Crushing Modes of Aluminium Tubes under Axial Compression

Transcription

1 5 th Austrlsin Congress on Applied Mechnics, ACAM December 2007, Brisbne, Austrli Crushing Modes o Aluminium Tubes under Axil Compression Florent Pled 1, Wenyi Yn 2 nd Cuie Wen 3 1 Déprtement de Génie Mécnique, Ecole Normle Supérieure de Cchn, Frnce 2 Deprtment o Mechnicl Engineering, Monsh University, Clyton, Victori 3800, Austrli 3 Center or Mteril nd Fibre Innovtion, Dekin University, Geelong, Victori 3217, Austrli Abstrct: A numericl study o the crushing o circulr luminium tubes with nd without luminium om illers hs been crried out to investigte their buckling behviours under xil compression. A crushing mode clssiiction chrt hs been estblished or empty tubes. The inluence o boundry s on crushing mode hs lso been investigted. The eect o om iller on the crushing mode o tubes illed with om ws then exmined. The predicted results would ssist the design o crshworthy tube components with the preerred crushing mode with the mximum energy bsorption. Keywords: inite element simultion, porous mterils, luminium om, crushing mode, crshworthy component. 1 Introduction In the design o metllic energy dissipting systems, thin-wlled circulr luminium tubes under xil loding s hve been identiied or severl decdes s very eicient impct energy bsorbing elements, becuse o their progressive xil olding [1, 2]. When circulr thin-wlled tube is crushed xilly, it collpses either xisymmetriclly or non-symmetriclly, depending primrily on the rtio o the dimeter to the thickness ( D / t ), but lso on the rtio o the length to the dimeter ( L / D ). The xisymmetricl mode is oten known s the ring mode or concertin mode, while the non-symmetricl mode is reerred to s dimond mode. The dimond mode is chrcterised by the number o lobes. For certin vlues o D / t rtio, tube my strt collpse with the ring mode nd then switch to the dimond mode, hence exhibiting mixed mode. Tubes cn lso collpse in Eulertype buckling mode, which is ctstrophic collpse involving lrge bending o the tube nd resulting in considerble loss o energy bsorbing cpcity. I the crushing modes o circulr tubes with given geometric dimensions cn be ccurtely predicted, then initition o the preerred concertin mode with the highest energy bsorption vlues cn be gurnteed without the need or physicl testing [3]. Recently, there hs been n incresing interest in using luminium oms s cores inside thinwlled metllic tubes or energy bsorption. Previous study indictes tht the energy bsorption o the om-illed tubes cn be improved signiicntly compred to empty tubes. Thereore luminium omilled tubulr structures oer signiicnt potentil or industry pplictions where crshworthiness nd weight re criticl due to their unique combintion o physicl nd mechnicl properties, nd excellent energy bsorption. Beore quntiying the energy bsorption by n empty tube or om-illed tube, it is importnt to understnd the crushing modes s dierent crushing mode will led to dierent energy bsorption cpcity. Finite element nlyses re crried out in this pper to ddress this issue. 2 Numericl Model Finite Element models were developed using the commercil pckge ABAQUS to simulte the crushing. A typicl FE model consists o cylinder clmped by two rigid compression pltes. Both 4- node shell nd 8-node solid elements re considered to model the empty tube. The om core ws modelled by using 8-node solid elements when modelling om-illed tube.

