, Voorzitter

Prof. dr. Z. G¨urdal, Technische Universiteit Delft, promotor Prof. dr. ir. S. van der Zwaag, Technische Universiteit Delft, promotor Prof. dr. ir. R. Benedictus, Technische Universiteit Delft

Prof. dr. ir. F. van Keulen, Technische Universiteit Delft Prof. dr. ir. A. de Boer, Universiteit Twente

Prof. dr. M.W. Hyer, Virginia Tech

ir. T.J. van Baten Technische Universiteit Delft

ISBN-10: 90-9020228-5 ISBN-13: 978-90-9020228-0

Keywords: crashworthiness, axial compression, crushing, multi-material tubes, specific energy absorption

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopy-ing, recording or otherwise, without the prior written permission of the author J.L.C.G. de Kanter, Delft University of Technology, Faculty of Aerospace Engineering, P.O. Box 5058, 2600 GB Delft, The Netherlands.

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Nomenclature

Latin symbols

A,B,C,D Johnson-Cook constants

dAB distance between points A and B

D tube diameter

eij strain deviator

E Young’s modulus of elasticity Ec E composite in fibre direction E∗ bending stiffness

f circumferential to longitudinal stiffness ratio Fd correction factor for (dynamic) displacement Fu ultimate load

Fy yield load

g gravitational constant h fold length (one leg) hi inclined hinge length

H height

i, j index number in length and thickness direction K Power function constant

l momentary element length l0 original element length L length or fold length: L = 2h L1/4 quarter fold length, L1/4= 1/2h m eccentricity factor

M bending moment or mass M0 collapse bending moment n Johnson-Cook or Power constant

N normal force or number of diamond lobes P mean (compressive) load

Ppeak peak load

PE11 tangential equivalent plastic strain PE12 shear equivalent plastic strain PE22 radial equivalent plastic strain q Cowper-Symond constant

r hinge radius

R tube radius

R0 momentary tube radius

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sij stress deviator S11 tangential stress S22 radial stress S33 out of plane stress Svm Von Mises stress t (momentary) thickness tc composite shell thickness tm metal shell thickness t0 original element thickness

T time

U energy

Uf riction energy dissipated by friction Uhourglass energy dissipated by hourglassing Uint internal energy

vc velocity at point C vf fibre volume percentage vm matrix volume percentage ¯

v average velocity

V velocity or shear force

w momentary strip element width

wct = Rεct

w0 original strip element width

W work

Wb hinge bending work

WBH horizontal hinge bending work WBI inclined hinge bending work Wcurve decurving work per fold

We external work

Wm membrane work

Wmc+ composite tensile membrane work

Wmc− composite compressive membrane work

Wmm metal membrane work

˙

W work rate

˙

We external work rate ˙

Wi internal work rate x, y cartesian coordinates xm fold thickness z thickness coordinate

Greek symbols

δe effective crush distance (per fold) δij Kronecker delta

δtot total displacement

δu displacement at ultimate load δy displacement at yield load

εct composite ultimate strain parallel to fibres εnom nominal strain

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Nomenclature xi

εtrue true strain εu ultimate strain

εθθ circumferential tube strain εz axial tube strain

˙ ε strain rate ˙ ε0 Johnson-Cook constant logarithmic strain ij strain tensor kk hydrostatic strain t tangential strain r radial strain s out of plane strain ¯

effective logarithmic strain η relative thickness ηs stroke efficiency φ proportionality factor θ angle κ curvature ¯ κ relative curvature, t/ρ µ coefficient of friction ν Poisson’s ratio ω relative width, w/w0 ρ density or bend radius ρa inner radius

ρb outer radius

ρm momentary cental layer radius ρn neutral layer radius

ρ0 un-elongated layer radius

σcr composite compressive strength perpendicular to fibres σcf composite compressive strength parallel to fibres σij stress tensor

σkk hydrostatic stress σmean mean stress σnom nominal stress σ0 static flow stress σ00 dynamic flow stress σpeak peak stress

σt tangential stress σtrue true stress σr radial stress σs out of plane stress σu ultimate stress σvm Von Mises stress σy yield stress σ0.2 0.2% yield stress ¯

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Abbreviations

BM bending moment CAE refers to Abaqus CAE CFRP carbon fibre reinforced plastic DOF degrees of freedom

EA energy absorption expl. explicit

FE finite element

FRP fibre reinforced plastic GFRP glass fibre reinforced plastic i.p. integration point

impl. implicit LL long transverse LR load ratio

P-C PAM-CRASH

Q.S. quasi static

RP reference point (FE)

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Chapter 1

Introduction

The presented thesis can be considered to be part of the large theme of traffic safety and in specific the ’crash’ safety. Within the theme this research is directed at the increase of the crashworthiness of mainly cars by improvement of structural components that absorb crash energy.

This introduction will firstly describe automobile crashworthiness with the paradoxes and contradictions, which are involved. Secondly the situation of the aerospace crash-worthiness is discussed with its relevance to the automotive industry and the presented thesis. Then the subject of the thesis, energy absorbing components, is introduced, followed by the objective of the research and the outline of the thesis.

1.1 Car crash safety

1.1.1 Automobile crashworthiness design

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Pre-tensioners and force limiters for seat belts Front and

side airbags Contoured foam interior panels 8 kph elastomeric foam or skinned composite bumpers Ultra-stiff passenger compartment to protect occupants from intrusion Energy absorption for

collisions with stationary objects or cars

Progressively stiffer elements designed to absorb energy in severe collisions

Figure 1.1: Simplified representation car safety layers

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1.1. Car crash safety 3

Figure 1.2 shows the structural elements of the different crash zones of a Volkswa-gen Golf. Clear is the bumper beam, the front crush zone existing out of two S-shaped beams, the longitudinals, and above that two small beams, the shotguns. The progres-sive zone includes the region from the shock towers to the safety cage. The safety cage has some added stiffness elements, like the B-pillars (beam between the two doors) and the door beams, both to prevent other cars from intruding in case of a side impact. The door beams were introduced for geometric compatibility with the bumper of a colliding car. Side collisions are a typical geometry compatibility problem, which is enhanced with the presence of SUV’s (Sports Utility Vehicles) or Pick-up trucks, due to their large ground clearance.

The interior of the safety cage should provide a soft shell for the occupants. Meaning that dashboard, door panels and other ’hard’ elements need to be relatively soft and cushion a possible impact with the body. Edges need to be rounded etc. The steering column was already discussed by Nader in 1964 [42] as a dangerous element. A lot has been done on this specific aspect. First issue was to prevent the intrusion of the column into the driver space and second to have a progressive collapsing column [30]. In later years the same was done for the pedals. Airbags have finalised the soft shell space around the occupants. A quantity of 6 airbags in a modern day car is quite standard. This comprises: two frontal airbags (driver and passenger), two side airbags and two curtain airbags protecting the window area. New additions are knee airbags, which replace the shortly introduced knee paddings. Some ’hard’ elements in the cage are the seat belts, the head restraints (and seats) and the windscreen. Seat belt development has seen to the development of the lap-shoulder belt with pre-tensioners and constant force extension. And now some ideas are present about introducing a four point seat belt, as known from racing cars.

Structural crash elements

The structure of the car and in specific the crush zones and safety cage as described previously form the foundation of the crash performance of the car. Add-on devices as airbags, seat belts and cushions must fine tune the crash behaviour and create the interface between (soft) occupant and (hard) car safety structure. Some of the crash elements will be described in more detail as they are possible areas for the application of this research work.

