Origin of the Anomalous f10 12g Twinning during Tensile Deformation of Mg Alloy Sheet

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1 Materials Transactions, Vol. 49, No. 2 (28) pp to 28 #28 The Japan Institute of Metals Origin of the Anomalous f 2g Twinning during Tensile Deformation of Mg Alloy Sheet J. Koike*, Y. Sato and D. Ando Department of Materials Science, Tohoku University, Sendai , Japan In order to understand the origin of the anomalous twinning of the f2g type, rolled sheets of AZ3 Mg alloy were deformed at room temperature in tension along the rolling direction. An excellent correlation was found between f2g twinning tendency and basal dislocation slip activity. alculation of strain tensor indicated that the diagonal strain components associated with the localized basal slip can be canceled completely by the f2g twinning. The results led to the conclusion that anomalous f2g twins were formed to accommodate the strain incompatibility caused by localized basal dislocation slip. [doi:.2/matertrans.mra28283] (Received August 9, 28; Accepted September, 28; Published November 2, 28) Keywords: magnesium, twin, deformation. Introduction Deformation twinning in Mg alloys is known to be an important deformation mechanism. Frequently observed twins are single twins of f2g, fg, f3g and double twins of fg f2g and f3g f2g. Among these twins, the f2g twins have been investigated in detail because of the following characteristics. In the first, the critical resolved shear stress (RSS) of the f2g twinning is only 2 to 3 MPa, ) the second smallest next to the RSS of basal dislocation slip. ecause of the small RSS, the f2g twins are formed in the initial loading stage and influences yielding behavior. 2 ) In the second, the f2g twinning induces tensile strain parallel to the c axis and compressive strain perpendicular to the c axis. Therefore, the f2g twins are formed abundantly when tensile (or compressive) stress is applied in the parallel (or perpendicular) direction to the c axis and can contribute to substantial strain softening. 5,6,) In the third, the f2g twins introduce additional interfaces which can act as barriers for continuous dislocation slip, and contributes to strain hardening. 5, 3) In the fourth, the f2g twinning accompany the rotation of the c-axis by 86. This crystallographic rotation is a major mechanism of deformation-induced texture. Almost all studies of f2g twins have been performed when external loading direction was conveniently chosen so as to match the sign of external strain with that of f2g twinning strain. However in some cases, the f2g twins can be formed despite that the twinning strain has an opposite sign to the external strain. For example, the f2g twins are frequently observed when tensile test is performed in the rolling direction () of rolled Mg sheets. In this case, because of the basal texture of the rolled sheets, the external loading imposes compressive strain along the c axis of the textured grains. Apparently, this is in conflict with the observation of the f2g twins that produces a tensile strain along the c-axis. To date, this anomalous type of the f2g twinning has been considered to be formed either by unloading or by complex internal stress state. In the present work, we focus on this anomalous twin and show that the *orresponding author, koikej@material.tohoku.ac.jp {2} twins are formed during tensile loading to accommodate the anisotropic strain due to basal dislocation slip. The previous reports on the f2g twinning are now summarized in relation to the present work. Their main focuses were () the agreement between the f2g twinning strain and the externally imposed strain, and (2) the agreement of the f2g twinning shear with the Schmid factor rule. Yang et al. 4) investigated twin orientation in extruded bars and rolled plates deformed at 97 by channel die compression. Using X-ray diffraction and ESP analysis, twin orientation was experimentally determined and compared with calculated strain tensor of all six twin variants. Good agreement was found between the calculated strain and the externally imposed strain with a few exceptions. Twins of the exceptions produced the opposite sign of strain from those expected based on the external stress state. These twins are categorized as anomalous twins. Meanwhile, Nave et al. 5) performed channel die compression test on pure Mg rolled samples. The deformed samples were analyzed with ESP to investigate the twinning modes and their effect on microstructure development. In this work, they observed the f2g twins