2 The xil compression is simulted by moving down the upper plte while the lower plte is sttionry. Severl boundry s relecting the connection between the empty tube nd the pltes re studied. To our knowledge, the inluence o the boundry s on crushing mode o empty tubes hs not been pid much ttention in literture yet. All the boundry s studied in the crushing simultions re listed in Tble 1. Tble 1 All the boundry s involved in the crushing simultion The vlue o the riction coeicient between the tube nd the om is chosen s 0.1. The mterils dt re obtined rom unixil compression tests [4]. ABAQUS contins n extensive librry to model the behviour o vrious engineering mterils. It includes models or metl plsticity nd crushble om plsticity. The tube is n nneled luminium Alloy A6063T5. Its Youngs modulus E t nd Poissons rtio ν t re 68.9 GP nd 0.33, respectively. The initil yield stress σ is 48 MP. The density o the luminium tube ρ t is 2700 kg/m 3. The tube mteril is ssumed to be elstic-plstic with isotropic strin hrdening, which is described by power-lw work hrdening. The hrdening dt were clibrted rom unixil compressive tests curves [4]. The om is treted s n isotropic homogeneous mteril. The reltive density o the om, deined s the rction o spce occupied by the solid, is determined by the porosity P r [3]: ρ ρ = = (1 Pr ), (1) ρ where ρ nd ρ re the densities o the om nd the luminium solid, respectively. In the pst, vrious nlyticl, experimentl nd numericl investigtions hve been crried out to derive mteril models or metl oms. It ws ound tht the mechnicl behviour o metl oms depends on the reltive density ρ signiicntly. The Youngs modulus o the om, E, cn be estimted rom the reltive density ρ by [5]: where E E Yt 2 = C1 ρ (2) E is the Youngs modulus o the luminium solid. In the previous experimentl study [4], the luminium om smples hve closed-cell structure. C is constnt relted to the cell geometry o 1 the om, which ws clibrted s 0.3 rom the experimentl results [4]. The solid cell wll mteril o luminium hve Youngs modulus E o 68.9 GP, yield stress σ o 170 MP nd density o 2700 kg/m 3, ccording to the experimentl dt. The om smple Y hs porosity P r o 91.5%, hence we cn estimte the reltive density ρ which is equl to nd the density o the om, ρ, which is equl to kg/m 3. According to Eq. (2), the Youngs modulus o the om, E, is obtined s 159 MP. The liner elstic regime in the stress-strin curve o the 1 st boundry 2 nd boundry 3 rd boundry 4 th boundry Bottom o the tube tie constrint om smple is chrcterized by n elstic Youngs modulus o 160 Mp. A vlue o ν.33 hs been chosen in our numericl simultion [6]. The crushble om plsticity model developed by Deshpnde nd Fleck [7] is pplied to simulte the plstic behviour o the luminium om. The yield unction or this model is deined s: Fixed Fixed u r u θ u r u θ Roller Top o the tube ixed roller u r u θ roller roller

3 where σ + α σ eq m Y ( α / 3) σ eq is the Von Mises eective stress, σ m is the men stress or pressure stress, α is the shpe ctor o the yield surce nd Y is the hrdening mteril unction. The vlue o α cn be determined by the unixil nd hydrosttic compressive test, which is chosen s The plstic p Poissons rtio ν.18. (3) 3 Results nd Discussion 3.1 Mode clssiiction chrt or empty tubes The crushing mode o n empty tube strongly depends on the geometry o the structure. First, the numericl simultions re compred with the vilble experimentl dt provided in Re. [3], nd nlyzed with respect to the plstic olding mode. The irst type o boundry s o Tble 1 hs been pplied s it is close to the experimentl s. Fig. 1 Comprison o crushed shpes o empty tubes () shell-element simultion, (b) experiments nd (c) solidelement simultion The concertin modes hve been correctly predicted. Fig. 1 shows the inl crushed shpe o n empty tube, compred with experimentl observtion. Both shell elements nd solid elements were pplied to model the tube nd cn successully simulte the crushed shpe. The shell element tube olds more requently due to the ese with which the elements cn rotte. The shell element model predicts 15 olds, which is exctly the number o olds observed in experiments, wheres the solid element model predicts only 12 olds. Secondly, prmetric study hs been crried out to understnd the inluence o geometric prmeters on the buckling modes o empty tubes. The rnge o geometries used ensured tht vrious crush modes were observed. Two hundred nd our tubes were considered with vrying L / D rtio (rom 1 to 12) nd D / t rtio (rom 10 to 500). Due to the smll thickness o the tubes, shell elements were pplied in the modelling. The crushed shpes hve been successully simulted or ll the dierent crushing modes. Fig. 2 Axisymmetric concertin mode nd Cut view o concertin mode Fig. 3 Dimond 3 lobe mode nd Bottom view o dimond mode