First the bumper beam. This beam has to redistribute the crash forces from the point of impact, ideally to both sides of the car. The core of the bumper is a profile made out of steel or aluminium (figure 1.3), which redistributes the lateral load, but is itself an energy absorber as well, by ’axial’ crushing of its webs and by global bending. A soft outer region must be present to absorb crash energy from pedestrian impact. This is mostly done by means of a thermoplastic cover containing ribs, which fold together.

Second the shotguns (two upper tubes) and the forward (straight) part of the lon-gitudinals or S-beams (figure 1.4). Both elements are tubular in nature and are loaded axially by the bumper beam, with or without an offset. These elements have the high-est energy absorbing potential and should crush progressively, from front to rear. The elements are generally made of steel or aluminium, where the aluminium outperforms the steel as will be shown later. These are also the elements, which are similar to the cylindrical tubes presented in this research.

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deforma-Figure 1.3: Aluminium bumpers from Norske Hydro

tion of the material. Stability of these S-shaped beams is also a main concern, especially in case of offset crashes.

Furthermore experiments are done with add-on energy absorbers. These are also tubular elements, which are axially loaded and crushed progressively. These elements can be situated between engine and fire wall for example and have a very high specific energy absorption, as that is the sole requirement of the element. In this case also composite materials are used instead of aluminium or steel. The geometry can be dedicated to the space available and varies from cylindrical and square tubes to cones.

1.1.2 Safety paradoxes and contradictions

Although there appears to be a consensus about what safe transport means, it contains paradoxes, and some ideas lead to contradictions. To arrive at a better understanding of what safe transport could mean some of the paradoxes and contradictions will be uncovered.

Local safety and global safety. Driving a tank or just a heavy Volvo sta-tion car will give a high level of local safety to the occupants. However when a collision with another small-sized car takes place the level of global safety of all involved may be impaired. Fortunately this paradox is acknowledged and amongst others Volvo engineers put a lot of effort to increase the mentioned global safety by harmonisation of their crash structures and softening the front end for e.g. pedestrian impacts.

Weight paradox. The core of vehicle crashworthiness is the absorption of ki-netic energy, which is proportional to the mass. Now safety devices add to the car a lot of mass, which means kinetic energy that has to be absorbed in addition. Furthermore the mass difference between cars gives a compatibility mismatch resulting in different injury levels for the parties involved. The mass incompatibility is often accompanied by geometry and stiffness incompatibility. Areas which get a lot of attention the last few years. See for example the reports from IIHS [28] or NHTSA [43] or their websites.

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1.1. Car crash safety 5

Figure 1.4: Aluminium Audi A8 structure with tubular S-beams and shotguns

Nader [42]. Only in 1956 the effect of car collisions was first studied and it was the introduction date of the safety belt. However this belt was only meant to prevent the occupant from ejecting out of the car. Since 1970 full scale car crash testing led to incorporating safety in the structural design of the car. It was also the date of the introduction of the airbag. Since then car design gradually evolved as well as the safety measures up to the early nineties. It was in the nineties that the customers got more informed about the cars environmental impact and safety. It was then that the car safety war for star ratings started (first EuroNCAP publication of crash test results was in 1997). Nowadays safety is bought by the informed customer. EuroNCAP claimed that ’today, more than ever before, safety sells cars. For car buyers it is a key element of their purchasing decision’ [16]. However how much safety is exactly bought is unclear nor what this safety means for ’your opponent’ in case of a car crash. First this raises the issue whether the tests really resemble actual accidents and whether the buyers (or occupants) resemble the crash test dummies. Second question it raises is whether the improved safety to the occupants gives rise to danger for the people outside the car.

It clearly seems that buyers are more concerned with safety than whether the car lives up to their transportation needs.

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and the abundance of e.g. airbags this is true as the edge is made to look far away, but never where exactly. And the car is not likely to tell you either, except when you do cross that ’edge’. Where one means by crossing the ’edge’ is having an accident.

In addition to this information gap these systems partly take over the function of the driver and in doing so take away the driver responsibility. However it is and will remain the driver that is (partly) responsible for 85 to 95% of all accidents.

Sustainability issues. The mid nineties have shown an increased environ-mental and safety awareness of both politicians and consumers. In case of transport vehicles, in specific cars, this has led to a paradox, which is generally not recognised by the public nor government. It is the paradox of increased safety versus increased environmental impact.

Where the efficiency of the petrol engine has increased the last two decades by 100%, the average car (e.g. VW Golf) still consumes, in practice, 10 litres per 100 km. Just like it did all those years ago. This offset is the result of both safety and luxury that have been incorporated the last decades. Safety and luxury measures increase not only the weight of the car but also the volume, which partly explains the increased outer size of the car with constant internal space. As an example: The Opel Astra increased 27% in mass from 1985 (820 kg) to 1998 (1040 kg) and the Opel Omega increased 38% in mass from 1985 (1120 kg) to 2000 (1540 kg) (both had a model change in this period, data from [15]). As a bold measure the weight of the car determines half the fuel consumption and the other half of the fuel consumption is determined by the aerodynamic drag, which is proportional to the frontal area of the car and the coefficient of drag.

Accident prevention potential. More than 90% of the possible measures that one can take to decrease the consequences of traffic accidents have already been installed in today’s car (according to Seiffert [49]). Seiffert showed an S-curve for the development of safety measures, which is now at 90%. This means that additional safety measures will be costly (both economically and environmentally) and will give marginal improvements. Possible improvements are for example the 4 point seat belt and knee airbags.

Aircraft safety. The safety paradoxes are less evident in the aircraft (or other commercial modes of transport). Some typical differences are that the user of the vehicle is a well educated professional, the occurrence of collisions is minimal (with much larger consequences) and safety regulations are more stringent. A final difference is the absence of consumer influence, which may change with the introduction of a black list in 2005 by some governments, which publishes less reliable airlines. Actually what is most important is that the aircraft industry has grown up with a major safety paradox: Flying ’unsafe’ with a certain risk, or not flying at all. And within this concept the risk is fully focused on prevention of crashes, assuming the ending is fatal when things do go wrong.

An important safety paradox in aviation is that of dangerous defences. Ac-cording to Reason [47] ’measures designed to enhance a system’s safety can also bring about its destruction.’ Proof is given by the fact that in commercial aviation quality lapses in maintenance are the second most significant cause of passenger deaths.

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1.2. Aerospace crashworthiness design 7

1.2 Aerospace crashworthiness design

The two safety design approaches of both the aerospace and the automotive seem to meet each other in the current decade. The aerospace focus is including crash protection next to the crash prevention and the automotive crash protection is being extended by crash prevention measures.

Figure 1.5: Experimental full scale crash of Boeing transport B720 [29]

Another factor in the car design is the integration of aerospace materials in the automotive industry. First of all aluminium and second also composite materials. The integration of composite materials in the aerospace industry already led to difficulties concerning crashworthiness and the description and behaviour of these materials are still a challenge for the simulation tools and for the engineers. There is however in the helicopter industry good experience with composites in crashworthy structures, which can be transferred to other industries, including the automotive.

