Anisotropy of wrought magnesium alloys:A focused overview

Bodong Shi ,Chong Yng ,Yn Peng ,Fuheng Zhng ,Fusheng Pn

a National Engineering Research Center for Equipment and Technology of Cold Rolled Strip,School of Mechanical Engineering,Yanshan University,Hebei-West-Str.438,Qinhuangdao 066004,PR China

b State Key Laboratory of Metastable Materials Science and Technology,Yanshan University,Qinhuangdao 066004,PR China

cNational Engineering Research Center for Magnesium Alloys (CCMg),Chongqing University,Chongqing 400044,PR China

Abstract In this paper,the relationship between anisotropic mechanical properties and the corresponding microstructure evolution of wrought magnesium alloys is critically reviewed.Experimental observations of the strong anisotropy (including the strength differential effect) induced by texture and twinning are discussed under different loading conditions (i.e.,monotonic,cyclic and multiaxial loading).An accurate constitutive model is essential to describe the mechanical responses and to predict the forming performance considering engineering applications.Therefore,macroscale constitutive modeling of the anisotropy of magnesium alloys with directional distortional hardening are comprehensively reviewed with different approaches.To clarify the origin of the anisotropic behavior,physics-based mesoscale modeling of the anisotropy is also compared in detail.

Keywords: Wrought Mg alloys;Mechanical anisotropy;Deformation mechanism;Constitutive modeling.

1.Introduction

Magnesium alloys have an enormous advantage over aluminium alloys and steels owing to their low density and high specifi strength,which is approximately 30% less than aluminium and one-quarter that of steel [1,2].The application of lightweight magnesium alloys is used to improve energy efficien y in the automotive,aircraft and aerospace industries[3-7].However,low ductility and poor formability at room temperature limit the widespread application of Mg products.Five independent easy slip systems are required for uniform plasticity deformation of a polycrystalline aggregate[8].Basal＜a＞slip and prismatic＜a＞slip offer a total of four slip systems,failing to satisfy the Taylor criterion.Pyramidal＜c+a＞slip can accommodate deformation along thec-axis,offering more independent slip systems [9,10].However,low ductility and high hardening during plastic deformation are ascribed to＜c+a＞dislocation,which undergoes a transition to basaldissociated immobile dislocation structures,eliminatingc-axis strain on easy-glide pyramidal II planes [11].In contrast,the＜c+a＞dislocations can be activated to accommodate plasticity,exhibiting plasticity in pure Mg samples with small sizes [12].At present,understanding the relationship between microstructure evolution and corresponding mechanical properties of Mg alloys are still the main challenge [13].

Many efforts have been made by material scientists in improving the ductility and formability of Mg alloys [14,15].Due to low symmetry of the HCP (hexagonal close-packed)crystal structure,a strong initial basal texture can be generated due to the activation of twinning during rolling and extruding,where thec-axis is predominantly aligned perpendicular to the rolling direction (RD) or the extruding direction (ED) [16],which results in poor formability.The crystal orientation and mechanical behavior of Mg alloy are determined by different processing techniques [17].Therefore,the method of texture optimization has been proposed to improve the mechanical properties of Mg alloys [18,19].Alloying [20-22] and deformation processing with deformation path [23,24] or stress state [25-27] changing have been used to alter the texture.The addition of rare earth elements [28-33] and calcium[34,35] can weaken the initial texture and enhance ductility.Specifi dilute solute additions can increase the＜c+a＞cross slip and multiplication rates to enhance the cross-slip mechanism [36].The shear component was introduced to weaken the texture strength by severe plastic deformation,such as in equal channel angular extrusion (ECAP) [37,38]and asymmetric rolling (ASR) [39].A high ductility can be achieved when the grain size is reduced to the order of one micron due to the strong influenc of grain boundary glide [40].

A series new forming processes have been proposed to improve mechanical properties of wrought Mg alloys,such as pulsed electric-assisted forming [41,42],accumulative extrusion [43,44] and rolling with combined side-rolling and torsion[45].However,the texture can still make a large influenc on the performance of products during the sheet forming process.Mechanical anisotropy can be produced from the texture during plastic deformation,which results in the earing of sheet material during deep drawing [46,47].In addition,mechanical anisotropy can be induced by strain path changes,such as the Bauschinger effect,and rapid transient strain hardening and softening [48,49].Anisotropy is ascribed to the heterogeneous dislocation substructure induced by previous strain.In addition to dislocation slip,twinning and detwinning are also important deformation mechanisms in Mg alloys.Mechanical twinning is a polar mechanism,where a simple shear occurs only in one direction rather than both forward and backward[50-54].Detwinning is an inverse process without nucleation stress and depends on the strain path.For example,detwinning can occur during tension following compression loading along the RD [55-58].Therefore,the anisotropic characteristic of plasticity deformation is strongly strain-path dependent for Mg alloys.

To understand the underlying mechanisms of anisotropy,mesoscale crystal plasticity models have been developed.The crystal plasticity models are divided into two groups:phenomenological models and physics-based models [59].The critical resolved shear stress (CRSS) of each slip system is considered as a state variable within the phenomenological models.In contrast,the microstructural state variable of the physics-based models is the dislocation density.Focusing on Mg alloys,twinning and detwinning play an important role in plastic deformation.Various twinning models have been built to predict the mechanical response and microstructure evolution,such as the predominant twin reorientation (PTR) and twinning and de-twinning (TDT) models.Mesoscale modeling is a straightforward approach to understand the origin of anisotropy corresponding to microstructures.However,computational efficien y limits application of mesoscale modeling in practical forming processes.In contrast,macroscale modeling is widely applied in industry for processing technologies,e.g.,to predict sheet forming.A series of yield functions were proposed to describe the initial anisotropy and tension-compression asymmetry of Mg alloys,such as CB2004 [60] and CPB2006 [61].Compared to traditional isotropic and kinematic hardening models,directional distortional hardening models were developed to capture the effect of strain path changes.Recently,the homogeneous anisotropic hardening (HAH) model [62,63] was extended to describe the anisotropy that occurred during strain path changes [64,65].

The details of mechanical behavior and constitutive modeling are still open to explore,especially under complex loading paths.In the following sections,anisotropic mechanical behavior is reviewed under different loading conditions,including monotonic/cyclic loading,strain path changes and multiaxial loading.The corresponding microstructure evolution is also discussed.Regarding constitutive modeling,the macroscale ones focus on anisotropic yield function with distortional hardening.The crystal plasticity models in mesoscale with twinning effects are briefl reviewed.

2.Experimental investigations of mechanical anisotropy under different loading modes

The mechanical behavior of Mg alloy exhibits strong loading path and orientation dependence due to the low symmetry of HCP.A typical example is tension-compression asymmetry induced by twinning.It is important to clarify the relationship between the macroscale mechanical behavior and the microstructure evolution for improving the plastic forming ability.In this section,the mechanical responses with respect to the microstructure is reviewed under different loading conditions (i.e.,monotonic loading,uniaxial cyclic loading,twostep loading and multiaxial loading),and the experimental determination of yield surface is also explored.

Fig.1.Stress-strain curves for rolled AZ31 Mg alloy (a) uniaxial tension and compression along the RD,45° and TD [55],(b) uniaxial tension along 0°,30°,45°,60°,and 90° from the RD [68].

Fig.2.Basal (0002) pole figure of rolled ZE10 and AZ31 Mg alloys [72].

2.1.Uniaxial loading

2.1.1.Monotonic tension and compression

For wrought Mg alloys,strong mechanical anisotropy and tension-compression asymmetry (also named as the strength differential effect,SD effect)are observed during uniaxial tension and compression along different directions [55,66-70].The tensile and compressive stress-strain curves along the RD,45°and TD are shown in Fig.1(a) for AZ31 rolling sheet[55].The stress-strain curves vary from a concave-up aspect to a concave-down appearance under uniaxial tension along 0°,30°,45°,60°,and 90° from the ND in the ND-RD plane,as shown in Fig.1(b) [68].The basal texture with basal poles spreading toward RD is generated during rolling of AZ31 alloy [71] as shown in Fig.2.While the basal poles spread toward transverse direction (TD) for ZK10 rolling sheet [72] as shown in Fig.2.The difference of texture leads to different planar mechanical anisotropy between AZ31 and ZE10 as shown in Fig.3 [72].An AZ31 extrusion bar exhibits typical fibr texture.The tensile and compressive stress-strain curves of the AZ31 extrusion bar along different directions from TD to ED are shown in Fig.4 [70].Stronger anisotropy of the extrusion bar is observed than that of the rolling sheet.The mechanical anisotropy depends on the initial texture,which is affected by previous forming process and alloying system.

The additions of rare earth lead to the tilt of basal poles away from the ND as shown in Fig.5 [73-75].ECAP-processed alloys show basal poles tilting towards the processing direction with angles of 35°-45° [76].A split texture can be produced by multi-pass caliber-rolling [77,78].The qual channel angular rolling and continuous bending process with subsequent annealing (ECAR-CB-A) process was performed to form a rare RD-split bimodal texture [79].Although the texture distribution and plastic anisotropy of Mg alloys are different for each forming process and alloying system,the nature of anisotropy is the same,i.e.,the different CRSS of deformation mechanisms under specifi loading direction.

More explicitly,the main deformation modes are basal＜a＞slip,prismatic＜a＞slip,pyramidal＜c+a＞slip and tension twinning during monotonic loading at RT,as shown in Fig.6.The CRSS values of different deformation modes have a wide disparity and are obtained from experimental measurement and modeling.In polycrystals,the effective CRSS value is the sum of the single-crystal CRSS and the internal reaction stress arising from grain boundaries [80].The corresponding CRSS values of basal＜a＞slip,prismatic＜a＞slip,pyramidal＜c+a＞slip and twinning are summarized,as shown in Table 1.Different CRSS values are reported in different studies due to various approaches.However,it is commonly believed that CRSSbas＜CRSSpri＜CRSSpyr.The CRSS depends on the solutes [81,82].The activation of each deformation mode follows the Schmid law.To be more precise,the Schmid factor (SF)mfor uniaxial deformation is define as:

Table 1 CRSS values of different deformation modes for Mg alloy at room temperature

Table 2 The corresponding shear rate and twin volume fraction evolution during twinning [178].