after compressive deformation along the normal direction of the rolled plate. Under this condition, external stress imposed compressive strain in the c-axis, while the f2g twinning strain was in the opposite direction. At the moment, this anomalous twin is considered to be formed during unloading or in response to anomalous internal stress states. Furthermore, arnett et al. 6) performed tensile test at room temperature on cast AZ3. ESP analysis was performed in situ during deformation in SEM. They observed a particular f2g twin that disappeared during loading and reappears during unloading. This twinning behavior was in opposite sense from the prediction according to the Schmid factor rule and was attributed to the inhomogeneity of the internal stress distribution. More recently, Jiang et al. 7) performed compression test at 5 on rolled AZ3 in the transverse direction (). Twinning behavior was observed and analyzed with ESP. Most twin variants could be explained consistently with the magnitude of the Schmid factor. However, when multiple twin variants were observed, some twins had much smaller Schmid factors than other

2 Origin of the Anomalous f2g Twinning during Tensile Deformation of Mg Alloy Sheet 2793 possible twin variants. This anomalous twinning was attributed to complex stress state of pre-formed twins. There have been some reports claiming a complete match with the Schmid factor rule. Godet et al. 8) used an extruded AM3 tube and performed compression test in the extrusion direction and tensile test in the hoop direction both at 2. ESP analysis of the twin orientation indicated that the f2g twinning satisfied the Schmid factor rule. Similar work was done in detail in AM3 and AZ3 at room temperature to 25 with the focus on texture development. 9) However, their claim is based on the change of the orientation distribution in pole figures before and after deformation. It is possible that the formation of the anomalous twin might be missed in the macroscopic texture representation with the pole plot. Otherwise, deformation temperature is high enough so that the inhomogeneity of the internal stress distribution might not exit. To date, there have been only indirect evidences of the formation of anomalous f2g twinning during tensile deformation parallel to the basal plane that produces the opposite strain to the external strain. It was not clear to claim whether the anomalous twins are formed during unloading or during loading. Incidentally, recent work by Jain et al. 2) provides a clear answer to this problem. They employed in situ neutron diffraction during tensile test of AZ3 rolled sheet samples. The development of the anomalous twins was indeed observed during tensile deformation in the. In order to consistently simulate the flow behavior with the visco-plastic self-consistent (VPS) model, the occurrence of the anomalous f2g twinning was indicated to be necessary. 2) Now, the anomalous f2g twinning appears to be caused by some unknown internal stress state. The present paper shows that the anomalous twinning is induced by anisotropic strain accumulation by basal dislocation slip in particular type of grains. 2. Experimental Procedure Samples were rolled sheets of AZ3-O (Mg-3Al-Zn in mass%) having an average grain size of 3 mm. The asreceived samples were further cold-rolled at room temperature with a thickness reduction of 2% at each rolling step to a total reduction of 6%. Final sample thickness was mm. Then the samples were annealed in Ar atmosphere at 4 for 4 h. After annealing, the grain size was increased to the range of 3 to 4 mm. The crystallographic texture of the rolled and annealed samples was obtained by taking X-ray pole figure of the (2) planes, using a Schultz reflection method in the angle range of 2 to 9. As shown in Fig., the basal pole of the annealed sample was centered around the ND direction, indicating a characteristic basal texture to rolled Mg sheets. Tensile-test samples were machined with the tensile direction aligned parallel to the rolling direction (). Tensile test was performed at room temperature at a constant cross-head speed with an initial strain rate of : 3 s. Tests were terminated at a nominal strain of 5,, 5, 2% and up to fracture. After tensile test, surface microstructure was observed with an optical microscope. The observed samples were mechanically ground with Si papers to #4 grit, followed Fig. by buffing with.25 mm diamond pastes. The samples were then chemically etched with a mixture of 6% picric acid ethanol solution 7 ml, acetic acid ml, purified