4 Fig. 4 Mixed mode nd Euler-type buckling mode Fig. 5 Mode clssiiction chrt or luminium tubes The results shown in Fig. 5 re consistent with the previous works. Our chrt covers suiciently lrge rnge o D / t nd L / D. For short nd thick tubes, concertin mode ppers. When the thickness o the tube is reduced nd the length o the tube is incresed, mixed mode occurs nd then dimond mode is present. For long nd very thin tubes, Euler-type buckling mode tkes plce. 3.2 Inluence o boundry con ditions on buckling modes or empty tubes Four empty tubes hve been selected to investigte the inluence o the boundry s on crushing modes. Under the irst type o boundry s, our dierent crushing modes re expected rom these our representtive tubes, see Tble 2. Tble 2 Dimension prmeters o luminium tubes Length (mm) Outer Dimeter D (mm) Wll thickness t (mm) L/D D /t Expected crush modes Concertin Dimond Mixed Euler Tube L For the tube 1, concertin (xisymmetric) mode ws observed or ll the boundry s in the simultion. Nevertheless or the two symmetric boundry s, (2) nd (4), the olds pper t both sides o the tube, s shown in Fig. 6. Fig. 6 Comprison o crushed shpes o Tube 1 or dierent boundry s (1) tie constrint-roller, (2) ixed-ixed (3) ixed-roller nd (4) roller-roller For the Tube 2, dimond mode were observed or the boundry s (1) nd (4) in the simultion, wheres mixed mode ws observed or the boundry s (2) nd (3), s shown in Fig. 7. A reinement o the mesh hs been done to see the inluence o boundry s or the Tube 2. We observed tht mixed modes ppered or the boundry s (1), (3) nd (4) nd concertin mode ppered or the boundry (2). Thereore the boundry (2) seems to be the best or obtining xisymmetric olds nd bsorbing higher mount o energy. Fig. 7 Comprison o crushed shpes o Tube 2 or dierent boundry s (1) tie constrint-roller, (2) ixed-ixed (3) ixed-roller nd (4) roller-roller

5 For the Tube 3, mixed mode ws observed or boundry s (1), (2) nd (4), but concertin mode ws observed or the boundry (3) in the simultion, s shown in Fig. 8. Thereore when one end o the tube is ixed, concertin mode ppers insted o mixed mode. Fig. 8 Comprison o crushed shpes o Tube 3 or dierent boundry s (1) tie constrint-roller, (2) ixed-ixed (3) ixed-roller nd (4) roller-roller For the Tube 4, Euler buckling (ctstrophic collpse) crush mode ws observed or ll boundry s in the simultion. When one end o the tube is ixed or tied to plte, this prt o the tube sty in contct with the plte nd when roller boundry is pplied t end o the tube, this end o the tube slides over the plte, s shown in Fig. 9. Fig. 9 Comprison o crushed shpes o Tube 4 or dierent boundry s (1) tie constrint-roller, (2) ixed-ixed (3) ixed-roller nd (4) roller-roller 3.3 Inluence o om illers on the crushing mode According to experiments, n interction eect usully exists between the tube wll nd the om core. The deormtion behviour o luminium oms provides resistnce to the inwrds deormtion o the tube wll. Hence it cuses n outwrds expnding o the progressive buckling o tube wll, or shit o deormtion mode rom dimond to concertin with om-illing. Numericl simultions o the Tubes 1, 2 nd 3 illed with luminium om hve been crried out under the irst set o boundry s. The numericl results crried out in this prt re o gret importnce since they oer the possibility to predict the potentil eects o illing on the buckling modes o tube. The eect o om-illing or Tube 1 hs two consequences: Fom-illing did not chnge the deormtion mode, but it cuses decrese in the plstic old length. Indeed the presence o luminium om results in shorter olding length in experiments [4]. Moreover, om illing lso cused the progressive buckling expnding outwrds noticebly compred to the tube without om, s shown in Fig. 10. () (b) Fig. 10 Comprison o crushed shpes o the empty Tube 1 nd the om-illed tube, showing the sme crushing mode nd decrese in the plstic old length (), but n outwrds expnsion o the buckling (b). Fom-illing chnged the deormtion mode rom dimond mode to Euler-type buckling mode nd concertin mode or the Tube 2 nd the Tube 3, respectively, see Fig. 11. The numericl model seems to cpture quite correctly the contribution o the om or the concertin mode. Indeed, the modelled deormed shpe o Tube 1 with om mtches the experimentl deormed shpe. The numericl results o Tube 3 show good greement with the experimentl observtion, s the progressive buckling mode chnged rom non-symmetric mixed mode into concertin mode. The numericl results demonstrte the deormtion shit o Tube 2 rom dimond mode to Euler-type buckling mode due to om illing. In the reported experimentl work [4],

