Unloading behaviors of the rare-earth magnesium alloy ZE10 sheet

Wiqin Tng,Jong Yon L,Humio Wng,d,?,Dirk Stglih,Dyong Li,d,?,Yinghong Png,Pidong Wu

aState Key Laboratory of Mechanical Systems and Vibration,Shanghai Jiao Tong University,800 Dongchuan Road,Shanghai 200240,PR China b Department of Materials Science and Engineering,Korea University,Seoul 02841,Republic of Korea

cInstitute of Materials Research,Materials Mechanics/ACE-Centre,Helmholtz-Zentrum Geesthacht,Max-Planck-Str.1,21502 Geesthacht,Germany

d Materials Genome Initiative Center,Shanghai Jiao Tong University,200240,China

e Department of Mechanical Engineering,McMaster University,Hamilton,Ontario L8S 4L7,Canada

Abstract Due to their low symmetry in crystal structure,low elastic modulus(～45GPa)and low yielding stress,magnesium(Mg)alloys exhibit strong inelastic behaviors during unloading.As more and more Mg alloys are developed,their unloading behaviors were less investigated,especially for rare-earth(RE)Mg alloys.In the current work,the unloading behaviors of the RE Mg alloy ZE10 sheet is carefully studied by both mechanical tests and crystal plasticity modeling.In terms of the stress–strain curves,the inelastic strain,the chord modulus,and the active deformation mechanisms,the substantial anisotropy and the loading path dependency of the unloading behaviors of ZE10 sheets are characterized.The inelastic strains are generally larger under compressive Loading–UnLoading(L–UL)than under tensile L–UL,along the transverse direction(TD)than along the rolling direction(RD)under tensile L–UL,and along RD than along TD under compressive L–UL.The basal slip,twinning and de-twinning are found to be responsible for the unloading behaviors of ZE10 sheets.? 2020 Chongqing University.Publishing services provided by Elsevier B.V.on behalf of KeAi Communications Co.Ltd.This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/)Peer review under responsibility of Chongqing University

Keywords:Inelasticity;Magnesium alloy;Rare-earth;Crystal plasticity;Twinning;Detwinning.

1.Introduction

As the lightest metal for structural components,magnesium(Mg)alloys have drawn much attention in the automotive,aerospace and electronics industries.The inelastic behavior,referred to the non-linear stress–strain relation during unloading,is the characteristic and crucial behavior for the precise design of Mg structural components[1–3].The damping coefficients,the apparent elastic constants,the springback,and the fatigue property etc.,are dramatically affected by the inelastic behaviors[4-6].This inelastic behavior is usually represented in terms of the chord modulusEchordthat is obviously less than the Young’s modulusE,and the significant Loading–UnLoading(L–UL)hysteresis loops(see Fig.1).

Such inelastic behavior has been experimentally investigated for Mg alloys by using in-situ neutron diffraction[3,7],metallographic observation[2],Electron BackScattered Diffraction(EBSD)[8]and a rare blend of advanced in-situ techniques with different spatiotemporal resolutions[9].Nagarajan et al.[10]and Ang et al.[11]investigated the L–UL inelastic behavior of Mg alloys in terms of solid solution softening/hardening of slip planes and their influence on tensile twinning.De-twinning was found to be partially responsible for the significant inelasticity of Mg alloys.Due to their hexagonal close-packed(HCP)crystallographic structure,Mg alloys have multiple deformation mechanisms available to accommodate plastic deformation,such as basal slip,non-basal slips,tensile twinning,and de-twinning.Therefore,besides tensile twinning,dislocation slip is also a potential deformation mechanism causing inelastic behavior.Lee et al.[7,12]suggested detwinning for unloading from compression,but dislocation slip for unloading from the tension of extruded Mg-8.5wt%Al alloy and rolled AZ31 plate.As demonstrated by Fallahi et al.[13],the inelastic strain of cast ZM21 increases to a maximum and levels off thereafter due to the saturation of twins.Through the crystal plasticity finite element simulation,Hama and his coworkers[14–16]simulated the L–UL behavior of AZ31 alloy sheet under tension.The sheet with a random texture exhibits a larger amount of inelasticity than that with a more pronounced texture.The authors demonstrated that this difference was due to the activation of the basal slip systems.Elastic viscoplastic selfconsistent(EVPSC)polycrystal plasticity model[17],implementing the Twinning and De-Twinning(TDT)scheme(denoted as EVPSC-TDT model[18,19]),was used to describe the inelastic behavior.The very distinct hysteresis loops between tensile and compressive L–UL of AZ31B extruded rod were quantified and ascribed mainly to basal slip and detwinning[20,21].However,there are still missing aspects to be investigated,such as the inelastic behavior of RE Mg alloys,and the associated anisotropy and loading path dependency.

