C.Haase et al.Acta Materialia 80 (2014)327-340 329 Table 1 orientation distribution functions (ODFs)were calculated Chemical composition of the investigated alloy. in MTEX.The volume fractions of the corresponding tex- Element Fe Mn Al Si N & ture components were calculated using a spread of 15 (wt.%) Bal. 0.32522.461.210.0410.0150.01 from their ideal orientation. The Vickers hardness (ASTM E384-10e2)of the cold- rolled and annealed samples was examined using a Shima- 2.1.2.Specimens and characterization techniques dzu HMV microhardness tester with a load of I kg(HV1). Specimens with the dimensions 10 mm x 12 mm(trans- Ten indents per sample were performed. verse direction (TD)and rolling direction (RD),respec- The mechanical properties of the material in deformed, tively)were cut from the cold-rolled and annealed sheets. recovered and recrystallized condition were evaluated by The samples were then mechanically ground with 800, uniaxial tensile tests at room temperature and a constant 1200,2400 and 4000 SiC grit paper and mechanically pol- strain rate of 10-3s-along the previous rolling direction ished using a 3 um and I um diamond suspension.For on a screw-driven Zwick 1484 mechanical testing device. scanning electron microscopy (SEM)and electron back- Flat bar tension specimens were used with a gauge length scatter diffraction(EBSD)the RD-ND (ND:normal direc- of 44 mm,gauge width of 12 mm,fillet radius of 20 mm tion)section was electropolished at room temperature for and variable thickness depending on the rolling degree. 20s at 22 V.For texture analysis and hardness testing, the middle layer of the RD-TD section was electropolished 2.2.Simulation setup for 2 min at 22 V.The used electrolyte consisted of 700 ml ethanol (C2HsOH),100 ml butyl glycol (CH1402)and The model used for simulations is a finite element model 78 ml perchloric acid(60%)(HCIO).The same electrolyte (FEM)implementation of the analytical model described in was used for preparing the samples for transmission elec- Ref.[66].To achieve this,a number of modifications simi- tron microscopy(TEM).In order to reveal the microstruc- lar to those described in Ref.[67]were made.First,the dis- ture by SEM,the specimens were additionally etched at location cell structure was neglected.Second,all evolution room temperature using a 2%Nital solution (95 ml equations were rewritten in a per slip/twin system formula- C2HsOH and 5 ml HNO3). tion.Since the twin volume fraction is in the focus of this SEM and EBSD analyses were performed in a LEO work,the evolution equations used to calculate the twin 1530 field emission gun SEM operated at 20 kV accelerat- volume fraction are recalled in the following.The equa- ing voltage and a working distance of 10 mm.EBSD map- tions for the dislocation part of the model are rather stan- pings were generated with a step size of 0.28 um.The HKL dard and can be found in Ref.[68]. Channel 5 software was utilized for data post-processing The twinning process is split into two parts,namely twin and removal of wild spikes and non-indexed points,taking nucleation and twin growth.For the nucleation,the model at least five neighbor points into account.Furthermore, of Mahajan and Chin [69]is adopted.It relies on the EBSD mappings were subdivided into subsets including reaction of two dislocations to form a twin nucleus: only recrystallized (RX),recovered (RC)or deformed 号(01i)+号(10i)=3×g(1I2).The twin nucleation rate (DEF)grains using an algorithm of the MATLAB-based (NB)per twin system B is calculated by multiplying the total MTEX package [63,64].The internal grain/subgrain mis- number density of potential twin nuclei per unit time (p), orientation was determined using the grain reference orien- by the probability that a sufficient stress concentration for tation (GROD-AO)technique,which takes the average grain/subgrain orientation as a reference.An internal grain the formation of the nucleus exists(p),and by the prob- misorientation threshold of RX<1.5<RC<6<DEF ability that one of those nuclei grows into a twin (p): was used [65].Grains containing fewer than 10 data points NB pPncsPtw (1) were disregarded. In order to prepare the TEM samples,the initial speci- In the FEM implementation p and p are calculated analogous to Ref.[66].However,as slip systems are differ- mens were ground to a thin layer of ~100 um,from which disks 3 mm in diameter were stamped out.The disks were entiated,pr can be calculated individually for each twin system B based on the slip activity of the slip systems then electropolished using a double jet Struers Tenupol-5 involved: with a voltage of 29 V at 15C.The RD-TD sections of the final specimens were analyzed in a JEOL JEM 2000 p22+2p1 (2) FX II analytical TEM operated at 200 kV. P= Lo The crystallographic texture was characterized by means where,2 are the shear rates on slip system / of X-ray pole figure measurements.Three incomplete (0-85)pole figures,(111),(200}and {220),were p,p are the dislocation densities on slip system acquired at the mid-layer of the sheet thickness on a Bruker and Lo is the length of the sessile partial dislocations D8 Advance diffractometer,equipped with a HI-STAR area detector,operating at 30 kV and 25 mA,using filtered IIn Ref.