a comparison of the deformation of magnesium alloys with aluminium and steel in tension, bending and...

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A comparison of the deformation ofmagnesium alloys with aluminium and steel in

tension, bending and buckling

Article in Materials and Design December 2006

Impact Factor: 3.5 DOI: 10.1016/j.matdes.2005.03.005

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Mark Alan Easton

RMIT University

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Wiliam Song

Monash University (Australia)

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Trevor Bruce Abbott

Monash University (Australia)

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A comparison of the deformation of magnesium alloyswith aluminium and steel in tension, bending and buckling

Mark Easton a,*, Wei Qian Song b, Trevor Abbott c

a CRC for Cast Metals Manufacturing (CAST), School of Physics and Materials Engineering, Monash University, PO Box 69, Victoria 3800, Australiab Industrial Research Institute of Swinburne (IRIS), Swinburne, University of Technology, Hawthorn, Victoria, Australia

c Advanced Magnesium Technologies, Monash University, Victoria 3800, Australia

Received 22 October 2004; accepted 2 March 2005Available online 18 April 2005

Abstract

A comparison was made between high pressure die cast and wrought magnesium alloys and formed mild steel and aluminium intensile, bending and buckling deformation. It was found that the energy absorption properties of magnesium alloys were particularlygood in bending and buckling, absorbing up to 50% more energy than the aluminium and over 10 times more energy than the mildsteel. The primary reason for the good performance of Mg alloys was that the low density means that sections of increased thicknesscan be made without increasing weight. This is particularly beneficial in bending as the strength of a section in bending is propor-tional to the square of the thickness. However, it was also observed that the high rate of work hardening of Mg alloys is particularlyimportant and this allows for considerably more energy to be absorbed. A simple analytical strut buckling model has been modifiedto incorporate work hardening and a good correlation has been observed between this model and the experimental results.2005 Elsevier Ltd. All rights reserved.

Keywords: A. Non-ferrous metals and alloys; F. Plastic behaviour; I. Buckling

1. Introduction

One of the potential applications of magnesium al-loys is in the components that make up the crash struc-ture in automobiles. The components of the energyabsorbing system include bumper bars, crash cans/boxesthat collapse at pre-specified loads, collapsible steering

columns, rollover protection, side impact protection indoors, and a collapsible engine cradle/mounting that di-rects the engine under the passenger compartment. Tofulfil their function, these systems rely on both thegeometry of the component and the material from whichthey are made. Traditionally, most of these systems have

been fabricated from stamped or roll formed steel sec-tions [1]. The desire to keep weight to a minimum hasled to the use of very thin gauges, which leads to local-ised rapid collapse and very sudden loss of properties. Anumber of alternative materials have been suggested forthese applications, but the most promising materials areultra high strength steels, aluminium alloys and, more

recently, magnesium alloys. This paper examines the po-tential for the use of magnesium alloys in theseapplications.

Currently, most magnesium alloys used are highpressure die cast (HPDC) MgAl alloys. If magnesiumalloys are to be used in the crash structure of an auto-mobile, then they are required to have energy absorp-tion properties comparable to formed steel andaluminium alloys, while ideally lowering vehicleweight. Wrought magnesium alloys are especially

0261-3069/$ - see front matter 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.matdes.2005.03.005

* Corresponding author. Tel.: +61 399 053 895; fax: +61 399 054940.

E-mail address:[email protected](M. Easton).

www.elsevier.com/locate/matdes

Materials and Design 27 (2006) 935946

Materials& Design
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promising for these applications; with the benefit ofgreater ductility than high pressure die cast alloysbut are presently limited by small production volumesand higher costs.

One of the barriers to the use of magnesium alloysin more components is the lack of understanding of

their deformation behaviour. A survey of the literature[25] shows that there are some measurements of en-ergy absorption during the deformation of magnesiumalloys in tension (Table 1) with the magnesium alloysbeing comparable, although slightly inferior to alumin-ium alloys and steel. There is also information on theimpact toughness of magnesium die castings [5,6].However, most deformation during impact occurs inbending and buckling modes under high rates ofstrain, where static tensile data is insufficient for acomparison of the performance of different candidatematerials.

Comparisons of the crashworthiness properties of

magnesium and aluminium steering wheel armatures[7] and cast profiles [8] have provided important infor-mation about the properties of magnesium alloys rela-tive to alloys of other metals. However, informationon the deformation behaviour of magnesium alloys is re-quired to identify applications, where magnesium alloyshave particular advantages. To this end, this paper com-pares the properties of mild steel, aluminium alloy 6061in the T6 heat treated condition, and high pressure diecast and wrought magnesium alloys to determinewhether the relative performances of these materialsvary with the particular deformation mode. Another

important factor to consider is the effect of strain rateof the deformation mode, which will be reportedseparately.

