International Journal of Solids and Structures

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1 Intenational Jounal of Solids and Stuctues 49 (2012) Contents lists available at SciVese ScienceDiect Intenational Jounal of Solids and Stuctues jounal homepage: Expeimental investigation and constitutive modeling fo the hadening behavio of 5754O aluminum alloy sheet unde two-stage loading Haibo Wang a,, Yu Yan a, Min Wan b, Xiangdong Wu b a College of Mechanical and Electical Engineeing, Noth China Univesity of Technology, Beijing, China b School of Mechanical Engineeing and Automation, Beihang Univesity, Beijing, China aticle info abstact Aticle histoy: Received 9 Apil 2012 Received in evised fom 25 June 2012 Available online 23 August 2012 Keywods: Stain path Bauschinge effect Tansient effect Hadening model The two-stage loading tests of 5754O aluminum alloy sheet wee caied out. In the fist loading stage, the uniaxial tensile tests of big sheets wee caied out. Then small specimens wee cut off fom the pe-stained big sheets along diffeent diections. And in the second loading stage, the uniaxial tensile tests of the small specimens wee pefomed. Fom the expeimental esults, it is found that the initial yield stess of each specimen in the second loading stage deceases when the stain path changes. In addition, when the stain path changes the tansient effect appeaed and no obvious pemanent softening was obseved. In this study, in ode to descibe the hadening behavio of 5754O aluminum alloy sheet unde two-stage loading, the Chaboche type combined isotopic kinematic hadening models wee adopted with Yld2000-2d and Hill48 as yield functions. It is poven that no pemanent softening can be descibed with Chaboche type model in two-stage loading. Thee methods fo detemining the paametes of the hadening models wee developed in ode to establish accuate isotopic kinematic hadening model descibing the hadening behavio of 5754O aluminum alloy sheet unde two-stage loading. The established constitutive models wee implemented into the commecial FEM code ABAQUS as a use mateial suboutine (UMAT) fo numeical simulations. By compaing the expeimental and simulated esults of the two-stage loading tests, the isotopic kinematic hadening models descibing the hadening behavio of 5754O aluminum alloy sheet unde two-stage loading wee accuately detemined. Also, the influences of the chaacteization method of Hill48 yield function on the accuacy of the esulting hadening models wee discussed. It is also found that the established isotopic kinematic hadening model descibing the hadening behavio unde two-stage loading can descibe easonably the spingback pofile of the thee point bending of the pe-stained specimen. Ó 2012 Elsevie Ltd. All ights eseved. 1. Intoduction Sheet metal foming is usually a complicated defomation pocess, duing which the stain path of the mateial points may change. The defomation behavio of mateials unde complex loading condition is diffeent fom that unde monotonic loading condition since the loading path change will affect the hadening behavio, flow ule and foming ability, etc. When the stain path changes, some phenomena such as tansient effect, incease o decease of the yield stess and the hadening ate of the stess stain cuves will appea. Among the stain path change loading conditions, evese loading conditions have been widely investigated fo a long time. Duing evese loading, Bauschinge effect, tansient effect and pemanent softening ae commonly obseved. In ode to investigate the effect of the evese loading on the hadening behavio of sheet Coesponding autho. Fax: addess: wanghaibo@ncut.edu.cn (H. Wang). metals, many cyclic tension compession loading tests have been pefomed in which some methods fo peventing buckling duing compession loading have been developed. Kuwabaa et al. (1995, 2009) caied out tension compession tests without buckling of the specimen using a specially designed equipment mounted with comb-type dies. In thei tension compession tests, a decease in the flow stesses due to the Bauschinge effect was clealy obseved. Chen et al. (1999) developed evese tosion tests to investigate the Bauschinge effect and multiaxial yield behavio of mild steel. Yoshida et al. (2002) caied out in-plane cyclic tension compession tests at lage stain by employing adhesively bonded sheet laminate specimens as well as a special device fo peventing the buckling of specimens. They found that the cyclic hadening was stongly influenced by cyclic stain ange and mean stain. The tansient softening and wok hadening stagnation duing stess evesals wee also obseved. Boge et al. (2005) pefomed continuous stain evesal tests with a special specimen geomety and solid flat plates as buckling constaints. The Bauschinge effect, oom-tempeatue ceep, and anelasticity afte stain evesals in commecial sheet alloys /$ - see font matte Ó 2012 Elsevie Ltd. All ights eseved.