Fig.1.Definition of the characteristic values during an unloading–loading cycle.

As mentioned above,most of the L–UL behaviors investigated are non-RE Mg alloys[2,3,7,12–21].The texture of these wrought alloys usually exhibits strong basal texture,which restrains their formability at room temperature.Mg alloys containing RE elements were developed to weaken or randomize the processed texture and consequently improve the ductility at room temperature[22].The RE-Mg alloy ZE10(Mg+1%Zn+0.3%Ce based mischmetal),studied in the present work,exhibits a significantly weakened texture with the(0002)peaks are orientated～45°away from the sheet normal direction(ND)towards the transverse direction(TD)as a spread double peak[23,24].Yi et al.[25]evaluated that the conventional AZ31 sheet has highr-value which caused by strong basal-type texture,while the ZE10 sheet shows lowerrvalues as a result of its weaker texture.Thus,the ZE10 sheets can be successfully deep-drawn at ambient temperature due to their superior formability.It is expected that the RE-Mg alloy has unique unloading behaviors compared to the conventional Mg alloys.The quantitative characterization,the anisotropy and tension-compression asymmetry of the inelastic behavior during L–UL of RE-Mg alloy are worthy to be systematically investigated,which is the goal of current work.

In present work,the in-plane L–UL behaviors of ZE10 sheets under tension and compression along rolling direction(RD),transverse direction(TD)and 45° with respect to RD are investigated to reveal the loading path dependency in terms of both experiments and simulations.The inelastic behavior is characterized in terms of the stress–strain curves,the inelastic strain,the chord moduli,and the activity of various deformation mechanisms.It is found that the loading path dependency on the inelastic behavior of the ZE10 sheet results from the activation of different deformation mechanisms,especially the different combinations of the mechanisms of basal slip,twinning,and de-twinning.

2.Material and mechanical tests

An Mg alloy ZE10(Mg+1%Zn+0.3%Ce based mischmetal)rolled sheet(O-temper)with a thickness of 2mm was prepared.The sheet has a fully recrystallized microstructure with an average grain size of 15μm(Fig.2a).The grains show a preferential alignment with a longer axis in RD instead of equiaxed grains,which indicates a delay of recrystallization during rolling.The texture was examined by using X-ray measurements of six incomplete pole figures in a diffractometer.Compared with conventional rolled magnesium alloys,such as AZ31,a significantly weakened basal texture was obtained with thec-axis deviating from normal direction(ND)towards the transverse direction(TD)(see pole figures in Fig.2).As a consequence,ZE10 alloy sheet showed improved ductility at room temperature[24,26].Such a texture preferring tensile twinning along TD under both tension and compression will cause weaker tension-compression asymmetry than that along RD.This will be discussed in detail in Sections 4.1 and 4.2.

Monotonic tensile tests(along RD and TD)were conducted on a Zwick Z050 test machine at room temperature,with a flat dog-bone specimen and a constant nominal strain rate of 1.0×10?4/s.In-plane compressive tests(along RD and TD)were carried out using the cubic specimens(10mm×10mm×10mm)glued with five layers of the ZE10 sheets to prevent buckling.A servo-hydraulic 1000kN machine was used,and a compressive force was applied in the plane of the sheets along RD and TD with a constant nominal strain rate of 1.0×10?4/s.

