Deformation mechanisms and differential work hardening behavior of AZ31 magnesium alloy during biaxial deformation

Yuanjie Fu ,Yao Cheng ,Yun Cui ,Yunhang Xin ,Shouwen Shi ,Gang Chen

a School of Chemical Engineering and Technology,Tianjin University,Tianjin 300072,China

b International Joint Laboratory for Light Alloys,College of Materials Science and Engineering,Chongqing University,Chongqing 400030,China

c School of Chemical Engineering and Technology,Ren’ai College of Tianjin University,Tianjin,China

Abstract Magnesium alloys are frequently subjected to biaxial stress during manufacturing process,however,the work hardening behavior under such circumstance are not well understood.In this study,the deformation mechanisms and differential work hardening behavior of rolled AZ31 magnesium alloy sheets under biaxial loading are investigated.The change of plastic work contours with increasing plastic strain indicates the differential work hardening behavior of AZ31 magnesium alloy under biaxial stress state,resulting in higher macroscopic work hardening rates of biaxial loading than uniaxial loading,with the elastic-plastic transition part of work hardening extended and stage III hardly emerged.Electron backscatter diffraction and Schmid factor analysis confir the low activation of non-basal 〈a〉 slip during biaxial loading tests.While the thickness strain is primarily accommodated by pyramidal 〈c+a〉 slip at the initial stage of biaxial deformation,{10-11} contraction twinning is activated at larger plastic strain.The low activation of non-basal 〈a〉 slip also retards the dynamic recovery and cross-slip of basal and prismatic 〈a〉 slips,leading to the differential work hardening behavior of AZ31 magnesium alloy under biaxial stress state.

Keywords: Magnesium alloy;Biaxial tensile tests;Differential work hardening;Cruciform specimen;Deformation mechanism.

1.Introduction

Magnesium is abundant in its reserves.Being the lightest engineering metal material,magnesium alloys offer a series of advantages such as low specific gravity,high specific strength and stiffness,good thermal conductivity,and strong electromagnetic shielding ability [1,2].It is an important lightweight material in automobile,aerospace,national defense and military industry[3,4].However,compared with other engineering metals,such as aluminum alloys and steels,the application of magnesium alloy is still limited by its poor compatibility of deformation and strong tension-compression anisotropy[5].This is because magnesium alloys are hexagonal closepacked (HCP) crystalline metals whose independent slip systems are not enough to coordinate plastic deformation under room temperature.High temperature does help improving the formability of magnesium alloy in manufacturing processes[6].But recovery and dynamic recrystallization processes are taking place reducing hardening effects of deformation and mechanical property [7,8].Therefore,the deformation mechanisms at ambient temperature are researched for a wide range of technological applications [9].When rolled magnesium alloy sheets with strong basal texture were uniaxial stretched along the normal direction (ND) or compressed along rolling direction (RD) or transverse direction (TD),{10-12} extension twinning was the main deformation mechanism [10].But when rolled magnesium alloy sheets were uniaxial stretched along RD or TD,the 〈a〉 slip only occurred in directions paralleling to the RD-TD plane.And,contraction twinning as well as pyramidal 〈c+a〉 slip were difficult to be activated to accommodate the shrinking straining along the normal direction [11].So,magnesium alloys are easily to be cracked and have poor stretch formability under room temperature,which leads to the high processing cost and limits the large-scale industrial application [12].

Metal sheets are frequently subjected to multiaxial stress or strain conditions during manufacturing processes such as stamping and rolling [13].Previous studies implied that the multiaxial stress state led to a decrease in the elongation of magnesium alloys,which was different from that under uniaxial stress state.For example,the uniaxial tensile elongation of magnesium alloy was similar to that of aluminum alloy,both can reach about 20%,but the in-plane forming ability of magnesium alloy was much worse [14].In addition,the Erichsen value of commercially pure titanium sheet,which was also a HCP metal,under room temperature could reach 11.6mm[15],which was much higher than that of AZ31 magnesium alloy(3.1mm)[16].Therefore,it is necessary to study the deformation mechanisms that decrease the elongation of magnesium alloy under multiaxial loadings.However,the research on deformation mechanisms of magnesium alloys under multiaxial loading are still limited.One important reason is that it is difficult to achieve and control multi-axial stress and strain states during experiments.For instance,the Erichsen test was widely used to introduce multiaxial stress states for metal sheets [17].However,the punch exerted extra stresses along ND on the specimen,which would seriously affect the activation of microstructure deformation mechanism.Even so,only equal biaxial stress and strain ratio can be approximately achieved in Erichsen tests [18].The hydraulic bulge test using tubular specimens could achieve various stress ratios [19],but the applied stress ratio could not be changed instantly during the test because the circumferential stress was indirectly controlled by inner hydraulic [20].So,the abrupt change of stress path is difficult to achieve.Besides,there are still some stress ratios that cannot be achieved by hydraulic bulge tests.More importantly,both test methods mentioned above cannot achieve in-plane deformation.By comparison,any arbitrary stress and strain ratios can be precisely achieved and immediately changed for biaxial tensile tests using cruciform specimens by adjusting each loading axis [21].The in-plane stress and strain fields of cruciform specimens are also uniform and easily to be measured.Therefore,this method is advantageous over others during investigation of deformation mechanisms of metal sheets under complex multiaxial stress or strain states.