The requirements for crashworthiness design are very tough to be setup in the aircraft industry and it takes many years from first consideration to their enforce-ment [6]. For the practice it means that aerospace crashworthiness design has to rely on the simulation of numerical models and not on extensive testing, which would involve too high costs. Still some large scale testing will be done and has been done in the past by governmental institutions [3, 29], figure 1.5.

A relative newcomer in the crashworthiness design is the space industry, where safety and lightweight design are even more important. Below some examples will be shown for the different aerospace disciplines.

1.2.1 Helicopter crashworthiness

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impact velocity. The vertical impact velocity limit is 12.8 m/s stated by MIL-STD-1290. To absorb the kinetic energy and protect the occupants multiple energy absorbing elements are present, just like in a car. These are presented in figure 1.6. To extend the crashworthiness it is sometimes possible to ’remove’ heavy items during the crash. Like the breaking of the tail section, as demonstrated in figure 1.7.

Figure 1.6: Simplified representation safety layers helicopter

As helicopters make extensive use of composite materials, here the crashworthiness of composites is well studied and used in the design. Typical composite energy absorbing structure is the subfloor. Figure 1.8 shows the experimental crash of a composite helicopter fuselage.

1.2.2 Aircraft crashworthiness

Nowadays aircraft crashworthiness is getting more attention. One of the problems is the increased number of passengers per aircraft and also the application of new (more brittle) materials. One of the possibilities to dissipate energy during a crash is deformation of the sub-floor area. An example is shown of the crushed sub-floor of a Boeing 707 in figure 1.9. However with new materials or new lay-outs this may require new design philosophies. This is demonstrated in figure 1.10 for a carbon fibre fuselage. Here also the sub-floor should dissipate the crash energy. Only due to the high stiffness and small failure strain this is much more critical. Also the resulting deformations are smaller and thus the accelerations higher.

1.2.3 Spacecraft crashworthiness

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1.2. Aerospace crashworthiness design 9

Figure 1.7: Full scale helicopter crash test, source NASA

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Figure 1.9: Experimental crashed fuselage structure [29]

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1.3. Energy absorbing components 11

Figure 1.11: Possible configuration Mars sample return vehicle after crash, source NASA

that the sample return vehicle is fully crashworthy. A possible concept is shown in figure 1.11. The concept has a deformable outer shell, which is supported by a cellular structure filled with foam, which encloses a rigid inner shell, which fits tightly around the spherical containment vessel (not shown in the figure).

1.3 Energy absorbing components

Following paragraphs describe the requirements and behaviour of energy absorbing components made of different materials.

1.3.1 Requirements for an energy absorber

An energy absorbing element dissipates kinetic energy. In order to do this a deformation takes place, which in general is plastic. First requirement therefore is that it absorbs all energy, at a minimum weight, volume or length (see weight paradox and sustainability issues). Second requirement is that in absorbing the energy, the force level does not exceed a predefined value. This is often based on acceleration levels defined by occupant injury criteria or instrument limitations. For comparison of different elements and materials the following indices are defined:

SEA = EA_M Specific Energy Absorption ηs = δtot_L Stroke efficiency

LR = Ppeak

P Load ratio

where EA is the absorbed energy, M is the mass of the element, δtot is the total axial deformation and L is the original element length, Ppeak is the peak load and P is the mean load.

In all cases of transport vehicles the mass is of utmost importance and therefore also for the energy absorbing components. The weight efficiency is expressed by the SEA. For dedicated energy absorbers the SEA may vary between 15 and 75 kJ/kg.

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Stroke efficiency is the deformation length divided by the element length. In case of axial collapse this is in the order of 70%. When the efficiency is 100%, the element is fully compressed with no residual length.

The load ratio is the peak load divided by the mean load. This is of importance for the maximum occurring force level, which would ideally be the mean load. Therefore a load ratio of one is ideal and the higher the ratio the worse the absorber is. For cars the impact force is proportional to the acceleration level the occupants have to withstand.

Secondary requirements are defined to enable the primary requirement of en-ergy absorption. These are reliability and predictability. Reliability comes from a consistency in performance as well as predictability. Stability and detail insensitivity of the element are required as the level, direction and character of the crash energy can have a wide spread.

A final requirement is the post-crash integrity. When the energy absorbing ele-ment is carrying another component, it may be required to keep doing so also after a crash. Other relevant issues are that during and after the deformation process no other elements (including occupants) may be damaged by the energy absorber.

1.3.2 Typical behaviour energy absorbers

Considering the most efficient energy absorbers, axially loaded thin-walled cylinders, one recognises two different collapse mechanisms. The metal progressive plastic col-lapse and the progressive fibre reinforced plastic (FRP) brittle colcol-lapse The colcol-lapse behaviour of metal cylinders was first described by Alexander [4]. A good review of structural impact and of metal tube folding is later given by Jones in his book [32] and an overview of Collapsible impact energy absorbers was presented by Alghamdi [5]. The fibre reinforced plastic tube collapse was already studied in 1979 by Thornton and Farley [18, 57] and is well described by Mamalis in his book [38]. Both collapse behaviours will be described shortly here.

Metal cylinders

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1.4. Objective 13

Fibre Reinforced Plastic cylinders

Fibre reinforced plastics have an elastic deformation up to failure at a few percent at most. In case of crushing the composite material will have a multitude of failure mech-anisms, all at a local level. These include fibre buckling, fibre pull-out, delamination and matrix cracking. This local damage growth disintegrates the complete tube in a progressive manner, figure 1.12. The result is a very high SEA. The SEA can get as high as 65 to 75 kJ/kg for Carbon Fibre Reinforced Plastic (CFRP) cylinders [19]. The SEA for Aramid-Epoxy cylinders is low at 9 to 23 kJ/kg and the SEA for Glass Fibre Reinforced Plastic (GFRP) cylinders is comparable to the metal tubes, 30 to 50 kJ/kg. The disintegration also results in a very high stroke efficiency, up to 90%, as all material is broken into small parts and scattered around. This in itself limits the post-crash integrity, which is nihil. While the failure is over very small distances the load rise and fall are limited over this distance. The load ratio is therefore close to 1, approximately 1.2. Often the mean load rises a bit with the deformation. The initial stability (buckling) of FRP is good, but once breached, there is limited resis-tance against further deformation. Also detail sensitivity is high due to the brittle nature. A hole in a tube will likely initiate local crushing, reducing the stability of a complete section. It is also required to initiate crushing at one end of the tube. This is done by a incorporating a ’trigger’ [57], a weak, cambered, edge for example. To limit delamination in the crushing process, stitching may be used [60].

Multi-material energy absorbers

Multi-material elements demonstrate both metal and composite characteristics in the crushing behaviour. First work on multi-material tubes was from Mamalis [39]. How-ever this was a combination of two ’plastically behaving’ materials (e.g. two metals or PVC and a metal), which will not be discussed here. The work on combined com-posite and metal crash behaviour can be found in references [11, 25, 56, 61]. Wang demonstrated both the plastic metal behaviour and the brittle composite behaviour of the multi-material tubes. A new phenomenon in the multi-material tube is the de-bonding between metal and composite, which can be compared to delamination within a composite. The de-bonding is a combination of peel and shear. Also typical for the multi-material tubes are the interactions of both metal and composite. First is the mode change of the metal tube due to the constraining fibres, and second is the extensional fibre fracture due to extension (’barreling’) of the metal tube.