The tensile L–UL tests were conducted by using the same settings and equipment as the monotonic tensile tests.The traverse speed of 0.5mm/min was set for both loading and unloading,leading to a nominal strain rate of 1.0×10?4/s for loading and unloading.A minimum of three samples was tested in order to ensure the consistency of the tests.The experimental data were collected approximately every 1s.The control program was set to reverse the displacement direction at a fixed displacement(engineering strains of 2%,4%,6%,8%,10%,12% and 14%).When the load dropped to near zero,the direction of displacement was again reversed to the original tension direction.

Fig.2.(a)Microstructure and(b)initial texture of the rolled ZE10 alloy sheet.

3.EVPSC-TDT model

The EVPSC-TDT model is briefly described in this section.Details about the model can be found elsewhere[18,19].The EVPSC-TDT model[17]treats each grain as an ellipsoidal inclusion embedded in a homogeneous effective medium(HEM)which is an aggregate of all grains.Interactions between grains and the HEM are described through the Eshelby inclusion formalism[27].The affine self-consistent scheme was found to give the best overall performance[28]and was employed in the present study.As part of the strain rate,elastic strain rateof a crystal is related to the stressσijthrough the elastic stiffness tensorLijkl,

whereandare slip/twinning direction and plane normal,respectively.In case of a slip,the shear rateis calculated in terms of the power law,

whereτα,,andmare the resolved shear stress,the critical resolved shear stress(CRSS),the reference shear rate,and the strain rate sensitivity,respectively.

For twinning and de-twinning,the shear rateof twinning systemαis where the resolved shear stressταis calculated according to the stress of the matrix for operation MR/MP and the stress of the twin for operation TR/TP respectively.is the critical resolved shear stress of twinning or detwinning systemα.

The associated twin volume fraction is determined in terms of the shear rates and characteristic shear of the twin.Two additional parameters,A1andA2,are used to terminate twinning if the twin volume fraction reaches the statistic threshold volume fraction ofVth=min(1.0,A1+A2Veff/Vacc),whereVeffandVaccare the weighted volume fraction of the twinned region and the volume fraction of twin-terminated grains,respectively.The evolution of CRSS is given by

whereΓis the accumulated shear strain in the grain,hαβis the latent hardening coefficient,andτ?αis the threshold stress and is defined by Voce hardening law

whereτ0,τ0+τ1,h0,andh1are initial,back-extrapolated CRSSs,initial and asymptotic hardening rates,respectively.

4.Results

4.1.Monotonic results

In the numerical simulation,the deformation mechanisms of basalslip,prismatic〈a〉slip,pyramidal〈c+a〉slip and extension twinning were employed to accommodate the plastic deformation[29,30].The reference slip/twinning rate,γ˙0=0.001 s?1,the rate sensitivity,m=0.1,and the roomtemperature elastic constants,C11=58,C12=25,C13=20.8,C33=61.2,C55=16.6(unit of GPa)[31]of the Mg single crystal were used:The hardening parameters,obtained by fitting both the uniaxial compression and tension along RD(RD-C and RD-T)(Fig.3a),are listed in Table 1.Additionally,the experimental stress–strain curves under compression and tension along TD(TD-C and TD-T)were simulated and well reproduced by the model(Fig.3b).Apparent tensioncompression asymmetry can be observed for both loadings along RD(Fig.3a)and TD(Fig.3b).Due to the distinct initial texture(Fig.2),the asymmetry along TD is not as significant as along RD,which is different from a conventional Mg alloy,such as rolled AZ31 sheet[28,29,32].Relative activity,which is the ratio of the shear strain rate of the respective deformation mode to the total shear strain rate,is used to evaluate the relative contribution of each deformation mode to the shear strain.A notable twinning activity was observed under both TD-C and TD-T,which explains the weakened tension-compression asymmetry(Fig.4).