Work hardening behavior can effectively reflect the plastic deformation properties of metals,including strength,ductility and deformability.Besides,work hardening behavior is also related to the deformation mechanism.The effects of deformation on macroscopic work hardening behavior of magnesium alloy under uniaxial stress state were extensively investigated.For example,Huang et al.[22] found that the{10-12} extension twinning of AZ31 magnesium alloy at the early stage of uniaxial tensile deformation limited the activity of non-basal slips,which depressed dynamic recovery and contributed to high work hardening rate.Wang et al.[23]reported that when plastic deformation was dominated by{10-12} extension twinning,the stress-strain curves of AZ31 magnesium alloy were usually S-shaped.The work hardening behavior is also different form the plastic deformation dominated by slip.During biaxial loading tests,the work hardening behavior of magnesium alloy will change the shape of plastic work contours,a phenomenon called differential work hardening behavior [24].At present,there are many systematic researches focusing on the experimental measurement and numerical simulation of differential work hardening behavior.However,the microstructure deformation mechanisms that result in the various work hardening behavior of magnesium alloy under biaxial loading are still not well understood.For instance,Steglich et al.[25] predicted the plastic work contours of AZ31 magnesium alloy under different stress ratios using crystal plastic model.Hama and Takuda [26] used a rate-dependent crystal plasticity finite-element method to predict the deformation mechanism and contours of plastic work of rolled magnesium alloy AZ31 sheets.However,microstructure observations that can correlate the differential work hardening behavior to the poor formability are still lacking.Alkan et al.[27]researched the deformation and fracture behavior of a magnesium alloy AZ31 sheet under uniaxial tension and biaxial loading tests.But there was no content about differential work hardening.Wei et al.[28] and Zhao et al.[29] studied the strain hardening behavior of Mg-Y alloys.Nevertheless,strain hardening behavior was only researched under uniaxial stress state.Therefore,it is necessary to clarify the work hardening behavior and deformation mechanisms of magnesium alloys under biaxial loading.

To this end,biaxial tensile tests with different stress ratios are conducted using optimized cruciform specimens.Plastic work contours and macroscopic work hardening rates are measured to investigate the differential work hardening behavior.In addition,the impacts of biaxial stress state on twin variants and slip systems are also evaluated by a combined analysis of electron backscatter diffraction(EBSD)results and Schmid law.

2.Experimental

2.1.Uniaxial and biaxial tensile tests

Commercial AZ31 magnesium alloy (Mg-3.6%Al-1.0%Zn-0.5%Mn) hot-rolled sheets homogenized at 400°C for 2h with the thickness of 2±0.05mm were used.Uniaxial and biaxial experiments were carried out using a biaxial cyclic testing system (IPBF-8000,CARE Measurement &Control Co.,Ltd).The test system (Fig.1) supports fully closed-loop control of load,displacement and strain in two loading directions.

Fig.1.Photograph of the biaxial cyclic testing system IPBF-8000.The inset shows a magnified view of the cruciform sample together with the load axes.

The geometry of specimens for uniaxial tensile tests is shown in Fig.2.The Young’s modulus (E),yield strength(σ0.2),tensile strength (C),ductility (Ef) and the Lankford value (r-value) of AZ31 magnesium alloy sheets in RD,TD and 45° from the rolling direction (45) were measured by uniaxial tensile tests.The strain was measured using a digital image correlation (DIC) system.Ther-values were measured using a special biaxial strain gauge(BF120-3BC).The biaxial strain gauge and the schematic diagram of bridge circuit are shown in Fig.3.Load control was adopted in tests,and the loading rate was 30N/s.

Fig.2.The geometry of the specimens used in uniaxial tensile tests.

Fig.3.(a) The schematic diagram of bridge circuit and (b) the cruciform specimen and biaxial strain gauge.

Fig.4.Microstructure of AZ31 magnesium alloy observed using an optical microscope.

