new model for twin-variant selection in hexagonal alloys

NEW MODEL FOR TWIN-VARIANT SELECTION IN HEXAGONAL ALLOYS

C. Schuman 1,a, L. Bao1,2, J.S. Lecomte1, M.J. Philippe 1, Y. Zhang 1,3, X. Zhao 2, L. Zuo2 , J.M. Raulot1 and C.Esling1. 1

LEM3, CNRS-UMR 7239, University Paul Verlaine - Metz, Ile du Saulcy, 57045 Metz, France 2 Key Laboratory for Anisotropy and Texture of Materials (Ministry of Education), Northeastern University, Shenyang 110004, China 3 Shenyang Aerospace University, Shenyang 110136, China a [email protected]

Keywords: titanium (Ti), plastic deformation mechanism, twinning, twin variant selection, Schmid factor. Abstract. A new selection criterion to explain the activation of the twinning variant is proposed. This criterion is based on the calculation of the deformation energy to create a primary twin. The calculation takes into account the effect of the grain size using a Hall-Petch type relation. This criterion allows to obtain a very good prediction for the variant selection. The calculations are compared with the experimental results obtained on T40 (ASTM grade 2) deformed by Channel Die compression. Introduction In the present work, we experimentally investigated the deformation process of commercially pure titanium with interrupted in situ SEM/EBSD orientation measurements [2,3]. Based on the experimental examination, we tried to work out the twin variant selection rule. This rule is not established by a mere statistical examination to have statistical representation but to reveal the physical and mechanical criteria (energy) of variant selection that would be useful for the modeling of the mechanical behavior of the alloys. Experimental procedure Material and Sample preparation The material was hot-rolled commercial pure titanium (ASTM grade 2) sheet of 1.5 mm thickness. Grain growth annealed was performed at 750°C for 2 hours to produce a fully recrystallized microstructure with a grain ranges from 150 to 250µm. After annealing, the samples were mechanically ground and then electrolytically polished (a 500×300 m2 area was carefully polished and marked out with four micro-indentations). The samples were channel die compressed with 8 % and 16% reduction. To follow the rotation of the individual grains during the deformation, the orientation of all the grains in this polished area (800 grains) was measured by SEM/EBSD before and after each deformation. Before compression, the two sample plates were firmly stuck together to avoid sliding during the compression in order to maintain a good surface quality [2,3]. Crystallography and identification It is well known that large grains favour the formation of twins during deformation. We have examined more than 80 grains in this individual follow-up and identified all the twin types, their variants and their order of appearance. We found only compressive twin ( 1122 or C Type) and tension twin ( 1012 or T1 type) (see table 1), and we did not find T2 or 1121 tension twin. The activation of these twins depends only on the initial orientation of the grain and the local stress tensor which is a function of the applied forces. Furthermore with the increase of the deformation, secondary twins appear inside primary twins: T1 inside C or C inside T1.

To identify the type of system and the active variants that accommodate the plastic deformation, trace analysis is used. The trace angles of all possible twin planes (Table1) on the grain surface are calculated with respect to the sample coordinate system. Then the trace angles of the observed twin planes are measured in the same coordinate system and compared with the calculated ones to identify the corresponding active twin system and twin Table 1: list of variants for two type of twins twin variant. Compression C Tension T1 1012 1011 We have followed step by step the onset of primary and CV1 1122 1123 T1V1 secondary twin. The results show that the twin variants 2112 2113 1102 1101 T1V2 that appear in the grain to accommodate the deformation CV2 are not necessary those with the highest Schmid factor CV3 1122 1123 1012 1011 T1V3 (SF). In fact, only less than 50% of the variants with the 0112 0111 T1V4 highest SF are selected. Often in the equiaxed grains, CV4 1212 1213 several twin variants can appear; whereas in those with 2112 2113 1102 1101 CV5 T1V5 elongated shape, only one variant appears, but it appears 0112 0111 repeatedly. This indicates that the shape of the initial CV6 1212 1213 T1V6 grain also influences the selection of the variants. We therefore decided to measure the length of each twin variant that appears and also calculate the maximum length of all variants that may occur to analyze the variant selection in terms of absorbed energy, as explained in the following. Deformation Energy [7] A variant will be activated if the energy of deformation which is used to create the twin was sufficient. We have considered here that the material is an ideal rigid- plastic body to calculate the energy of deformation. The plastic energy of deformation (by unit of volume) is calculated by the equation: WTwin   c

tw inf rame

  ij'  ij

sampleframe

.