Fig.3.Experimental and simulated stress–strain curves of ZE10 sheets under monotonic tension and compression along(a)RD and(b)TD.

Table 1Parameters associated with the EVPSC-TDT model for ZE10 alloy.

The developed textures at the strain of 0.15 under tension and compression were predicted and presented in Fig.5.The developed texture of RD-T is similar to the initial texture with strengthened intensity as the grain orientations are not favorable for twinning(Fig.5a).However,the developed textures under RD-C and TD-C are very different and exhibit two peaks in{0001}pole figures,which indicate that the grains have been oriented nearly 90° by twinning(Fig.5b and d).The texture of TD-T falls in between the previous two cases(Fig.5c).Some of the grains were developed withc-axes approximately perpendicular to the sheet plane due to slip(similar to RD-T),while other grains withc-axes nearly parallel to the sheet plane due to twinning(similar to RD-C or TD-C,though the amount is not as significant).This is different than the AZ31 sheet with strong basal texture,where twinning is seldom active under TD-T[28,29].

4.2.Tensile L–ULs

Fig.6 presents the L–UL stress–strain curves of the ZE10 sheet under tension along RD and TD.The EVPSC-TDT model successfully reproduced the experiments,which is the natural outcome of the model without using any additional tuning parameters.Unloading–reloading loops are apparent in L–ULs along both RD and TD.Close observation reveals that the inelastic strain for the L–UL along TD is greater than that along RD.In order to better characterize the unloading behavior of the ZE10 sheet,the maximum loop widthδmax,and the chord modulusEchordwere computed.

As illustrated in Fig.1,the inelastic strain,characterized as maximum L–UL loop widthδmax,is the summation of unloading widthδUand the loading widthδL,i.e.,δmax=δU+δL.The chord modulusEchordis the slope of the line connecting the cross and endpoints of the unloading curve,which is apparently smaller than Young’s modulusE.The variation of L–UL inelastic strain and chord modulus with respect to the loop number(or strain)are presented in Fig.7.With increasing the strain upon unloading,the inelastic strain decreases gradually,while the chord moduli increase gradually,under both L–ULs along RD and TD.The L–UL along TD exhibits larger inelastic strain than that along RD.Consistently,the chord moduli of L–ULs along RD are higher than those along TD,which can be attributed to the more twinning activity under TD-T than RD-T(Fig.4a and c).The predictions by the EVPSC-TDT model agree well with the experimental results for both larger inelastic strain and chord modulus,except for the first few loops.The reason is possibly because that the average stress is used to represent the stress inside a grain.In fact,defects,dislocation segments,and therefore stress are spatially distributed over a grain.This distribution will lead to two facts,especially when the driving force(average stress)is released:the first one is that though the average stress is released,the local stress is somewhere high and contributes to the inelastic deformation;and the second one is that the stress fluctuation will lead to the rearrangement of the dislocation segments,which does not change the dislocation density(i.e.,no hardening),but contributes to the inelastic deformation[33,34].

Fig.4.Relative activities of deformation mechanisms under(a)RD-T,(b)RD-C,(c)TD-T and(d)TD-C.

Fig.5.Predicted textures in terms of(0001)and(100)pole figures at the strain of 0.15 under(a)RD-T,(b)RD-C,(c)TD-T,and(d)TD-C.

Fig.6.Simulated stress–strain responses under monotonic and L–UL tension along(a)RD,(b)TD together with the experimentally measured results.

Fig.7.Experimental and predicted(a)inelastic strain and(b)chord modulus as a function of hysteresis loop under tensile L–UL along RD and TD.