For biaxial loading tests,a modified ISO cruciform specimen is adopted with a 12mm×12mm uniform gauge area.Each arm of the specimen has five slits that are 12mm long and 0.2mm wide.The interval between two slits is 2.2mm.All slits are made by laser cutting.The stresses along RD and TD were define asσxandσy,respectively;and the maximum strains along RD and TD were define asεxmaxandεymax,respectively.Biaxial loading tests were carried out with load control and a series of loading ratios (σx:σy=1:1,1:2,2:1,1:4,4:1).The strain was also measured using biaxial strain gauges.To measure Young’s modulus (Eb) and yield strength (σb0.2),biaxial loading-unloading tests for all stress ratios were also carried out.The tensile strength (Cb) was determined as the maximum stress in the loading direction when the cruciform specimen fractured.

2.2.Microstructure characterization

Specimens were firstly observed using an optical microscope(KEYENCE).Prior to observation,specimens were polished with a series of SiCs papers and etched with an aceticpicral solution[30].Fig.4 shows the optical microstructure of AZ31 magnesium alloy sheet,exhibiting a fully recrystallized microstructure with an average grain size of 30μm.Electron backscatter diffraction (EBSD) observations were performed using a step size of 1μm on a scanning electron microscope(FEI Nova 400) with an HKL channel 5 system.All EBSD maps are measured in the surface of gauge areas of fractured specimens after uniaxial and biaxial loading tests.The specimens for EBSD mapping were mechanically grounded and electrochemically polished in an AC2 electrolyte.All EBSD data were analyzed using a channel 5 software.

2.3.Calculation of Schmid factor

The Schmid law is given in Eq.(1) [31]:

whereτis resolved shear stress,Fis the force loaded on the material andAis the cross-section area of the specimen.Besides,λis the angle betweenFand slip/twinning direction,andφis the angle betweenFand the normal direction of slip/twin plane.According to the Schmid law,the twinning or slip system can be activated if the resolved shear stress applied to the material is higher than the critical resolved shear stress (CRSS) of the specific twinning or slip system.Thecosλ·cosφis define as the Schmid factor.It is clear that the slip or twinning system with the highest Schmid factor has the highest possibility to be activated [18].Therefore,the Schmid factor has been widely used to evaluate the activation of deformation modes of magnesium alloys under uniaxial loading [32].

It should be noted that Eq.(1) can be used only for uniaxial stress state.Recently,the global Schmid factor (GSF)was introduced to investigate the activation of deformation mechanism under complex stress states [31],such as rolling process or plane strain state.For the calculation of GSF,a stress tensor is used to represent the complex load during deformation,which usually can be simplified into Eq.(2) whereσ1,σ2,andσ3are three principle stresses.

Thus,the GSF can be calculated using Eq.(3):

where b=(bx,by,bz) represents the specific slip or twinning direction,and n=(nx,ny,nz) represents the normal direction of slip or twin plane.

For orthogonal biaxial plane stress state,the stress tensor can be written as Eq.(4):

whereαandβcan be determined from the stress ratio applied to the cruciform specimen.Thus,the GSF under biaxialtensile tests can be calculated by Eq.(5):

mxand mycan be calculated from Eqs.(6) and (7):

whereλ(RD)andλ(TD)are the angles between applied force along RD/TD and the slip/twinning direction,respectively.φ(RD)andφ(TD)are the angles between applied force along RD/TD and the normal direction of the slip/twin plane,respectively.

3.Results

3.1.Uniaxial and biaxial tensile behavior

Fig.5 shows the true stress-strain curves and the variation ofr-value within 2% true plastic strain in RD,TD and 45.The E,σ0.2,C and Efvalues of AZ31 Mg alloy sheet on RD,TD,and 45 are listed in Table 1.

Table 1Experimental results of uniaxial tensile tests.

The true stress-strain curves for biaxial tensile tests with different stress ratios are shown in Fig.6,with Eb,σb0.2,Cb,εxmaxandεymaxlisted in Table 2.Compared with uniaxial loading,the maximum achievable strain in biaxial loading tests is much smaller.When the stress ratio isσx:σy=1:1,the strains of RD and TD are positive.Changing the stress ratios toσx:σy=1:2 andσx:σy=2:1,the strain caused by the lower stress is almost zero,which is close to a plain strain state.Further changing the stress ratios toσx:σy=1:4 andσx:σy=4:1,the strain in the axis with lower stress is negative.It is noteworthy that when the maximum strain is zero or negative,the yield strength and Young’s modulus cannot be measured.Considering the fact that the yielding of metal sheets is not direction-related [33],the earlier achieved yield point of two loading directions is considered as the yield point for both loading directions.

Fig.5.(a) True stress-strain curves and (b) the variation of r-values within 2% true plastic strain.