Fig.3.Flow curves of uniaxial tension for (a) ZE10 and (b) AZ31 at room temperature [72].

Fig.4.Stress-strain curves of an extrusion AZ31 bars along different loading direction from TD to ED,(a) tensile and (b) compression [70].

Fig.5.The tilt of basal pole from the ND of ZM21,ZK10,ZE10,ZEK100,ZEK410 and ZW41 sheets toward the rolling direction [73].

Fig.6.The schematic diagram of available deformation mechanisms at room temperature in magnesium alloys.

whereλis the angle between the stress axis and the slip/twinning direction,andφis the angle between the stress axis and slip/twinning normal plane.In line with the geometry of HCP,the SF can be calculated as a function of the angle between the loading direction and thec-axis [83].Further,the deformation mechanism can be determined according to the angle between the loading direction and thec-axis [84-87].

Fig.7.Activation stress for different deformation modes as a function of the basal pole tilting angles (Φ) with respect to the ND in AZ31 plate or to the ED in AZ31 rod [95].

The ideal basal texture isc-axis parallel to the normal direction (ND) in the sheets or perpendicular to the extruded direction in the rods.However,some grains’c-axis deviate from the ideal direction.The basal poles of AZ31 sheet are tilted towards the rolling direction [93].Koike and Ohyama[94] found that when the basal planes were tilted by more than 16.5° towards the tensile axis,the basal＜a＞slip was the dominant deformation mechanism rather than prismatic＜a＞slip.The relationship between the dominant deformation modes and the tilting angle of thec-axis is shown in Fig.7 [95].It is believed that basal＜a＞slip with low CRSS is activated due to the large angular spread of the basal pole.In contrast,when the angular spread of the basal pole is less than 13°,basal＜a＞slip exhibits a low Schmid factor and prismatic＜a＞slip is activated with lower activation stress.The different dislocation slips were observed along different tension directions of 0°,45° and 90° using TEM by Koike and Ohyama [94] as shown in Fig.8.The basal＜a＞dislocations occur during tension along 0°.For 45° and 90°,the dislocations appear to cross-slip from the basal planes to the prismatic planes.It is confirme that prismatic＜a＞slip is the dominant deformation mechanism for 45° and 90°.The basal slip can result in the basal pole altering toward the compressive direction;a rotation of the lattice around its＜c＞axis can be induced without changing the＜c＞axis position[96].Different from AZ31,however,the basal poles of ZE10 are tilted towards the TD direction,as shown in Fig.2 [72].Consequently,the yield strength and fl w stress along the TD are lower than that along the RD.Strong in-plane anisotropy originates from the initial texture and non-basal cross-slip of dislocation with＜a＞type Burgers vectors [54,97,98].For ECAR-CB-A sheet with RD-split bimodal texture in Fig.9(a),both dislocation slip and tensile twinning are activated during tension along RD,resulting in the gradual diffusion and rotation of tilted basal poles to ND.A new TD-component texture was developed during uniaxial tension as shown in Fig.9(b) [79].Overall,the deformation mechanisms along different directions are determined by the distribution of the initial basal poles and lead to strong in-plane anisotropy.

Fig.8.Bright-fiel images for the tensile directions of (a) 0°,(b) 45° and (c) 90° to RD in rolled AZ31 Mg alloy [94].

Fig.9.Pole figure of (a) ECAR-CB-A AZ31 alloy and (b) at tension degree of 12% [79].

Tension-compression asymmetry (SD effect) is observed due to the pole nature of twinning at RT[99].Extension twinning can be activated under tensile loading parallel to thecaxis or compression loading perpendicular to thec-axis [53].A typical sigmoidal curve appears with tension twinningdominated deformation.The relationship between the tensile stress axis and thec-axis changes from parallel to perpendicular along 0°,30°,45°,60°,and 90° of ND to RD of the initial rolled plate.The dominant deformation modes vary from tension twinning to prismatic slip [69,100].Therefore,strong anisotropy of Mg alloy plate is exhibited during tension along thickness direction.

For Mg alloys,tension twins can lead to uniform deformation;in contrast,contraction twins can decrease the uniform elongation [101,102].However,the mechanical behavior is different between compression perpendicular to thec-axis and tension parallel to thec-axis due to the effect of twin variants[85],as shown in Fig.10.The strain hardening rate drastically increases after 4% straining under compression along the RD;in contrast,the strain hardening rate under tension along the ND increased slowly.The twin morphology is different from different loading modes as shown in Fig.11.The twins are almost parallel to each other during compression along RD,while the twins show the different orientations and intersections between each other.This is ascribed to the activation of strain path-dependent twin variants [103-105].More precisely,{10-12} extension twin exhibits six equivalent twin variants with a specifi shear direction of＜10-11＞to accommodate deformation,i.e.,(10-12)[-1011],(01-12)[0-111],(−1102)[1-101],(−1012)[10-11],(0-112)[01-11]and(1-102)[−1101].The activation of twin variants commonly follows the Schmid law[106].Thus,twin variants with a high Schmid factor will be activated.For compression along the RD,the result for the angle ofθbetween the loading axis and thea-axis in the range 30-60° is identical to that in the range of 0-30° due to the unique angle relationship between the threea-axes of the HCP lattice,as shown in Fig.12(a).One twin variant exhibits the highest SF value.In contrast,the compressive stress axis distributes randomly in the RDTD plane,as shown in Fig.12(b).Consequently,all six twin variants exhibit an equal SF value under tension along the ND.Therefore,the activation of specifi twin variants can result in anisotropic work hardening.However,the SF fails to predict the non-SF twins [107,108].For twin variant selection,a composite Schmid factor (CSF) that uses both the SF and the geometric compatibility factor (m’) was recently proposed to interpret common-boundary twins to describe Mg alloys [91]:

wherem’is the geometry compatibility factor between two twinning systems T1 and T2;mAandmBare the SFs for T1 and T2,respectively;andηis the local shear stress factor.

A rapid hardening is presented in the twinning-dominated plastic deformation.The different hardening mechanisms induced by tensile twinning have been proposed:(1) Hall-Petch effect resulted from grain refinemen induced by twinning boundaries [109];(2) orientation hardening due to crystallographic reorientation induced by twinning [110,111];(3) the Basinski effect,i.e.,a glissile-to-sessile transformation of dislocation in the twinned region [112,113];(4) the thin contraction twins restricting the slip length associated with pyramidal＜c+a＞slip [110];(5) a composite response associated with elasticity of a large volume fraction[114].However,Knezevic et al.found that tensile twins are not very effective in strain hardening induced by Hall-Petch effect due to fast growth[110].The strain hardening is attributed to texture due to orientation changing by tensile twins.The rapid hardening was attributed to dislocation associated mechanism[112].The role of tensile twins in strain hardening is unclear for Mg alloy.The activation of twinning depends on grain size.With grain size decreasing,a clear transition from non-basal to basal-slip dominated fl w takes place under tension at room temperature [115] and a transition from twinning to basal slip takes place under compression [116].The decreasing of grain size increases levels of connectivity between favourably oriented grains,which facilitate slip transfer across grain boundaries.However,the accurate prediction of twin transmission at the grain boundary remains a challenge [117].

2.1.2.Simple shear or torsion

Simple shear and torsion tests are another effective approach to investigate large deformation behaviors under monotonic loading.For AZ31 Mg alloy sheets,anisotropy was detected during in-plane simple shear,i.e.,the lowest yield stress was observed along the RD;in contrast,the highest yield stress was obtained along the TD [118].The deformation mechanisms along the RD,45° and TD are similar during in-plane simple shear [55,118].An almost linear hardening was reported by Khan et al.[118] and Kang et al.[119].However,a concave-down shape at small strain and a concave-up shape at large strain were observed under in-plane shear [55,120].In contrast,linear strain hardening was observed during through-thickness shear [120].The Bauschinger effect is strong during reversal shear [55,121].The anisotropic behavior of ZEK100-O Mg alloy during shear loading is ascribed to different twinning modes [122].The fraction of twinning is affected by shear loading direction,which leads to change of principle stresses directions.The different simple shear test methods were compared in detail by Yin et al.[123].The deformation along the axial direction is homogenous without strain localization during torsion.Thus,the material parameters of plasticity models at high strain can be calibrated by torsion tests.Free-end and fi edend torsion loading are two different boundary conditions.An almost linear hardening was detected during pure torsion for Mg alloy [124-127].

Fig.10.Deformation characteristics along the RD and ND for rolled AZ31 Mg alloy:(a) stress-strain curve and (b) strain hardening rate curve [85].

Fig.11.Optical microstructures with −2% strain along the (a) RD and (b) ND for rolled AZ31 Mg alloy [85].

Experiments and modeling confir that basal slip,prismatic slip and tensile twinning are activated and plastic deformation is dominated by dislocation slip during in-plane simple shear and torsion [119-121].The basal slip and twinning accommodate the plastic strain throughout torsional deformation [128].The dominant mechanism depends on the loading direction.Torsion tests or simple shear tests can be performed under two different loading conditions -with the shear direction either parallel or perpendicular to thec-axis.For through-thickness shear,basal slip controls the deformation,and the activity of twinning decreases [120].During torsion along the RD at large strain the main deformation mode is prismatic＜a＞slip;comparably,basal＜a＞slip dominates plastic strain during torsion along the ND [126].The activation of prismatic slip can lead to reduction of strain hardening rate decreasing at large shear strain [127,128] and change of grains orientation [129].The prismatic plane perpendicular to extruded direction is transformed from{11-20}plane to{10-10} plane [129].In addition to tensile twinning,compressive twins,double twins,even tertiary twins are observed during torsion along ND of AZ31 Mg plate as shown in Fig.13[128].Although twinning occurs during torsion,the texture and strain hardening are not affected by twinning due to low twin volume fraction [127,128,130,131].Two type reorientation of grains occurs during torsion.One is the basal poles rotate toward radial direction.The other is that basal poles rotate toward extruded direction in most grains [131].