In a study by Shin [51] also the bending behaviour of multi-material tubes was stud-ied, which is typical for a car longitudinal at the curvature of the wheel bays. There he concluded that the hybrid (Aluminium/GFRP) tube has a little better energy ab-sorption capability than the sum of aluminium and composite tubes. Longitudinal and lateral composite splitting of the 0° and 90° plies was prevented by ±45° ply resulting in collapse with tearing failure of the composite.

1.4 Objective

1.4.1 Challenges multi-material energy absorbing elements

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1.4. Objective 15

As both composite and metal energy absorbers do not fulfill all requirements of the ideal absorber, but both are complementing each other, one could aim at an energy absorber close to the ideal one by integrating both materials in one absorber. This can be achieved by taking three approaches:

reinforce the metal cylinder with FRP reinforce the FRP cylinder with metal

create a multi-material (metal/FRP) component as one

The first two ways will be described below, the latter is not obvious and requires the knowledge of material combinations, which are mostly unexplored so far.

Reinforce metal cylinder with FRP

Shortcomings of the metal cylinder are the cyclic force deformation profile, with load ratio of 1.6. A moderate stroke efficiency of 70% and a SEA of approximately 30 to 50 kJ/kg. The reinforcement of the FRP should reduce the maxima and/or raise the minima. Changing the stroke efficiency is not required and will be very difficult, while maintaining the desired post-crash integrity. Furthermore an increase in the SEA is valuable.

Reinforce FRP cylinder with metal

Shortcomings of the FRP cylinder are the disintegration during collapse, the detail sensitivity and the global post-buckling behaviour. By reinforcing with metal, the plasticity of the metal layer should limit the complete disintegration to a minimal residual structure, and provide a load path for detail disturbances as well as a stable global post-buckling behaviour.

1.4.2 Success factors

Basically six success factors were described in the previous section: 1. Specific energy absorption

2. Load ratio

3. Detail insensitivity 4. Stroke efficiency 5. Post crash integrity

6. Reliability, stability, predictability, consistency

Go or no go factors can be applied on the success factors. For the SEA a target of +20% is reasonable, meaning a weight reduction of 20% for a crash component. Values are derived from the development of composite materials in the aerospace, where 20% weight reduction is a good reason for the introduction of a composite component, while cost remain within a 10% increase. New metals have a target of 10% weight reduction together with a 10 to 20% cost reduction. Ultimate goal for the introduction of new materials would be 30% weight reduction and 40% cost reduction. However cost will not be part of this investigation.

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Stroke efficiency improvement is not one of the targets of the research, and should therefore kept at a comparable level to the metal cylinders.

The other success factors are difficult to quantify and will be considered in a quali-tative way.

1.4.3 Boundary conditions of research

Research can easily get out of bands. Therefore some limiting factors or boundary conditions on the research are stated below.

First the approach will be to reinforce the metal tube with fibre reinforced plastic. The reason for this approach comes from the extensive know-how on metal tubes, which presents a solid base for the extension and due to experience with fibre metal laminates, where also the fibre layer was an add-on.

Second the metal reinforcement approach results in a choice for aluminium as metal for its good performance in energy absorption compared to other metals (steel, copper) and also due to the experience with aluminium in the aerospace environment. As a reinforcement the baseline chosen is glass fibre reinforced plastic. With a standard epoxy as matrix material and S2-glass fibres. Glass fibres were selected for their stiffness close to aluminium and their large failure strain. For production the filament winding process is best suited to place the continuous fibres at an angle. Only for longitudinal fibre placement an alternative will be used.

Third the shape of the specimens will be circular as this is the most efficient geometry for energy absorption in axial compression. Finally cost will not be considered in this research.

1.5 Outline

The baseline behaviour of the metal tube will be described first on the basis of the symmetric static folding in chapter two and the symmetric dynamic folding in chapter three. Both chapters are based on the pillars of analytical modeling, numerical modeling and experiment. Chapter four gives an extension of the analytical theories, to gain insight in the energy absorbing behaviour of the metal tubes and to bridge the gap between the analytical and numerical models by means of a finite difference method. Chapter five broadens the research of the aluminium tube towards non-symmetric folding (diamond mode folding), as will occur with the reinforced tubes. Also this chapter includes experimental, numerical and analytical research.

Chapter six presents the results of the circumferential fibre reinforced aluminium cylinders. The reinforcement results in a modified folding behaviour and energy absorbing potential. This research is extended in chapter seven with variation of the fibre orientation and fibre material.

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Chapter 2

Static axisymmetric folding of

aluminium cylinders

2.1 Introduction

Metal components may absorb energy by folding progressively in a kind of buckling pattern (figure 2.1). This folding process of thin-walled circular tubes has been studied for many years. Results of these studies are mainly analytical theories and a huge amount of experimental data. Also the numerical field has progressed enormously and offers powerful tools, e.g. finite element analysis, for these kind of analysis [8, 9, 41].

Figure 2.1: Progressive folding

In order to get a grip on the behaviour of multi-material components it is deemed essential to cover the mono-material behaviour of metal tubes first. In the first place because the multi-material behaviour is partly governed by the metal behaviour. And in the second place because the metal behaviour will highlight the advantages and disadvantages, which may be influenced for the better by introducing multi-materials. In this chapter the focus is therefore on understanding of the progressive folding process and the validation of available tools for parameter and design studies on the energy absorption of thin-walled cylinders. The collapse in the axisymmetric, or con-certina mode is studied here as this failure mode is the most common and illustrative case.

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good understanding of the plastic energy dissipation during folding. Secondly a test program is introduced to get a physical meaning of the folding and thirdly a finite element analysis is discussed. The test program covers static testing on three different tube geometries made of aluminium. The FE model is validated by these test, based on deformation profile, mean, maximum and minimum loads and fold length. Also the analytical theories are compared to both test and FE data.

2.2 Theory

2.2.1 Alexander model

The basic theory of the folding of metal cylinders is given by the early work of Alexander [4]. Alexander assumed stationary simple hinges with only outward folding or internal folding, where the folds have zero thickness. The amount of plastic work was calculated as being due to two processes: bending at the hinges and stretching of the sheet between the hinges. The material model was a rigid perfectly plastic material. The Alexander model is visualised in figure 2.2 and the analysis follows hereafter.

h R Hinges t P P θ D

Figure 2.2: Alexander stationary hinge model xm xm t/2 t/2 2xm+t=2h-δe P P R t

Figure 2.3: Abramowicz and Jones model

During an increment dθ of the angle θ, the increment of bending work done at the hinges is:

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2.2. Theory 19

M

-σ

₀

σ

₀

σ

₀

σ

-σ

₀

-σ

t

+

-Figure 2.4: Development of the plastic zones in an elastic, perfectly plastic 2D strip subjected to a pure bending moment

strain occurs. This gives, using von Mises yield criterion and plastic Poisson’s ratio of 0.5 (for incompressible flow), the following collapse moment:

M0= σ0t2/4 = (2/ √

3)σyt2/4 (2.2)

The fold leg length is indicated by h and is the length between two hinges, figure 2.2. This is the same for the curved fold in figure 2.3.