2 3694 H. Wang et al. / Intenational Jounal of Solids and Stuctues 49 (2012) wee investigated. Lee et al. (2005) caied out in-plane uniaxial cyclic tension compession tests whee solid plates made of the hadened steel wee placed along the side of the sheet specimen in ode to pevent buckling of the thin sheet specimen. With thei expeimental esults, the mateial paametes of the combined isotopic kinematic hadening law (Chung et al., 2005) wee detemined. Cao et al. (2009) developed a fixtue with a egula tension compession machine, whee the nomal suppot was povided to the entie length of the specimen duing the tension compession cycle so that the potential buckling of sheet specimen can be pevented. The cyclic tension compession behavio duing evese loading and the non-symmetic behavio duing eloading wee obseved and descibed. Since sheet metal foming is a complicated pocess, the stain path change of sheet metals duing foming pocess is vaious. Besides in-plane uniaxial cyclic tension compession tests, some two/multi-stage loading tests such as tension tension, tension sheaing and two-stage biaxial loading have also been pefomed in ode to investigate the hadening behavio of sheet metals unde complicated loading paths. Wagone and Laukonis (1983) caied out stain path-change tests (plane stain tension followed by uniaxial tensile tests) to evaluate the esidual wok-hadening behavio. They found that the subsequent hadening cuves depended pimaily on the elative diection between majo stain axes in the two defomation stages and vey little on the specific pe-stain pocedue. Doucet and Wagone (1989) pefomed tensile tests along the tansvese diection on specimens which ae pe-stained along the olling diection unde plane stain condition. Khan and Wang (1990) developed a non-popotional biaxial compession expeiment in which a ectangula block fist undegoes uniaxial compession and then, afte finite defomation, was subjected to biaxial compession. Hu et al. (1992), Thuillie and Rauch (1994) pefomed two-stage expeiments (including shea and tension tests) to investigate the defomation behavio of metals. Hiwatashi et al. (1997, 1998) pefomed non-popotional biaxial loading expeiments to evaluate the cystal plasticity. Kuwabaa et al. (2000) caied out stain path-change expeiments and found the impotant diffeences between the yield suface shapes found by the stain path change pocedue and the shapes found by pobing the yield points fom the elastic egion. Balat et al. (2003b) obtained non-linea defomation paths using uniaxial tension followed by simple shea tests. The flow stess was epesented as a function of the plastic wok in ode to eliminate the influence of the initial plastic anisotopy and to compae the esults in a consistent manne. Based on expeimental and simulation esults, the elative contibutions of the cystallogaphic textue and dislocation micostuctue evolution to the anisotopic hadening behavio of the mateial wee discussed. Bouvie et al. (2005) caied out two-stage tests (shea shea and tension shea) to investigate the defomation behavio. Bouvie et al. (2006) investigated the isotopic, kinematic and distotional hadening of olled metal sheets at modeate and finite stains using twostage non-popotional loadings involving sequences of simple shea and uniaxial tensile defomations. Kim and Yin (1997) pefomed thee-stage tests, whee in the fist loading stage, the sheets wee stetched in the olling diection, in the second loading stage the sheets wee stetched at cetain angles to the olling diection, and in the thid loading tensile test wee pefomed along diffeent diections. Fom tensile tests, effects of the twostage pe-stains on the subsequent yielding wee investigated. They obseved that the oientations of the othotopic axes change dastically ove a few pecent of tensile stain. Kuwabaa et al. (2002) caied out the same test on an IF steel sheet. Fom the thee-stage tests, Hahm and Kim (2008) found that the -value distibution is hadly affected by the pe-stain. Wu et al. (2005) caied out uniaxial tensile tests along diffeent angles fom the olling diection fo both as-eceived and pe-stained sheet to investigate the effect of pe-stain on mateial anisotopy. They concluded that the conventional methodology fo detemining mateial anisotopy oveestimated the effect of the pe-stain. Taigopula et al. (2008) investigated the elasto-plastic behavio of sheet metals unde two-stage loading and obseved Bauschinge effect, tansient effect and pemanent softening phenomena. Manninen et al. (2009) caied out two-stage tests and investigated the effect of pe-stain on mateial anisotopy. Consideable Bauschinge effect, tansient effect and pemanent softening wee obseved. Khan et al. (2009, 2010a,b) investigated the initial and subsequent yield sufaces of aluminum alloy. The subsequent yield sufaces wee detemined duing vaious loading paths. It is found that the initial yield suface is vey close to the von-mises yield suface and the subsequent yield sufaces undego tanslation and distotion. On subsequent yield suface a nose in the loading diection and flattened shape in the evese loading diection wee obseved. They found that the yield sufaces afte unloading depict stong anisotopy, positive coss-effect and exhibits diffeent popotion of distotion in each loading condition. Vema et al. (2011) pefomed two-stage uniaxial tests along with uniaxial cyclic tests and biaxial tests to evaluate the effect of moe geneal stain path changes on the defomation behavios of sheet metals. They poposed the combined isotopic kinematic hadening model which can easonably descibe vaious expeimental phenomena unde moe geneal stain path changes such as the hadening stagnation, coss-effect and asymmety in tension and compession, Bauschinge effect and the tansient behavios. In ode to descibe the hadening behavio of sheet metals unde stain path change loading, many hadening models including micostuctual and phenomenological models have been poposed. Teodosiu and Hu (1995) poposed a micostuctual model which can easonably descibe the macoscopic hadening behavio unde stain path change loading condition (Haddadi et al., 2006; Oliveia et al., 2007). Bouvie et al. (2005) poposed a physically-based phenomenological model using fou intenal state tenso vaiables based on Teodosiu and Hu s model. The accuacy and the efficiency of the model wee evaluated with the spingback simulations of the stamping of a cuved ail. Howeve, compaed to the phenomenological models, the micostuctual models ae computationally expensive and the paametes identification is vey complex so that they have not been widely used. Two kinds of phenomenological models have been used in ode to descibe the in-plane uniaxial cyclic tension compession hadening behavio: one is based on kinematic hadening (shifting of a single-yield suface), and the othe one involves multiple yield sufaces (Khan and Huang, 1995). Fo the fist kind of models, Chaboche type combined isotopic kinematic hadening model (Chaboche, 1986) genealized fom the Amstong Fedeick type model (Amstong and Fedeick, 1966) has been used widely fo a long time. In ode to moe accuately descibe the hadening behavio duing stain path change loading, mainly fo evese loading, some new hadening models and some new methods fo detemining the kinematic hadening paametes have been developed based on Chaboche type model (Chaboche, 1991; Chun et al., 2002; Geng et al., 2002; Geng and Wagone, 2000, 2002; Khan and Huang, 1995; Ohno and Wang, 1993a,b; Yoshida et al., 2002). Based on the genealized plastic wok equivalence pinciple, Chung et al. (2005) poposed one modified Chaboche type combined isotopic kinematic hadening law, which can accuately descibe the tansient behavio by consideing the slope of the loading and unloading cuves. Since no pemanent softening can be pedicted by this model (Kim et al., 2006), Ahn et al. (2009), Chung et al. (2010) and Lee et al. (2006) impoved it by intoducing softening paamete. Some two-suface models (Dafalias and Popov, 1976; Kieg, 1975; Lee et al., 2007; Yoshida and Uemoi, 2002) have also