water ml. After manually tracing twin interfaces, the traced images were digitized to calculate the area fraction of twins using an NIH software, Image J. Note that the twin area fraction include not only the f2g type but other types. Orientation images were obtained with a field-emission scanning electron microscope (FE-SEM) attached with a TexSEM OIM system by analyzing electron backscatter patterns (ESP). The gage section of the tensile tested samples was cut to a small piece of mm in length and mechanically ground and buffed as was done for optical microscope observation. Surface was then chemically polished with a solution of hydrochloric acid ml and absolute ethanol 2 ml. Samples were polished in the solution for approximately 2 seconds, rinsed in absolute ethanol and dried with an air blower. ESP orientation maps were recorded in an area of 2 2 mm 2 with a beam step of.5 mm. When necessary, detailed ESP maps were recorded with a beam step of. to.25 mm. Grain dilation procedure was carried out only once so as not to artificially alter the orientation of fine microstructural features. 3. Results X-ray intensity distribution of () pole of AZ3 Mg alloy. Figure 2 shows optical micrographs of deformed samples to nominal strains of.5,.,.5 and.2. Wide lenticular twins are formed up to a strain of. and their area fraction increases with increasing strain. Notice that most of the lenticular twins are elongated approximately in the direction, corresponding to the fact that the twinning planes are facing towards the direction. Further deformation to.5 and.25 induces the formation of narrow banded twins. These banded twins have their twinning planes in different directions from the lenticular twins. In our previous works, 6,) we identified the lenticular twins as f2g twins. The narrow twins are mainly fg twins and fg f2g double twins. Since the present work focuses its attention to the formation mechanism of {2} twins, the

3 2794 J. Koike, Y. Sato and D. Ando ε =.5. 5µm.5.2 Fig. 2 Optical micrographs of deformed samples in tension along the to the indicated values of strain. following analysis is performed in the samples deformed to " ¼ :5 ad.. Figure 3 shows ESP orientation maps of the deformed samples to (a) " ¼ :5 and (b) " ¼ :. The maps are taken in regions including twinned grains. The inset triangle shows the color code of various pole directions. A strong red color represents the basal poles being parallel to the paper plane normal direction (ND). As the red color weakens or changes to different colors, the basal pole deviates from the ND according to the color code. Notice that the grain dilation procedure is performed only once, so that grain boundaries and highly strained regions can be identified by multiple arrays of small hexagonal dots having various colors. For easy 3-D viewing of the orientation distribution, a hexagonal unit cell is shown in correspondence with the measured Euler angle. In Fig. 3(a), the central grain has a f2g twin shown by green color. The matrix of the twinned grain has a light beige color indicating the deviation of the basal pole from the ND direction. In Fig. 3(b), a few grains contain f2g twins indicated by green color. Though it may not be as straightforward as in Fig. 3(a), the matrix color of the twinned grains indicates the deviation of the basal pole from the ND. These results together with the results of many other observations suggest that the f2g twinning occurs preferentially in the grains with large c-axis tilting from the ND when these grains are surrounded by grains with small c-axis tilting. It is noted that Nave et al. 5) also found that the anomalous twinning occurs within grains that have initial orientations very different from the dominant texture. The crystallographic information of the twinned grains is investigated in ten to twenty different grains with the observed directions parallel to the ND and the. In the case of observation along the ND, the ND surface of the deformed sample was mechanically and chemically polished to reveal the surface information. In the case of observation along the, the deformed sample was cut in half along the, followed by mechanical and chemical polishing to reveal the information inside of the sample. Figure 4 shows () pole figures of the deformed samples to " ¼ :5 and to., indicating the distribution of the basal pole of the matrices of the twinned grains with open symbols and the twins themselves with closed symbols. In addition, circle symbols are for the observation along the ND direction to reveal the information from the surface area, while triangle symbols are for the observation along the direction to reveal the information from the internal area. The obtained results provide two important informations. In the first, the matrix of the twinned grains have their basal poles deviated extensively from the center of the pole figure, namely the ND. omparison of Figs. and 4 clearly shows that the basal poles of the twinned grains do not represent the majority of grains. They are substantially tilted along the that is parallel to the tensile loading direction. The obtained results suggest that the f2g twinning occurs preferentially in the