6 om iller o the tube converts the dimond mode into concertin mode, but the geometry o the tube is dierent rom tht o Tube 2. Furthermore the om used is light (the porosity o the om core ws 91.5 %) compred to tht used in the experiments (90 % o porosity). Thereore it could be interesting to chnge the om properties nd to see the inluence on the crushing mode o dimond. The interction eect between the tube nd om might be ected by not only the tube dimension prmeters, but lso the properties o the om. As om hs been deined s n isotropic homogeneous mteril, only the eect o chnging the density o om hs been investigted. The intention ws to know i the shit rom dimond mode to concertin mode due to om illing mode could occur or tube illed with om o higher density. The porosity o the om hs been chnged rom 91.5% to 88%. Finlly the Tube 2 illed with denser om collpses in Euler-type buckling mode nd not concertin mode. Further systemtic work is required to completely understnd the inluence o om iller on the crushing mode. Tube 2 Tube 3 Fig. 11 Crushed shpes o the Tube 2 nd the Tube 3 illed with om o 91.5 % o porosity, showing tht progressive crushing mode chnges rom non-symmetric mode into Euler-type buckling mode nd concertin mode or the Tube 2 nd Tube 3, respectively 4 Conclusions Numericl simultions o the crushing o luminium tubes with nd without om iller hve been crried out. The experimentl observtions o the post-buckling deormtion shpe hve been successully predicted. The inluence o geometric prmeters on buckling modes o empty tubes hs been investigted nd mode clssiiction chrt hs been estblished. It shows quite good greement with previous experimentl chrts. Dierent boundry s or the tube hve been studied. It shows tht when t lest one end is ixed is the best set o boundry s or energy bsorption becuse it cretes more xisymmetric olds, wheres when the two ends o the tube re ree is the worst set o boundry s or energy bsorption. Our numericl observtions give excellent results compred to the experiments or the concertin mode. Numericl results or the dimond mode show tht Euler-type buckling mode could occur or om-illed tube. Indeed progressive crushing mode did not chnge rom dimond mode to xisymmetric concertin mode numericlly, but luminium om converted the dimond mode into the Euler-type buckling mode, even with om o higher density. Thereore or thick nd short tubes, om-illing seems to be eective in enhncing the energy-bsorption perormnce, wheres i tube is too thin, it my hve strong tendency to undergo Euler-type buckling, gretly reducing the energy bsorption. Reerences [1] N.Jones: Structurl impct, (Cmbridge University Press, Cmbridge, 1989) pp [2] G. Lu nd T. Yu: Energy bsorption o structures nd mterils (Woodhed Publishing Limited, Cmbridge, UK, 2003) pp [3] W. Yn, E. Duri, Y. Ymd nd C. Wen: Mter. Trns. 48 (2007) [4] Y. Ymd, T. Bnno, Z. Xie nd C. Wen: Mter. Trns. 46 (2005) [5] N. Fleck: Cellulr Metls nd Polymers, ed. By R. F. Singer, C. Koerner, V. Altsedt nd H. Muenstedt (Trns Tech Publictions, Zuerich, Switzerlnd, 2005) pp [6] E. Andrews, W. Snders nd L. J. Gibson: Mterils Science nd Engineering A270 (1999) [7] V. S. Deshpnde nd N. Fleck: Journl o Mechnics nd Physics Solids 48 (2000)