2. Experimental methods

Quasi-static tensile, bending and buckling tests wereperformed on mild steel grade HA300 (1006), alumin-ium alloy 6061 in the T6 heat treated condition, HPDCmagnesium alloys AM20, AM50 and AM60, and the

wrought magnesium alloy AZ31.Plates of magnesium alloys were high pressure diecast with dimensions of 125 75 3 mm. The AZ31plates were machined from flat extrusions. The tensilespecimens were also machined from the extrusions par-allel to the extrusion direction. Uniaxial compressiontests were also performed on the magnesium alloys.The anisotropy typically observed in wrought magne-sium alloys, e.g. [9], is not considered in detail here.However it should be noted that typically, the compres-sion behaviour of AZ31 shows a considerably loweryield stress and higher work hardening rate in compres-sion than in tension and this was found to be important

in this study. The amount of anisotropy observed inHPDC magnesium alloys was found to be very smalland therefore were considered isotropic. The aluminium6061-T6 extrusions were obtained commercially. Boththe AZ31 and 6061-T6 were machined to the samedimensions as the HPDC plates. The steel plates werecut from 3 mm thick steel sheet. To enable comparisonsfor the same mass of the materials as well as the samethickness, 2 mm thick plates were also machined from6061-T6 extrusions and plates were sectioned from0.75 mm mild steel sheet. Flat 3 mm thick tensile sam-ples were machined from the cast plates for the HPDC

magnesium alloys and from the AZ31 and 6061-T6extrusions and mild steel sheet. The sample dimensionswere 3 mm thick, 6 mm wide and 25 mm gauge length.

Table 1Energy absorption of different materials in a tensile test from various sources in the literature

Material Geometry tested Strain rate (s1) Specific energy absorption (kJ/kg)

1.2 mm Auto Steel [2] Pressed and spot welded hollow section NR 14.52.0 mm AA5754-O[2] Pressed and spot welded hollow section NR 22.4-25.85056-O[3] Tensile test with modified Hopkinson bar 1.7 103 42.8*

7075-T6[3] Tensile test with modified Hopkinson bar 1.0 103 23.1*

Pure Mg[3] Tensile test with modified Hopkinson bar 1.1 103 9.4*

WE43[3] Tensile test with modified Hopkinson bar 1.1 103 15.947.3*

DC04 mild steel[4] Tensile test 2.0 102 22ZstE340 high strength steel[4] Tensile test 2.0 102 29.5Dual phase steel[4] Tensile test 2.0 102 2335TRIP steel[4] Tensile test 2.0 102 42Austenitic stainless steel[4] Tensile test 2.0 102 69.575AM20[5] Tensile test Quasi-static 17.1AM50 (4.8Al)[5] Tensile test Quasi-static 17.7AM50 (5.3Al)[5] Tensile test Quasi-static 16.6AM60 (5.8Al)[5] Tensile test Quasi-static 16.0

Compared to all the other reported values, this appears to be a typographical error.NR means not reported.* Please note: the original reference stated the units of J/kg, instead of kJ/kg.

936 M. Easton et al. / Materials and Design 27 (2006) 935946
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Tensile testing and three point bend testing were per-formed on an Instron 4505 testing machine. The tensiletesting was performed at a rate of 1 mm/min. The spanin the three point bending was 90 mm and plates werebent at a rate of 5 mm/min up to 50 mm displacement.During three point bend testing, the plates were allowed

to slide over the rounded supports.Buckling tests were performed on a high strain rateMTS testing machine. Buckling testing was performedby crushing a plate between two platens. To guide thebuckling, an initial kink or imperfection was put intothe plate using a 5-mm displacement in three pointbending mode and the plate was placed in a ledge engi-neered into the platens. Testing of the plates occurred ata rate of 1 mm/s over a travel of 100 mm.

Multiple tests were performed to check reproducibil-ity. It was found that for the wrought materials 2 or 3tests were sufficient, whilst for the HPDC material 35tests were performed at each condition to obtain the

necessary confidence in the data.The work hardening coefficient, n, was calculated

using

n d log rtd log et

; 1

where rtis the true stress and etis the true strain at stres-ses above the yield point. The energy absorbed duringtesting was determined by measuring the area underthe loaddeflection graphs.

3. Results

3.1. Tensile

A comparison of the engineering stressstrain behav-iour (Table 2) shows that mild steel had the greateststrength, slightly higher than aluminium 6061-T6. Allthe magnesium alloys had significantly lower strengths,with AZ31 performing best. The steel had the highestelongation followed by the wrought alloys, AZ31 and6061-T6. The elongations of the HPDC AM series alloyswere lower again. However, AZ31 had the highest spe-cific strength of all the alloys (Table 2), 6061-T6 had

the second highest specific strength, followed by the

AM series magnesium alloys and the steel had the lowestspecific strength.