The“Swift effect”is a typical mechanical phenomenon during torsion (i.e.,axial displacement under free-end torsion or axial force under fi ed-end torsion)that results from texture development and deformation-induced anisotropy [132,133].For pure torsion with free-end,axial elongation is observed if the torsion direction is parallel to thec-axis,whereas axial contraction appears if the torsion direction is perpendicular to thec-axis in Mg alloys [122,126,134].According to EVPSC modeling,it is believed that axial strain decreases at firs and then increases,with shear strains greater than 1.7 under freeend torsion of Mg alloy extruded bar [135].The activation of twinning with elongation of thec-axis is responsible for axial strain for Mg alloy [126].Twinning can result in increasing axial stress or decreasing axial strain at small strains [135].The Swift effect of Mg alloys is sensitive to the texture evolution and twinning during torsion at RT [135,136].However,the Swift effect of Mg alloys is only weakened and not disappearing without twinning [136].Recently,it is proposed that the origin of Swift effect of Mg alloy is attributed to dislocation slips [128].At elevated temperatures,the axial strain shortening stopped beyond a shear strain of 1.35 at 250 °C[39].The transition of Swift effect is induced by axial stress state,which is ascribed to prismatic slip [137].The detailed relationship between the microstructure (e.g.,texture evolution,twinning,dislocation slips) and the Swift effect is still unclear and further investigation is undergoing for Mg alloys.

Fig.12.SF analysis of a {10-12} twinning system:(a) compression perpendicular to the c-axis;(b) tension parallel to the c-axis [85].

To clarify the deformation mechanism,the shear stress state can be equivalent to the biaxial normal stress state based on the axial transformation in the principal stress space[121,127,138].The tensile stress component results in the activation of the slip system with the highest SF.Extension twinning can be activated by a compressive stress component.Recently,Chen et al.[139] proposed an effective Schmid factor that combined shear stress and normal stress to determine the deformation mechanism in magnesium alloys.The twin mode was found to be sensitive to the sign (or direction) of the shear stress,which changed the effective Schmid factor of the twin mode.The twin volumes and rate of twinning activity are affected by stress states [128].

In summary,the yield strength,strain hardening behavior and deformation mechanism of Mg alloys strongly depends on the initial orientation and the stress axis direction under monotonic tension,compression,simple shear and torsion in Mg alloys.In addition to the anisotropy during plastic deformation,anisotropic fracture behavior due to distribution of texture are also reported in Mg alloy [140].The prediction of anisotropic fracture has been taken account into constitutive modeling [141-143].The strong texture results in resistant to shear localization in AZ31 Mg alloy with high Lankford value.However,a susceptible shear localization is presented in ZEK100 Mg alloy with relatively low Lankford value[144].The anisotropic fracture behavior of ZEK100 Mg alloy depends on the stress triaxiality and Lode parameter [145].It was also reported that both loading conditions and strain history make effect on the fracture behavior [146,147].More precisely,various loading conditions with tension,compression and shear were performed in AZ31 Mg alloy by Jia and Bai [148].The anisotropy leads to variation of fracture strain with orientation changing under a specifi loading condition.The fracture strain is independent on pre-strain under shear state during uniaxial and biaxial loading [146].It is found that contraction twins are active during biaxial tension loading,which results in transgranular fracture with pronounced shear-type fracture.In contrast,it should be noted that tensile twinning activity during uniaxial tension leads to intergranular fracture [149].Therefore,the contribution of twinning and dislocation slip to macro-deformation and texture evolution is still unclear,especially under multiaxial loading.

Fig.13.Microstructure evolution of a rolled AZ31B magnesium alloy plate during free-end torsion along normal direction.(a) SF pole figure for tensile and compressive twinning.(b-e) The microstructure at plastic shear strains of 0.036,0.069,0.131,0.191,and 0.254,respectively [129].

Fig.14.Stress-strain curves of single-cycle loading for rolled AZ31 Mg alloy,(a) TCT and (b) CTC [55].

2.1.3.Uniaxial cyclic loading

Tension-compression-tension (TCT) and compressiontension-compression (CTC) are two common strain paths for cycle loading.In both cases,an inflecte hardening curve and low fl w stress were observed in the tension after initial compression for rolled AZ31 Mg alloy,which indicated that hydrostatic pressure has little influenc on the asymmetry between monotonic tension and compression [55],as shown in Fig.14.The yield strength decreases during reverse tension after compression.While,the yield strength during subsequent compression can be increased after pre-tension [55,150].

The hardening behavior of AZ31 Mg alloy under cyclic loading includes three dominant deformation mechanisms,i.e.,slip,twinning and de-twinning [151].The microstructure evolution of an extruded AZ31 Mg alloy during a cyclic loading were observed using EBSD by Yin et al.[152],as shown in Fig.15.A large number of twins are activated during compression as shown in Fig.15(a).The twins showed by white and red arrows became narrower and smaller after unloading from compression as shown in Fig.15(b).The twins with black and red arrows disappeared during reversal tension as shown in Fig.15(c).However,the twins marked by blue arrow become thinner after unloading from tension as shown in Fig.15(d).The twinning-detwinning-retwinning of Mg single crystal are detected directly during cyclic tensioncompression along[0001]direction as shown in Fig.16[153].The deformation mechanisms depend on loading direction and crystallographic orientation [154].For the TCT loading,the twinning-detwinning-pyramidal slip-retwinning are depicted alongc-axis.The mechanisms are basal slip+twinningdetwinning-retwinning+basal slip when loading direction is deviated from thec-axis by 45°.And the mechanisms are double twinning-detwinning-retwinning-detwinning when loading direction is deviated from thec-axis by 90°.For the CTC loading,the pyramidal slip-twinning-detwinning are depicted alongc-axis.The mechanisms are basal slip-twinningdetwinning-retwinning when loading direction is deviated from thec-axis by 45°.And the mechanisms are prismatic slip-double twinning-detwinning-retwinning when loading direction is deviated from thec-axis by 90°.In cyclic loading,twinning and detwinning appear alternately,resulting in a sigmoidal aspect for Mg alloys [155].Detwinning is a contraction of twinned regions without nucleation [156].

Fig.15.Microstructure evolution of extruded AZ31 Mg alloy during a cyclic loading.(a) compressive strain～0.5%,(b) unloading from compression,(c)reversal tension at tensile strain～0.7%and(d)unloading from tension[134].Black arrows show that the twins remained stable during unloading then disappeared after tensile loading;red arrows mark that the twins became smaller during unloading then disappeared after tensile loading;blue arrows show that the twins kept stable during unloading then become thinner under tensile loading;white arrows mark that the twins become smaller during unloading then become much smaller after tensile loading [152].

Fig.16.The stress-strain curve and microstructure evolution of Mg single crystal corresponding in situ observation [153].

Fig.17.Cyclic testing of AZ31B Mg alloy sheets:(a) tension-compression-tension test cycle,(b) compression-tension-compression test cycle [160].

The multiplication of dislocations in tension can retard twin nucleation at the very early stage of plastic deformation,but it makes little effect on twin growth and detwinning.For CTC loading,the second compressive yield stress increases with increasing reverse tensile stress,which is dominated by slip.Therefore,the compressive yield strength is related to the accumulated dislocation density.Moreover,the deformation mechanisms of TCT and CTC tests along the RD and TD directions were calculated by means of an EVPSC-TDT model[157].For CTC,the dominant deformation mechanisms during compression and tension were twinning and detwinning.Basal slip was always activated throughout deformation,and prismatic slip was activated under uniaxial tension or the tension after compression with detwinning exhausted.

Temperature effect are also investigated for AZ31B under TCT and CTC single-cycle loading [158-160],as shown in Fig.17.The inflecte stress-strain curve disappeared between 125 °C and 150 °C during compressive loading.Twin nucleation was observed from 20 to 200 °C by the acoustic emission(AE)technique[161].Recently,Sim et al.[162]also found that extension twinning can be activated at and below 100 °C and suppressed above 150 °C for Mg single crystals.This phenomenon is related to the pyramidal＜c+a＞slip with typical thermal activation [163,164] and twinning without temperature sensitivity[165].The difference in CRSS values between pyramidal＜c+a＞slip and twinning decreases with increasing temperature.Thus,there is a critical temperature with an equal CRSS value for pyramidal＜c+a＞slip and twinning [166].The transition from twinning to pyramidal＜c+a＞slip takes place when the testing temperature is higher than the critical one for the strain rate and grain size sensitivity.

Due to the effect of springback on the design of structural components [167-171],inelastic behavior has been investigated under cyclic loading-unloading experiments of magnesium and its alloys [55,155,172-175].This nonlinearity exhibits a significan unsymmetrical strain hysteresis loop.The inelastic strain increases up to a plastic strain of approximately 1-2% and slightly decreases at larger strains for cast Mg alloys [172,173].It is found that the inelastic behavior is more pronounced during compression than during tension for extruded AZ31 magnesium alloy due to the occurrence of twinning-detwinning processes.In contrast,the inelastic behavior gradually disappears with dislocation slip-dominated plastic deformation under large compressive strains [174-176].The inelastic behavior of Mg alloys during unloading has been investigated by the crystal plasticity finit element method (CP-FEM) and the EVPSC-TDT model [177,178].The inelastic behavior was found to strongly depend on the initial texture and loading history.For deformation dominated by slip dislocation,the size of the hysteresis loop decreases with increasing strength of the initial texture due to weak activation of basal slip.De-twinning can be activated during unloading for deformation dominated by twinning.In this case,the size of the hysteresis loop increases with increasing strength of the initial texture.The high strain level can reduce the size of the hysteresis loop due to a stable texture and grain stress.It should be mentioned that the inelastic behavior of Mg alloys also exhibits tension-compression asymmetry.Comparably,no apparent detwinning activities were observed during unloading after compression [179,180].