The mean circumferential strain in extending the metal between the hinges, during the incremental change dθ, is

π(D + h sin(θ + dθ)) − π(D + h sin θ) π(D + h sin θ) =

hdθ cos θ

D + h sin θ (2.3)

The stress between the hinges equals the yield stress, so that the increment of work done in stretching will be

dW2= σy

hdθ cos θ

D + h sin θπ(D + h sin θ)2ht = 2πσyh 2

tdθ cos θ (2.4)

The total work done in collapsing one convolution, i.e. for θ increasing from 0 to 90 degrees, is therefore W = Z (dW1+ dW2) = Z π/2 0 [M04π(D + h sin θ) + 2πσyh2t cos θ]dθ (2.5)

This must equal the external work, the mean axial load P multiplied by its total displacement 2h (neglecting the thickness of the tube). Hence integrating and rewriting,

P M0 = π2D h + 2iπ + 2π √ 3h t (2.6)

where i=+1 for the demonstrated outward folding and i=-1 for inward folding. The fold length is determined by minimising the mean load, equation 2.6, to give

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Alexander assumed a true deformation to lie somewhere between the two modes (inward and outward) and adopted the mean value of equation 2.6 with i=+1 and i=-1, i.e. P M0 = π2D h + 2π √ 3h t (2.8)

This becomes a function of t and D only, after substitution of the fold length:

P M0 = 20.73 r D t (2.9)

2.2.2 Modifications

Abramowicz and Jones [1] modified the Alexander solution for axisymmetric crushing by considering the circumferential strain as a function of its location between the hinges, where Alexander used a mean circumferential strain. Furthermore they applied an effective crush distance, equation 2.10. This is the fold length minus thickness of the tube and the fold thickness (xm), see figure 2.3.

δe= 2h − 2xm− t (2.10) xm≈ 0.28(h/2) (2.11) which gives: h = 0.88√Dt (2.12) P M0 = 20.79 q D t + 11.9 0.86 − 0.568 q t D (2.13)

where the nominator in equation 2.13 is the factor resulting from the effective crush distance.

Another improvement from Abramowicz and Jones [2] was a result of the deformed state of a lobe (fold), where the hinge deforms more than an angle of π. This is implemented by integrating the work W1in equation 2.5 over an angle larger than π/2. The governing equations then result in:

h = 0.893√Dt (2.14) P M0 = 25.30 q D t + 14.7 0.86 − 0.568 q t D (2.15)

Wierzbicki and Bhat [63] reconsidered the Alexander theory in 1986 and based their theory on the movement of hinges with respect to the shell material. This theory describes the shape of the deformed tube, i.e. the folds, much better than before. Further the solution gives good predictions for folding length, effective crush distance and mean load.

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2.3. Finite element model set-up 21

2H

m2H n2H D

Figure 2.5: Wierzbicki model, on left super-element model and on the right the sim-plified model

In 1992 again a new model (shown in figure 2.5) is introduced by Wierzbicki et al. [64]. The model allows the fold to move both inward and outward. The ratio of outward folding is defined by m, the eccentricity ratio. This geometry introduces several new features like finite values for the peak loads (however not accurate), unequal distances between the first peak and consecutive peaks and a realistic final shape (super-element). Most important new feature is the intermediate peak introduced, which is also present in most of the experimental curves. The resulting relationships are defined by: h = 1.31√Dt (2.18) P M0 = 31.74 r D t (2.19)

Further extensions of the Alexander theory have been made over the years by amongst others Grzebieta [20], Singace [54] and Gupta [24]. These are dedicated refinements to derive, amongst other things, the cyclic force-displacement characteristics, the eccen-tricity of the fold and material plasticity effects. All of the mentioned theoretical studies indicate the proportionality of the normalised mean crush load, P/M0, to (D/t)1/2for axisymmetric folding. Only the experimental program from Guillow [21] indicates that P/M0is proportional to (D/t)1/3, which questions the theory used. For our goal to have a general understanding of the folding process and for design purposes the Alexander, Abramowicz and Wierzbicki models suffice.

Results of both Alexander, Abramowicz and Wierzbicki’s theory are presented in table 2.5, section 2.5.4, for a cylinder similar to the tested specimen.

2.3 Finite element model set-up

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The analysis done concerns the static cases, using the implicit solver. The implicit code gives more accurate results compared to the explicit code at the expense of computing time.

The model consists of an axisymmetrical cylinder (as the collapse mode for investi-gation is the symmetric concertina mode) and two rigid compression plates, figure 2.6. The compression plates are modeled as analytical rigid elements. The cylinder is a de-formable body. Cylinder and compression plates interact through contact definitions. The upper compression plate is moved downward, with no allowance for rotation and lateral movement. The lower plate is held fixed in all directions. Contact between the cylinder folds, is done through self-contact definition.

RP

1

2

3 R

Figure 2.6: FE model definition

The analysis is a static general non-linear analysis, which is stabilised by dissipated energy fraction (0.0002), the default in Abaqus. Stabilisation is required for the post-buckling character of the problem. Minimum increment size is 1e-8. A linear 4 node element is taken, which uses a reduced integration. Quadratic elements should not be used for this type of analysis as the distortions are too large [17]. When the quadratic elements are used, the global force deformation profile is reasonable, however the ele-ment stresses are not. The plane stress/strain thickness of the eleele-ment is 1.

The cylinder material is elastic-plastic with isotropic strain hardening. Material data can be found in table 2.1. The presented nominal stresses and strains, σnom and εnom, are engineering values and should be converted to true stress and strain values by following equations

σtrue= σnom(1 + εnom) (2.20)

εtrue= ln(1 + εnom) (2.21)

for isotropic material. The true strain is converted to plastic strain by subtracting the elastic strain

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2.4. Test set-up 23

Table 2.1: Material data

Aluminium alloy E [GPa] ν [-] σy[MPa] σu[MPa] εu[%]

6063-T6 69 0.3 215 240 12

6060-T66 69 0.3 180 212 12

The plasticity data are put into the model as tabular input (appendix A.5). An additional point is added to the table, which is beyond the ultimate strain, with a barely higher stress level (approximately 1 to 2 MPa). This is required for Abaqus, which extrapolates the tabular material data when required beyond the last data point. No damage models are incorporated.

The cylinder which will be discussed, AL4541, has a median diameter of 43 mm and a thickness of 2 mm. Length of the cylinder is 90 mm. Interaction properties are defined for three couples, cylinder to bottom plate contact, cylinder to top plate contact and cylinder to cylinder contact. The interaction is defined through tangential and normal contact. The tangential contact is implemented through a penalty function with a friction coefficient µ. The normal contact is defined as hard, no penetration will occur, with separation allowed. The friction coefficient was chosen as 0.35 for the cylinder to top plate interaction and 0.5 for the cylinder to cylinder contact and the cylinder to bottom plate contact. The difference in friction for the top plate and the bottom plate has two reasons. One is to introduce an imperfection in the model, which prevents the squashing of the model and initiates folding on one side of the model only instead of on both sides. And second the difference in kinetic and static friction is thus applied. The coefficients were chosen based on a small FE study as data for friction coefficients are limited and have a wide spread [50].

The minimum number of elements is determined from a sensitivity analysis in which the element size was varied. The refinements varied from 1 element over the thickness of the tube to 12 elements over the thickness. The outcome is visually checked for the complete force-displacement graph (figure 2.10), which converges at 4 elements over the thickness (720 elements in the whole model). The element size is then 0.5 x 0.5 mm2. When convergence was checked for the first peak load only, this would have resulted in the most basic model with 45 elements, which after first peak shows no agreement with the other models.