3 H. Wang et al. / Intenational Jounal of Solids and Stuctues 49 (2012) been poposed in ode to descibe the hadening behavio duing evese loading accuately. As to two-stage loading fo some mateials, hadening behavios such as Bauschinge effect and tansient behavio have been obseved (Gadey et al., 2005; Hahm and Kim, 2008; Manninen et al., 2009; Taigopula et al., 2008, 2009; Yoshida et al., 2011). Some attempts have been made to descibe the hadening behavio when stain path changes. Hahm and Kim (2008) made an attempt to appoximate the obseved yield and flow behavio unde two/ multi-stage tests based on a modified isotopic kinematic hadening model with Hill48 yield function. The yield stess distibutions wee pedicted to some degee. Taigopula et al. (2008) used Chaboche type combined isotopic kinematic hadening model with a high-exponent yield function (Heshey, 1954). The paametes of the hadening model wee deduced by numeical tests and coelation with cetain expeimental stess-plastic stain cuve. This model can descibe the emakable yield stess decease and the tansient effect when stain path changes with easonable accuacy. The geneal tends of the expeimental esults wee descibed easonably. Then Taigopula et al. (2009) evaluated the pefomance of the established Chaboche type hadening model fo othe defomation modes of dual-phase steel subjected to non-popotional loading. It is found that the Chaboche type model pedicted the geneal tends of the expeimental esults such as tansitoy hadening and oveall wok hadening. Howeve, the tansient hadening behavio subsequent to stain path changes cannot be descibed accuately. Manninen et al. (2009) analyzed the evolution of kinematic and isotopic hadening components in twostage tests using Chaboche type combined isotopic kinematic hadening model with Mises yield function, while no pedicted esults of expeimental cuves wee pesented. In this study, the hadening behavios of 5754O aluminum alloy sheet unde a two-stage loading (tension tension) wee investigated. The pimay objective is to establish an isotopic kinematic hadening model that can descibe the hadening behavio of 5754O aluminum alloy sheet unde two-stage loading accuately. In ode to descibe the hadening behavio unde two-stage loading, the Chaboche type combined isotopic kinematic hadening model with Yld2000-2d yield function (Balat et al., 2003a) was adopted, and thee methods fo detemining the paametes of the hadening model wee established. The Chaboche type combined isotopic kinematic hadening model with Hill48 yield function was also used to descibe the hadening behavio. The established constitutive models wee implemented into ABAQUS softwae and wee veified by compaing the theoetical and expeimental esults. Consideing moe geneal stain path changes, thee point bending tests of pe-stained specimens wee pefomed to veify futhe the established hadening model futhe. Fig. 1. The simple tension hadening cuves along vaious diections. Fig. 2. The nomalized stesses-plastic wok along vaious diections. along the 45 off and tansvese diections) can be descibed easonably when the hadening cuve along the olling diection is adopted as the efeence cuve, which will be shown in Section 6.1 (Figs. 8 and 13). The -value was calculated with Eq. (1) (Chung et al., 2011): de ¼ de yy =de zz ¼ de yy =ðde xx þ de yy =de xx yy Þ¼ ð1 þ de yy =de xx Þ whee e xx, e yy and e zz ae the tue stains along the longitudinal, width and thickness diections in simple tension test, espectively. The value of de yy /de xx was detemined by linealy intepolating the measued stains along the width and the longitudinal diections, espectively (e yy and e xx ) and the measued de yy /de xx of simple tension tests along vaious diections ae shown in Fig. 3. ð1þ 2. Mateial chaacteization Simple tension tests of the test mateial 5754O aluminium alloy sheet along the olling, 45 off and tansvese diections wee caied out. The measued hadening cuves along the thee diections ae shown in Fig. 1. In this study, the simple tension along the olling diection is adopted as the efeence state. As fo anisotopy, the nomalized stesses along the olling, 45 off and tansvese diections ( 0 / 0, 45 / 0, 90 / 0 ) ae calculated at diffeent plastic wok as shown in Fig. 2. As shown in Fig. 2, the nomalized stesses 45 / 0 and 90 / 0 vaies with the plastic wok obviously. In ode to chaacteize the anisotopy of stess in the whole defomation pocess easonably, the aveages of the nomalized stesses at the plastic wok fom 10 Mpa to 40 Mpa is adopted as the mechanical popeties and to calculate the coefficients of the yield functions pesented in Section 4 instead of those at the initial yield point. Then the anisotopy of stess (e.g., the hadening cuves Fig. 3. The stain along the width diection and that along the longitudinal diection.

4 3696 H. Wang et al. / Intenational Jounal of Solids and Stuctues 49 (2012) Table 1 The mechanical popeties of the test mateial 5754O aluminium alloy sheet. 0 / 0 45 / 0 90 / 0 b / The measued nomalized stesses and the -values of 5754O aluminum alloy ae listed in Table Expeimental 3.1. Expeimental pocedue The two-stage loading tests wee made up of two steps. Fist, the uniaxial tensile tests wee pefomed in one goup of big sheets. Afte some pe-stains, small specimens wee cut off along diffeent diections fom the cente of the pe-stained specimens and then uniaxial tensile tests of the cut off specimens wee pefomed. The dimensions and diections of the specimens in the fist and the second loading stages ae shown in Fig. 4. Thee goups of two-stage loading tests with stain path changes wee pefomed: Goup (1): As shown in Fig. 4(a), the fist and the second loading stages ae along the olling and the tansvese diections, espectively. The plastic pe-stains in the fist loading stage wee 0.013, 0.02, 0.028, 0.047, and 0.107, espectively. Goup (2): As shown in Fig. 4(a), the fist and the second loading stages ae along the tansvese and the olling diections, espectively. The plastic pe-stains in the fist loading stage wee and 0.077, espectively. Goup (3): As shown in Fig. 4(b), the fist and the second loading stages ae along the olling and the 45 diections, espectively. The plastic pe-stains in the fist loading stage wee and 0.057, espectively. In addition, the tensile tests without stain path change wee pefomed and the dimensions of the specimens ae shown in Fig. 4(c) Expeimental esults The expeimental stess stain cuves in the second loading stage ae shown in Fig. 5. As shown in Fig. 5(a) (c), Bauschinge effect and tansient effect appea due to stain path changes, i.e. the initial yield stess deceases and the slopes of the stess stain cuves change apidly. Afte some defomation, the stess stain cuves in the second loading stage convege to that in monotonic loading, i.e. thee is no pemanent softening. As shown in Fig. 5(d), when the fist and the second loading stages ae along the same diection (i.e., no stain path change), the stess stain cuves coincide with those in monotonic loading. In othe wods, no above phenomena such as Bauschinge effect and tansient effect appea when the stain path does not change. Fo the two-stage loading whee the fist and the second loading stages ae along diffeent diections, the expeimental stess stain cuves can only be analyzed qualitatively in the same coodinate system due to anisotopy. In ode to obtain a easonable equivalence fo the stess stain cuves in diffeent diections, the stess should be epesented as a function of the plastic wok pe unit volume (Khan and Huang, 1995), and then a diect compaison between the assumption of isotopic hadening and the eal hadening exhibited by the mateial can be caied out (Balat et al., 2003b). Fig. 4. Dimensions and diections of the specimens (mm). (a) Second loading stage oiented 90 to the diection of the fist loading stage. (b) Second loading stage oiented 45 to the diection of the fist loading stage. (c) Second loading stage along same diection of the fist loading stage. The stess-plastic wok pe unit volume cuves in the second loading stage ae plotted in Fig. 6. As shown in Fig. 6(a) (c), when stain path changes, Bauschinge effect and tansient effect wee obseved by compaing the stess-plastic wok cuves in the second loading stage and those in monotonic loading. Afte some plastic wok, the stess-plastic wok cuves in the second loading stage convege to that in the monotonic loading, i.e., no pemanent softening appeas. As shown in Fig. 6(d), when stain path does not change, no Bauschinge effect o tansient effect appeas, and duing plastic defomation pocess the stess-plastic wok cuves coincide with that in monotonic loading. Theefoe, isotopic hadening will not be sufficient to descibe the obseved Bauschinge effect and tansient effect unde two-stage loading, and kinematic hadening should be consideed to model the obseved hadening behavio unde two-stage loading. 4. Isotopic kinematic hadening model 4.1. Yield function In this study, Yld2000-2d yield function poposed by Balat et al. (2003a) was adopted to establish the isotopic kinematic

5 H. Wang et al. / Intenational Jounal of Solids and Stuctues 49 (2012) Fig. 5. Stess stain cuves in the second loading stage with diffeent pe-stains. (a) Cuves along the tansvese diection with pe-stains along the olling diection. (b) Cuves along the 45 diection with pe-stains along the olling diection. (c) Cuves along the olling diection with pe-stains along the tansvese diection. (d) Cuves along the olling diection with pe-stains along the olling diection. Fig. 6. Stess-plastic wok pe unit volume cuves in the second loading stage. (a) Cuves along the tansvese diection with pe-stains along the olling diection. (b) Cuves along the 45 diection with pe-stains along the olling diection. (c) Cuves along the olling diection with pe-stains along the tansvese diection. (d) Cuves along the olling diection with pe-stains along the olling diection.