(1) Where 'ijis the critical resolved shear stress required to activate the twinning system and ij is the corresponding twinning deformation (in the sample frame). In the case of channel die compression compressive force is applied in the sample normal direction (the third axis). In a grain, the stress applied to a twinning system is composed of the macroscopic applied stress and an additional local stress resulting from the interaction of the considered grain with the neighboring grains. Since we restrict to relatively small deformation degrees, we neglect the local stress resulting from the interaction with the neighboring grains. The stress applied to a twinning system is thus restricted to the macroscopic compressive stress 33, which corresponds to the Sachs (or static model) hypothesis. The twinning system will be active when the resolved shear stress reaches the corresponding critical value'33. When the twinning system is active, the corresponding deformation energy expressed in the macroscopic coordinate system is given by the above Eq. 1. We introduce the grain size effect by expressing the critical resolved shear stress according to a Hall Petch (HP) type equation [4]. Where  0 represent the stress when the length of the grain is infinite. L is the free path of the twin before encountering an obstacle (grain boundary, precipitate or other twins). Then the deformation energy can be expressed as: W  ( 0  k

L

) 33   0 33  k

L

 33

. (2) Taking into account that twinning is activated when the size of a grain exceeds a certain value below which only crystal glide can be activated, we can deduce that the term ε33 / 𝐿 of the equation is dominant. In the right hand term of Eq. 2,  33 and L , are accessible to the experiment. In the following we will focus on this term, ε33 / 𝐿 . Clearly, the length (L) of the free path of a twin lamella in a grain can be visualized with its boundary traces on the sample

observation plane. The maximum longitudinal length of the twin lamella appearing on the sample observation plane is determined as L for each twin variant. In the present work, the ε33 / 𝐿 term is calculated in the sample coordinate system. For the sake of simplicity, the displacement gradient tensor 𝑒𝑖𝑗 (which is restricted in titanium, for the two types of twins, [1] to e13= 0.218 in compression and e13= 0.175 in tension twinning) was first expressed in an orthonormal reference frame defined by the related twinning elements [5,6]. Through coordinate transformation, this displacement gradient tensor can be expressed in the crystal coordinate system. With the Euler angles measured by SEM/EBSD this tensor can be further transformed into the macroscopic sample coordinate system to obtain strain tensor (ij). Results and Discussion The plastic energy of deformation (by unit of volume) (Eq. 1) is always positive. In the case of compressive stress, to have Table 2: Deformation for two type of twin Grain (111.5; 20.5; 34.9) Grain (32.4; 77; 39.1) positive deformation energy, it is necessary to 33 33 33 33 have negative value of the deformation CV1 -0.104 TV1 0.07 CV1 0.057 TV1 -0.031 component. Table 2 shows the calculation of CV2 -0.084 TV2 0.072 CV2 0.001 TV2 -0.007 deformation for two families of twins and CV3 -0.048 TV3 0.076 CV3 0.096 TV3 -0.026 two different grains (In yellow, the variant CV4 -0.053 TV4 0.071 CV4 0.067 TV4 -0.073 which is actually active in the grain). In CV5 -0.089 TV5 0.077 CV5 -0.006 TV5 -0.005 CV6 -0.103 TV6 0.061 CV6 0.035 TV6 -0.080 addition, within a family, the variant which is active is that having the highest absolute ratio ε33 / 𝐿 (Table 3 and 4), but not necessary the highest SF (in red). In this table 4, secondary twin is treated as a primary twin. The twin can be regarded as a new grain with its own dimension and crystallographic orientation. In fact, SF gives less than 50% of prediction. With our criteria, we have more than 85 % of good prediction (95% in the case of the secondary twin). Table 3: Results for primary twin C twin CV1 CV2 CV3 CV4 CV5 CV6 T1 twin T1V1 T1V2 T1V3 T1V4 T1V5 T1V6

Orientation of grain:{111.5, 20.5, 34.9} SF Length(µm) 33 ε33 / 𝐿 -0.479 -0.104 85 -11.3 -0.383 -0.084 70 -10.0 -0.222 -0.048 95 -5.0 -0.242 -0.053 90 -5.6 -0.407 -0.089 70 -10.6 -0.475 -0.103 75 -11.9 Orientation of grain:{142.7, 85.9, 12.4} SF Length(µm) 33 ε33 / 𝐿 -0.022 -0.004 145 -0.3 -0.272 -0.048 145 -4.0 -0.019 -0.003 165 -0.3 -0.442 -0.077 140 -6.5 -0.263 -0.046 145 -3.8 -0.454 -0.079 130 -7.0

Table 4: Results for secondary twin T1C twin CV1 CV2 CV3 CV4 CV5 CV6 CT1 twin T1V1 T1V2 T1V3 T1V4 T1V5 T1V6

Orientation of grain:{64.4, 161.1, 11.7} Length(µm) 33 ε33 / 𝐿 -0.064 125 -5.7 -0.078 128 -6.9 -0.103 110 -9.8 -0.106 160 -8.4 -0.096 110 -9.2 -0.05 180 -3.7 Orientation of grain:{32.4, 77, 39.1} SF Length(µm) 33 ε33 / 𝐿 -0.175 -0.031 15 -7.9 -0.042 -0.007 15 -1.9 -0.151 -0.026 25 -5.3 -0.417 -0.073 35 -12.3 -0.028 -0.005 10 -1.6 -0.455 -0.08 50 -11.3