The HPDC AM series magnesium alloys have workhardening coefficients almost three times that of the alu-minium alloy and double that of the steel (Fig. 1). AZ31,however, shows a large degree of anisotropy. With

increasing strain in tension, the work hardening rate ofAZ31 decreases from about 0.3, just above the yieldpoint, to approximately 0.1, which is similar to the alu-minium alloy. However, Fig. 1 shows that the workhardening rate of AZ31 in compression increases rapidlyas the tensile work hardening rate decreases. In com-pression the n value is above 1 at the point of highestwork hardening. Consequently AZ31 will also behaveas a material with high work hardening rates in bendingand buckling modes. The anisotropy in wrought magne-sium alloys needs to be considered in more detail; how-ever this is beyond the scope of this paper.

The high work hardening rates of the HPDC magne-

sium alloys mean that the tensile strength is approxi-mately twice the yield strength (Table 2). The tensilestrength of the aluminium alloy and steel was within20% of the yield strength. For extruded AZ31 the tensilestrength was about 50% greater than its tensile yieldstress, although five times the compressive yieldstrength.

Fig. 2compares the energy absorbed during the ten-sile tests. When equal cross-sections are compared, themagnesium alloys absorb much less energy than alumin-ium alloy 6061-T6 and steel during a tensile test (Fig.2(a)). The mild steel absorbs by far the most energy,

mainly due to its high elongation (Fig. 2(a) and Table2). However AZ31 absorbs as much energy per unitmass as the mild steel (Fig. 2(b)) because it has a higherspecific strength, which compensates for its lower ductil-ity. Hence for a high strength, energy-absorbing compo-nent, AZ31 is superior to the mild steel on an equalweight basis. Aluminium 6061-T6 behaved very simi-larly to AZ31 in terms of specific strength ( Table 2)and energy absorption (Fig. 2). The HPDC magnesiumalloys absorbed the least amount of energy, per unitmass of material, which is not surprising as high pres-sure die castings tend to contain many more defects,which reduce the elongation, than wrought alloys. Ofthe HPDC magnesium alloys, AM20 absorbed the most

Table 2Values of the tensile properties of mild steel and 6061-T6 compared with AM20, AM50 and AM60 flat tensile samples. Data for uniaxial tension andcompression of AZ31 are also included on the table

AM20 AM50 AM60 AZ31 comp. AZ31 tens. Steel 6061-T6

Yield strength (MPa) 87.6 115.6 122.1 84.8 174.8 314.4 286.8Tensile strength (MPa) 194.5 229.3 242.2 433 267.4 384.2 328.9Elongation (%) 10.5 9.6 9.6 16.1 13.1 37.3 13.2Work hardening rate 0.37 0.32 0.33 0.3 ! 1.2 0.33 ! 0.10 0.19 0.16Specific strength (m3MPa/kg) 48.7 64.2 67.8 47.1 97.1 43.7 106.2

M. Easton et al. / Materials and Design 27 (2006) 935946 937
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energy because of its high elongation, although a largerange in the elongation to failure leads to a high uncer-tainty in the average value. AM60 absorbed the leastamount of energy due to its low ductility.

3.2. Bending and buckling

The force deflection curves from the three point bend-ing tests are compared inFig. 3. The 3 mm thick alumin-ium 6061-T6 had the highest peak load of the alloys.Mild steel had the second highest peak load. Of the mag-nesium alloys, AM60 had the highest peak load. How-ever, the magnesium alloys carried significantly moreload than 6061-T6 of the same weight (2 mm thick)and steel of the same weight (0.75 mm thick). The0.75 mm thick steel performed extremely poorly com-pared with the other materials, with less than 10% ofthe load carrying capacity of the magnesium alloys(Fig. 3).

Some of the AM60 samples tested fractured at lessthan 50 mm displacement. Fracture was generally initi-ated at the extremity of the width of the sample, wheretypical anticlastic behaviour was observed. This crackslowly grew to the other side of the sample indicatinga relatively ductile fracture. Sometimes a few crackswere observed, which joined during subsequent defor-mation. Whilst there was evidence of cracking in someof the samples, the AM20 and AM50 alloys did not fullyfracture during testing. The AZ31, 6061-T6 and steelsamples did not show any signs of cracking orfracturing.

Loaddisplacement curves during buckling testingare shown in Fig. 4. Since the pre-bend in the sampleswas relatively small (


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it failed earlier in the deformation process before thetwo sides of the plate became parallel. Cracking wasnot observed in the aluminium alloy or the steel,because the radius of curvature during deformationwas lower and therefore this configuration was notattained.

The performance of the magnesium alloys atabsorbing energy during bending and buckling wasmuch better than the tensile results would indicate(Fig. 7). For sections of the same thickness, the prop-erties of the magnesium alloys were only slightly infe-rior to the steel and aluminium alloys. The steel and6061-T6 of the same thickness absorbed approximately4050% more energy than the best of the magnesiumalloys. However, when sections of the same mass wereconsidered, the magnesium alloys absorbed 50% moreenergy during deformation than 6061-T6 and over 10times the energy absorbed by the steel. Therefore, onan equal weight basis, the steel performed by far the

worst of all the materials in the bending and bucklingtests.