2.1.4.Uniaxial two-step loading

Taking deep drawing as an example,the strain path changes from pure shear to biaxial tension in the material from the flang area to the die cavity [181,182],as shown in Fig.18.More precisely,the mechanical behavior depends on the strain history,especially for anisotropic materials.Therefore,to capture the effect of the strain path during practical sheet forming processes,the uniaxial two-step loading tests are widely employed,namely,subsequent uniaxial loading along different directions after pre-uniaxial loading [183-188],as shown in Fig.19.Reverse loading with coaxial testing is a typical test for strain path changes.For DP590 steel,three main features,including (a) the Bauschinger effect,(b)rapid transient strain hardening (yielding at low stress and a rapid change of the working hardening rate) and (c) longterm or“permanent”softening,can be detected during reverse loading [48],as shown in Fig.20.Transient hardening and permanent softening with the strain hardening rate approaching monotonic loading takes place during non-coaxial loading in low-carbon and DP steels [50,185,189].

Fig.18.Schematic diagram of stress state changes in circular cup drawing.

For tension-tension loading,the mechanical behavior of Mg alloys during strain path changes is similar to that of steels.Transient hardening and permanent softening are also observed in AZ31 [65,190].The yield stress of reloading decreases with the reloading angle changing from 0° to 90°along the pre-loading direction due to the increase of backstress associated with the dislocations produced during the pre-loading [190],as shown in Fig.21.Upon increasing the reloading angle,additional softening is detected along the pre-loading direction during pseudo cross-loading for tensiontension loading [65],as shown in Fig.22.However,tensioncompression asymmetry due to different deformation mechanisms are exhibited in Mg alloys.Stress-strain curves with pronounced sigmoidal shapes were observed for the anglesθ=0 and 30° along the rolling direction;in contrast,less pronounced curves were exhibited at the anglesθ=45,60 and 90° during compression-tension loading [191],as shown in Fig.23.This is ascribed to the decrease of {10-12} detwinning activity when the angle increases from the RD to the TD.The deformation mechanism transforms from detwinning to slip-dominated deformation.In the two-step compression of extruded AM30,the twin growth is restricted during re-compression in the direction parallel to the firs loading direction.However,the formation of new twins and detwinning can co-exist due to multiple sets of texture after prestraining [192].For brass,the yield behavior of reloading relates to the stacking fault energy (SFE) [193].The hardening mechanism is dislocation latent hardening with moderate-tohigh SFE,whereas dislocation-twin interactions describe the hardening mechanism for high-to-moderate SFE.The strain path change was performed to investigate the interactions between {10-12} twinning and slip [194].AfterRDc5-TDc9(pre-compression 5% along RD then recompression 9% along TD)almost all the matrix and the primary twins have been reoriented with＜c＞close to TD in rolled AZ31 Mg alloy.In the 45RDc5-TDc7 (pre-compression 5% along 45° between ND and RD then recompression 7% along TD) sample,a few twins are oriented with＜c＞between RD and ND,and most of the twins are oriented with＜c＞close to TD as shown in Fig.24.The self-hardening of twinning is stronger than the latent hardening for {10-12} by slip modes during strain path change.It is believed that the inhomogeneous dislocation structure generated by pre-strain hinders the dislocation movement,resulting in transient hardening during preloading.The contribution of the anisotropy associated with texture is small due to strong strain gradients produced by the pre-strain [190,195,196].The transient hardening in the earlier reloading stage depends on the amount of pre-strain and on the amplitude of the strain path change;the amplitude is much stronger when the pre-strain and reloading angle along the pre-loading direction are increased.Permanent softening may result from the interactions between the pre-induced dislocation and the subsequent dislocation during second tension.However,it is not clear whether the transient responses result from the heterogeneous dislocation substructures (i.e.,anisotropy on the mesoscopic level) or from the nature of the dislocations themselves and their interactions on the slip system level (i.e.,anisotropy on the microscopic level) [49].

Fig.19.Schematic diagram of two-step loading.

Fig.20.Mechanical responses of DP590 under monotonic and reverse compression-tension test with three characteristic regions:the Bauschinger effect,rapid transient strain hardening and“permanent”softening [48].

Fig.21.Experimental true stress-strain curves of tension-tension tests with different reloading angles from the RD of rolled AZ31 Mg alloy [190].

Fig.22.Stress-strain curves during second loading along (a) 35° and (b) 90° for two-step tension of rolled AZ31 Mg alloy [65].

Fig.23.True stress-logarithmic strain curves of rolled AZ31 Mg alloy obtained in the second loading at different angles in (a) virgin sheet,(b) 6% pre-tension sheet,(c) −3% pre-compression sheet,and (d) −6% pre-compression sheet for rolled AZ31 [191].

Fig.24.The EBSD map of (a) RDc5-TDc9 and (b) 45RDc5-TDc7 in rolled AZ31 Mg alloy [190].

In summary,wrought Mg alloys are inherent strong mechanical anisotropy,a consequence of their hexagonal closedpacked (hcp) lattice.Several reasons contribute to this effect.First,at room temperature,the critical resolved shear stresses (CRSSs) of basal and non-basal slip systems exhibit very different values,spanning several orders of magnitude;second,twinning,a very common deformation mechanism in these materials,exhibits pronounced polarity effect,i.e.its activation depends on the relative orientation between thecaxis and the applied stress;finall,both hot and cold deformation processing textures are often quite sharp and the way that how the activation of different slip systems is influence by the local texture and grain boundary network is not clear.Consequently,these factors lead to a dependence of the dominant deformation mechanisms on the texture,grain size,testing mode (tension or compression) and the testing direction,resulting in large differences in yield strength and strain-hardening responses.The pre-strain loading and corresponding texture control also provide a way to improve the formability of wrought Mg alloys [197,198].

2.2.Biaxial loading

In practical forming processes (e.g.,drawing) and service conditions,sheet metal and alloy materials are under multiaxial stress.The plastic fl w of metals is determined by mechanical responses along all dimensions (i.e.,σxx,σyy,σzz,σxy,σyzandσzx) instead of uniaxial loading.The mechanical behavior and microstructure evolution are different from that obtained under uniaxial stress conditions.For instance,equiaxial loading facilitates the martensitic transformation fromγ-martensite toα’-martensite for metastable austenitic 304 stainless steel [199].Therefore,the investigation on the relationship between mechanical behavior and corresponding microstructure evolution under multiaxial loading is crucial for sheet forming process.

2.2.1.Yield surface and equivalent plastic work contour(EPWC)

To probe the complete evolution of the mechanical behavior,the six-dimensional evolution of yield surfaces must be clarified However,due to experimental limitations,the mechanical responses are usually studied in two dimensions.In particular,the stress state distribution is presented in theσxxandσyyspace,as shown in Fig.25.Thus,biaxial tension,axial loading combined with external pressure,and combined tension-torsion loading tests are usually subjected as multiaxial loading tests [200].The hydraulic bulge test [201],Marciniak punch test [202],thin-walled tubes with combined axial loading and internal/external pressure [203-205] and cruciform specimens [206,207] were used to study the mechanical behavior of the biaxial stress state.Biaxial cruciform testing is usually employed owing to the ability to defin stress ratios and strain path changes without interruption under in-plane loading modes [208-213].

Both the yield surface evolution [214-220] and equivalent plastic work contours [221-224] are typically employed to study the multiaxial mechanical behavior in theσxx-σyyorσxx-σxystress space.The subsequent yield surface evolution strongly depends on pre-loading path.Monotonic,intermittent,proportional and angle loading paths were employed to study the subsequent yield surfaces [216-219,225,226].The effect of loading path change on subsequent yield surface evolution for anisotropic AZ31 Mg alloy was investigated under non-proportional loading [220,227],as shown in Fig.26.The distortion and rotation of the subsequent yield surfaces are observed with an offset strain of 500με.In contrast,the opposite evolution of yield surfaces was also detected,i.e.,a high curvature in the compression direction and flattenin in the tension direction due to tensioncompression asymmetry resulted from the occurrence of twinning,as shown in Fig.26(a).It is believed there is a ‘‘nose”in the loading direction and a flattene shape in the reverse loading direction for yield surface evolution.The distorted yield surface in the firs quadrant is attributed to the competition between dislocation hardening and orientation hardening.The anisotropic expansion and rotation of yield surface is observed for AZ31 sheet with strong basal texture,as shown in Fig.26(b).Due to inhibition of twinning nucleation,the pretension leads to an expansion of yield surface in the compressive direction instead of contraction presented for FCC and BCC metals.However,the effect of pre-tension on reversal compressive yield strength is weak.

The definitio of yielding is crucial in the experimental identificatio of yield surfaces.Three common definition are often employed:(a) the stress at a proportional limit,(b) the stress at a strain offset of a given amount,or (c) the stress obtained by a backward extrapolation from the stress-strain curve to the elastic line [216],as shown in Fig.27.Determination of the proportional limit depends on the precision of the instrument and is affected by the judgment of the observer.Recently,Chen et al.[228,229] found that the proportional limit is 0 MPa for various alloys,i.e.,no linear elastic region was detected.The criticism of the offset definitio of yielding is that it is arbitrary and its physical meaning is unclear.Similarly,no physical nature can be found for the backward extrapolation definition which is a gross indication of yielding.The offset strain is usually employed to determine the yield stress.Strong distortion of yield surface was detected with 10μεoffset strain for an annealed extruded AZ31 alloy under proportional and non-proportional loading paths [230,231].Therefore,the shape of the yield surface depends on the definitio of offset strain,which has been selected from 5μεto 2000με.A ‘‘nose”in the loading direction and a flattene shape in the reverse loading direction are usually observed with relatively lower offset strains,from 5 to 1000με[216-218,232].With a definitio of yielding as 2000μεoffset strain,“isotropic”yield surfaces can be observed even for“anisotropic”materials.The effect of the offset strain has been reported,e.g.,in Khan’s works [218,219].Several specimens have been employed to probe subsequent yield surfaces,with different offset strains define between 5 and 1000με.In Kabirian and Khan’s work [233],smaller offset strains were measured by complete unloading.However,a physically sound definitio of yield strength remains missing.