4 Origin of the Anomalous f2g Twinning during Tensile Deformation of Mg Alloy Sheet 2795 (a) (a) 2 µm (b) (b) 2 µm Fig. 3 ESP orientation maps of the deformed samples to (a) " ¼ :5 and (b).. grains where basal dislocation slip occurs more easily than in the surrounding grains. In the second, the basal poles of the twins are distributed around the. Exception can be found only in two cases in the observation along the ND. These exceptional cases are indicated by arrows. The twinning accompanies the rotation of the c-axis from the ND to the (ND- twins), which brings about compressive strain in the and tensile strain in the ND. The ND- twins were also observed by Jain et al. 2) during tensile test of AZ3 rolled sheets and confirmed by in-situ neutron observation to be formed during tensile loading but not unloading. The exceptional cases accompanies the rotation of the c-axis from the ND to the (ND- twins), which brings about compressive strain in the direction and tensile strain in the ND direction. onsidering the direction of the compressive strain, the ND- twins are in the opposite sense, and most likely formed during unloading after tensile test. Since we are interested in the deformation twins, further consideration is given only for the ND- twins. In order to better understand the relation between twinning tendency and basal slip tendency, the number fraction of twinned grains was determined by ESP analysis. At the Fig. 4 asal pole distribution of the twinned-matrix (open symbols) and the twins (closed symbols) of the deformed samples to (a) " ¼ :5 and (b).. same time, the matrix of the twinned grains was further analyzed to obtain Schmid factors for basal slip. One may consider that Schmid factor for f2g twinning in the matrix grains should also be evaluated. However, as mentioned in the introduction, resolved shear stress of the external tensile stress is in the opposite direction to induce f2g twinning. Since external stress does not induce the {2} twinning, it is meaningless to evaluate the Schmid factor for the f2g twinning. The correlation of the twinning tendency and the Schmid factor is shown in Fig. 5. The horizontal axis represents the Schmid factor of basal slip in the matrix of the twinned grains. The vertical axis represents the number fraction of the twinned grains of a given value of the Schmid factor with respect to the the total twinned grains. In the deformed sample to " ¼ :5, the number of twinned grains were 65 among the total analyzed grains of 998. In the deformed sample to " ¼ :, the number of twinned grains were 66 among the total analyzed grains of 553. If the vertical axis was multiplied by 65/998 or by 66/553, the vertical axis would represent the number fraction of the twinned grains of

5 2796 J. Koike, Y. Sato and D. Ando Fraction of twinned grains.3.2. ε= Schmid factor of basal slip ε=.5.5 Fig. 5 Fraction of twinned grains having a given value of basal-slip Schmid factor with respect to the total number of the twinned grains at tensile strain of.5 and.. Star symbols are the calculated results derived in the discussion section. (a) (b) τ τ twin twin Fig. 6 (a) Slip-induced twinning in the neighboring grain by concentrated stress of the piled-up dislocations (after Meyers et al. 24) ). (b) Slip-induced twinning in the same grain to accommodate strain compatibility with the surrounding grains. a given Schmid factor with respect to the total analyzed grains, instead of with respect to the total twinned grains. Whichever the case, the tendency is the same and we choose to use the number of all twinned grains as our reference standard. Later in discussion, we will find it useful to understand the origin of the twinning. As shown in Fig. 6, the twin fraction increases with increasing the basal-slip Schmid factor, m. A rapid increase in the twin fraction is observed at m > :3 for " ¼ :5 and at m > :5 for " ¼ :. Twinning occurs in the grains of the smaller Schmid factors with increasing strain from.5 to.. The obtained results indicate that twinning tendency increases with increasing the activity of the basal slip in the corresponding grain. 