The AZ31, AM50 and AM60 absorbed the greatestamount of energy of the magnesium alloys in the threepoint bending test (Fig. 7(a)). AM60 carried more loadthan the other alloys but failed during testing. The en-ergy absorbing properties of AM20 were lower becauseof its lower strength. In the buckling testing, AZ31 per-formed less well, while AM50 and AM60 absorbed rel-atively more energy (Fig. 4(a)). This appeared to bebecause AZ31 carried lower loads in buckling, whichmay be due to the compression yield strength being low-

er than its tensile strength. However, this observation re-quires further investigation.

0

500

1000

1500

2000

2500

3000

0 5 10 15 20 25 30 35 40 45 50

extension (mm)

Loa

d(N)

6061-T6 3 mm

6061-T6 2 mm

AM50

AM20

AM60

steel 0.75mm

steel 3mm

AZ31

Fig. 3. Typical forcedeflection curves of three point bending for the alloys.

0

10

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40

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80

AM20 AM50 AM60 AZ31 6061-

T6

steel

EnergyAb

sorbed(J)

0

5

10

15

20

25

AM20 AM50 AM60 AZ31 6061-

T6

steel

EnergyAbsorbed(k

J/kg)

(b)

(a)

Fig. 2. A comparison of the energy absorption during tensile testing.

(a) The data as tested, i.e., energy per volume. (b) A comparison of theenergy absorbed per unit mass.

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4. Discussion

It was found that, despite performing not as well intension, the energy absorption of magnesium alloys inbending and buckling was far superior when comparedto aluminium and steel. The purpose of this discussionis to explain the reasons for this behaviour.

The energy absorption of a material is maximisedwhen the combination of strength, which is affected bythe work hardening behaviour, and ductility is maxi-mised. Often factors that increase the strength of an al-loy reduce its ductility, therefore the best energyabsorbing alloys generally have an intermediate combi-nation of both strength and ductility. A comparison ofthe energy absorbing properties of some of these alloyshas been previously presented[10]and is expanded uponbelow.

4.1. The importance of geometry in bending and buckling

The energy absorption in bending and buckling high-lights the importance of thickness effects in the develop-

ment of strong, energy absorbing components forautomotive applications. Whilst steel appears to haveadvantages in energy absorption in tensile testing, evenfor equal weights (Table 1andFig. 2), the performanceof mild steel in the bending and buckling tests was extre-mely poor.

A simple analysis clarifies the reason for the poorperformance of steel in comparison to magnesium al-loys during bending and buckling testing. As a firstapproximation, the energy absorbed is related to theload that can be carried and the distance of deforma-tion. The stress in the outer fibre, r, is related to themoment of inertia of a section, I, the bending mo-ment, M, and the distance from the neutral axis tothe outer fiber, y, [11]in elastic bending by the generalequation

r My

I . 2

It needs to be mentioned at this point that the neutralaxis can move during plastic deformation in bending,particularly for materials with anisotropic materials

0

500

1000

1500

20002500

3000

3500

4000

4500

5000

0 20 40 60 80 100Displacement (mm)

Forc

e

(N)

AM60AZ31

AM50

AM20

0

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2000

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3000

3500

4000

4500

5000

0 20 40 60 80 100

Displacement (mm)

Force

(N) 6061T6

Steel

AM50

6061-T6 2mmSteel 0.75mm

(a)

(b)

Fig. 4. (a) A comparison of typical forcedeflection curves for buckling testing for the magnesium alloys. (b) A comparison of the AM50 with the

buckling behaviour of mild steel and 6061-T6 samples for the same thickness and the same weight.

https://www.researchgate.net/publication/291869586_A_comparison_of_the_energy_absorbing_properties_of_magnesium_alloys_with_steel_and_aluminium_alloys?el=1_x_8&enrichId=rgreq-9e1dd44e43b891a047d1259ed7957ced-XXX&enrichSource=Y292ZXJQYWdlOzIyMzQ3NzI0MTtBUzoxMzU2OTU1MDI5NDIyMDhAMTQwOTM2MzczNzE1Ng==http://-/?-http://-/?-http://-/?-https://www.researchgate.net/publication/291869586_A_comparison_of_the_energy_absorbing_properties_of_magnesium_alloys_with_steel_and_aluminium_alloys?el=1_x_8&enrichId=rgreq-9e1dd44e43b891a047d1259ed7957ced-XXX&enrichSource=Y292ZXJQYWdlOzIyMzQ3NzI0MTtBUzoxMzU2OTU1MDI5NDIyMDhAMTQwOTM2MzczNzE1Ng==http://-/?-http://-/?-http://-/?-http://-/?-


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behaviour like AZ31, but, in order to simplify the anal-ysis, it will be assumed that the neutral axis remains inthe centre of the section.