2.4 Test set-up

Quasi-static axial compression tests were performed (figure 2.1) on thin-walled alu-minium tubes. Table 2.1 shows the material data of the tubes. Testing was done on a static 25 ton Zwick testing machine. Compression plates, with a spherical joint, were installed, which provide a simply support on top and bottom. Zwick TestXpert 9.0 was used as controller. Compression speed was 10 mm/min and loading was displacement controlled. The test was manually arrested at approximately 20% of the tube length remaining. Data acquisition was taken from the crosshead displacement and the force sensor.

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Table 2.2: Specimen dimensions

Material Median Thickness Length D/t diameter [mm] [mm] [mm] [-] AL4541 6063-T6 42.9 2.0 89.9 22 AL5045 6060-T66 47.5 2.5 120.0 19 AL5047 6060-T66 48.5 1.5 100.0 32

For this part of the work, folding in concertina mode was desired, to enable com-parison between experimental, analytical and FE analysis. Gupta [22], Andrews and Singace [53] show D/t regions between 10 and 40 give concertina mode, where Jones [32] limits the region to D/t<80.

2.5 Results and discussion

In this section the results of the analytical theories, the experiments and the finite ele-ment model are presented and discussed individually. After that the three approaches are compared and discussed in the last section. However first the typical force defor-mation profile of progressive tube folding is presented to give better insight into the results.

2.5.1 Force-deformation profile

The actual folding process, visualised in figures 2.7 and 2.8, is the following for two consecutive folds: At points 1 and 6 due to instability a region of the cylinder is pushed outward. Due to this geometry change the force drops as compressing of the now deformed cylinder is easy, points 2 and 7. The folding is arrested when the first leg contacts the previous fold (or compression plate), just after points 3 and 8. This contact gives rise to the loading. Again a small stability problem is reached here, point 4, where the lower hinge is forced inwards and due to this geometry offset, some reduction in loading follows. Then again just after points 5 and 9 the lower leg also makes contact to the previous leg and a rise in loading will occur up to points 6 and 10.

In the theory of Alexander the folding starts at three hinges at the same time col-lapsing at the buckling load (first peak, figure 2.8). After collapse the load is reduced drastically until just before the folds make contact, the minimum. Then the load im-mediately rises again to the buckling load, which again initiates the folding. Main difference between the actual folding and the theoretical folding of Alexander’s theory is that the actual folding has no stationary hinges (the plasticity is spread over the fold curvature) and does not have symmetrical folding, i.e. the fold legs do not make contact simultaneously. One proof of this is the intermediate peaks, e.g. point 4 in figure 2.8.

2.5.2 Experimental results

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2.5. Results and discussion 25

Figure 2.7: Folding process stages, numbers coincide with points in figure 2.8

minima, the fold lengths, coincide as well. Typical results of the experiment are shown in table 2.3, only the AL4541 specimen will be discussed here, the other specimen show equal behaviour.

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0 10000 20000 30000 40000 50000 60000 0 10 20 30 40 50 60 70 Displacement [mm] Compressive force [N] Alexander FE 1 2 4 5 6 10 7 8 9 3

Figure 2.8: Force displacement profile with indication points, FE curve and Alexander model 0 10000 20000 30000 40000 50000 60000 0 10 20 30 40 50 60 70 Displacement [mm] Compressive force [N] 1 2 3 Mean load

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2.5. Results and discussion 27

Table 2.3: Experiment results

Mean load [kN] Peak load [kN] Effective stroke [mm]

AL4541 35.5 59.1 13.1

AL5045 55.2 81.4 18.8

AL5047 24.9 50.2 13.1

(crosshead displacement), which is not accurate enough for the initial elastic loading. This was confirmed by stiffness measurements from a LVDT. The effect on the post-buckling behaviour can be ignored.

2.5.3 Finite element results

The effect of some variables in the FE model are discussed here, as they influence the FE outcome considerately. These variables are the model or element size, the material data and the interaction properties (mainly friction).

0 10000 20000 30000 40000 50000 60000 0 10 20 30 40 50 60 70 Displacement [mm] Compressive force [N] 4 3 2 1

Figure 2.10: Effect of element size on force displacement profile (1 has 1 element, 2 has 4 elements, 3 has 6 elements and 4 has 12 elements over the thickness of the tube)

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0 10000 20000 30000 40000 50000 60000 0 10 20 30 40 50 60 70 Displacement [mm] Compressive force [N] 1 ₂ 3

Figure 2.11: Effect of friction (1: µ = 0.1, 2: µ = 0.35 and 3: µ = 0.5)

However only at 4 elements, number 2 in the figure, a desirable convergence is reached amongst the models. Number 3 represents the model with 6 elements over the thick-ness and number 4 the model with 12 elements over the thickthick-ness. This shows that the convergence occurs latest for the minima in the graph.

The interaction effect is studied for different friction coefficients. The friction coef-ficients for the interaction between top plate and cylinder (dynamic site) were varied from 0.1 for model 1 to 0.35 for model 2 and 0.5 for model 3. The effect, which is largest at the first minimum can be seen in figure 2.11. The minimum is raised and moved to a larger displacement. The intermediate peak on the other hand is reduced. The friction effect dampens out after the first fold and is not discernible anymore after the second fold.

The material strain hardening raises the force level of the complete folding process as can be seen in figure 2.12 and thus increases the mean crush force. Curve one represents the ideal elastic-plastic model, with yielding at 215 MPa and curve two represents the strain hardening model with yield starting at 215 MPa and an ultimate stress of 240 MPa (nominal) at 12% strain.

FE validation by Test

In order to validate the FEM crush behaviour, typical FEM output is compared to the experimental data.

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2.5. Results and discussion 29 0 10000 20000 30000 40000 50000 60000 0 10 20 30 40 50 60 70 Displacement [mm] Compressive force [N] 1 2

Figure 2.12: Effect of material strain hardening (1: ideal elastic plastic, 2: strain hardening) 0 10000 20000 30000 40000 50000 60000 0 10 20 30 40 50 60 70 Displacement [mm] Compressive force [N] III I II IV V VI Experiment FEM

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Table 2.4: Evaluation of maxima and minima FEM compared to experiment AL4541

1st 2nd 3rd 4th FEM-maximum 102% 108% 110% 111% FEM-minimum 90% 97% 91% 91% FEM-mean 105%

maxima and minima in table 2.4, where the ratios are given of the FE model results with the test results. Also the mean load ratio is presented.

The deformation profile of both FEM and test agree well as seen in figure 2.7. The effective crush distance, between two peaks, confirms the model as well: 15.4 and 15.2 mm respectively for the test and the FE model, a difference of only 1%.

Variations exist first of all at the elastic stiffness at the beginning of the compression. The apparent stiffness from the experiment is much too low as was stated earlier, due to the limited measurement accuracy at the initial elastic loading. The higher maxima of the FE model may be the result of too high values for the material data, as the implementation of strain hardening raised the force level over the whole deformation. This cannot be traced further without the exact stress-strain data of the tested specimen material. The difference in minima may be improved by refining the FE model, i.e. smaller element sizes. This was visualised in figure 2.10, where the refined models have higher minima.

2.5.4 Analytical results

The analytical formulas give fast results, with limited accuracy. Results of the analytical theories presented are collected in table 2.5 for an aluminium cylinder with a mean diameter of 43 mm and thickness of 2 mm, similar to the tested specimen. Such results from the experiment and the FE analysis are presented in the table as well.