5.Discussion

5.1.Anisotropy of inelastic behavior

As suggested,de-twinning will be responsible for the inelastic behavior if twinning is activated[17].Simulations that excluding detwinning were performed for both tensile and compressive L–ULs along RD,TD and 45° with respect to RD.The L–UL behaviors with and without de-twinning are compared in Fig.8.The capability of de-twinning during unloading depends strongly on the twinning in the previous loading since de-twinning is the reverse motion of twinning.Therefore,de-twinning has a negligible effect on the inelastic strain during tensile L–ULs along RD,TD and 45° since none of them are favorable for twinning(Fig.8a–c),which is confirmed by the small twinning activity in Fig.9a–c.P1,P2,and P3correspond to the unloading stress for Loop_1,Loop_2,and Loop_3.Therefore,the intervals of P1(P2or P3)to 0 and 0 to P2(or P3)correspond to unloading and loading,respectively.Close observation reveals that de-twinning on the inelastic behavior along TD-T or 45°-T is more visible than along RD-T(Fig.8a–c).

In contrast,large loops are obtained under compressive L–ULs along RD,TD and 45°,while the loops shrink significantly once de-twinning is excluded(Fig.8d–f).Apparently,de-twinning plays an important role in compressive L–UL behavior as a consequence of the strong twinning activity during previous loading(Fig.9d–f).The results indicate that the effect of de-twinning on the L–UL behavior is more significant under compression than tension,and more significant along the RD-C(Fig.8d)than the TD-C and 45°-C(Fig.8d–f).In addition to twinning,basal slip is active for all the L–ULs investigated.Relatively,basal slip is even more active during unloading in L–UL tension tests(Fig.9a–c),which reveals that basal slip is responsible more for the inelastic behavior of the ZE10 sheet during tensile L–UL.Apparently,the inelastic strain due to basal slip is less significant than that due to detwinning.

Fig.8.Simulated stress–strain responses with and without de-twinning under(a)RD-T,(b)TD-T,(c)45°-T,(d)RD-C,(e)TD-C and(f)45°-C.

Fig.10 compares the inelastic strains of both tensile and compressive L–ULs along RD,TD and 45°.The inelastic strain increases with straining during compressive L–UL,which is contrary to tensile L–ULs where the inelastic strain decreases with straining.When de-twinning is excluded,the inelastic strain of RD-C decreases more than TD-C and 45°-C because RD-C favorites more twinning than TD-C and 45°-C.However,the inelastic strains for tension show minimal variation whether the de-twinning is considering or not.Apparently,de-twinning does play an important role in the inelastic L–UL behaviors only if twinning plays a noticeable role in plastic deformation.Fig.10 also exhibits the strong anisotropy of the inelastic behavior as the difference among the inelastic strains along RD,TD and 45 is large under both tensile and compressive L–ULs.

5.2.Empirical analysis of L–UL

As demonstrated above,crystal plasticity modeling can connect the inelastic behavior to the underlying mechanisms of the ZE10 sheet.However,crystal plasticity modeling tools may not be feasible for all readers.Alternatively,an empirical analysis is provided in this Section to quantify the inelastic behaviors of the ZE10 sheet.The first unloading–loading loop(Loop_1)along TD is selected to demonstrate the empirical analysis.The experimentalΔσ?Δεcurves under unloading and reloading are presented in Fig.11,whereΔσandΔεare the absolute stress and strain variations with respect to the values at the beginning of unloading or reloading.TheΔσ?Δεrelations show a remarkable resemblance between unloading and loading along TD(Fig.11).As suggested by Chen et al.[35],a single-parameter relation is adopted to correlate the stress variationΔσto the strain variationΔεduring unloading and reloading:

whereE=45GPa is the Young’s modulus,and the modulus reduction rateAis the only parameter to be adjusted,which is 258 for the first loop(Fig.11).

According to Eq.(8),the inelastic strain can be calculated as

whereσuis the stress at the crossing point of unloading and loading curves,εpis the true plastic strain,εis the total strain,andEis the Young’s modulus(see Fig.1).According to Eq.(9),the maximum value of inelastic strain for loop_1 is 0.002.