For a thin plate in elastic three point bending, as usedin this analysis the equation relating the load at yielding,Ly, the yield stress, ry, and the dimensions of the sample,length L, width b and thickness h is

Ly 2bh2ry

3P . 3

It should be pointed out that Eqs. (2) and (3) are forelastic behaviour. Therefore these equations can beused only as a first approximation when consideringplastic deformation as is being considered here. How-ever, using the elastic equations allows for simplecomparisons to be made for materials with different

densities. In this discussion, the energy absorbed willbe the load determined by Eq. (3) multiplied by thedeflection, d.

The geometric advantage, R, for a thicker, lessdense material can be determined by considering theratios of the stress in Eq. (2). By assuming the appliedforces are the same, the geometric advantage can bedefined as

R rsl

rMgIMg yslIsl yMg

. 4

This equation can be applied to assist with choosing thematerial with the best properties for a particular

component.Since I bd

3

12, where b is the width and d is the

thickness of a solid rectangular cross-section andy= d/2, a consideration of Eq. (4) shows that if thethickness can be increased so that the weight of amagnesium section and a steel section is the same,the load being carried will be multiplied by the squareof the ratio of the densities of the materials, i.e., (7.9/1.8)2. This means that magnesium alloys have a 19times geometric advantage over the steel for rectangu-lar sections of the same weight. So even if the yieldstrength of the steel is 2.5 times higher than magne-

sium alloys (i.e., mild steel compared with AM60 inTable 2) the magnesium alloys will still carry 8 timesas much load as the steel before yielding in bendingfor equal weight cross-sections.

The reason for the good tensile energy absorptionproperties of steel in tension is a combination of goodyield strength and high elongation. However, in thebending and buckling tests performed here, the extraductility of the steel is not required. Most of the magne-sium alloys, especially the wrought alloy, AZ31, had en-ough elongation to deform as much as required in thetests without fracturing. The reason for this is that athinner cross-section requires less inherent ductility todeform as much as required in bending and buckling,as the amount of strain in the outer fibre is higher fora thicker section than a thinner section at the same ra-dius of curvature. Hence the requirement for steels, evenhigher strength steels, to be made thinner and thinner tosave weight, means that the natural advantage of steelsin energy absorption high ductility is often not uti-lised. Also, as Eq. (3) shows, the load at deformation,and therefore the energy absorbed, has a greater depen-dency on the section thickness (inverse square) than theyield stress (linear). Therefore magnesium alloys, and toa slightly lesser extent aluminium alloys, by nature of

Fig. 5. Pictures of the deformed samples of (a) AM20, (b) mild steel

and (c) 6061-T6 after buckling testing.

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their low density, will have excellent properties for appli-cations that require bending and buckling strength andenergy absorption.

However, in sections that require more ductility andhave less of a geometric disadvantage, steels and inparticular high strength steels may still have an advan-

Fig. 6. Stages in the deformation of AZ31 during buckling deformation.

0

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120

AM20

AM50

AM60

AZ31

steel

0.75

mm

ste

el3mm

6061T6

2mm

6061T6

3mm

Energyabsorbed(J)

0

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AM20

AM50

AM60

AZ31

6061

-T62m

m

6061

-T63m

m

Steel0

.75m

m

Steel3

mm

Energyabsorbed(J)

(a)

(b)

Fig. 7. Comparison of the energy absorbed during (a) three point bend testing and (b) buckling testing. To compare equal weights the 0.75 mm thickmild steel and 2 mm 6061-T6 need to be compared with the magnesium alloys. The 3 mm thick mild steel and 6061-T6 are for comparing equalthicknesses with the magnesium alloys.



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tage. A calculation of R, for a comparison of hollowsquare tubes made out of magnesium to one madeout of steel of the same weight with constrained outerdimensions is shown in Fig. 8 (calculations are shownin Appendix A). It shows that for a thin walled largesection, the geometric advantage of magnesium oversteel is 4.4. This means a comparison of the specificstrengths is adequate in these applications. However,as the wall thickness increases with respect to the tubedimensions, the geometric advantage of the magne-

sium decreases below 4.4. Therefore, steels, especiallyhigher strength steels, may still be a better materialchoice for large thin walled sections over the lightmetals, magnesium and aluminium.

4.2. Work hardening behaviour

The shape of the deformation of the plastic hinge isrelated to the work hardening behaviour. The magne-sium alloys had the highest work hardening rate andthe plastic hinge with the largest radius of curvature.The aluminium alloy had the smallest radius of curva-ture and the lowest work hardening rate. Mild steelhad behaviour that lay between these two extremes. Acomparatively high rate of work hardening of magne-sium alloys means that they become stronger as defor-mation continues.