4. Discussion ε slip A slip ε + ε A grain grains ε slip A twin Fig. 7 Schematic illustration showing the difference of slip-induced strain (" slip ) between the A grain and the surrounding grains. The difference can be accommodated by the formation of twinning-induced strain (" twin ). The present results clearly indicate that the f2g twinning is induced by inhomogeneous activity of basal dislocation slip. The f2g twinning occurs when basal dislocation slip occurs more easily in the twinning grain than in the surrounding grains. The slip-induced twinning has been investigated extensively in cubic metals. ) Figure 6(a) shows a schematic illustration of slip-induced twinning after Meyers et al. ) In this model, they considered dislocation pile-up at grain boundaries, giving rise to a concentrated stress field in a neighboring grain. When the magnitude of stress concentration becomes larger than the nucleation stress of a twin, a twin is considered to be formed at an adjacent grain. In other words, deformation twins are formed in accordance with the external stress aided by the concentrated stress due to dislocation pile-up. Thereby, shear strain is continuously transferred from the dislocation-slipped grain to the twinned grain. Since the number of the piled-up dislocations increases with grain size, this model can provide qualitative explanation of the grain-size dependence of deformation twinning. In contrast to this model, the present results indicate that the twins are formed in the grain where basal dislocation slip is active, as shown in Fig. 6(b). Moreover, the twins are formed with no reasonable accordance with the sign of external stress. An alternative model is necessary to explain the f2g deformation twinning in Mg alloys. A new model has to be consistent with three major observation of the f2g twinning. In the first, the f2g twinning gives rise to no strain in the direction of external tension. In the second, the f2g twinning accompanies the rotation of the c-axis direction from the ND to the, which gives rise to compressive strain in the and tensile strain in the ND. In the third, the f2g twinning occurs in a grain of active basal slip surrounded by grains of nearly nonactive basal slip. A possible model is given as follows. As

6 Origin of the Anomalous f2g Twinning during Tensile Deformation of Mg Alloy Sheet 2797 schematically illustrated in Fig. 7, let us suppose one grain (A) having a tilted c-axis towards the tensile direction and surrounding grains () having their c-axis parallel to the ND. In the A grain, basal slip and prismatic slip are active and produces a strain tensor of " slip A. On the other hand in the grains, basal slip is not active or active to a limited extent and prismatic slip is mostly active to produce a strain tensor of " slip. The difference in the basal-slip activity brings about strain difference between " slip A and " slip at grain boundaries. This strain difference may be accommodated by the formation of the f2g twins in the A grain with its strain tensor of " twin A. The total strain in the A grain of "A slip þ "A twin is now fit with the strain in the grains of " slip. Using this model, we attempt to qualitatively explain the relationship and its origin between the twinning tendency and the basal slip tendency shown in Fig. 5. In addition to the basal slip and the twinning, other active deformation modes should be taken into consideration. In a different aspect from twinning, it has also been reported that stress concentration caused by the strain incompatibility can enhance the activity of prismatic dislocation slip. 24) The enhanced prismatic slip has been experimentally observed and quantitatively discussed. 