Different anisotropic evolutions could be probed with different definition of yield stress for anisotropic materials under biaxial loading.The yield stress is usually define based on the von-Mises equivalent formulation during proportional loading.Strictly speaking,this method is not reasonable for anisotropic materials because the anisotropic yield criterion is unknown prior to the experiment [234].In line with the classical elasticity and plasticity theory [235],the effect of the loading path on the mechanical behavior during elastic deformation can be neglected.It is believed that the loading paths will not make influenc on the initial yield surface.Therefore,loading paths with combined tension and torsion in the elastic domain were designed to probe the yield point.The different definition of yield stress lead to different shapes of the initial yield surface.Therefore,the definitio of the yield strength under biaxial loading is still under investigation.

Fig.25.Stress state distribution of a yield surface in σxx and σyy space,with 0 ＜a ＜b. The complete yield surface can be probed by combined tension/compression and torsion tests with internal/external pressure for a tube specimen.

Fig.26.The subsequent yield surface evolution for AZ31 Mg alloy,(a) in σxx-σxy space [220] and (b) in σxx-σyy space [227].

Fig.27.Schematic diagram of the yield point definition (a) proportional limit,(b) offset strain,(c) backward extrapolation.

Fig.28.The evolution of equal plastic work/strain contours.(a) equal Equal plastic work contours for rolled AZ31 Mg alloy [224] and (b) equal plastic strain contours for extruded AZ31 Mg alloy [232].

A theoretical framework of plastic work contours was proposed by Hill and Hutchinson [236] to capture the differential hardenings with accumulated plastic deformation due to texture and predict the evolution of anisotropic behavior [237].For Mg alloys,Andar et al.[224] performed proportional biaxial cruciform tensile tests to capture the anisotropy of AZ31 Mg alloy sheets by means of EPWC evolution,as shown in Fig.28(a).Moreover,the equal plastic strain contours (EPSCs) were examined by using a combined tension-torsion proportional tests in Kabirian and Khan [233],as shown in Fig.28(b).The remarkable tension-compression symmetry of EPWC is measured in theσxx-σxystress space.A larger difference between successive EPWCs of Mg and steel was observed[205]due to the strong basal texture.Hama and Takuda[238] predicted the contours of equal plastic work for an AZ31 magnesium alloy sheet in the firs quadrant of the stress space by using the crystal plasticity finit element method.The deformation during equibiaxial tension was found to be dominated by basal slip,which eventually leads to the differential work-hardening behavior.Steglich et al.found that basal and pyramidal slips are the main deformation mechanisms in equi-biaxial tension from VPSC modeling[239].The differential work hardening behavior during biaxial loading was studied by Fu et al.[240].It is found that the pyramidal＜c+a＞slip is activated at the initial stage during biaxial loading.With plastic strain increasing,the {10-11} contraction twinning is activated which leads to low plastic strain during biaxial loading.The activation of non-basal＜a＞slip is low,which retards the cross slip and dynamic recovery,resulting in the strongly differential working hardening during biaxial loading.

Theoretically,the initial plastic work contour with Wp=0(εp=0) is identical to the initial yield surface without plastic deformation.Therefore,it is believed that for sufficientl smallεp,the plastic work contour is equivalent to the initial yield surface[224,241].According to the associated fl w rule,the plastic potential is define identical to the yield function,i.e.,the direction of the plastic strain rate (or plastic velocity gradient for finit strain plasticity) is normal to the yield surfaces [205,236].However,the difference between yield surface and equal plastic work contour should be clarified First,a yield point on a specifi yield surface is determined by a unique loading path.In contrast,the loading paths are not unique for the points on a specifi plastic work contour[220,242].Second,the yield surfaces must be convex,according to Drucker [243],whereas the plastic work contours could be non-convex for anisotropic materials due to directional strain hardening rate changes during plastic deformation[220,221,241].

In addition,the twinning can occur in the second,third and fourth quadrants of yield surface for HCP metals.There are two distinct features of yield surface due to twinning[244].A high curvature of plastic work contours is formed under the balanced biaxial compression in the third quadrant.A negative lateral strain ratio results in the shift in plane-strain compression conditions from the third to the second and fourth quadrants of the yield surface [244].In the second quadrant of yield surface,a turning point corresponding to pure shear or equibiaxial tension and compression is observed with the transition of deformation mechanisms from basal slip and tensile twinning to non-basal slip [245].Due to limitations of biaxial specimen,the whole shape of yield surface in all four quadrants is still unclear.

The application of cruciform testing is limited due to the drawback of small plastic strain at the centre area [246,247].The fracture of a biaxial specimen usually takes place in the arms before the centre section reaches the forming limit under normal stress [248].The ideal cruciform specimen is maximized plastic deformation and uniform strain at the centre of the gauge section.Therefore,different cruciform geometries have been designed to achieve the plastic strain,e.g.,the ISO standard geometry [249],the modifie ISO standard with thinned geometry [246],arc cross-section thinned geometry with and without slits [250,251].In ISO standard-type geometry with a straight cross-section,the coupling between the forces in the arms is weak,and the gauge stress can be computed as the loading force divided by the area.However,high stress concentrates at the slit ends [250].Although arc crosssection can achieve large plastic strains,the gauge stress is difficul to determine and a non-linear coupled biaxial stress is obtained at the centre of the gauge area.A standard established by researchers was recently reviewed;specimens with a straight cross-section shape are expected to show a stressstrain relationship and demonstrate yield models;specimens with arc cross-section shapes are used for forming limit determinations [247].The cruciform specimen geometry is expected to be optimized using finit element simulation.

2.2.2.Microstructure evolution during biaxial loading

For cold-rolled AZ31 Mg alloy,biaxial tension along RD and TD can promote dislocation slip and suppress twinning with basal texture.The impediment of dislocation motion by the dislocation accumulation produced during pre-strain results in strengthening of the yield stress during reloading[252].The mechanical response strongly depends on the loading path change.Dislocations are not free to run backwards;instead they act as barriers to forward dislocation motion.The majority of the strengthening is due to dislocation-based hardening in the grains from the pre-loading [252].Biaxial loading tests of AZ80 Mg alloy with combined axial tension or compression with internal pressure were performed with two different initial textures[253].It is found that the evolution of texture is strongly dependent on stress state.The Erichsen test was employed to investigate the effect of equal biaxial stress state on texture evolution by Xia et al.[254,255].The Schmid factor is only related to the angle betweenc-axis and ND.It is found that the tensile twins lead to difference between uniaxial loading and biaxial loading [254].More precisely,the tensile twins make a positive influenc on the activity of＜a＞basal during biaxial loading,which accommodate strain and release stress concentration.The strong basal texture and biaxial tensile stress limit the compatibility between slip systems and tensile twins [255].

Twin nucleation is more pronounced for uniaxial deformation,and the twinned volume fraction is higher for equibiaxial compression along RD and TD [256,257].The equibiaxial compression was preformed along RD and TD of AZ31 Mg plate by Xia et al.[258].The texture evolution during biaxial compression is different from that during uniaxial compression as shown in Fig.29.The distribution of basal poles after uniaxial compression is 30° away from RD.While the basal poles distribute more randomly in the RD-TD plane.The occurrence of tensile twinning variants inhibits the activation of prismatic slip during uniaxial compression,in contrast,the prismatic slip is promoted during biaxial compression.

Fig.29.The evolution of pole figure during uniaxial compression (a-d) and biaxial compression (e-h) for rolled AZ31 Mg alloy [258].

The biaxial tension of AZ31 plate was performed with different proportional loading along ND and TD by Cheng et al.[259].It is found that the behavior of {10-12} twinning strongly depends on stress ratioσND:σTD.The texture evolution is presented as shown in Fig.30.The six twin variants are activated during tension along ND.With the stress ratio decreasing,the activation of twin variants becomes (0-112)[01-11] and (01-12)[0-111].However,a stress-ration-mediated detwinning occurs under biaxial loading with a higher ratio ofσTD.Two loading orientation with combined pressure and shear stress state was performed to study the anisotropic behavior of a rolled AZ31B alloy under multiaxial loading by Zhao et al.[138],as shown in Fig.31.However,the tensile twinning is activated under loading orientation (a) with thec-axes primarily undergoing contraction and no twinning is observed under loading orientation (b) with compressive stress along the TD.The spread of texture plays a more important role in understanding the mechanical behavior under multiaxial stress state [138,260].The microstructure evolution under complex multiaxial stress state cannot be predicted by simple CRSS analysis in uniaxial loading,which is affected by loading sequences under combined loading condition [261].Therefore,the microstructure evolution under nonproportional loading will be focused to optimize the mechanical properties.In-situdiffraction [262],high-resolution digital image correlation[209,263]and three-dimensional dislocation dynamics (DD) simulations [261] provide new approaches to bridge the multiaxial micromechanics and microstructure evolution of materials.

3.Macroscale constitutive modeling of anisotropy

Macroscale constitutive models are usually built using a phenomenological approach.Due to the computational effi ciency in metal forming,this type of model is widely applied in engineering and is combined with finit element methods.In line with classical plasticity theory,a constitutive model is comprised of a yield function,fl w rule and hardening law.Traditionally,yield functions are define based on the Tresca[264] and von Mises yield criterion [265].In recent years,the Hill type [266-268] and Cazacu-Barlat series [269,270]have been widely employed.Two common hardening laws are applied,including isotropic and kinematic hardening,which capture the expansion and translation of yield surfaces,respectively.The hypothesis of associated [271,272] and nonassociated [273-276] fl w rules is employed to capture the evolution of plastic strain.However,the initial anisotropy,SD effects,distortion and rotation of yield surfaces are observed experimentally,especially for Mg alloys.This strong anisotropic behavior cannot be captured by only traditional isotropic and kinematic hardening.Consequently,an accurate macroscale anisotropic constitutive model is required to characterize the anisotropic mechanical behavior of Mg alloys in forming processes.

3.1.Anisotropic yield functions

Fig.30.Pole figure of biaxial tension with different proportional loading for rolled AZ31 Mg alloy. σND: σTD=(a) 1:0,(b) 4:1,(c) 3:1,(d) 2:1,(e) 1:1,(f)1:2 and (g) 1:4 [259].