To illustrate the advantages of work hardening onbuckling loads, a comparison was made between theexperimental buckling data and the predictions for plas-tic flow from a model developed by Grzebieta and Mur-ray[12]for strut buckling. They developed the followingequation for plastic flow during deformation of a plastichinge during buckling behaviour assuming rigid plasticbehaviour (i.e., no work hardening) was

dP

M2PDP2

1 P

Py

2" #2; 5

where dP is the plastic vertical deflection, Mpis the plas-tic bending moment (ry D) is the span length, P is theload and Py is the squash load (ry b h). The model

was verified with experimental observations and a goodcorrelation was found.Using the yield strength obtained from the tensile

testing (Table 2) and the dimensions of the samples,Eq.(4) can be used to obtain a prediction of the behav-iour of the alloys tested in bending from their tensileyield behaviour. There was reasonable correlation be-tween the model and the experimental data for alumin-ium 6061-T6 and steel in this work, although the modelbegins to underestimate the behaviour at high deflec-tions. In contrast, the magnesium alloys outperformthe model by almost 300% (Fig. 10).

The discrepancy between the model and experimen-

tal results appears to be due to work hardening. In amaterial that work hardens, the stress at which furtheryielding occurs increases as the deformation continues.This means that the assumption of perfect plasticbehaviour will cause the load to be underestimatedas deformation continues as seen in Fig. 10(b). Workhardening also spreads the deformation over a greatervolume causing magnesium alloys to deform with alarger radius of curvature than the other alloys ( Fig.5). This means that a larger amount of the materialis involved in plastic deformation, leading to a greateramount of energy being absorbed. Also, the deforma-

tion is not concentrated at one point, which meansthat the amount of strain on the outer fibre of theplastic hinge is lower, allowing for extra deformationof the section before fracture (as long as the hinge isnot crushed). Finally, the length of the strut is de-creased slightly, which decreases the force applied tothe hinge and decreases the amount of stress on theplastic hinge.

To determine whether work hardening could ex-plain the discrepancy between the experimental dataand the model results of Grzebieta and Murray [12],an analytical model has been developed using the soft-ware package MathCad which incorporates workhardening into the strut-buckling model. Because ofthe way the equations have been developed workhardening cannot be incorporated directly into theequations. Consequently, an iterative approach wasused, where an initial estimation of the deflection fora particular load is made assuming perfect plasticbehaviour. From this estimate, a strain is determinedfrom the displacement; a corresponding stress is thendetermined from tensile stressstrain behaviour thatis incorporated into the original equation. It wasfound that convergence is observed after between 10and 30 iterations.

0

1

2

3

4

0 100 200 300 400 500

D/tst

B/tst

R

2

5

10

100

Fig. 8. Graphical representation of a scaling factor,R, for comparingthe strength of magnesium alloys and steel in bending for a hollowrectangular tube for different geometries when the outer dimensions ofthe tube remain the same, i.e., the thickness of the wall,tsl, is increasedtowards the centre of the tube.

http://-/?-http://-/?-https://www.researchgate.net/publication/236952090_The_static_behaviour_of_struts_with_initial_kinks_at_their_centre_point?el=1_x_8&enrichId=rgreq-9e1dd44e43b891a047d1259ed7957ced-XXX&enrichSource=Y292ZXJQYWdlOzIyMzQ3NzI0MTtBUzoxMzU2OTU1MDI5NDIyMDhAMTQwOTM2MzczNzE1Ng==http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-https://www.researchgate.net/publication/236952090_The_static_behaviour_of_struts_with_initial_kinks_at_their_centre_point?el=1_x_8&enrichId=rgreq-9e1dd44e43b891a047d1259ed7957ced-XXX&enrichSource=Y292ZXJQYWdlOzIyMzQ3NzI0MTtBUzoxMzU2OTU1MDI5NDIyMDhAMTQwOTM2MzczNzE1Ng==https://www.researchgate.net/publication/236952090_The_static_behaviour_of_struts_with_initial_kinks_at_their_centre_point?el=1_x_8&enrichId=rgreq-9e1dd44e43b891a047d1259ed7957ced-XXX&enrichSource=Y292ZXJQYWdlOzIyMzQ3NzI0MTtBUzoxMzU2OTU1MDI5NDIyMDhAMTQwOTM2MzczNzE1Ng==https://www.researchgate.net/publication/236952090_The_static_behaviour_of_struts_with_initial_kinks_at_their_centre_point?el=1_x_8&enrichId=rgreq-9e1dd44e43b891a047d1259ed7957ced-XXX&enrichSource=Y292ZXJQYWdlOzIyMzQ3NzI0MTtBUzoxMzU2OTU1MDI5NDIyMDhAMTQwOTM2MzczNzE1Ng==http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


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A number of assumptions were made in thecalculation:

1. It is important to decide how to relate a deflectionto a value of strain in the outer fibre of the deform-ing material. It was assumed that at the plastic

hinge deformed at a point to simplify the calcula-tion (rather than being rounded as is really thecase, particularly in magnesium alloys). A calcula-tion was made to determine the extra length ofthe fibre that would be required so that fracturewould not occur (see Fig. 9). Initially, it wasassumed that the whole strut length accommodatedthe lengthening of that fibre, i.e., strain is Dl/L,where L is the half strut length. However, one ofthe factors that could be and was varied was thedistance over which the plastic hinge was formed.Finally a function that described the growth ofthe plastic hinge during deformation was used to

best fit the experimental data.