SF -0.295 -0.357 -0.471 -0.488 -0.441 -0.229

Each time when a twin forms, the size of the parent grain is modified and thus the apparent stress on each possible twin variant will change according to the HP law. As a result, variants which did not have a sufficient level of stress through the SF can nevertheless be active in a newly created domain (fig1a and table 5), despite the orientation of the initial grain not having changed. This explains why in equiaxed grains, several variants can appear. However, in the elongated grains, although the activation of the twin variant changes the dimension of the grain, it does not change the length of the free path of this variant. Thus this variant can form repeatedly as long as it does not create conditions more favorable for another variant. In such conditions, the twins activated can continue to grow until the whole grain is twinned out. A twin can thus be regarded as a new grain, which is a slightly different point of view than that of secondary twin. Generally, this new grain presents an elongated shape (at least at the early stage of its formation). Thus there will generally be one activated variant in this existing twin. In fact, this notion strongly depends on the size of the twin. Indeed, it was seen that when a grain is fully

consumed by twinning, there can be several variants thereafter. When several twin variants that do not cross the grain right through, form in one grain, the grain is divided into domains (fig1b). Table 5 : effect of domain ( fig 1) on the choice of the variant Orientation of grain:{65.3, 173, 49.2} SF Length(µm) 33 ε33 / 𝐿 -0.3864 -0.084 110 -8.03 CV1 -0.4607 -0.1 70 -12.00 CV2 -0.4891 -0.107 110 -10.17 CV3 -0.4755 -0.104 80 -11.59 CV4 -0.4263 -0.093 70 -11.11 CV5 -0.4072 -0.089 85 -9.63 CV6 Orientation of grain:{65.3, 173, 49.2} domain 2 Fig 1 C twin SF Length(µm) 33 ε33 / 𝐿 -0.3864 -0.084 55 -11.36 CV1 -0.4607 -0.1 70 -12.00 CV2 -0.4891 -0.107 55 -14.38 CV3 -0.4755 -0.104 50 -14.66 CV4 -0.4263 -0.093 70 -11.11 CV5 -0.4072 -0.089 50 -12.56 CV6 C twin

A domain is thus limited by twin boundaries and grain boundaries. Each domain has its own crystallographic orientation (generally that of the matrix) and its own dimensions. Currently, it is not possible to have an absolute prediction because interactions between neighbors (local field stresses and deformations) is not taken into account. In addition, we have a surface vision of a volume phenomenon.

Conclusion With the experimental examination, an energetic twin selection rule in titanium during deformation has been proposed. The prediction is correct in 85% of the cases (95 % in the case of secondary twin), whereas that according to Figure 1a): Domain effect on the b)CV2 : On the left domain 1, on variant selection the right domain 2 and the twin SF only in 50%. which is the domain 3. At the current stage, this rule has been verified with the (1012) and (1122) twins that are frequent in titanium and its alloys. The rule is : 1) The selection of the active twin system is made by the condition W>0. 2) If several twin families would be eligible (e.g. tension twins : T1 and T2), the system selects the family with the less microscopic work. Since the work to create T2 twin is higher than T1 according to the twinning shear (sT2=0.63 in comparison to sT1=0.175), T1is generally activate 3) Within a twin family, the variant selection is determined by the criterion that the absolute value of the ratio ε33 / 𝐿 be maximal. Taking account of the grain size seems to be one of the essential keys to explain the variant selection. More precisely, it will introduce the concept of domain, free path and a twin will be regarded as a-new-grain characterized by its size and crystallographic orientation. Acknowledgement: This work was supported by the Federation of Research for Aeronautics and Space (project OPTIMIST (optimisation de la mise en forme d’alliages de titane)). References [1] M. Yoo, Metallurgical and Materials Transactions A, (1981) 12(3): pp. 409-418. [2] L. Bao, J.-S. Lecomte, C. Schuman, M.-J. Philippe, X. Zhao, C. Esling, Advanced Engineering Materials, (2010) 12(10): pp. 1053-1059. [3] L. Bao, C. Schuman, J.-S. Lecomte, M.-J. Philippe, X. Zhao, L. Zuo, C. Esling, Computers, Materials, & Continua, (2010) 15(2): pp. 113-128. [4] M.A. Meyers, O. Vöhringer,V.A. Lubarda, Acta Materialia, (2001), 49(19): pp. 4025-4039. [5] É. Martin, L. Capolungo, L. Jiang, J.J. Jonas, Acta Materialia, (2010), 58(11): pp. 3970-3983. [6] J.J. Jonas, S. Mu, T. Al-Samman, G. Gottstein, L. Jiang, É. Martin, Acta Materialia, (2011), 59(5): pp. 2046-2056. [7] C.Schuman, L. Bao, J.S. Lecomte, M.J. Philippe, Y. Zhang, X. Zhao, L. Zuo, J.M. Raulot, C.Esling, Accepted in Advanced and Engineering Materials (2011).