25) It is noted that the prismatic slip produces tensile strain in the tensile direction and compressive strain in the direction. This strain condition is in accordance with macroscopic strain produced by uniaxial tension, and is an independent deformation mode from twinning. Thus, " slip A and " slip are now considered to contain both basal slip and prismatic slip as " slip ¼ " basal þ " prism : ðþ Here, and are constants and will be explained later. The superscripts of A and are omitted for now to represent strain tensor in both types of grains. Although pyramidal slip may occur, its contribution has been reported to be very small 3) and is neglected from further consideration. If the condition, " slip ¼ " slip "A slip ¼ "A twin, holds, strain difference disappears at the grain boundary between A and. For convenience, we introduce a strain tensor ratio of ½RŠ ij as, ½RŠ ij ¼ ½"A twin Š ij : ð2þ ½" slip Š ij When the strain tensor ratio, ½RŠ ij, is equal to unity, strain difference disappears. In order to calculate the strain tensor ratio, the following conditions are assumed, based on the common knowledge of rolled Mg sheets as well as on the obtained results in the present work. () The a-axis and c-axis of the A grain are deviated from the and the ND, respectively, by various angles of to45 rotated around the. (2) The a-axis of the grains is parallel to the and the c-axis is parallel to the ND. (3) The f2g twinning accompanies tilting of the c-axis towards the by 86. (4) The volume fraction of the twins in the A grain is varied in the range of c ¼ =, /5, /3 and /2. To begin with, we define an orthogonal coordinate of ½2Š as the x-axis or, ½Š as the y-axis or, and [] as the z-axis or ND. This geometry represents a typical texture of the rolled Mg sheets. In this case, strain tensor of the six variants of the f2g twinning is given for a unit shear strain by, ½twinŠ ij :998 :645 A; :645 :998 :748 :432 :432 :25 :3 A; :559 :3 :998 :748 :432 :432 :25 :3 A :559 :3 :998 Next, strain tensor of the basal slip is given for a unit shear strain by, :5 ½basalŠ ij :866 A; :5 :866 :866 A: ð4þ :5 :866 Finally, strain tensor of the prismatic slip is given for a unit shear strain by, :866 :5 ½prismŠ ij :5 :866 A; :866 :5 :866 A: ð5þ Although six variants are available for twinning, basal slip and prismatic slip, only one strain tensor is sufficient for each deformation mode under the condition of the rotation by twinning and of the tensile deformation along the x-axis. Thus the relevant strain tensors for the calculation are ½twinŠ ij :998 :645 A; :645 :998 ½basalŠ ij A; :866 :5 ½prismŠ ij :5 :866 A ð6þ Now, the tilting of the c-axis towards the direction in the A grain can be considered by introducing a rotation matrix around the y-axis by an angle A as, ½rotð A ÞŠ ij cos A sin A sin A cos A ð3þ A: ð7þ

7 2798 J. Koike, Y. Sato and D. Ando Schmid factor, mbasal Strain tensor ratio, [R] ij c =/ 2 3 c =/ Tilt angle, θa 2 3 c =/5 2 3 c =/2 / degree Fig. 8 Strain tensor ratio at a tensile strain of " ¼ :5 for various tilting angle of the c-axis in the A grain. The gray dotted circles indicate ½RŠ ij ¼. Then, the twinning strain in the A grain is given as, ½" twin A Š ij ¼ ½rotð A ÞŠ mn ½twinŠ pq : ð8þ where is a twinning strain of.29. The slip strain in the A grain is given by the sum of the basal slip strain and the prismatic slip stain as, ½" A slip Š ij ¼ A m A basal ½rotð AÞŠ mn ½basalŠ pq þ A mprism A ½rotð AÞŠ mn ½prismŠ pq : ð9þ Here, m is the Schmid factor of either basal or prismatic slip system under a given rotation angle of A. The parameters of and are some constants and will be quantitatively determined later. The superscript of A represents the A grain. If twinning occurs in the A grain by a volume fraction of c, the total strain in the A grain is written as follows by assuming that the twinning strain is evenly distributed in the A grain. ½" A Š ij ¼ c½" twin A Š ij þð cþ½" slip A Š ij ðþ In the case of the grains, no twinning is considered and the slip strain is equivalent to the total strain in the grains. For the texture in consideration, the c-axis in the grains being perfectly parallel to the ND, the Schmid factor for the basal slip becomes zero. The total stain in the grains is, thus, given as, ½" Š ij ¼½" slip Š ij ¼ mprism ½prismŠ ij ðþ With regard to the parameters of A and A in the A grain, they are related to the magnitude of the basal and the prismatic slip. Agnew et al. quantitatively analyzed the contribution of the basal slip and the prismatic slip during uniaxial tension of an AZ3 rolled sheet using in situ neutron diffraction and reported their relative activity to be nearly the same after macroscopic yielding. 3) The substantial activity of the prismatic slip was also reported in AZ6 rolled sheets at room temperature. 25) Therefore, we can assume the following relation for the same relative dislocation activity, A mbasal A =A mprism A ¼ : ð2þ For example, when the c-axis of the A grain is tilted by 45, the Schmid factors become m A basal ¼ :5 and ma prism ¼ :43. This gives the relation, A = A ¼ :43=:5. Therefore, depending on the tilting angle of the c-axis in the A grain, the Schmid factors are calculated and the A = A ratio is determined accordingly.