2. Whilst stressstrain behaviour from tensile testscould be used to determine a stress at a particularstrain, the data was insufficient for these calculations.In a tensile test, any flaw along the gauge length willcause fracture to occur. However, in a buckling testonly a small region is stressed in tension to a stress

where fracture may occur. So the likelihood that acritical defect, that may cause fracture, is present ina highly stressed region is lower than in a tensile test,the stresses achieved locally before fracture are likelyto be higher than the fracture stress observed in a ten-sile test. This means that an assumption must bemade about the behaviour of the flow curve after fac-ture occurs. In this case it was assumed that the workhardening rate continues as the deformationcontinues.

The results from the Mathcad simulation are showninFig. 10. The addition of work hardening to the model

gave a very good fit to the experimental data.Some consideration has been given to incorporating

work hardening into material models in the literature[13]. However, this increases the complexity of the mod-el considerably. Furthermore, the mechanical behaviourof both steel and aluminium alloys can be approximatedreasonably well by an assumption of perfect plasticbehaviour [14], which has been confirmed here. Forthese reasons perfect plastic behaviour is generally as-sumed. However, with the invention of finite elementmodelling, work hardening can be incorporated intomodels much more easily. Some research[15]has indi-

cated that, in materials with high work hardening rates,the peak deflection upon loading and the final deflectionvary substantially compared with materials with perfectplastic behaviour. Therefore it is important to considerthe work hardening behaviour of magnesium alloys dur-

l

central axisL

Fig. 9. Model used for plastic hinge bending, showing the deformationoccurring at a point and how the change in length of the outer fibrewas determined.

0

500

1000

1500

2000

2500

3000

0 20 40 60 80 100

Displacement (mm)

Force(N)

Model with work hardening

Model without work hardening

Fig. 10. A comparison of the experimental data with the Grzebieta model and the Mathcad model where work hardening has been incorporated intothe model.

http://-/?-http://-/?-https://www.researchgate.net/publication/239361341_Influence_of_strain-hardening_and_strain-rate_sensitivity_on_the_permanent_deformation_of_impulsively_loaded_rigid-plastic_beams?el=1_x_8&enrichId=rgreq-9e1dd44e43b891a047d1259ed7957ced-XXX&enrichSource=Y292ZXJQYWdlOzIyMzQ3NzI0MTtBUzoxMzU2OTU1MDI5NDIyMDhAMTQwOTM2MzczNzE1Ng==https://www.researchgate.net/publication/245371354_Recent_Studies_on_the_Dynamic_Plastic_Behavior_of_Structures?el=1_x_8&enrichId=rgreq-9e1dd44e43b891a047d1259ed7957ced-XXX&enrichSource=Y292ZXJQYWdlOzIyMzQ3NzI0MTtBUzoxMzU2OTU1MDI5NDIyMDhAMTQwOTM2MzczNzE1Ng==https://www.researchgate.net/publication/245150170_Effects_of_strain_hardening_and_strain_rate_sensitivity_on_the_transient_response_of_elastic-plastic_rings_and_cylinders_Int_J_Mech_Sci?el=1_x_8&enrichId=rgreq-9e1dd44e43b891a047d1259ed7957ced-XXX&enrichSource=Y292ZXJQYWdlOzIyMzQ3NzI0MTtBUzoxMzU2OTU1MDI5NDIyMDhAMTQwOTM2MzczNzE1Ng==https://www.researchgate.net/publication/245371354_Recent_Studies_on_the_Dynamic_Plastic_Behavior_of_Structures?el=1_x_8&enrichId=rgreq-9e1dd44e43b891a047d1259ed7957ced-XXX&enrichSource=Y292ZXJQYWdlOzIyMzQ3NzI0MTtBUzoxMzU2OTU1MDI5NDIyMDhAMTQwOTM2MzczNzE1Ng==https://www.researchgate.net/publication/239361341_Influence_of_strain-hardening_and_strain-rate_sensitivity_on_the_permanent_deformation_of_impulsively_loaded_rigid-plastic_beams?el=1_x_8&enrichId=rgreq-9e1dd44e43b891a047d1259ed7957ced-XXX&enrichSource=Y292ZXJQYWdlOzIyMzQ3NzI0MTtBUzoxMzU2OTU1MDI5NDIyMDhAMTQwOTM2MzczNzE1Ng==https://www.researchgate.net/publication/245150170_Effects_of_strain_hardening_and_strain_rate_sensitivity_on_the_transient_response_of_elastic-plastic_rings_and_cylinders_Int_J_Mech_Sci?el=1_x_8&enrichId=rgreq-9e1dd44e43b891a047d1259ed7957ced-XXX&enrichSource=Y292ZXJQYWdlOzIyMzQ3NzI0MTtBUzoxMzU2OTU1MDI5NDIyMDhAMTQwOTM2MzczNzE1Ng==http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


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ing deformation. Since magnesium alloys appear to ben-efit from a high work hardening rate in global bucklingsituations it is also likely that steels with high workhardening rates, such as dual phase steels, may also beused advantageously in components that undergo buck-ling deformation.