6 3698 H. Wang et al. / Intenational Jounal of Solids and Stuctues 49 (2012) hadening model that can descibe the hadening behavio unde two-stage loading accuately. Yld2000-2d yield function is given by / ¼ / 0 þ / 00 ¼ 2 m ð2þ whee exponent m is a mateial coefficient and ( / 0 ¼jX 0 1 X0 2 jm / 00 ¼j2X 00 2 þ X00 1 jm þj2x 00 1 þ X00 2 jm Hee, u is the sum of two isotopic functions, which ae symmetic with espect to X 0 1 and X0 2 as well as X00 1 and X00 2. The pincipal values of the matices, X 0 and X 00 ae 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi >< X 0 1 ¼ 1 2 ðx0 11 þ X0 22 þ ðx 0 11 X0 22 Þ2 þ 4X 02 12Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4aþ >: X 0 2 ¼ 1 2 ðx0 11 þ X0 22 ðx 0 11 X0 22 Þ2 þ 4X 02 12Þ and 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi >< X 00 1 ¼ 1 2 ðx00 11 þ X00 22 þ ðx X00 22 Þ2 þ 4X Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi >: X 00 2 ¼ 1 2 ðx00 11 þ X00 22 ðx X00 22 Þ2 þ 4X 002 Þ 12 ð3þ ð4bþ Components of X 0 and X 00 ae obtained fom the following linea tansfomation of the Cauchy stess: X 0 ¼ L 0 ; X 00 ¼ L 00 whee L =3 0 0 L =3 0 0 L 0 21 ¼ 0 1= L = L 0 66 L L L L L b 1 b 2 b ¼ ð5þ ð6þ In Eqs. (5) (7), is Cauchy stess and b 1 b 8 ae eight anisotopy coefficients. The pocedue fo solving b 1 b 8 numeically was developed accoding to the method poposed by Balat et al. (2003a). In this study m = 8 since 5754O aluminum alloy is an FCC mateial. Hill48 yield function is also adopted to establish the isotopic kinematic hadening model and unde a plane stess condition it is given by b 3 b 4 b 5 b 6 b ðg þ HÞ 2 x 2H x y þðf þ HÞ 2 y þ 2N2 xy ¼ 2 whee F, G, H and N ae mateial coefficients which can be calibated with nomalized stesses ( 0 / 0, 45 / 0, 90 / 0, and b / 0 ) o the -values along diffeent diections ( 0, 45 and 90 )(Pak and Chung, 2012). In this study, fo convenience, the Hill48 yield function chaacteized with -values is called Hill48 (1) while that chaacteized with nomalized stesses is called Hill48 (2). Then the established isotopic kinematic hadening models with diffeent yield functions (Yld2000-2d, Hill48 (1) and Hill48 (2)) will be compaed with expeimental esults, and the influences of chaacteization method of Hill48 on the accuacy of the esulting hadening model will be analyzed. ð7þ ð8þ 4.2. Hadening model Chaboche type model The kinematic hadening law in Chaboche type model, i.e., Amstong and Fedeick evolution ule (Amstong and Fedick, 1966), can be expessed as da ¼ c ð aþ de p cade p whee is Cauchy stess tension, a is backstess tenso, c and c ae mateial paametes, e p is the equivalent plastic stain and is the equivalent stess (the isotopic hadening). Hee only one tem of the backstess in Chaboche type model was adopted. Chaboche type model as well as its modified model has been widely used fo descibing the hadening behavio of sheet metals unde cyclic tension compession loading. The kinematic hadening paametes can be assumed to be constant o be epesented as the functions of the plastic defomation (Ahn et al., 2009; Cao et al., 2009; Chung et al., 2005, 2010; Geng et al., 2002; Geng and Wagone, 2002; Khan and Huang, 1995; Kim et al., 2006; Lee et al., 2005) accoding to actual needs. Chaboche type model can descibe the Bauschinge effect and the tansient effect duing evese loading easonably wheeas it cannot descibe the pemanent softening in the uniaxial cyclic tension compession loading (Chun et al., 2002; Chung et al., 2005; Yoshida et al., 2002), and Kim et al. (2006) pesented the detailed theoetical explanation. By eplacing the stess tenso in Yld2000-2d yield function with a, the Chaboche type combined isotopic kinematic hadening model with Yld2000-2d yield function can be expessed as /ð aþ ¼2 m ð10þ Fo this two-stage loading, thee ae only two pinciple stesses, which ae along the olling and the tansvese diections, espectively. So we have /ð 1 a 1 ; 2 a 2 Þ¼2 m ð9þ ð11þ whee 1 and a 1 ae the stess and backstess vaiables along the olling diection and 2 and a 2 ae those along the tansvese diection. In this study, the simple tension along the olling diection is adopted as the efeence state. Then the pefomance of Chaboche hadening model fo the pemanent softening in two-stage loading is analyzed in Appendix. It is poven that no pemanent softening can be descibed with Chaboche type model in two-stage loading. Coincidentally, no obvious pemanent softening was obseved in the expeimental esults in this study. Theefoe, we make an attempt to descibe the hadening behavio in the two-stage loading using Chaboche type model Calculation of the backstess and the isotopic hadening The method fo detemining the paametes of the hadening models in uniaxial cyclic tension compession loading cannot be diectly used fo that in two-stage loading since the loading paths ae diffeent. Also take the two-stage loading whee the fist and the second loading stages ae along the olling and the tansvese diections espectively as an example. Accoding to the Chaboche type combined isotopic kinematic hadening model, besides the isotopic expansion, the yield suface will also shift along the loading diection in the fist loading stage. At the end of the fist loading stage (uniaxial tension in the olling diection), 2 = 0 and a 2 = 0. Then fom the isotopic kinematic hadening model we have 1 a 1 ¼ ð12þ