Several anisotropic yield functions were available to capture the initial anisotropy without an SD effect[266,267,269,270,277-279].The accuracy of anisotropic functions can be improved by introducing more anisotropic coefficient [280].However,strong SD effects (tensioncompression asymmetry) were detected for Mg alloys due to different deformation mechanisms under tension and compression as mentioned in previous sections.Thus,pressure-sensitive and -insensitive anisotropic yield functions were developed to capture the SD effects for metals [281].More precisely,the firs invariant of a stress tensor was introduced into the von Mises yield function to capture the effect of hydrostatic pressure on the SD effects at early time[282,283].Later,the pressure-sensitive yield criterion was extended to capture the SD effect [284-287] and anisotropic hardening [288,289].For HCP metals,the SD effects are related to the direction-sensitive twinning,which are essentially hydrostatic pressure-insensitive [55,60,290].Therefore,the pressure-insensitive criterion is assumed in anisotropic yield functions for HCP metals.

Fig.31.Two loading orientations of rolled AZ31B Mg alloy in pressure-shear plate impact experiments corresponding to Microstructure.Orientation (a)compression along the ND and shear in plane,orientation (b) compression along the TD and shear along the ND [138].

A simple effective approach to generalize isotropic yield functions into anisotropic ones is the linear transformation of the stress tensor [277].According to the assumption of pressure insensitivity,a modifie Hill’s [268] criterion was proposed by adding linear stress terms without shear components[291].

whereA,B,F,G,andHare material constants.The elliptical yield loci are predicted by the yield function.However,the highly asymmetry of tension-compression cannot be captured.Comparably,generalized invariants of stress tensor approach are developed to extend isotropic yield function to anisotropic one[292,293].In particular,an asymmetric yield function was proposed in a more concise form based on Drucker’s isotropic yield criterion [243] that involves both second and third invariants of the stress deviator [60]:

whereare the generalization to orthotropy ofJ2andJ3:

where all the coefficient ofai(i=1…,6)andbj(j=1…,11)are the anisotropic material constants.The Drucker yield function is based on the three invariants of the stress tensor,which can be used for both plane stress and spatial loading.Compared to polynomial functions,the convexity of Drucker yield function is fulfille by setting a coefficien in a specifie range [294].A macroscopic orthotropic yield criterion capturing both the anisotropy and the tension-compression asymmetry insensitive to the hydrostatic pressure was developed with the eigenvalues of the stress deviator by Cazacu et al.[61].Here,a new prototype model of the yield condition was introduced:

where Si(i=1,2,3) are the eigenvalues of the stress deviator.The transformed stress tensorΣwas define by using a linear transformation:

Where C is a constant 4th-order tensor capturing the anisotropy.Similarly,the orthotropic criterion is define as:

whereΣi(i=1,2,3) are the eigenvalues ofΣ.This type of yield function has been employed to describe the mechanical response of AZ31B under proportional loading at room temperature [295].

The linear transformation of a stress tensor into a new space was further extended to describe anisotropic behavior[296].More precisely,a tensorwas define by a linear transformation ofσ,=M:σ.M is a 4th-order mapping tensor and is represented by a 6 × 6 matrix due to material symmetry,which contains the anisotropic coefficient in the material frame.Later,an extended linear transformation was incorporated into the CPB06 criterion as [297]:

wherekand k’ are material parameters that characterize strength differential effects,andais the degree of positive homogeneity.The convexity and incompressibility of yield function can be maintained by linear transformations.The extended CPB06 yield criterion with non-associated fl w rule is applied to capture the evolving anisotropy of ZEK100-O Mg alloy under proportional loading [298].And,the high anisotropy can be captured using multiple transformations[299].The CPB06 criterion is improved to capture anisotropic and asymmetric behavior of AZ31B and ZK61M [300].More precisely,the anisotropic yielding can be captured by updating the anisotropic coefficient of Hill’s and CB’s series yield functions.However,the orthotropic yield criterion is a function of principle stresses,resulting in complicated computation for spatial loading.

3.2.Prototype distortional hardening models for BCC/FCC metals

Focusing on distortional hardening,at least three existing model series are widely applied:the Teodosiu model,the Levkovitch &Svendsen model and the Feigenbaum &Dafalias model.A typical Teodosiu model starts from the Hill-type yield function with direction-dependent“isotropic hardening”terms [301]:

with the equivalent stress define as:

where H is a constant fourth-order tensor,and the fourth-order tensor S captures the distortional hardening effect.The classical Armstrong-Frederick evolution equations are employed for isotropic and kinematic hardening:

wherecisoanddefin the saturation ratio and limit of isotropic hardening.Different from the traditional Armstrong-Frederick kinematic hardening,within the Teodosiu model,the saturation limitdepends on the fourth-order distortional hardening tensor S and captures the cross-hardening effect by:

with S is decoupled further as:

Here,the dynamic hardeningSDcontributes to the overall increase of hardening,as it is associated with the dislocation density along the active slip systems.In contrast,SLcontributes to the latent hardening due to partial designation of the preformed dislocation structures.TheSDand SLterms are governed by the following evolution equations:

whereCSDandCSLcharacterize the saturation rate ofSDand SL,respectively,andSsatdenotes the saturation limit ofSD.It should be emphasized that isotropic,kinematic and distortional hardening are fully coupled in the original Teodosiu model.Later,it is found that the original Teodosiu model is over-determined from a mathematical point of view [302].Consequently,this deficien y is eliminated in a straightforward manner by definin an extended evolution equation for the whole distortional hardening tensor S:

where

In contrast toh,the functiongdepends on the polarity tensorPcapturing the texture effect,which is introduced by the traditional Armstrong-Frederick type rule:

Here,cPis a material parameter that characterizes the saturation ratio of polarity,andPevolves in the direction of the current plastic fl w.However,Pdoes not change its direction spontaneously due to the memory termP˙λin Eq.(22).Thus,the loading path effect is realized by the following projection:

By this projection,the functiongis define as:

Here,nPis a material parameter.As discussed in Feigenbaum and Dafalias [271],the function ofgwas designed to characterize the work hardening stagnation and resumption for steel.

Similar to Teodosiu’s work,the Hill-type yield functions are also employed in the original Levkovitch &Svendsen model and in the Feigenbaum &Dafalias model.In particular,a positive homogeneity of degree two yield function was redefine in Feigenbaum’s work [271]:

where H is a fourth-order stress-like internal variable that captures the distortion of the yield surface by:

where H0is the initial value of H,and the fourth-tensor A is introduced by a similar evanescent memory-type hardening rule:

wherec1andc2are material parameters.It should be mentioned that the Feigenbaum &Dafalias model is thermodynamically consistent because each of the dissipation terms is assumed to be a non-negative quadratic form,as discussed in Feigenbaum and Dafalias [271].Different from the equal contribution of isotropic hardening and anisotropic hardening into dissipation in the original Feigenbaum model [271],a weighting parameter is introduced to balance the contribution of isotropic and anisotropic hardening into dissipation in Khan et al.[219].This weighting parameter captures the loading mode for Mg alloys.

Slightly different from the Feigenbaum model,a positively homogeneous one-degree yield function is employed in the Levkovitch &Svendsen framework [303]:

where standard Armstrong-Frederick type evolution equations are adapted again.Isotropic hardening and kinematic hardening are define by their respective evolution equations,similar to the Teodosiu model (see Eqs.(14) and (15)).The effect of currently active hardening and latent dislocation hardening are described by the fourth-order tensor H,with the evolution equation define as:

where HD=(N·HN)N⊗Nand HL=H −HDdefine dynamic and latent hardening.Consequently,isotropic,kinematic and distortional hardenings are uncoupled in the Levkovitch &Svendsen model.

Hill-type yield function can capture initial yielding of BCC/FCC metals.However,not all fundamental principles are fulfilled e.g.,thermodynamic consistency.Moreover,considering the strong SD effect observed in HCP metals,which cannot be captured by a traditional Hill-type yield function,three existing distortional hardening model series were extended using the CB2004-type yield function.In this way,both the initial anisotropy and SD effect can be captured.

3.3.Distortional hardening models for HCP metals

3.3.1.A prototype model with distortional hardening for Mg alloys

The CB2004 yield function was reformulated in tensor form by Mekonen et al.[69]:

The distortional hardening is governed by the fourth-order tensors H1and H2.The components of Hiare define by an exponential function of equivalent plastic strain:

Fig.32.Evolution of normalized yield surfaces predicted by the extended models for AZ31 Mg alloy under orthogonal loading path changes,(a) the modifie Teodosiu model,(b) the modifie Levkovitch &Svendsen model and (c) the modifie Feigenbaum &Dafalias model [304].

whereAj,BjandCjwithj={1…6} are the model parameters.It should be noted that the effect of strain path changes cannot be captured by this constitutive model becauseis employed.In contrast,the strain path significantl changes in practical forming processes as shown in Fig.18,and the loading path effect must be taken into account for anisotropic materials.

3.3.2.Extended Teodosiu,Svendsen and Feigenbaum models for Mg alloys

The Teodosiu model,the Levkovitch &Svendsen model and the Feigenbaum &Dafalias model were extended for magnesium alloys using the CB2004-type yield function of the original Hill-type in Shi and Mosler [304].An evolving fourth-order tensor governed by a differential equation with the classical Armstrong-Frederick rule was employed to capture the mechanical behavior in magnesium alloys.Evolution of subsequent yield surfaces under uniaxial tension test followed by reverse loading was predicted by the extended models for the magnesium alloy AZ31 as shown in Fig.32[304].The Teodosiu model can capture the cross hardening and the effects of work hardening stagnation,softening and resumption.The distortion of yield surface disappears,because distortional hardening is coupled with isotropic and kinematic hardening.To be more precise,distortional hardening has been divided into isotropic and kinematic hardening.The distortion of the yield surface can be captured by the extended Levkovitch &Svendsen model for AZ31 alloy.The extended Feigenbaum and Dafalias model can predict the high curvature of the yield surface in the loading direction and the flattenin in the opposite direction.However,such a response is not observed for most magnesium alloys.Neither the extended Teodosiu model nor the Levkovitch&Svendsen model fulfil the thermodynamical consistency.Only the extended Feigenbaum &Dafalias model can fulfil the second law of thermodynamics numerically.Therefore,a novel constitutive model was proposed for magnesium alloys.