Fig.6.True stress-strain curves of AZ31 Mg alloy sheets under biaxial tensile tests at stress ratios of σx: σy=(a) 1:1,(b) 1:2,(c) 1:4,(d) 2:1,(e) 4:1.

Table 2Experimental results of biaxial tensile tests.

3.2.EBSD analysis

Fig.7 shows the inverse pole figure maps and pole fig ures of as-received AZ31 magnesium alloy.The (0001) pole figure shows that the as-received material exhibits a strong basal texture with an angle of 20° tilting to RD.The intensity of (0001) pole is 14.04.The (10-10) pole figure shows that thea-axis is randomly rotated in the RD-TD plane and perpendicular to thec-axis.Fig.8 shows the pole figure of uniaxial loaded specimens.After uniaxial loading along RD and TD,the intensity of the (0001) pole (24.67 and 19.11)becomes higher compared to the as-received material (14.04),and RD shows more alignment than TD.The (10-10) pole figure show preferred distribution and form a six-fold symmetry.{10-12} extension twins are marked by red lines in twin boundary misorientation maps while {10-11} contraction twins are marked by green lines.The inverse pole fig ure maps and twin boundary misorientation maps (Fig.9)show that the specimen loaded along RD has more {10-12} extension twins (Fig.9(a)),whereas more {10-11} contraction twins are found in the specimen loaded along TD(Fig.9(b)).

Fig.7.(a) Inverse pole figure maps and (b) pole figure of as-received material.

Fig.8.Pole figure of uniaxial loaded specimen along (a) RD and (b) TD.

Fig.9.Inverse pole figure maps and twin boundary misorientation maps of uniaxial loaded specimen along (a) RD and (b) TD.

The pole figure of biaxial loaded specimens at different stress ratios are shown in Fig.10.The intensity of(0001)pole figure also shows more alignment than the original material.However,the preferred distribution and six-fold symmetry of(10-10) pole figure do not appear in biaxial loaded specimens for every stress ratio.The inverse pole figure maps and twin boundary misorientation maps at different loading ratios (Fig.11) show that only a few {10-12} extension twins are found in biaxial-loaded specimens.Besides,hardly any{10-11} contraction twins are found in Fig.11.

Fig.10.Pole figure of biaxial loaded specimen at stress ratio of σx: σy=(a) 1:1,(b) 1:2,(c) 1:4,(d) 2:1,(e) 4:1.

Fig.11.Inverse pole figure maps and twin boundary misorientation maps of biaxial loaded specimen at stress ratio of σx: σy=(a) 1:1,(b) 1:2,(c) 1:4,(d)2:1,(e) 4:1.

4.Discussion

4.1.The effect of cruciform specimen geometry

There are two alternative cruciform specimen geometries in this study,SP1 and SP2,as shown in Fig.12.The SP1 geometry follows the ISO standard 16,842.Compared with standard ISO cruciform specimens,SP1 has a smaller size and fewer slits in order to achieve a balance between the achievable maximum plastic strain and uniform stress/strain distribution within the gauge area[24,34].SP2 has a locally thinned gauge area to achieve more plastic strain within the gauge area [35].The slits in each specimen arm can also avoid the shear loading and deformation restriction to the perpendicular loading direction [29].Finite element analysis (FEA) was used to analyze the coupling between loaded forces and in-plane gauge stresses as well as the restriction induced by geometries of cruciform specimens to the deformation of gauge area.This is because stress coupling makes it difficult to calculate the gauge stress as a function of the applied force,and the restricted deformation will affect microstructure evolution.

Fig.12.Geometries and FEA models of (a) modified ISO cruciform specimen (SP1) and (b) specimen with a locally thinned gauge area (SP2).

The simulation adopts the method proposed by Upadhyay et al.[36].All geometries are loaded in one direction and keep free in the other in order to analyze the stress coupling and deformation restriction.According to the results of uniaxial loading tests in Fig.5(a),the anisotropy of rolled AZ31 magnesium alloy sheet is not obvious when it is stretched within RD-TD plane,which is in well agreement with Xia et al.[18].Andar et al.[37] found that the initial yield surface of AZ31 magnesium alloy within RD-TD plane agreed with that calculated by von Mises yield function.But when the deformation under biaxial stress state became larger,the differential work hardening behavior changed the shape of plastic work contours.Considering the fact that only uniaxial tension is processed,and there are no strain path changesduring differential work hardening behavior simulation,material models are assumed to be isotropic in both elasticity and plasticity in uniaxial loading.Two specimen geometries are uniaxially loaded inx-axis withy-axis kept free.The simulated stresses are define asσsxandσsy,and the simulated plastic strains areεsxandεsy.Simulations are performed under load control with a linearly ramped force.When the load onx-axis reaches 5kN,the simulation stops.During the elastic deformation stage,the Young’s modulus is set as 45GPa and the Poisson ratio is 0.35.In the plastic deformation part,the von Mises yield criterion is used.And the work hardening behavior is simulated using the Chaboche model.The back-stress components are confirmed and listed in Table 3.A quarter of the 3D finite element models were built for both specimen models.