[68]a slightly different formulation is used for the twin volume iron radiation and polycapillary focusing optics.The evolution.2.1.2. Specimens and characterization techniques Specimens with the dimensions 10 mm  12 mm (trans￾verse direction (TD) and rolling direction (RD), respec￾tively) were cut from the cold-rolled and annealed sheets. The samples were then mechanically ground with 800, 1200, 2400 and 4000 SiC grit paper and mechanically pol￾ished using a 3 lm and 1 lm diamond suspension. For scanning electron microscopy (SEM) and electron back￾scatter diffraction (EBSD) the RD–ND (ND: normal direc￾tion) section was electropolished at room temperature for 20 s at 22 V. For texture analysis and hardness testing, the middle layer of the RD–TD section was electropolished for 2 min at 22 V. The used electrolyte consisted of 700 ml ethanol (C2H5OH), 100 ml butyl glycol (C6H14O2) and 78 ml perchloric acid (60%) (HClO4). The same electrolyte was used for preparing the samples for transmission elec￾tron microscopy (TEM). In order to reveal the microstruc￾ture by SEM, the specimens were additionally etched at room temperature using a 2% Nital solution (95 ml C2H5OH and 5 ml HNO3). SEM and EBSD analyses were performed in a LEO 1530 field emission gun SEM operated at 20 kV accelerat￾ing voltage and a working distance of 10 mm. EBSD map￾pings were generated with a step size of 0.28 lm. The HKL Channel 5 software was utilized for data post-processing and removal of wild spikes and non-indexed points, taking at least five neighbor points into account. Furthermore, EBSD mappings were subdivided into subsets including only recrystallized (RX), recovered (RC) or deformed (DEF) grains using an algorithm of the MATLAB-based MTEX package [63,64]. The internal grain/subgrain mis￾orientation was determined using the grain reference orien￾tation (GROD-AO) technique, which takes the average grain/subgrain orientation as a reference. An internal grain misorientation threshold of RX < 1.5 < RC < 6 < DEF was used [65]. Grains containing fewer than 10 data points were disregarded. In order to prepare the TEM samples, the initial speci￾mens were ground to a thin layer of 100 lm, from which disks 3 mm in diameter were stamped out. The disks were then electropolished using a double jet Struers Tenupol-5 with a voltage of 29 V at 15 C. The RD–TD sections of the final specimens were analyzed in a JEOL JEM 2000 FX II analytical TEM operated at 200 kV. The crystallographic texture was characterized by means of X-ray pole figure measurements. Three incomplete (0–85) pole figures, {1 1 1}, {2 0 0} and {2 2 0}, were acquired at the mid-layer of the sheet thickness on a Bruker D8 Advance diffractometer, equipped with a HI-STAR area detector, operating at 30 kV and 25 mA, using filtered iron radiation and polycapillary focusing optics. The orientation distribution functions (ODFs) were calculated in MTEX. The volume fractions of the corresponding tex￾ture components were calculated using a spread of 15 from their ideal orientation. The Vickers hardness (ASTM E384-10e2) of the cold￾rolled and annealed samples was examined using a Shima￾dzu HMV microhardness tester with a load of 1 kg (HV1). Ten indents per sample were performed. The mechanical properties of the material in deformed, recovered and recrystallized condition were evaluated by uniaxial tensile tests at room temperature and a constant strain rate of 103 s 1 along the previous rolling direction on a screw-driven Zwick 1484 mechanical testing device. Flat bar tension specimens were used with a gauge length of 44 mm, gauge width of 12 mm, fillet radius of 20 mm and variable thickness depending on the rolling degree. 2.2. Simulation setup The model used for simulations is a finite element model (FEM) implementation of the analytical model described in Ref. [66]. To achieve this, a number of modifications simi￾lar to those described in Ref. [67] were made. First, the dis￾location cell structure was neglected. Second, all evolution equations were rewritten in a per slip/twin system formula￾tion. Since the twin volume fraction is in the focus of this work, the evolution equations used to calculate the twin volume fraction are recalled in the following. The equa￾tions for the dislocation part of the model are rather stan￾dard and can be found in Ref. [68]. 1 The twinning process is split into two parts, namely twin nucleation and twin growth. For the nucleation, the model of Mahajan and Chin [69] is adopted. It relies on the reaction of two dislocations to form a twin nucleus: a 2 h0 11i þ a 2 h1 01i ¼ 3  a 6 h1 12i. The twin nucleation rate (N_ b) per twin system b is calculated by multiplying the total number density of potential twin nuclei per unit time (pb st), by the probability that a sufficient stress concentration for the formation of the nucleus exists (ptw), and by the prob￾ability that one of those nuclei grows into a twin (pncs): N_ b ¼ pb stpncsptw ð1Þ In the FEM implementation ptw and pncs are calculated analogous to Ref. [66]. However, as slip systems are differ￾entiated, pst can be calculated individually for each twin system b based on the slip activity of the slip systems involved: pb st ¼ c_ a1qa2 þ c_ a2qa1 L0 ð2Þ where c_ a1, c_ a2 are the shear rates on slip system a1=a2, qa1, qa2 are the dislocation densities on slip system a1=a2 and L0 is the length of the sessile partial dislocations Table 1 Chemical composition of the investigated alloy. Element Fe C Mn Al Si N P (wt.%) Bal. 0.325 22.46 1.21 0.041 0.015 0.01 1 In Ref. [68] a slightly different formulation is used for the twin volume evolution. C. Haase et al. / Acta Materialia 80 (2014) 327–340 329