7 H. Wang et al. / Intenational Jounal of Solids and Stuctues 49 (2012) whee a 1 and 1 ae the backstess and the measued stess along the olling diection at the end of the fist loading stage. At the initial yield point of the second loading stage, 1 = 0 and a 2 = 0. Then fom Eq. (11) we have /ð a 1 ; 2 Þ¼2 m ð13þ whee 2 is the measued yield stess at the initial yield point of the second loading stage. Then the isotopic hadening and the backstess a 1 can be obtained by solving Eqs. (12) and (13) numeically. The esults based on the Chaboche type model with Hill48 yield function can also be obtained just by eplacing Yld2000-2d with Hill48 yield function in Eqs. (12) and (13) Detemination of the paametes of the hadening model With seveal two-stage loading tests with diffeent pe-stains in the fist loading stage, a goup of and a 1 can be obtained fom Eqs. (12) and (13). Afte the calculation of the backstess and the isotopic hadening, thee methods fo detemining the paametes of the hadening model ae pesented below utilizing the expeimental data of goup (1) (the fist and the second loading ae along the olling and the tansvese diections, espectively). Method (1) The kinematic hadening paametes of Chaboche type model, c and c, ae assumed to be constants. Then in the fist loading stage of goup (1), fom Eq. (9) we have a 1 ¼ c c ð1 expð c e p ÞÞ ð14þ Voce type hadening function was adopted to descibe the isotopic hadening : ¼ 0 þ qð1 expð be p ÞÞ ð15þ whee 0, q and b ae mateials paametes. c and c wee obtained by fitting a goup of data (a 1 ; e p ) while 0, q and b wee obtained by fitting data (; e p ). Method (2) The kinematic hadening paametes of Chaboche type model, c and c, wee detemined with Method (1). As fo the isotopic hadening, we use the expession below: ¼ 1 a 1 ¼ 0 þ qð1 expð be p ÞÞ c c ð1 expð c e p ÞÞ ð16þ whee 0, q and b wee obtained by fitting the uniaxial tensile test data, ( 1 ; e p ), in the olling diection. Method (3) Voce type hadening function was adopted fo the isotopic hadening and the paametes 0, q and b wee detemined with Method (1). Fo the fist loading stage of goup (1), fom Eqs. (9) and (12) we have da 1 de ¼ c ca p 1 ¼ d 1 d ð17þ de p de p whee d 1 de can be obtained fom the expeimental cuve in the fist p loading stage of goup (1), and de d can be obtained fom Eq. (15) o p fom the esulting data (; e p ) if they ae sufficient. At the initial yield point of the second loading stage of goup (1), a 2 = 0 and 1 =d 1 = 0. Then accoding to Eq. (9) Fo the fist loading stage of goup (1), accoding to Ducke s postulate and plastic wok equivalent theoem de p 2 de p 1 a 1 Þ ð20þ whee de p 2 is the plastic stain incement along the tansvese diection. Then we have d 2 de ¼ d 2 de p p de p 2 de ¼ d 2 2 p de p 1 a 1 Þ ð21þ The diffeentiation of the Chaboche type combined isotopic kinematic hadening model with Yld2000-2d yield function (Eq. (10)) leads 1 da 1 d 2 da 2 d ð22þ Fo the second loading stage of goup (1), accoding to Ducke s postulate and plastic wok equivalent theoem de p 2 de 2 p ð 1 a 1 a þ ð 2 a a 2 Þ ð23þ Since at the initial yield point of the second loading stage of goup (1), a 2 = 1 = 0 we have de p 2 de 2 p a 2 Þ Then we have d 2 de ¼ d 2 de 2 p de p de ¼ d 2 2 p de p 2 2 a 2 Þ 1 þ 2 Fom Eqs. (18), (19), (22), and (24) we c a 1 þ c þ d 2 ð@/=@ 2 Þ 2 de p a a 2 Þ de p Then fom Eqs. (17) and (26) we have c ¼ P I 1 c ¼ P I K M 1 a 1 1 ð@/=@ 2 Þ ð24þ ð25þ ð26þ ð27þ ð28þ whee de d, K ¼ d p 2 de p a þ M ¼ d 1 1 p de d, p 2 a 2 Þ 1 P a Hee, d 2 can be obtained fom the expeimental de p 2 2 cuve in the second loading stage. With the test data of six two-stage loadings of goup (1), a set of data (c; e p ) and (c; e p ) can be obtained fom Eqs. (27) and (28). Hee, the kinematic hadening paametes, c and c, wee epesented as the exponentially decaying functions of the equivalent plastic stain e p : cðe p Þ¼c 1 þ c 2 e c 3 ep ð29þ da 1 de ¼ c ð 1 a 1 Þ c ca p 1 ¼ a 1 þ c and da 2 de ¼ c ð 2 a 2 Þ ca p 2 ¼ c 2 ð18þ ð19þ cðe p Þ¼c 1 þ c 2 e c 3 ep ð30þ By fitting the obtained data (c; e p ) and (c; e p ), the paametes of Eqs. (29) and (30) (c 1, c 2, c 3 and c 1, c 2, c 3 ) can be detemined. The esults based on the Chaboche type model with Hill48 yield function can also be obtained fom the above equations just by eplacing Yld2000-2d with Hill48 yield function.