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The same analysis was applied to other unloading–reloading loops along both RD and TD,where a good match was obtained by the analytical results from Eq.(8)(Fig.12).Similar to the first cycle along TD(Fig.11),the unloading and loading stress–strain curves of other cycles show great resemblance along both RD and TD.For the sake of brevity and readability,the unloading stress–strain curves were omitted from Fig.12.Fig.13 presents the corresponding values of the modulus reduction rateA,which are approximately linear to the stress upon unloading(σu).Therefore,the empirical Eq.(6)can well address the stress–strain relation during both unloading and reloading.

Fig.9.Relative activities of deformation mechanisms under tensile L–UL along(a)RD,(b)TD,and(c)45°,and compressive L–UL along(d)RD,(e)TD,and(f)45°.

Fig.10.Inelastic strain as a function of loop number under(a)tensile and(b)compressive L–ULs along RD,TD and 45° WRT RD.

Fig.11.Stress–strain responses in terms ofΔσ～Δεduring unloading and loading along TD for Loop_1.

Fig.12.FittedΔσ～Δεresponses during loading for 6 hysteresis loops along(a)RD and(b)TD.

Fig.13.Fitted modulus reduction rate A as a function of unloading stress.

6.Conclusions

In this work,the inelastic behaviors of rolled ZE10 sheet under tensile and compressive L–ULs along RD,TD and 45°were systematically investigated.The effect of various deformation mechanisms on the evolution of the hysteresis loop and chord modulus are carefully analyzed.The main conclusions are drawn as follows:

(1)The ZE10 sheet exhibits a weak basal-type texture with thec-axis tilted towards the TD,therefore exhibits different behavior under tension and compression along RD and TD.During tension along RD,most initial grain orientations are not favorable for twinning.During tension along TD,some grains withc-axes nearly parallel to the loading direction are undergoing twinning,though the amount of such grains is not very significant.This is different from the AZ31 sheet with a strong basal texture,where twinning is not active under tension.During in-plane compression,the reorientation of 86° induced by twinning leads to dramatic texture change.

(2)The inelastic behavior of the ZE10 sheet is highly dependent on the loading paths,showing obvious anisotropy and tension-compression asymmetry of inelasticity.Compressive L–UL loops are generally larger than tensile ones.The tensile L–UL loops along TD and 45° are larger than those along RD,while the compressive L–UL loops along RD are larger than those along TD and 45°.

(3)With the aid of the crystal plasticity modeling,the relatively small inelastic behavior of the ZE10 sheet under tensile L–UL was mainly ascribed to basal slip.Larger inelastic strains(or smaller Chord moduli)along TD and 45° than RD is due to their difference in twinning activities,where a small portion of grains underwent twinning for the first two loading paths.

(4)The relatively large inelastic strains(or small Chord moduli)under compressive L–ULs were ascribed to the favorable mechanism of twinning and de-twinning,which is confirmed by excluding de-twinning in the crystal plasticity modeling.Contrary to tensile L–ULs,de-twinning contributes more to compressive L–ULs along RD than along TD and 45°.

Declaration of Competing Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The authors from SJTU would like to acknowledge the support of the National Natural Science Foundation of China(Nos.51775337,51675331 and 51975365),Major Projects of the Ministry of Education(No.311017),and the Program of Introducing Talents of Discipline to Universities(Grant No.B06012).HW was sponsored by the Shanghai Pujiang Program(18PJ1405000)and the University of Sydney-Shanghai Jiao Tong University Partnership Collaboration Awards.Peidong Wu was supported by the Natural Sciences and Engineering Research Council of Canada(Nos.RGPIN-2016-06464).This paper is partly supported by the Materials Genome Initiative Center,Shanghai Jiao Tong University.The University of Michigan and Shanghai Jiao Tong University(UM-SJTU)joint research project(AE604401)is acknowledged.