Table 3Material parameters of the Chaboche model.

The contour plots ofσsxandσsyas well asεsxandεsyof SP1 and SP2 are shown in Fig.13.The contour plot of SP1 (Fig.13(a)) shows that although there is stress and strain concentration at the top of each slit and the edge of the gauge area,the distributions ofσsxandσsywithin the gauge area are uniform.The distributions ofεsxandεsyalong path 1 and path 2 also reflect this phenomenon (Fig.14(a) and (b)).However,in the contour plots of SP2 in Fig.13(b)(marked by a red line),the thick frame of the thinned gauge area is bent into a circle shape due to thex-axis loading,which causes an uneven distribution ofεsxalong they-axis of the thinned gauge area.The top of the circle frame causes more serious stretch to the thinned gauge area than the two sides.Thus,εsxin the middle of the thinned gauge area is larger than that in the edge along they-axis,and the distribution of negativeεsyis similar.The simulated strain distributions along path 3 and path 4 reflect this phenomenon (Fig.14(c) and (d)).

Fig.13.Contour plots of (a) SP1 and (b) SP2,illustrating the distributions of σsx and σsy as well as εsx and εsy.

Fig.14.The distribution of simulated (a) εsx (b) εsy of SP1 along path 1 and path 2,and the distribution of simulated (c) εsx (d) εsy of SP2 along path 3 and path 4.

To assess the restriction to the deformation of gauge area of SP1 and SP2,a parameterνis proposed and given by Eq.(8):

Fig.15(b) shows thatνof SP1 along path 1 and path 2 are greater than SP2 along path 3 and path 4,which indicates that the contraction of SP1 gauge area alongy-axis is greater than the SP2 thinned gauge area.Considering the fact that theεsxof SP2 is much greater than SP1 (Fig.14(a)and (c)),it can be concluded that they-axis contraction of SP2 is restrained.This phenomenon is caused by the thick frame surrounding the thinned gauge area.As for SP2,when loading alongx-axis,the contraction of thinned gauge area alongy-axis is much more than the thick frame because the frame is thicker and have higher stiffness.Thus,they-axis contraction of thinned gauge area is restricted by the thick frame;the closer to the thick frame,the lower ofν,indicating more server restriction.By contrast,theνdistribution of SP1 is much uniform,ignoring the sharp change caused by the edge of slits.It seems that SP2 applies more restriction on the thinned gauge area.

Fig.15.The distribution of (a) σsy and (b) ν of SP1 and SP2.

Fig.16.The fractured specimens in biaxial loading σx: σy=1:1 (a) SP1 (b)SP2.

The coupling between loaded forces and in-plane gauge stresses is evaluated byσsy,which is illustrated in Fig.15(a).For SP1,the distributions ofσsyalong path 1 and path 2 are quite uniform and stabilize at -10MPa.Hanabusa et al.[34] also found in-plane compressive stresses normal to the uniaxial loading direction during the FEA simulation of ISO type cruciform specimen.As for SP2,althoughσsyis 0 at the center of gauge area,σsyincreases rapidly along path 3 and path 4 and reaches 70MPa at the edge of thinned gauge area.It can be concluded that the stress coupling of SP2 is more serious and complex than SP1.The experiment results also support the simulation results,as is shown in Fig.16.In Fig.16(a),the fracture position of SP1,marked out by red ellipse,is at the edge of gauge area,which illustrates that the stress state within gauge area is uniform and the fracture is resulted by the stress concentration caused by slits.Fig.16(b) shows that because the thinned gauge area causes uneven stress distribution as well as stress concentration,the fracture location of SP2 appears within thinned gauge area,which is also marked out by red ellipse.In conclusion,the stress coupling and deformation restriction of SP1 are much lower than SP2.And according to the FEA simulation of Hanabusa et al.[34],the stress measurement error of ISO cruciform specimen was estimated to be less than 2%.Thus,SP1 is more suitable for this study.