5. Conclusions

Magnesium alloys, and to a lesser extent aluminiumalloys, have distinct advantages over mild steel in termsof strength and energy absorption for deformation inbending and buckling of thin sections compared withsteel on an equal weight basis. In some situations, mag-nesium alloys absorb over 1000% more energy than steelof equal weight. This is mainly due to the low density ofmagnesium alloys, allowing for much thicker sections tobe used for an equal weight, increasing the load carried

by the square of the thickness. Also, in thin sections, thehigh ductility of steels is not fully utilised duringdeformation.

The high work hardening rate of magnesium alloys,compared with the steel and aluminium, also increasesthe amount of energy absorbed during bending andbuckling by increasing the strength of the section duringdeformation and spreading the plastic hinge. An analyt-ical plastic buckling model was modified to incorporatework hardening behaviour and it was found that thisprovided a good correlation with the experimental data.

This work has indicated that the applications that

most suit magnesium alloys are those where failure oc-curs by bending and buckling of relatively thin sectionsas magnesium alloys can be made thicker while stillobtaining weight savings. Such applications includebumper bars or door inners. Whilst magnesium alloysmay also be used in thin sections, the geometric advan-tage is lower.

Acknowledgements

Dietlinde Siebenrock from the University of AppliedScience, Aalen, is thanked for her help with the experi-mental work. Dr. Chris Davies of Monash Universityis thanked for his provision of the AZ31 extrusionsand mechanical data and his comments on the text.Dr. Roman Schmidt is acknowledged for the collationof some of the literature data used in this work. The highpressure die casting was performed at CSIRO Centre forManufacturing and Infrastructure Technology, whoalong with Monash University and IRIS are core partic-ipants in the CRC for Cast Metals Manufacturing(CAST) which was established under and is supportedin part by the Australian Governments Cooperative Re-search Centres scheme.

Appendix A. Calculation of strength comparison ratio

using the moment of inertia for rectangular tubes

Outer dimensions of the tube remain the same.

Steel Magnesium

B

Dtsl/2

tMg/2

Neutral

axis

y

Where tMg= 4.4 tslFor steel tube

IBD3

12

B 2tsl D 2tsl

12 .

For Mg tube

IBD3

12

B 2tMg D 2tMg

12 .

Since r MyI

,

rsl

rMgIMg yslIsl yMg

.

In this case yMg= ysl. Therefore

rsl

rMg BD

3 B 4.4tsl D 4.4tsl3

BD3 B tsl D tsl3

.

The variation ofB, Dand tsl are shown inFig. 8.

Appendix B. Buckling model

The steps for incorporating work hardening into thebuckling model of Grzebieta and Murray [12] are asfollows:

1. Calculate the angle at the hinge

yd L d

L ;

hd sin1yd.

2. Calculate the strain in the hinge

Dld H

tanhd;

ed ln 1 Dld

PdL

.

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3. Relate true stress to the deflection using extrapolatedtensile stressstrain data

ryd fed.

4. Iterate to obtain solution using Eq. (4).

dp1;Pry

;0BH2

4 2

1 P

ry;0BH 2 2" #

LP2 ;

dpi1;P

ry;iBH2

4

21 P

ry;iBH

2 2" #

LP2 ;

where L is the strut length, dthe displacement in theouter fibre (seeFig. 9), eis the strain, rythe stress atwhich yielding occurs at the strain calculated, dp thedisplacement of the cross head,Pthe applied load, Bthe width and H the thickness.

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[10] Easton MA et al.. A comparison of the energy absorbingproperties of magnesium alloys with steel and aluminium alloys.In: Light metals technology 2003. Brisbane: CRC for CastMetals Manufacturing; 2003. p. 31520.

[11] Jones N. Structural impact. Cambridge: Cambridge UniversityPress; 1989. p. 575.

[12] Grzebieta RH, Murray NW. The static behaviour of struts withinitital Kinks at their centre point. Int J Impact Eng1985;3(3):15565.

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[14] Jones N. Recent studies on the dynamic plastic behaviour ofstructures. Appl Mech Rev 1989;42(4):95115.

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a comparison of the deformation of magnesium alloys with aluminium and steel in tension, bending and...

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