3.3.3.A new model with directional distortional hardening for Mg alloys

The three extended models are not qualifie due to the energy minimization principle or thermodynamical consistency.Therefore,a new thermodynamically consistent constitutive model was built to capture distortional hardening for magnesium alloys based on a CB2004-type yield function [60,305]:

whereJ2andJ3are the second and third invariants of the modifie effective stresses:

with the linear transformations mapping:

Here,uncoupled isotropic,kinematic and distortional hardening can be obtained.A new plastic potential was proposed with quadratic terms to describe the distortional hardening:

Based on the generalized normality rule,the evolution of strain-like internal variables for distortional hardening is:

The corresponding evolution equations of stress-like internal variables for distortional hardening are derived as:

and

with the abbreviation:

The dissipation can be conveniently computed in closed form.It eventually results in a non-negative value:

Fig.33.Evolution of the normalized yield surfaces as predicted by the distortional hardening model for extruded AZ31 Mg alloy,(a) uniaxial tension test followed by reverse loading and (b) uniaxial tension followed by orthogonal loading [227].

The distortional hardening model is successfully applied to capture the yield surface evolution of AZ31 Mg alloy under strain path changes compared with experimental data as shown in Fig.33 [220,227].The rotation and high curvature of subsequent yield surfaces is captured along loading direction.

3.3.4.Homogenous anisotropic hardening (HAH) model for Mg alloys

Focusing on the prediction of the Bauschinger effect during strain path changes,a homogenous anisotropic hardening(HAH) model was originally proposed by Cazacu et al.[61].In the HAH model,the yield function was define by combining a stable termφand a fluctuatin termφh:

whereq,f1andf2are coefficients andis the microstructure deviator representing a given set of active slip systems.f1andf2are define as a function of the state variablesg1andg2:

The state variables are define as functions of the effective strain increment in the following manner:

To account for permanent softening during the strain path change,two additional state variables,g3andg4,were introduced [62].The evolutions of the state variables g1,g2,g3and g4are define as follows:

The HAH model and its extension have been applied to capture the mechanical response of strain path changes in EDDQ and DP steels [49,61,63,306].

To capture the mechanical behavior of Mg alloys during strain path changes,another two fluctuatin components were introduced to the original HAH model by Liao et al.[64]:

whereR(θ) is the in-plane transformation matrix and is expressed as:

The evolution of state variables g5-g8is similar to that of g1and g2:

wherek6-k11are material constants.The evolution of yield loci during reversal loading and re-loading was captured by the extended HAH model inπ-plane as shown in Fig.34[65].The softening of yield strength induced by strain path change can be described by the extended model.

The anisotropy and tension-compression asymmetry in AZ31 Mg alloy can be predicted by CBP06 yield function with distortional hardening [307].A new distortional hardening model within HAH framework was proposed with CBP06 yield function by Lee et al.[308].To be more explicit,two different isotropic hardening laws are employed to capture the asymmetric behavior during tension and compression for AZ31 Mg alloy.For uniaxial tension:

For uniaxial compression:

WhereA1,B1,C0,C1,D1,ε0,ε1are hardening parameters.The twin strain rate is considered to model the effect of nonproportional loading conditions.The total strain rate is define as follows:

Recently,the distortional hardening model was further extended to predict the mechanical behavior under cyclic loading at elevated temperatures [309].

However,the HAH model does not rely on kinematic hardening and distortional hardening to capture the Bauschinger effect and the distorted yield surface shape.The isotropic hardening and fluctuatin components are employed to capture the mechanical anisotropy.Recently,an extended HAH model (HAH20) considering the influenc of the hydrostatic stress was proposed to describe the distortional plasticity during strain path changes by Barlat et al.[310].The state variable evolution equations are improved to describe the crossloading and permanent softening.The HAH20model is suitable for two reversals,e.g.,T-C-T,which capture the permanent softening and distortion during strain path changes.In order to capture mechanical behavior with strain path change,a constitutive model based on slip,twinning and detwinning mechanisms was built with isotropic and nonlinear kinematic hardening by Li et al.[311].However,the distorted shape of yield surface can be induced by strain path changes,which cannot be described by isotropic and kinematic hardening.

The anisotropic and tension-compression asymmetric behavior is still key to construct accurate constitutive model for Mg alloys.Although phenomenological modeling is computationally efficient the underlying physical mechanism is not clarified From a physical metallurgical point of view,the macroscale deformation behavior is determined by the corresponding microstructure.Consequently,physics-based mesoscale modeling on the anisotropy of Mg alloys is crucial.

4.Physics-based mesoscale modeling of the anisotropy

At the mesoscale,texture and mechanical anisotropy can be studied by polycrystal plasticity models [46].A prototype crystal plasticity model was proposed by Sachs [312] based on the assumption of a homogeneous stress state throughout a polycrystal,regardless of intragranular strain compatibility.In contrast,Taylor [8] assumed that the strain of both the grains and the sample were equal,i.e.,intragranular strain compatibility,regardless of the stress equilibrium.Extensions of the Sachs and Taylor models have been applied to capture large elastic-plastic and viscoplastic deformation [312-315].The Taylor polycrystal plasticity model has been employed to simulate sheet metal forming with hardening laws at the slip system level [316-318].Because of the actual heterogeneities that occur during the deformation of polycrystals,a lower and an upper bound estimate of the material response are achieved by the Sachs and Taylor models,respectively.

Fig.34.Evolution of yield loci for AZ31 Mg alloy in π-plane predicted by the extended HAH model (a) reversal loading and (b) re-loading [65].

Recently,crystal plasticity-based finit element method(CP-FEM) [319-321],crystal plasticity-based fast Fourier transform (CP-FFT) method [322],and self-consistent (SC)models [323] are the main approaches targeting to describe the mechanical response of polycrystals.The grain morphology and the interaction between a grain and its neighbours can be computed in the CP-FE and CP-FFT models [59].A single crystal is treated as an element with more computational intensity.The polycrystalline aggregate is regarded as a homogenous effective medium (HEM) with the average properties of the aggregate in self-consistent polycrystal models.Each grain is considered as an ellipsoidal inclusion embedded into a HEM.The stress and strain coincide with the average of the aggregate.The viscoplastic self-consistent (VPSC)[323,324],elastic-plastic self-consistent(EPSC)[325,326]and elastic-viscoplastic self-consistent (EVPSC) polycrystal models [327] have been successfully applied to predict the mechanical anisotropy and texture evolution of magnesium alloys[16,55,328,329,330].

Dislocation slip and twinning are the two important deformation mechanisms for Mg alloys.Twinning can lead to grain reorientation and hardening.The mechanical behavior of Mg alloys cannot be fully captured by a polycrystal model with only dislocation slip [8,303,331,332].Thus,the deformation twinning was considered as a“pseudo-slip”mechanism and incorporated into the Taylor model by Houtte[333];Kalidindi[334];Staroselsky and Anand [335,336] and Wu et al.[337].To capture the grain reorientation and hardening behavior associated with twinning,different twinning models have been developed within a polycrystal plasticity framework.

4.1.Kalidindi model

The single-crystal plasticity model was based on the general framework proposed by Chen et al.[228],Hill and Rice[338,339],Asaro and Rice [340],and Asaro and Needleman[313].The slip systemαis represented by the slip direction (sα) and the direction normal to the slip plane (nα).The plastic velocity gradientLpfor the crystal is define as:

wherePαis the Schmid tensor for slip systemα.

To capture the contribution of twinning,the plastic velocity gradient was extended by Roters et al.[59] as:

whereNslipandNtwinare the number of slip and twin systems,respectively,andfis the volume fraction of the twinned region.The subsequent dislocation slip in the twinned regions was introduced into the plastic strain rate by Kalidindi[334,341]:

whereQis a transformation/rotation matrix that moves a vector of the initial lattice into its new position in the twin:

Here,nis the unit normal of the twin plane,andδijis Kronecker’s symbol.However,secondary twinning and grain reorientation of the twinned region are not considered in Kalidindi’s model.This model is named as“all twin variants(ATV) model”recently by Sahoo et al.[342].

A volume transfer scheme for twinning was proposed by Kalidindi [334].The volume associated with twinning during a time increment is define as [342-344]:

Here,Vparentis the volume of the parent grain,is the total slip rate of all slip and twin-slip rates andγtwinis the characteristic shear strain for extension twinning in Mg.The ATV model was incorporated into VPSC framework by Sahoo et al.[342].Compared to experiment,the strain hardening,texture and twin volume fraction evolution were reproduced with high precision.However,the detwinning and re-twinning are not captured in the Kalidindi’s framework,which usually occur during cyclic loading.

4.2.Predominant twin reorientation (PTR)

The predominant twin reorientation (PTR) scheme was proposed using the most active twin system in Tome et al.[345].In this scheme,the entire grain is completely reoriented when the twin volume fraction exceeds a critical value define by:

where Ath1and Ath2are two material constants,and Veffis the volume of the“effective twinned fraction”associated with the fully reoriented grains.Vaccis the“accumulated twin fraction”in the aggregate for the particular twin mode:

Here,the associated volume fractionVα,Ωis define as the ratio of the twinning shear strain to the characteristic twin shear strain:

Vaccand Veffare updated accordingly when the grains are reoriented.However,only the predominant twin reorientation in each grain is considered in the PTR scheme.A volume fraction transfer (VFT) scheme was proposed to account for all twinning reorientation and the associated twin volume fractions [345].The grain reorientation is updated by the transfer of twin volume fraction from one cell to another in the Euler space when the strain is incremented.A single grain can be represented by each cell,which enables one to obtain exact twinned volume fractions.A disadvantage of the PRT and VFT schemes is that the ‘‘grain’’ identity is lost and a realistic hardening scheme cannot be implemented because it fails to capture the mechanical response of strain path changes.

4.3.Composite grain (CG) model

To capture realistic twin structures and twin-parent interactions,a composite grain (CG) model [346] was proposed based on the lamellar grain model [347].A predominant twin system (PTS) was define using the maximum volume fraction of the grain contributed by twinning:

The thicknesses of twin lamellae and matrix are:

Here,dcis the separation between the twin centre planes.In order to account for the effect of directional Hall-Petch hardening,a mean free path is introduced for each system:

Here,αis the angle between the PTS plane and the slip or twin plane.