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4.2.Work hardening behavior of AZ31 Mg alloy

The results of biaxial tensile tests (Table 2) are used for the calculation of contours of plastic work in AZ31 Mg alloy.Fig.17(a)shows the contours of plastic work measured at true plastic strain range of 0.2% ≤εp≤0.6%.The straight line at each stress point represents the direction of the plastic strain rate at each stress point in the stress space.Fig.17(b)shows the plastic work contours normalized by the true stress of RD uniaxial loading (σ0).It is evident that the shape of the normalized plastic work curves changes significantly with the increase of plastic strain,reflectin the differential work hardening behavior [37].But the normalized contours of plastic work can only exhibit the differential work hardening of AZ31 Mg alloy.In order to build the link between the differential work hardening behavior and microstructure deformation mechanisms,macroscopic work hardening ratesθare further calculated by Eq.(9):

Table 4Results of average Schmid factor calculation.

whereσandεare the macroscopic true stress and true strain under uniaxial or biaxial loading,respectively.

The macroscopic work hardening rates curves for uniaxial and biaxial loading tests in RD and TD are shown in Fig.18.The abscissa of macroscopic work hardening rates curves represents macroscopic true stressσsubtracting corresponding yield stressσ0.2.Only true stress-strain curves with positive plastic strain are used for calculating the macroscopic work hardening rates to compare with the results of uniaxial loading.It is also worth noting that whenσx:σy=1:1 the plastic strain of TD is too small to calculate theθ～σ-σ0.2curve.

In the work hardening of uniaxial loaded hexagonal metals,stages I,II and III are observed,as in face-centered cubic (fcc) crystals [38].Stage I is an elastic-plastic transition process.In stage II,the work hardening rate is kept almost constant;while the work hardening rate decreases linearly with stress during stage III.During uniaxial loading,theθ～σ-σ0.2curves of RD and TD are similar.The work hardening rates decrease linearly with increasingσ-σ0.2after the elastic-plastic transition,which implies the suppression of work hardening in stage II.The slope of stage III is much smaller than stage I.It was reported that the liner decrease of work hardening rate during stage III was closely related to the dynamic recovery originating from the cross-slip of basal and prismatic 〈a〉 slips [22].As for rolled AZ31 Mg alloy sheet with strong basal texture that is uniaxial loaded along RD or TD,the width strain results from prismatic 〈a〉 slip while the thickness strain is induced by pyramidal 〈c+a〉 slip [12].The prismatic 〈a〉 slip exhibited greater relative activity compared with the pyramidal 〈c+a〉 slip during in-plane uniaxial tension at room temperature [39],which led to the softening behavior of stage III after the elastic-plastic transition.

Fig.17.(a) Contours of plastic work composed by stress points and directions of plastic strain rates measured at plastic strain of 0.2%,0.4%,0.5% and 0.6%,and (b) normalized plastic work contours at plastic strain of 0.2%,0.4%,0.5% and 0.6%.

Fig.18.θ～σ-σ0.2 curves for uniaxial and biaxial loaded specimens along (a) TD (b) RD.

Though the tendency ofθ～σ-σ0.2curves for biaxial loading tests are similar to that of uniaxial loading tests,biaxial stress state has greater influence on the work hardening behavior of AZ31 magnesium alloy.As shown in Fig.18,the macroscopic work hardening rates of biaxial loading tests are much higher than that of the uniaxial loading.It is worth noting that compared with uniaxial loading tests,the stage I for biaxial loading tests is extended,and stage III is short in everyθ～σ-σ0.2curve.Besides,the stress ratios of biaxial loading tests also affect the work hardening behavior.Whenσx:σy=1:1,the macroscopic work hardening rate is the highest and theθ～σ-σ0.2curve is almost straight with stage III the shortest.Changing biaxial ratio toσx:σy=1:4 andσx:σy=4:1,the macroscopic work hardening rate decreases significantly .And the change of slope in stage III also becomes pronounced.The tendency ofθ～σ-σ0.2curves inσx:σy=1:2 andσx:σy=2:1 are similar to that inσx:σy=1:4 andσx:σy=4:1,but less pronounced.

Fig.19 illustrates that the stress applied in biaxial loading tests is high enough to generate plastic strain and active stage III inθ～σ-σ0.2curves.However,the plastic strain in biaxial loading tests is quite small,and stage III is quite short.It is also worth noting that when stress ratios get closer to uniaxial loading,such asσx:σy=1:4 andσx:σy=4:1,clear stage III appears.Therefore,it can be speculated that the basal and prismatic〈a〉cross-slip which are closely related to the change of slope in stage III ofθ～σ-σ0.2curve are restricted in biaxial stress states.Next,in order to prove this speculation,the deformation mechanisms are discussed in Chapter 4.3.

Fig.19.Comparison of true stress-strain curves of uniaxial and biaxial loading tests (a) TD (b) RD.