8 3700 H. Wang et al. / Intenational Jounal of Solids and Stuctues 49 (2012) Fo convenience, the esulting Chaboche models with the above methods (1), (2) and (3) ae called Chaboche (1), (2) and (3), espectively in this study. 5. Numeical simulations The established Chaboche type combined isotopic kinematic hadening models with Yld2000-2d and Hill48 yield functions wee implemented into ABAQUS/Standad codes using the use-suboutines UMAT and the Backwad Eule method. The basic equations of the numeical fomulations fo the Chaboche type isotopic kinematic hadening model with Yld2000-2d yield function ae as follows. ð nþ1 a nþ1 Þ¼ð n a n þ D nþ1 Da nþ1 Þ¼q Da nþ1 ¼ a nþ1 a n ¼ c nþ1 a nþ1 ca nþ1 D nþ1 ¼ nþ1 n ¼ CðDe De p nþ1 Þ De p nþ1 ¼ D e p nþ1 q ¼ qðe p nþ1 Þ¼qð e p n þ D e p nþ1 Þ c ¼ cðe p nþ1 Þ¼cð e p n þ D e p nþ1 Þ c ¼ cðe p nþ1 Þ¼cð e p n þ D e p nþ1 Þ De p nþ1 ð31aþ ð31bþ ð31cþ ð31dþ ð32aþ ð32bþ ð32cþ whee e p is plastic stain, C is elastic matix and q is the efeence hadening cuve. The subscipt denotes the pocess time step. The pedicto-coecto scheme based on the Newton Raphson method was used to solve De p nþ1 in Eq. (31). To effectively solve the non-linea equation, the solution was obtained pogessively by adding seveal yield sufaces between the tial elastic stess ( T nþ1 ¼ n þ CDe) and the initial stess n (Yoon et al., 1999). As fo the isotopic kinematic hadening model with Hill48 yield function, only Chaboche (3) was implemented into ABAQUS since Hill48 yield function and the Chaboche type model with constant paametes ae aleady included in ABAQUS. Then the finite element simulations of the above two-stage tests wee pefomed whee the S4R shell elements wee adopted. 6. Results and discussions 6.1. Veifications of the hadening models The anisotopic coefficients of Yld2000-2d ae calculated with the seven measued mateial popeties (in Table 1) ae shown in Table 2 assuming b 3 = b 6 (theefoe, L ¼ L00 21 ) in Eq. (7) by solving a goup of non-linea equations (Balat et al., 2003a; Lee et al., 2005). The anisotopic coefficients of Hill48 yield functions ae listed in Table 3. InTable 3, the coefficients of Hill48 yield functions ae calculated with two diffeent elationships (Pak and Chung, 2012) with -values and nomalized stesses pesented in Table 1 espectively as shown in Section 4.1. The esulting paametes of Chaboche type model combined isotopic kinematic model with Yld2000-2d and Hill48 ae listed in Tables 4 9. The values in Tables Table 2 The anisotopy coefficients of Yld2000-2d yield functions. b 1 b 2 b 3 b 4 b 5 b 6 b 7 b Table 3 The anisotopy paametes of Hill48 yield function. F G H N Hill48 (1) Hill48 (2) and 5 wee obtained with Yld2000-2d, those in Tables 6 and 7 wee obtained with Hill48 (1) and those in Tables 8 and 9 wee obtained with Hill48 (2). The pedicted efeence cuve, isotopic hadening and backstess based on Chaboche (1), (2) and (3) with Yld2000-2d yield function ae compaed with the expeimental esults in Fig. 7. In Fig. 7, the expeimental efeence cuve is the simple tension cuve along the olling diection. The expeimental isotopic hadening and backstess ae obtained by solving Eqs. (12) and (13), whee 1 and 2 ae measued fom the two stage loading test of Goup (1) as pesented in Section 3.1, i.e. the fist loading and the second loading ae along the olling and the tansvese diections, espectively. Fo each two-stage loading test, one pai of stesses 1 and 2 ae measued fom the fist loading and the second loading tests, espectively. Then one pai of values of the isotopic hadening and backstess can be obtained by solving Eqs. (12) and (13). Theefoe six pais of values of the isotopic hadening and backstess ae obtained since thee ae six tests in the two-stage test of Goup (1). It can be seen that all the thee models can descibe the measued isotopic hadening and backstess easonably. Chaboche (2) and (3) can descibe the uniaxial tensile test cuves in the olling diections accuately wheeas Chaboche (1) cannot. Fo Chaboche (1), thee is a bifucation point between the simulated and expeimental cuve in the olling diection, afte which the eo of Chaboche (1) becomes lage with the incease of stain. This is because that only the limited expeimental data including the backstess and the isotopic hadening wee used fo detemining the paametes of Chaboche (1). When detemining the paametes of the isotopic hadening of Chaboche (2), the whole expeimental efeence cuve (uniaxial tensile cuves in the olling diection) was used so that the simulated and expeimental efeence cuves in the olling diection coincide with each othe. Besides the expeimental isotopic hadening and backstess, the slopes of the expeimental cuves wee also used fo detemining the paametes of Chaboche (3) which may lead to the easonable pediction of the expeimental efeence cuve. The uniaxial tensile test cuves simulated based on Chaboche (1), (2) and (3) with Yld2000-2d yield function along the 45 and the tansvese diections ae shown in Fig. 8. In this study, the expeimental uniaxial tensile cuves along the olling diection was adopted as the efeence cuve and the stess anisotopy changes duing defomation pocess wee consideed in Yld2000-2d yield function. Theefoe, as shown in Fig. 8, Chaboche (2) and (3) can descibe the uniaxial cuves along 45 and the tansvese diections well since the efeence cuves based on them coincide with the expeimental ones. Chaboche (1) can not descibe the expeimental cuves along 45 and the tansvese diection easonably since it can not pedict the efeence cuve easonably. The simulated and expeimental stess stain cuves in the second loading stage of the two-stage loading, whee the fist and the second loading stages ae along the olling and the tansvese Table 4 The paametes of Chaboche (1) and Chaboche (2) with Yld2000-2d yield function. 0 q b c c Chaboche (1) Chaboche (2)

9 H. Wang et al. / Intenational Jounal of Solids and Stuctues 49 (2012) Table 5 The paametes of Chaboche (3) with Yld2000-2d yield function. 0 q b c 1 c 2 c 3 c 1 c 2 c Table 6 The paametes of Chaboche (1) and Chaboche (2) with Hill48 (1). 0 q b c c Chaboche (1) Chaboche (2) Table 7 The paametes of Chaboche (1) and Chaboche (2) with Hill48 (2). 0 q b c c Chaboche (1) Chaboche (2) Table 8 The paametes of Chaboche (3) with Hill48 (1). 0 q b c 1 c 2 c 3 c 1 c 2 c Table 9 The paametes of Chaboche (3) with Hill48 (2). 0 q b c 1 c 2 c 3 c 1 c 2 c Fig. 7. Isotopic hadening, backstess and efeence cuve based on Chaboche type model with Yld2000-2d yield function. diections, espectively, ae compaed with each othe as shown in Fig. 9. The simulated esults based on the isotopic hadening model ae also pesented whee the monotonic tension cuve based on Chaboche (3) along the olling diections is adopted as the efeence cuve. It can be seen that the stess stain cuves simulated based on Chaboche (3) with Yld2000-2d yield function ae in good ageement with the expeimental esults. Chaboche (3) can descibe the tansient effect of the entie expeimental esults well, because the slopes of the expeimental cuves in the second loading stage wee consideed. When the pe-stain is not lage (0.013, 0.02 and 0.028), thee is no significant diffeence between Chaboche (2) and Chaboche (3). And when the pe-stains ae 0.013, 0.02 and 0.028, Chaboche (1) and Chaboche (2) descibe the tansient effect easonably wheeas they do not act well on othe pe-stains. Fo all the thee models, the initial yield stess of the second loading stage can be pedicted accuately, because the expeimental backstess and isotopic hadening wee adopted to detemine the paametes of the isotopic kinematic hadening model. As shown in Fig. 9, neithe the initial yield stess of the second loading stage no the tansient effect can be descibed easonably by the isotopic hadening model. It can also be seen that the eo of the isotopic hadening model becomes lage with the incease of the pe-stain. Howeve, fo each cuve, afte some defomation the esults based on the isotopic hadening model convege to those based on Chaboche (3) gadually. This is because the monotonic tension cuve along the olling diection based on Chaboche (3) is adopted as the efeence and no pemanent softening exists in Chaboche model (shown in Appendix). If the simple tensile stess stain cuve based on anothe Chaboche model (Chaboche (1) o Chaboche (2)) is adopted as the efeence cuve, the simila esult based on the isotopic hadening will be obseved which will not be shown hee. Figs. 10 and 11 shows the expeimental and the pedicted backstess, isotopic hadening, and the efeence cuve based on Chaboche (1), (2) and (3) with Hill48 (1) and Hill48 (2) espectively. The esults ae obtained with the same method as above (as those with Yld2000-2d yield function). As shown in Fig. 10, fo the case with Hill48 (1) chaacteized with -values, all the thee models can descibe the isotopic hadening easonably. Chaboche (2) descibes the efeence cuve bette than Chaboche (1) and Chaboche (3). At the initial stage of defomation, Chaboche (3) undeestimates the backstess and the efeence cuve. Afte some defomation, the eo of the efeence cuve based on Chaboche (1) with Hill48 becomes lage with the incease of stain. As shown in Figs. 11 and 7, the pedicted esults based on the thee Chaboche models with Hill48 (2) chaacteized with nomalized stesses ae simila to those with Yld2000-2d, espectively which will no longe be discussed. The uniaxial tensile cuves in 45 and the tansvese diections simulated based on Chaboche (1), (2) and (3) with Hill48 yield function ae shown in Figs. 12 and 13. Fo the case with Hill48 (1) with -value stesses, as shown in Fig. 12, all the thee models with Hill48 yield function cannot descibe the cuves in 45 and the tansvese diections easonably. Rathe, as shown in Fig. 13. fo the case with Hill48 (2) chaacteized with nomalized stesses, all the thee models descibe the expeimental cuves easonably as well as those with Yld2000-2d yield function. Figs. 14 and 15 shows the esults in the second loading stage of the two-stage loading of goup (1) (the fist and the second loading ae along the olling and tansvese diections) based on Chaboche (1), (2) and (3) with Hill48 (1) and Hill48 (2). By compaing Figs. 9 and 15, the pefomances of the thee Chaboche model and isotopic hadening model with Hill48 (2) (chaacteized with nomalized stesses) ae same as those with Yld2000-2d yield function. The esults based on Chaboche (3) with Hill48 (2) ae as good as those based Chaboche (3) with Yld2000-2d. As fo the case with Hill48 (1) (chaacteized with -values), as shown in Fig. 14, none of thee Chaboche models can descibe the expeimental cuves of the second loading stage easonably though the initial yield stess can be pedicted well. As seen fom the above esults, Chaboche (3) with Yld2000-2d and Hill48 (2) (Chaacteized with nomalized stess) have the highest accuacy in descibing the expeimental esults unde two-stage loadings. In this study, the paametes of the isotopic kinematic hadening models wee detemined with the expeimental data unde the two-stage loading whee the fist and second loading stages ae along the olling and the tansvese diections, espectively. In ode to futhe evaluate the accuacy of the established hadening model, the simulated and expeimental esults unde