8 Origin of the Anomalous f2g Twinning during Tensile Deformation of Mg Alloy Sheet 2799 Schmid factor, mbasal Strain tensor ratio, [R] ij c =/ c =/5 3 c =/ Tilt angle, θa c =/2 / degree Fig. 9 Strain tensor ratio at a tensile strain of " ¼ : for various tilting angle of the c-axis in the A grain. The gray dotted circles indicate ½RŠ ij ¼. In addition, strain in the tensile direction should be homogenous and equal in both the A grain and the grains. This requires the following equality condition of the strain tensor components between the A grain and the grains at any given twin volume fraction of c. Moreover, the strain components should be equal to tensile strain. For quantitative evaluation, the tensile strain values of.5 and. are chosen. Since ½" twin A Š ¼ as in eq. (6), the equality condition is given as, ð cþ½" slip A Š ¼½" slip Š ¼ :5 or :: ð3þ Using eqs. (2) and (3), definite values of and can be calculated for a given twin volume fraction of c. Once and are determined, slip strain in eq. (6) can be calculated for the A grain and the grains for various tilt angles of the c axis, which enables us to calculate the incompatibility strain of " slip. Then using eq. (2), we can calculate the strain tensor ratio ½RŠ ij. The calculated results are shown in Fig. 8 for ½"Š ¼ :5 and in Fig. 9 for ½"Š ¼ :. The four figures at each case are the results of the different twin volume fractions of /, /5, /3 and /2. In the figures, diagonal strain components of " and " as well as non-diagonal strain components of " 2, " 3, and " are plotted as a function of the c-axis tilt angle in the A grain. The corresponding Schmidt factors are given in the upper horizontal axis. As shown in Fig. 8 at a tensile strain of.5, the strain tensor ratio takes a value of for the diagonal strain components but not for the non-diagonal strain components. This result indicates that the f2g twinning makes the diagonal strain tensor components compatible between the A grain and the grains. On the other hand, the f2g twinning cannot make the non-diagonal components equal. This incompatibility of the non-diagonal components may be accommodated by the emission of other dislocations or may become a driving force to induce de-twinning upon unloading. It is noted that the compatibility condition of ½RŠ ii ¼ is satisfied at the Schmid factor of.29 for the twin volume fraction of / and of.43 for the twin volume fraction of /5. Figure 9 shows the calculated results for a large tensile strain of.. In agreement with Fig. 8, the compatibility condition is satisfied only for the diagonal strain components but not for the non-diagonal strain components. The compatibility condition is satisfied approximately at a Schmid factor of.2 for the twin volume fraction of / and changes to larger values of the Schmid factor with increasing the twin volume fraction. The obtained values of the Schmid factor under the given values of twin volume

9 28 J. Koike, Y. Sato and D. Ando fraction are plotted in Fig. 5 for tensile strain of.5 with four-point stars and. with five-point stars. An excellent agreement can be found between the experiment and the calculation. In a qualitative manner, the above discussion can be understood as follows. In the A grain, both basal slip and prismatic slip can occur, while in the grain only prismatic slip can occur. The basal slip reduces the prismatic slip in the A grain to a smaller magnitude than in the grain. Note that the basal slip induces tension in the and a contraction in the ND. Prismatic slip induces tension in the and contraction in the. Therefore with respect to the grains, basal slip in the A grain induces ND contraction and lesser magnitude of prismatic slip induces tension. When twinning occurs in the A grain, it can induce reverse strain of ND tension and contraction, which can cancel the difference of diagonal strain components. 