4.3.Crystallographic microstructure evolution

For uniaxial loading tests shown in Fig.8,the pole figure illustrate that the alignment of (0001) pole is more serious compared with the as-received specimen.And the (10-10)pole figure exhibits preferred distribution as well as six-fold symmetry.This phenomenon can be explained by the activation of basal 〈a〉 slip and prismatic 〈a〉 slip [40].It is worth noting that basal〈a〉slip also contributed to thickness strain to a less extent [41].As mentioned above,the basal texture has a 20° tilt angle to RD,which is in favor of basal 〈a〉 slip and{10-12} extension twining [12].So basal 〈a〉 slip contributes more to the thickness strain in RD loading,and more{10-12}extension twins are also observed in inverse pole figures By contrast,more{10-11} contraction twins are generated in TD in order to accommodate thickness deformation,as shown in Fig.9.So,it is hard to generate thickness strain when loading along TD,which causes the highestσ0.2and lowest Efin TD among three loading directions.And,the Lankford value(r-value) of TD which is a ratio of thickness strain to width strain in a deformed sheet material,is higher than RD and 45,as shown in Fig.5.

Although the maximum strain in biaxial loading test with the stress ratio ofσx:σy=1:1 is much smaller than that of uniaxial loading in RD or TD,the (0001) pole intensity of equi-biaxial loading (21.44) is quite close to that of uniaxial loading tests (24.67 and 19.11) in Figs.8 and 10.This phenomenon can be explained by the effect of biaxial stress state.Previous study reported that basal 〈a〉 slip had higher relative activity in biaxial stress states [18].However,when the stress state gets closer to uniaxial,the texture evolution becomes similar with uniaxial loading tests.When the stress ratios areσx:σy=1:2 andσx:σy=1:4,the (0001) pole intensity (15.45 and 19.46) is weaker than that ofσx:σy=2:1 (21.51).This is because more basal 〈a〉 slip is activated when uniaxial loaded along RD,as mentioned above.It is also worth noting that whenσx:σy=4:1,the (0001) pole intensity becomes much weaker (14.70).This phenomenon might be ascribed to the weakened texture cause by deformation [42],because the strain for stress ratio ofσx:σy=4:1 is the maximum for all biaxial stress ratios.Besides,the(10-10)pole in all stress ratios distributes randomly around c-axis.It can be speculated that the relative activity of prismatic 〈a〉slip in biaxial stress state is low.

According to the inverse pole figure maps in Fig.11,less twins are found in biaxial loading than uniaxial loading.Only {10-12} extension twins are found while hardly any{10-11} contraction twins can be observed,which is quite different from results in previous reports using Erichsen tests [17,43].In their study,many {10-11} contraction twins and {10-11}-{10-12} double twins were found in biaxial loading tests.Considering the fact that punch can apply extra load along ND in Erichsen tests,and no extra stress are applied along ND during biaxial loading tests using cruciform specimens in our study,these extra {10-11} contraction twins and {10-11}-{10-12} double twins found in Erichsen tests maybe activated by the pressing of punch.In addition,the maximum strain achieved in biaxial loading tests using cruciform specimens is smaller compared with Erichsen tests.Therefore,{10-11}contraction twinning or{10-11}-{10-12}double twinning are not the main deformation mechanisms,at least in the early stage of biaxial deformation.Besides,for{10-12} extension twins,when the stress ratio isσx:σy=1:1,almost all {10-12} extension twins appear in grains whose c-axis is perpendicular to ND;while {10-12} extension twins are activated in grains with random orientation for other stress ratios.Such a difference can be ascribed to the fact that under biaxial stress state,the Schmid factor(SF) for {10-12} extension twins is high for grains with high angle between c-axis and ND [18].And in other stress ratios,the twinning behavior will get closer to uniaxial stress state.

4.4.Analysis of Schmid factor calculation

The average Schmid factor (SF) for various deformation mechanisms under uniaxial and biaxial loading tests are calculated,as listed in Table 4.For uniaxial loaded specimens,non-basal slips,such as prismatic 〈a〉 slip,pyramidal 〈a〉 slip and pyramidal〈c+a〉slip,have higher Schmid factors(＞0.4),while the SF of basal slip is relatively low (approximately 0.2).As discussed in Chapter 4.3,non-basal 〈a〉 slips (especially prismatic 〈a〉 slip) are activated to generate width strain,and basal slip also contributes to thickness strain to a less extent.The calculation results agree well with microstructure observations.Besides,pyramidal〈c+a〉slip and{10-11}contraction twins both have high SFs (＞0.4),which indicates that they will be activated to accommodate thickness strain.It is worth noting that even though the SF of {10-12} extension twinning is quite low (approximately 0.03),several {10-12}extension twins are still observed in uniaxial loading tests.This is because {10-12} extension twinning is easily activated at room temperature compared with other deformation modes.It was reported that {10-12} extension twinning can still be activated even if the SF was very low under uniaxial loading test [44].Besides,the average SFs of basal slip and{10-12} extension twinning in RD uniaxial loading are a bit higher than TD,which is because of the tilted basal texture towards RD,as discussed in Chapter 4.3.