Table 2.5: Results aluminium cylinder, AL4541, with concertina collapse mode

Eq. 2.9 Eq. 2.13 Eq. 2.15 Eq. 2.17 Eq. 2.19 Ex- FE Alexan- Abramo- Abramo- Wierz- Wierz- peri-der [4] wicz [1] wicz [2] bicki [63] bicki [64] ment

Ppeak[kN] - - - - 103.4 59.1 59.3

P [kN] 23.1 35.2 43.0 34.0 35.4 35.5 37.3 h [mm] 8.73 8.07 8.19 12.2 12.0 9-10 9-10 δe [mm] 17.5 11.9 12.1 19.8 19.5 15.4 15.2

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2.6. Conclusion 31

difference between fold length and effective stroke. In case of the Alexander model twice the fold length (h) equals the effective stroke for one fold (the stroke of the compression plate). In the later models and in reality the effective stroke is less than the material compressed inside the fold. The fold length is then half the length of the complete curved fold. The modifications by Wierzbicki [63] and [64] show good agreement in the mean load, however the fold length clearly deviates from the other models. The experimental value can be found somewhere in between the predicted values. The only model that presents a peak value is from Wierzbicki, this value however is much too high. A maximum might be stated in the form of the squash load, which is equal to the yield stress times the cross-sectional area. In this case the squash load is approximately 58.1 kN.

Analytical results versus FE and experiment

The congruence between test and FEM is also found for the mean load of most of the analytical theories. The 1984 theory of Abramowicz, the 1986 and 1992 theories of Wierzbicki differ less than 5% with the experimental values. The effective crush distance already has a large variation over the different models, and the experimental value is somewhere between the values of Abramowicz and Wierzbicki. Variation is approximately 25%.

2.6 Conclusion

Axisymmetric progressive folding of aluminium tubes was investigated experimen-tally, analytically and numerically. The load-compression curves of the specimen were recorded in the experiment and numerical simulation. The numerical and experimental results match well. This validates the numerical simulation as a useful tool for rapid analysis and parameter studies of the investigated phenomenon.

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Chapter 3

Dynamic axisymmetric folding of

aluminium cylinders

3.1 Introduction

The analysis in previous chapter showed a good way for understanding the static metal progressive folding behaviour and for the validation of the implicit Abaqus code. Now this chapter will show the influence of dynamics on the folding behaviour, as is the actual case for energy absorbing structures, and how it relates to the static case.

To this purpose an explicit FE code is required, which again will be validated. For the dynamic testing, up to a velocity of 10 m/s, a drop tower will be used.

The set-up of this chapter is equal to the previous one. The theory behind the dynamic folding is discussed first, then the used numerical models and after that the test set-up. The latter gets extra attention due to the complexity of dynamic measurements. As last the outcome of all three parts is presented and discussed.

3.2 Analytical model

The static analytical models, presented the previous chapter, can be extended to dy-namic states. The extension is based on the strain rate effect, which is the sensitivity of a materials plastic flow behaviour to strain rate, which is defined as

˙

ε = dε/dt = dδ/L/dt = V /L (3.1)

The influence of material strain rate sensitivity generally manifests itself as a strength-ening effect, which in some cases provides an additional safety factor, but for crashwor-thiness this might impart unacceptable forces and accelerations. Substitution of the static flow stress by the dynamic flow stress in the bending moment definition (equation 2.2) results in an expression for the dynamic bending moment, which can be used to determine the dynamic crush load, equation 2.9.

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average strain rate for the first wrinkle follows as

˙

εθ = εθ/T = V0/2D (3.2)

The strain rate regime of our interest is from quasi-static, ˙ε ' 0.001/s to a dynamic drop from 6 metres (V0 ' 10 m/s), ˙ε ' 100/s for a 50 mm diameter specimen. Based on the Cowper-Symond data for AL6061-T6 the ratio of dynamic to static flow stress is 1.09 at the impact velocity of 10m/s (equation 3.3). However the mass is decelerated over the specimen and therefore the velocity and strain rate decrease. If one assumes a constant deceleration the average deformation velocity is 5 m/s, which gives an average dynamic flow stress of 1.08 times the static flow stress.

This extension to the theory only puts in the dynamic material effect, as in flow stress. However, more issues are going on in a dynamic process. An important one is the shock wave traveling from the impacted end to the other side and being reflected there as a tensile stress wave, which will give a different stress state than the static case. Furthermore there is inertia, e.g. the specimen top has to be accelerated to the velocity of the impactor creating high inertia loads as well as possible vibrations. These issues will be considered in the test and finite element evaluation.

3.2.1 Material data

To implement the material strain rate sensitivity into a model, constitutive material equations must be used, like the Cowper-Symond or Johnson-Cook equations. The Cowper-Symond constitutive equation gives a simple, quasi empirical, relation between dynamic flow stress (σ00) and associated static flow stress (σ0)

σ00 σ0 = 1 + ˙ ε D 1/q (3.3)

where D and q are constants for a particular material. For mild steel D=40/s and q=5, where aluminium alloys have on average D=6500/s and q=4 [32]. Pure aluminium has the highest rate sensitivity, where the high strength alloys (2xxx, 6xxx and 7xxx series) are much more insensitive, especially in the heat treated state (figure 3.1, AL6061-T6: D=1288000/s and q=4). Abaqus explicit does not offer the Cowper-Symond model and therefore the Johnson-Cook model is used with the Abaqus code, which is defined as follows: σy= (A + Bεn) 1 + Cln ε˙ ˙ ε0 1 − T − Troom Tref− Troom m (3.4)

The first part between brackets represents the static stress strain curve, the second part the strain rate influence and the last part the thermal effects, which are fully neglected in the analysis. The strain rate constants, C and ˙ε0, are derived from the Cowper-Symond constants. The values D=1288000/s and q=4 for AL6061-T6 give C=0.0084 and ˙ε0=0.0040. To simulate a strain rate insensitive material the value of C can be put equal to zero and a strain rate sensitive value of D=6500/s gives C=0.0312.

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3.2. Analytical model 35

Figure 3.1: Percentage increase in plastic flow stress from quasi static ( ˙ε = 10−3/s) to dynamic ( ˙ε = 103/s) at 6% strain

Table 3.1: Material data 6060-T66 extrusion

True stress [MPa] plastic strain [%]

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3.3 FE model

For the dynamic analysis the explicit Abaqus solver (Abaqus 6.3.1) is used as well as PAM-CRASH (PAM-CRASH 2G version 2003.0.1). The dynamic Abaqus model is an axisymmetrical cylinder equal to the static model, except for using explicit elements and interaction definitions.

The PAM-CRASH model is a 3 dimensional model, which is built of solid brick elements. Solid elements are required as the element dimensions required for a good description of the folding behaviour are of the same size as the element thickness. The same maximum element dimensions, as for the 2D Abaqus model are taken, which means at least 3 elements over the thickness of the tube. The aspect ratio of the elements is kept equal or smaller than 4. Same friction coefficient is implemented in the PAM-CRASH and Abaqus model.

The dynamic impact is simulated by having a rigid body, with the dropweight mass of 80 kg, impacting the cylinder at a predefined velocity. The rigid body has only one degree of freedom, which is in the deformation direction . Also a gravitational field is applied to the system, where g = 9.81 m/s2_.