5. Summary The rolled sheets of AZ3 Mg alloy having a strong basal texture were deformed in uniaxial tension at room temperature. The formation behavior and the crystallographic information of the f2g twins were investigated with an optical microscope and with an SEM-ESP. The f2g twins were found to be formed in grains whose c-axis was tilted by larger angle from the ND direction than the surrounding grains. The f2g twinning accompanied the rotation of the c-axis from the ND to the direction. The f2g twinning tendency was found to increase with increasing the Schmid factor of the basal slip. This indicated that the f2g twins are formed to minimize strain incompatibility caused by the greater activity of basal slip and the less activity of prismatic slip in the twinning grain than in the surrounding grains. This notion was confirmed by performing the calculation of strain tensors of basal slip, f2g twinning as well as prismatic slip for various values of the c-axis tilt angle and of the twin volume fraction. The calculated results indicated that there can be an appropriate twin volume fraction in a given value of c-axis tilting. Under the appropriate condition, diagonal strain components can be perfectly canceled by the f2g twinning. REFERENES ) R. E. Reed-Hill and W. D. Robertson: Acta Metall. 5 (957) 7. 2) S. R. Agnew, M. H. Yoo and. N. Tome: Acta Mater. 49 (2) ) S. R. Agnew,. N. Tome, D. W. rown, T. M. Holden and S.. Vogel: Scr. Mater. 48 () ). H. aceres, T. Sumitomo and Viedt: Acta Mater. 5 () ) M. R. arnett, Z. Keshavarz, A. G. eer and D. Atwell: Acta Mater. 52 (24) ) J. Koike: Metall. Mater. Trans. A 36A (25) ) D. W. rown, S. R. Agnew, M. A. M. ourke, T. M. Holden, S.. Vogel and. N. Tome: Mater. Sci. Eng. A 399 (25) 2. 8) S.. Yi,. H. J. Davies, H. G. rokmeier, R. E. olmaro, K. U. Kainer and J. Homeyer: Acta Mater. 54 (26) ) S. H. hoi, E. J. Shin and. S. Seong: Acta Mater. 55 (27) ) Y. N. Wang and J.. Huang: Acta Mater. 55 (27) ) L. Jiang, J. J. Jonas, A. Luo, A. K. Sachdev and S. Godet: Scr. Mater. 54 (26) ) L. Jaing, J. J. Jonas, A. Luo, A. K. Sachdev and S. Godet: Mater. Sci. Eng. A (27) ) X. Y. Lou, M. Li, R. K. oger, S. R. Agnew and R. H. Wagoner: Int. J. Plasticity (27) ) P. Yang, Y. Yu, L. hen and W. Mao: Scr. Mater. 5 (24) ) M. D. Nave and M. R. arnett: Scr. Mater. 5 (24) ) M. R. arnett, Z. Kechavarz and M. D. Nave: Metall. Mater. Trans. A 36A (25) ) L. Jiang, A. Godfrey, W. Liu and Q. Liu: Scr. Mater. 58 (28) 25. 8) S. Godet, L. Jiang, A. A. Luo and J. J. Jonas: Scr. Mater. 55 (26) ) L. Jiang, J. J. Jonas, R. K. Mishra, A. A. Luo, A. K. Sachdev and S. Godet: Acta Mater. 55 (27) ) A. Jain, O. Duygulu, D. W. rown,. N. Tome and S. R. Agnew: Mater. Sci. Eng. A 486 (28) ) A. Jain and S. R. Agnew: Mater. Sci. Eng. A A462 (27) ) D. Ando and J. Koike: J. Japan. Inst. Metals 7 (27) ) M. A. Meyers, O. Vohringer and V. A. Lubarda: Acta Mater. 49 (2) ) J. Koike, T. Kobayashi, T. Mukai, H. Watanabe, M. Suzuki, K. Maruyama and K. Higashi: Acta Mater. 5 () ) J. Koike and R. Ohyama: Acta Mater. 53 (25)