Biaxial stress states affect the activity of basal slip and non-basal 〈a〉 slip.Whenσx:σy=1:1,the SFs of prismatic〈a〉 slip and pyramidal 〈a〉 slip are 0.0754 and 0.1772,respectively,which are quite low compared with the values under uniaxial stress state.By contrast,the SF of basal slip is 0.2958,which is higher than that of uniaxial loading.But the SFs of pyramidal 〈c+a〉 slip (0.4786) and {10-11} contraction twinning (0.4669) are still high,especially the {10-11}contraction twinning whose SF becomes even higher.When AZ31 magnesium alloy sheet is biaxial loaded,positive strains are generated in both RD and TD,and more negative strain in ND is needed to compensate for positive deformation.Therefore,the activity of pyramidal 〈c+a〉 slip and {10-11} contraction twinning as well as the basal slip is higher under biaxial stress state.Recalling the discussion about {10-11}contraction twinning in Chapter 4.3,it can be inferred that the pyramidal 〈c+a〉 slip is mainly activated to accommodate thickness strain at the early stage of plastic deformation under biaxial stress state.When plastic strain becomes larger,{10-11}contraction twinning will be activated.Xia et al.[18]also found that {10-11} contraction twins were not activated until the strain reached 0.07 in biaxial tension.

As for basal slip and prismatic 〈a〉 slip,previous research reported that the {10-12} extension twins activated in biaxial loading tests made basal slip more active than prismatic 〈a〉slip.Consequently,prismatic 〈a〉 slip was seriously restrained by biaxial stress state and basal slip showed relative higher activity [18].In addition,the average SF of {10-12} extension twinning for every stress ratio (approximately 0.0131)becomes much lower than uniaxial stress state.This is because for basal texture,most grains have low angles between c-axis and ND,which causes low SF of {10-12} extension twinning in biaxial loading tests,as discussed in Chapter 4.3.It is also worth noting that when stress ratios get closer to uniaxial loading,SF of non-basal 〈a〉 slip increases significantly while that of basal slip decreases.Whenσx:σy=1:4 andσx:σy=4:1,the SFs of basal slip and non-basal＜a＞slips are quite close to those of uniaxial stress state.So,it can be concluded that biaxial stress state has significant impact on the activity of basal slip and non-basal 〈a〉 slips.In terms of the work hardening behavior under biaxial stress state,the restriction of prismatic 〈a〉 slip also suppresses the cross-slip of basal and prismatic 〈a〉 slips,leading to the suppression of stage III.Besides,the low activity of prismatic 〈a〉 slip also leads to high macroscopic work hardening rates.

5.Conclusions

In this study,a series of biaxial loading tests using cruciform specimens are conducted to investigate the differential work hardening behavior of AZ31 magnesium alloy sheets.The twin variants and slip systems under biaxial loading tests are systematically studied,with the aim of revealing how biaxial stress states change the deformation mechanism and result in differential work hardening.Several conclusions can be drawn as follows:

(1) The pyramidal 〈c+a〉 slip mainly accommodates the strain along ND at the beginning of biaxial deformation.When plastic strain becomes larger,{10-11} contraction twinning will be activated to accommodate the ND strain together.However,both pyramidal〈c+a〉slip and {10-11} contraction twinning are difficult to be activated at room temperature.So AZ31 magnesium alloy shows higher yield strength and lower maximum strain in biaxial loading tests.

(2) Biaxial stress state decreases the activity of non-basal〈a〉 slip,which further depresses the dynamic recovery.So,the macroscopic work hardening rates of biaxial loading tests are much higher than that of uniaxial loading,and the stage III of work hardening is also suppressed.

(3) Basal slip shows higher activity in biaxial stress state.And it also contributes to ND strain to a less extent in both uniaxial and biaxial deformation.{10-12} extension twinning is preferred to active in grains whose caxis is perpendicular to ND when the stress ratio isσx:σy=1:1.Though the SF of {10-12} extension twinning is very low in uniaxial and biaxial deformation,it can still be activated in both conditions.

(4) Biaxial tensile tests using the cruciform specimens with uniform gauge area can achieve more accurate biaxial stress state.While for the cruciform specimens with a locally thinned gauge area,the thick frame surrounding the thinned gauge area not only causes uneven distribution of strain and stress but also restrains the deformation of thinned gauge area.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors gratefully acknowledge the financial support from the National Key R&D Program of China(2018YFC0808800),the National Natural Science Foundation of China (Nos.51875398 and 51471116),and the Sichuan Science and Technology Program (2019ZDZX0001).