3.4 Test set-up

Two drop towers were used, one with data acquisition of both acceleration and displace-ment during the test (Polytechnic Milan) and one without (Delft). The one without data acquisition during testing is only used for the determination of the global prop-erties like impact energy, mean load and permanent deformation and for investigation of the physical deformation of the specimen. Both drop towers have a similar set-up. The one from Delft is described here. The drop tower has a height of 6 metres (fig-ure 3.2), which results in velocities of up to 10 m/s. The drop-weight consists out of three elements, a steel block (68.7 kg), a load-cell (5.75 kg) and the impactor head (6.2 kg), which are bolted together, see figure 3.2. On the sides of the steel block four roll bearings are fixed, which slide between the two U-profiles of the tower. The impactor head is a flat cylinder, with diameter of 100 mm. The specimen is positioned, with-out constraints, onto a steel block, which stands on a flat steel plate. Release of the drop-weight is done by an electronic clamping device.

The accelerometer of the Milan drop tower is a standard piezoresistive element, which has a range of ±200g. It is installed inside the impactor head. A long cable is connected to the accelerometer and moves with the impactor. Displacement of the impactor is measured by means of a cable which comes off a drum, where the rotation of the drum is measured.

3.4.1 Specimen

Two different geometries are used. Tubes with outer diameter of 50 mm and inner diameter of 45 mm (designated as AL5045 specimen) with length 120 mm and tubes with outer diameter 50 mm and inner diameter 47 mm (designated as AL5047 specimen) with length 100 mm. Both tubes are made of extruded Aluminium 6060-T66.

3.4.2 Impact velocity measurement

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3.4. Test set-up 37 Load cell acc. Weight Head gage 6 m tower specimen Drop-weight A B C

Photo transducers A & B

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3.2. The time between the darkening of the transducers is measured by a counter, Universal Counter 9835 by RACAL instruments, with a frequency of 1 MHz. The distance between the two pairs A and B is 30.0 mm (dAB). The counter measures the time between the darkening of the two photocells. This means that an average velocity ¯

v is measured between A and B. It can be derived that the relationship between ¯v and the actual impact velocity (at point C) vcis:

¯ v = gp dAB v2 c− 2gdBC− p v2 c− 2gdAC (3.5)

with dBCthe distance between points B and C and dACthe distance between the points A and C. It assumes friction is negligible and thus the acceleration of the impactor is equal to one g, while passing the transistors. When the distance dBCbetween the lower transducer and the specimen is smaller than 5 cm (typical 2 to 4 cm) and the impact velocity is larger than 5 m/s (corresponding to an impact height of approximately 127 cm) no correction of the impact velocity is necessary. In this case the error from equation 3.5 is smaller than 2%.

The theoretical free fall impact velocity is derived with the expression:

V =p2gH (3.6)

The measured velocities differ less than 2% from the theoretical impact velocity. The impact velocity of the Milan drop weight is derived from the displacement signal. The average is taken over the last 12 data points of the displacement derivative just before impact.

3.4.3 Acceleration measurement

The acceleration transducer (Piezoresistive accelerometer) measures the acceleration of the complete drop-weight. The mounted resonance frequency of the transducer is 8000 Hz. The amplitude response of ±1 dB is given for frequencies between 0 and 1800 Hz. Non-linearity and hysteresis of the transducer is ±2% of the maximum (absolute non-linearity is at most 4g). Range of the transducer is ±200g.

An analog-digital converter (ADC), samples the signal with a frequency of 16 kHz. The data acquisition is started at release of the drop-weight.

3.4.4 Discussion of problems associated with the

instrumenta-tion

Vibration and noise in the signals

The contact force between dropweight and specimen is always measured indirectly, either as strain in the load cell or in an accelerometer. The output of these signals is not smooth but covered with noise due to the amplification system and to resonance.

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3.4. Test set-up 39 0 1000 2000 3000 4000 5000 6000 7000 8000 0 100 200 300 400 500 600 700 800 900 Frequency [Hz]

Power Spectral Density

Figure 3.3: Specimen A (AL5047) PSD curve, after filtering solid line and original signal dot-ted line 0 1000 2000 3000 4000 5000 6000 7000 8000 0 1000 2000 3000 4000 5000 6000 Frequency [Hz]

Power Spectral Density

Figure 3.4: Specimen D (AL5045) PSD curve, after filtering solid line and original signal dot-ted line

velocities of 1 m/s, or below, the amplitude of the resonance can be ignored (a long experiment time is achieved). Where above 7 m/s the resonance becomes a standard problem (experiment time short) [58].

The following sources are distinguished here: vibrations in the impactor

vibrations of the support vibrations of the specimen

inertial loading: high peak loads in the initial phase of impact

Resonant frequencies are determined from Power Spectral Density plots of the ac-celerometer signals, figures 3.3 and 3.4. A dominant drop weight resonant frequency in all tests is 5.4 kHz, resulting from the drop weight/transducer system. This frequency is sufficiently high and isolated to be fully filtered out. The lowest resonant frequencies of 0.2 and 0.3 kHz are the response of the specimen and should not be filtered. The AL5047 specimen further have an interaction resonance of 1.2 kHz. The AL5045 spec-imen have a similar resonance frequency at 1.3 kHz. Special attention must be paid to the effect of the intermediate frequencies (˜1 kHz) on the data and the data filtering, as it may be both the result of the specimen response and vibrations of the system. The high frequency is solely the result of system vibrations and will be filtered out completely.

Initial inertial loading

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Inertia load Mechanical load Total load Ve lo cit y Tu b load Time V0 Tub Specimen

Figure 3.5: Load time and velocity time curves during first micro seconds after initial contact [48]

The high initial acceleration present during approximately the first 20 to 30 micro seconds of the impact process thus causes high peak loads and can also cause load drops with a temporary loss of contact between drop weight and specimen. As the inertial force rapidly rises it will not always be measured due to the limited frequency response of the measuring system. Due to this limited frequency response high frequency oscil-lations and loads with a short rise time may lead to inaccurate measurements. The frequency of the sampling used is 16 kHz, as such per data point a mere 62 micro sec-onds pass. This is too low to visualise the actual initial loading, though, as mentioned, measurements will be influenced by the initial loading vibrations.

3.4.5 Data processing

Filtering

Energy Absorption of Monolithic and Fibre Reinforced Aluminium Cylinders

Energy Absorption

of

Energy Absorption

of

Monolithic and Fibre Reinforced

Aluminium Cylinders

Contents

Nomenclature

Latin symbols

Greek symbols

Abbreviations

Chapter 1

Introduction

1.1

Car crash safety

1.1.1

Automobile crashworthiness design

1.1.2

Safety paradoxes and contradictions

1.2

Aerospace crashworthiness design

1.2.1

Helicopter crashworthiness

1.2.2

Aircraft crashworthiness

1.2.3

Spacecraft crashworthiness

1.3

Energy absorbing components

1.3.1

Requirements for an energy absorber

1.3.2

Typical behaviour energy absorbers

1.4

Objective

1.4.1

Challenges multi-material energy absorbing elements

1.4.2

Success factors

1.4.3

Boundary conditions of research

1.5

Outline

Chapter 2

Static axisymmetric folding of

aluminium cylinders

2.1

Introduction

2.2

Theory

2.2.1

Alexander model

M

-σ

σ

σ

σ

-σ

-σ

t

+

+

+

2.2.2

Modifications

2.3

Finite element model set-up

RP

1

2

3

R

2.4

Test set-up

2.5

Results and discussion

2.5.1

Force-deformation profile

2.5.2