10 3702 H. Wang et al. / Intenational Jounal of Solids and Stuctues 49 (2012) Fig. 8. Uniaxial tensile stess stain cuves simulated based on Chaboche type model with Yld2000-2d yield function. (a) Cuves along the 45 diection. (b) Cuves along the tansvese diection. Fig. 9. Compaison of the expeimental and simulated stess stain cuves based on Chaboche type model with Yld2000-2d yield function (fo the second loading of the twostage tests whee the fist loading is along the olling diection and the second loading is along the tansvese diection). (a) Pe-stain of 0.013, (b) Pe-stain of 0.02, (c) Pestain of 0.028, (d) Pe-stain of 0.047, (e) Pe-stain of 0.077, (f) Pe-stain of

11 H. Wang et al. / Intenational Jounal of Solids and Stuctues 49 (2012) descibe the entie expeimental esults easonably. Fo the isotopic hadening model, neithe the initial yield stess no the tansient effect of the second loading stage can be descibed easonably wheeas the pedicted esults coincide with the expeimental esults gadually with the incease of the defomation due to no pemanent softening Pefomance of the yield functions on the esulting hadening model Fig. 10. Isotopic hadening, backstess and efeence cuve based on Chaboche type model with Hill48 (1) (chaacteized with -values). Fig. 11. Isotopic hadening, backstess and efeence cuve based on Chaboche type model with Hill48 (2) (chaacteized with nomalized stesses). othe two-stage loadings ae compaed with each othe. Figs. 16 and 17 show the compaisons between the expeimental and simulated esults based on Chaboche (1), (2) and (3) with Yld2000-2d and Hill48 (2) (chaacteized with nomalized stess). In Fig. 16(a), (b) and Fig. 17(a), (b) the fist and the second loading stages ae along the olling and the 45 diections, espectively, while in Fig. 16(c), (d) and Figs. 17(c), (d) the fist and the second loading stages ae along the tansvese and the olling diections, espectively. As shown in Figs. 16 and 17, the pefomances of the thee Chaboche models based on Hill48 (2) ae the same as those based on Yld2000-2d. Chaboche (3) with Hill48 (2) and Yld2000-2d Fo the hadening behavio unde in-plane uniaxial cyclic tension compession test, the pedicted esults do not depend on the choice of the yield function since the loading is always along the same axis (fowad o evese loading). Howeve, fo the two-stage loading, the loading paths ae along two diffeent diections, the anisotopy also plays an impotant ole on the pedicted esults as well as the kinematic hadening. In this study, the objective is to chaacteize the stess stain cuves and the stess anisotopy is dominant. Theefoe, when Hill48 is chaacteized with nomalized stess the pedicted esults ae vey close to those with Yld2000-2d function and Chaboche (3) model with both of them descibe the expeimental esults easonably. Rathe, when Hill48 is chaacteized with -values, the esults can not be descibed easonably. The above esults show that a pope chaacteization method is as impotant as the yield function itself. Theefoe, it is also impotant to choose a pope chaacteization method of the yield function when establishing the combined isotopic kinematic hadening model fo two-stage loading besides a pope kinematic hadening law. Fig. 18 shows the plastic wok contous of Yld2000-2d, Hill48 (1) and Hill48 (2) at diffeent equivalent plastic stain. As shown in Fig. 18, the plastic wok contous of Hill48 (1) and Hill48 (2) with the two chaacteization methods ae emakably diffeent. The diffeence of the pedicted stess in the tansvese diection will lead to the diffeences of the backstess and the isotopic hadening solved fom Eqs. (12) and (13). The stesses in the tansvese diection with Hill48 (2) always coincide with those with Yld2000-2d yield function which lead to the almost same esults. In this study, the isotopic hadening and the backstess wee obtained by solving Eqs. (12) and (13), fom which it can be concluded that the end loading point of the fist loading stage (A1) and the initial yield point of the second loading stage (A2) should be on the esulting subsequent yield suface as shown in Fig. 19, whee the equivalent plastic stain is OY, OH1 and OH2 denote the esulting centes of the subsequent yield sufaces of Yld2000-2d, Hill48 (1) and Hill48 (2), espectively, which demonstates the diffeence of the backstesses based on Hill48 (1) and Hill48 (2). Fig. 12. Uniaxial tensile stess-plastic stain cuves simulated based on Chaboche type model with Hill48 (1). (a) Cuves along the 45 diection. (b) Cuves along the tansvese diection.