In particular,three hardening mechanisms -statistical dislocations,geometrically necessary dislocations(GND)and the directional Hall-Petch effect associated with twinning -are considered in the CG model:

The contribution of statistical dislocations to hardening is captured by a classical saturation Voce law:

whereΓis the accumulated shear in the grain,represents the shear increment in the slip or twinning system s＇,andis the latent hardening coefficient coupling the hardening of s due to the activity of s’.The effect of the GNDs on the threshold stress depends on the directional mean free pathdmfp:

and the directional Hall-Petch effect:

Fig.35.Schematic representation of twinning and de-twinning in a grain [178].

whereHGNDandHHPare the strength parameters of the corresponding hardening mechanism.For Mg alloys,the hardening of twinning system is define by a ‘deformation modespecific law:

where M represents all the slip modes activated during deformation,including basal,prismatic and pyramidal slip,andΓMis the accumulated shear of all slip systems.The de-twinning during strain path changes can be captured by a high CRSS value of the other twin systems inside the twinned region to activate the predominant twin system (PTS) during reloading.The transformation of the twin volume occurs from the original twin to the matrix when the PTS is activated inside the twin [332].

4.4.Twinning and de-twinning (TDT) model

A physics-based model with stress driving the matrix and twinning was proposed to characterize the twinning and detwinning processes of Mg alloys [178,348].In this model,twins are regarded as separate,and all twin variants are allowed,which is different from the predominant twinning system in the CG model.Twinning process is divided into four steps:twin nucleation due to parent reduction,twin growth due to the child propagation,twin shrinkage due to the parent propagation and re-twinning inside the twinned region,as shown in Fig.35.

In the TDT model,the processes of twinning can be captured using the shear rate and volume fraction associated with twin nucleation (TN) and twin growth,including matrix reduction(MR)and twin propagation(TP),as shown in Table 2.Here,ταandare the resolved shear stress and the CRSS of the twinning systemα,respectively.γtwis the characteristic twinning shear strain of magnesium alloys.

De-twinning is the inverse processes of twinning [156].Twin shrinkage and re-twinning can occur during reversal loading.Twin shrinkage includes two parts:matrix propagation (MP) into the twin and twin reduction (TR).The shear rate and twin volume fraction evolution associated with the de-twinning mechanisms are summarized in Table 3.

Table 3 The corresponding shear rate and twin volume fraction evolution during de-twinning [178].

However,the evolution of the twin volume fraction for twinning systemαonly considers the contribution of twin growth (MR and TP) and twin shrinkage (MP and TR) in the TDT model due to the small twin volume fraction and the plastic strain induced by twin nucleation and re-twinning:

wherefM=1 −Σfαis the volume fraction of the matrix.To terminate twinning,a threshold twin volume fraction is define as:

where A1and A2are material constants,andVaccandVeffare the accumulated twin fraction and the effective twinned fraction,respectively.

The plastic strain rate of grains during twinning or detwinning is represented as:

Here,are the plastic strain rate induced by twinning in the matrix grain and theαthtwinned grain,re-spectively:

Fig.36.The contour of twin volume fractions for V1-V6 of extruded AZ31 Mg alloy at a strain of approximately 0.014,(a) a soft grain surrounded by a soft grain and (b) a soft grain surrounded by a hard grain.V1-V6 are the six twin variants:V1:(1-102)[11-01],V2:(11-02)[1-101],V3:(1-012)[101-1],V4:(101-2)[1-011],V5:(01-12)[011-1],V6:(011-2)[01-11]) [328].

The TDT model was implemented in the EVPSC polycrystal framework [327] and has successfully captured the mechanical response under cyclic loading [178,349,350],bending[351]and combined torsion and axial loading[352]of Mg alloys and strain rate sensitive [353].Recently the physicsbased twinning-detwinning model was incorporated into the CPFE framework [354].The cyclic response of ZK60A alloy and twin variation were predicted within a multiscale framework compared with experimental observations by Yaghoobi et al.[354].The macroscopic anisotropy during dynamic loading was predicted by CPFE method by embedding strainrate sensitivities inherent to deformation mechanisms[355].A TNPG (twin nucleation,propagation and growth) model was proposed with twin propagation taken into account by Wu[356].The stress relaxation is resulted from twin nucleation and twin propagation.The TNPG model was implemented into CPFE framework to study twinning behavior of Mg alloys [357,358].It is found that the activation of low SF twin variants is related to the interaction among twin variants due to a very hard propagation from a soft grain to a hard grain as shown in Fig.36 [358].Recently,a new twinning-detwinning law with residual twin effect according to PTR scheme was proposed by Bong et al.[359].More precisely,the mechanical behavior of AZ31 Mg alloy during cyclic loading at various temperatures is predicted.

For magnesium,the Hall-Petch-like effect of twin boundaries is weak due to the growth of twins,which only increase the stress of nucleation[360].Basal-prismatic(BP/PB)boundaries can be created by dislocation-twin interactions in AZ31 Mg alloy [361,362].The matrix dislocation can be transmuted to a twin prismatic dislocation;in contrast,the dislocation absorption occurs in the twin boundary during the Burgers vector non-parallel to the zone axis [363].In particular,the{11-21}twinning can be produced during interactions between prismatic slip and {10-11} twins [364].The twintwin interaction is profuse,with an internal stress fiel during sequential deformation [365].The deformation twinning in HCP metals is briefl reviewed where the mechanisms of twin-twin interactions and junctions should be further clarifie in HCP metals [366].In addition,the stress inside the twinned region,which drives slip and detwinning,is different from the stress of the twin-free region [367,368].The twinning behavior is affected by local stress [369,370].In addition to the nucleation and growth of twinning and detwinning,the natures of twins should be considered in a twinning model within a crystal plasticity framework.Crystal plasticity modeling is a useful tool to investigate complex twinning behavior in HCP metals [371].The effect of twinning on mechanical behavior of Mg alloy can be clarifie byin-situexperiments [372-376].Therefore,the advanced experimental technologies,such asin-situEBSD/ neutron diffraction,3D high energy synchrotron X-ray techniques,and digital image correlation under scanning electron microscopy (SEM-DIC),can provide more detailed information to validate crystal plasticity modeling [377].A multi-scale method can provide an accurate modeling of anisotropy during forming,which takes microstructure evolution into account [378-380].

5.Conclusions and outlook

A sharp overview of the anisotropic mechanical behavior with respect to microstructure evolution under different loading conditions and associated constitutive models at the macro and meso levels is critically reviewed for wrought Mg alloys.The mechanical responses and deformation mechanisms can be interpreted based on the Schmid law during monotonic loading and captured by existing constitutive models.The anisotropic behavior originates from the difference between slips and twinning,which depends on loading direction and crystallographic texture.

However,the anisotropic behavior is more complex due to the activation of twinning and detwinning during strain path changes.Moreover,the material is usually loaded under multiaxial stress state during practical forming processes.The microstructure evolution during multiaxial loading is different from that during uniaxial loading.Therefore,the anisotropic behavior under multiaxial stress state with strain path changes will be an emerging focus in the future research.Plastic strain is limited by the current multiaxial specimens,which results in very limited information of microstructure under multiaxial loading.Thus,large plastic strain will be another target to clarify the deformation mechanisms and corresponding microstructure evolution under multiaxial stress state.

Both the evolution of yield surfaces and equal plastic work contours play important roles in capturing the anisotropy under multiaxial loading.Due to limitations of biaxial specimens and biaxial testing technologies,the complete evolution of yield surfaces for Mg alloy is still a challenge.In particular,when the loading path/history effect is considered,the concepts of yield surfaces and equal plastic work contours are different,which will be clarifie for anisotropic materials in the future work.This concept will be further investigated using both experimental and numerical approaches,which is related to strain hardening induced by strain path changes.

Although the anisotropy and SD effect of Mg alloys can be captured by phenomenological modeling at macroscale with computational efficien y,the underlying physical mechanisms for the evolution of anisotropy are not quantitatively characterized.Crystal plasticity modeling is an effective approach to describe microstructure evolution.Within a crystal plasticity framework,however,the models that consider reorientation and the Hall-Petch-like effect induced by twins cannot accurately predict macro-level mechanical responses because the hardening effects of twins during interactions between matrix slip and twin boundaries are affected by the occurrence of dislocations transmutation and absorption.Dislocation dynamics and molecular dynamics simulations are effective tools to investigate the interactions between slip dislocations and twinning at nanoscale.However,the anisotropic behavior cannot be clarifie at single scale.Multiscale modeling andin-situexperiments will be focused in the future research in order to bridge macro-scale anisotropic behavior and microstructure evolution.More explicitly,the link of constitutive models from macro-to nano-scale will be developed based on integrated computational materials engineering (ICME).Moreover,the contribution of the interaction between slip and twinning to anisotropy will be clarifie via advancedin-situexperiments and simulation tools during strain path changes.

Data availability

The raw data required to reproduce these finding are available to download from [https://zenodo.org/record/3723062#.XnYFWPmsd-U].

Acknowledgments

Financial support from the projects by the NSFC[51771166],Chongqing Special Project of Science and Technology Innovation (cstc2020yszx-jcyjX0001),the Hebei Natural Science Foundation [E2019203452,E2021203011],the talent project of human resources and social security department of Hebei province [A202002002],the key project of department of education of Hebei province [ZD2021107],project of the central government guiding local science and technology development[216Z1001G]and Cultivation Project for Basic Research and Innovation of Yanshan University[2021LGZD002] is gratefully acknowledged.The work was supported by the State Key Laboratory of Materials Processing and Die &Mould Technology,Huazhong University of Science and Technology [P2020-013].B.Shi is grateful for fruitful discussions with Prof.Jörn Mosler from TU Dortmund University,Dr.-Ing.Dirk Steglich and Dr.-Ing.Ingo Scheider from Helmholtz-Zentrum Hereon.The helpful comments by anonymous reviewers are gratefully acknowledged.