12 3704 H. Wang et al. / Intenational Jounal of Solids and Stuctues 49 (2012) Fig. 13. Uniaxial tensile stess-plastic stain cuves simulated based on Chaboche type model with Hill48 (2). (a) Cuves along the 45 diection. (b) Cuves along the tansvese diection. Fig. 14. Compaison of the expeimental and simulated stess stain cuves based on Chaboche type model with Hill48 (1) (fo the second loading of the two-stage tests whee fist loading is along the olling diection and the second loading is along the tansvese diection). (a) Pe-stain of 0.013, (b) Pe-stain of 0.02, (c) Pe-stain of 0.028, (d) Pe-stain of 0.047, (e) Pe-stain of 0.077, (f) Pe-stain of

13 H. Wang et al. / Intenational Jounal of Solids and Stuctues 49 (2012) Fig. 15. Compaison of the expeimental and simulated stess stain cuves based on Chaboche type model with Hill48 (2) (fo the second loading of the two-stage tests whee fist loading is along the olling diection and the second loading is along the tansvese diection). (a) Pe-stain of 0.013, (b) Pe-stain of 0.02, (c) Pe-stain of 0.028, (d) Pe-stain of 0.047, (e) Pe-stain of 0.077, (f) Pe-stain of Fig. 20 shows the esulting isotopic hadening and backstess vs. the equivalent plastic stain cuves based on Chaboche (3) with Yld2000-2d, Hill48 (1) and Hill48 (2), fom which the diffeence between Hill48 (1) and Hill48 (2) can be seen clealy. As shown in Fig. 20, the esulting cuves based with Yld2000-2d and Hill48 (2) almost coincide with each othe. Though the esulting isotopic hadening and the backstess based on Yld2000-2d, Hill48 (1) and Hill48 (2) ae diffeent, they both satisfy Eqs. (12) and (13). Theefoe, both esulting initial yield stess of the second loading stage based on them coincide with the expeimental ones, as shown in Figs. 9, 14 and 15. The established combined isotopic kinematic hadening model Chaboche (3) with Yld2000-2d and Hill48 (2) can descibe the Bauschinge effect and the tansient effect unde two-stage loading easonably, but it can not descibe pemanent softening effect. In ode to descibe the hadening behavio with softening effect, a modification such as adding softening paametes is needed. 7. Thee point bending tests of the pe-stained specimen 7.1. Expeimental and numeical simulation In ode to futhe evaluate and veify the established hadening model consideing moe geneal loading path changes, the thee point bending test of the pe-stained specimen was pefomed. As shown in Fig. 21, in the fist step, a ectangle sheet with the length of 190 mm was stetched along the length diection up to the stain of 0.1. Then in the second step, a specimen with the length of 110 mm was cut off fom the pe-stained sheet with which the thee point bending test was pefomed.

14 3706 H. Wang et al. / Intenational Jounal of Solids and Stuctues 49 (2012) Fig. 16. Compaisons of the expeimental and the simulated stess stain cuves based on Chaboche type model with Yld2000-2d yield function. (a) Cuves along the 45 diection with pe-stain of along the olling diection. (b) Cuves along the 45 diection with pe-stain of along the olling diection. (c) Cuves along the olling diection with pe-stain of along the tansvese diection. (d) Cuves along the olling diection with pe-stain of along the tansvese diection. Fig. 17. Compaisons of the expeimental and the simulated stess stain cuves based on Chaboche type model with Hill48 (2). (a) Cuves along the 45 diection with pestain of along the olling diection. (b) Cuves along the 45 diection with pe-stain of along the olling diection. (c) Cuves along the olling diection with pestain of along the tansvese diection. (d) Cuves along the olling diection with pe-stain of along the tansvese diection.

15 H. Wang et al. / Intenational Jounal of Solids and Stuctues 49 (2012) Fig. 18. Plastic wok contous of Yld2000-2d, Hill48 (1) and Hill48 (2) at diffeent equivalent plastic stain (0.002, 0.019, and 0.098). Fig. 21. Schematic diagam of the thee point bending test (mm). (a) Fist step: Stetch. (b) Second step: Thee point bending. Table 10 The anisotopic elastic popeties. E 0 (GPa) E 45 (GPa) E 90 (GPa) m 0 m Fig. 19. Subsequent yield sufaces of Hill48 (1), Hill48 (2) and Yld2000-2d yield functions at the equivalent plastic stain of diection. R3D4 discete igid element was adopted fo the punch and die and the fiction coefficient l = 0.2 was adopted. Fo the thee point-bending test, elasticity (and its anisotopy) is as impotant as plasticity (Chung et al., 2011). Theefoe, othotopic elasticity was used unde the planes stess condition: E x E xm yx 30 1 xx 1 m xym yx 1 m xym yx 0 e xx B yy A ¼ 6 E ym xy E y 4 1 m xym yx 1 m xym yx 0 7B C 5@ e yy A ð33þ xy 0 0 G xy 2e xy whee m yx E x = m xy E y, G xy ¼ vxy E 45 Ex 1 Ey In Eq. (33), x, y, z ae the mateially embedded pincipal anisotopic axes: x fo the olling, y and z fo tansvese and thickness diections, espectively. The elastic modulus along the olling, 45 off and tansvese diections E 0, E 45 and E 90 ae shown in Table 10. Two models wee used in the simulations: Fig. 20. Isotopic hadening, backstess and efeence cuve vs. the equivalent plastic stain cuves. The width of the ectangle sheet befoe the fist step is 40 mm. Two specimens with the length along the olling and the tansvese diections espectively wee used. The diametes of the punch and die of thee point bending ae 20 mm and the span of the die is 40 mm as shown in Fig. 21(b). The displacement of the punch in thee point bending is 4.5 mm. The thee point bending test of no pe-stained specimen with the same dimensions as those of the pe-stained specimen was also pefomed. In the simulation of thee point bending, S4R shell element was adopted fo the blank with 5 integation points though thickness Isotopic kinematic hadening model, Chaboche (3), with Yld2000-2d yield function. Isotopic hadening model with Yld2000-2d yield function Results and discussions Fom the simulated esults of thee point bending, it can be seen that the mateial points at the edges of the specimen (with o without pe-stain) ae unde uniaxial loading while those at the cente of the specimen ae nealy unde plane stain loading. Duing bending pocess of the pe-stained specimens, the mateial points unde compessive loading (i.e. the mateial points above the neutal laye of the specimen) will expeience evese loading since they undego tensile loading in the fist loading stage. Fig. 22 shows the compaisons between the expeimental and the simulated spingback pofiles of the thee point bending