- Email: [email protected]
Elastic properties of lamellar Ti–Al alloys
Computational Materials Science 47 (2009) 206–212
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Elastic properties of lamellar Ti–Al alloys Michael M. Gasik * Helsinki University of Technology - TKK, P.O. Box 6200, FIN-02015 TKK, Finland
a r t i c l e
i n f o
Article history: Received 9 June 2009 Accepted 23 July 2009 Available online 27 August 2009 Keywords: Elastic properties Anisotropy Ti–Al Rotation Micromechanical modelling
a b s t r a c t Ti–Al intermetallic alloys have a great potential for high-temperature lightweight applications. They also have rather complicated microstructure and sophisticated behaviour under thermal–mechanical loading. In many cases, Ti–Al component optimization is sufficient when using effective elastic properties of these alloys. Computation of such effective properties is difficult due to various orientations of the phases micro-constituents. Here a simple micromechanical model is applied to calculate effective stiffness and other elastic parameters of Ti–Al alloys as function of composition, orientation and temperature. The dependence of these properties is computed and possibility of simplified, quasi-isotropic or orthotropic properties application is discussed. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Modern aero-engines and their key components are expected to respond to extreme requirements concerning reliability, minimum weight, high performance, cost-efficiency and long durability with reduced noise and pollutant emissions. There is growing interest in new solutions like light intermetallic materials due to their low density and enhanced temperature capability. Their recent major drawback is brittleness especially at low temperatures [1]. Since in the early stages of development of TiAl alloys the application targets have been set for low-temperature usage, such as low-pressure turbine blades for jet engines, and exhaust valves for passenger vehicle engines [2]. Gamma-based two-phase titanium aluminides induces four different types of microstructures, i.e. near gamma, duplex, near lamellar, and fully lamellar [1,2]. Each of these microstructures exhibits its unique mechanical properties, which greatly depend not only on its solidification structures but also on the respective heat treatment. The presence of peritectic phase at an interdendritic region of primary a phase is mainly responsible for the development of near gamma or duplex microstructures [1]. The mechanical properties of these structures are greatly influenced by a dispersion of interdendritic c-phase [1,2]. The cast c-based (a2 + c) two-phase titanium aluminide alloys are the best potential candidates for engineering materials for high-temperatures applications. These alloys normally have a lamellar type microstructure composed of the c-TiAl phase (tetragonal L10) and a2-phase (non-stoichiometric Ti3Al, hexagonal D019) [3]. Due to crystallographic orientations, six major alignments of these two * Tel.: +358 9 4512769; fax +358 9 4512799. E-mail address: [email protected].fi. 0927-0256/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2009.07.009
phases are possible [3,4]. Due to variety of microstructural features and different scale of the constituents, design of components requires a proper engineering approximation of effective elastic properties on macro- and mesoscale. In this work, evaluation of elastic modulus and Poisson ratios was performed for different simulated microstructures with the aim of calculation of quasi-homogeneous properties, which might be used in further 3D FEM computations.
2. Modelling approach In the present work a simplified micromechanical model was used which is based on the compliance between strains under different virtual loading schemes [3,5,6]. Such models are being efficiently used in composites and multi-phase materials, including anisotropic ones like FGM [5,7,8], where apparent or averaged properties of the material are needed and detailed computation on the lattice or microscopic level is not feasible. It is known [6] that 2D models cannot generally represent 3D behaviour of heterogeneous composites and thus explicit 3D analysis is needed. Earlier attempts of applying explicit micromechanical models to TiAl-type alloys were mainly based on crystal plasticity (a single domain analysis) based on Schmid’s law [3,9] which has to be solved with numerical methods. Orientation variation of elastic properties was either experimentally measured using single crystals of c-TiAl [10] or combined with other numerical methods [4]. Explicit calculations for c-TiAl were made for single parameter such as Young modulus (1/S1111) for some symmetry directions only [10] and for whole set of the rotation in 3D [11]. Such analytical analysis for arbitrary ‘‘composite” lamellar structure was not performed.
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M.M. Gasik / Computational Materials Science 47 (2009) 206–212
Fig. 1. Lamellae layout in the LRVE used for modelling in this work.
Table 1 Starting elastic properties of single phases used from [4] as non-rotated and used after orientation rotation in this work. Phase
c-TiAl
Condition
Non-rotated
a2 Rotated
Non-rotated
Rotated
Elastic modulus, GPa 140 Exx 140 Eyy 135.4 Ezz
173.5 197.5 221.2
125 125 190.9
125.0 125.0 189.3
Shear modulus, GPa Gxy 78 105 Gyz Gzx 105
80.2 60.0 64.9
43 62 62
43 62 62
0.284 0.298 0.298
0.454 0.101 0.101
0.454 0.101 0.101
Poisson’s ratio
mxy myz mzx
0.284 0.2977 0.2977
In this work the focus is on providing engineering solution for predicting elastic properties of lamellar alloys in the whole applicable range of composition, rotation angles and temperatures. For lamellar structures the model could be simplified to transversely orthotropic material [12]. Here Y-direction is adopted as through the thickness of lamellae in the LRVE – local representative volume element (Fig. 1). Dark and light phases represent in the case of duplex TiAl alloys c-TiAl and a2-Ti3Al, respectively. The original (non-rotated) values of the elastic moduli and Poisson ratios are shown in Table 1 for room temperature for separate phases [4]. Based on these values, respective non-rotated compliance tensor could be calculated using standard procedures [12]. Data of Table 2 show that calculated stiffness values are well cor-
related with experimental data [10,13] (He et al. [10] have discussed the reason for differences between room temperature and the phases 0 K data for single crystals c-TiAl). The co-orientation of in the LRVE is specified by relationships: h111ic h0001ia2 , h110i c h1120ia2 [3,4,14,15]. Assembling of the effective compliance matrix does not generally possess a problem, but in this case the crystallographic orientation should be taken into account since both phases (LRVE constituents) are anisotropic. For the assembly of LRVE both lamellae should be property rotated [14,15]. Doing proper rotation of the compliance tensor in 3D space requires some care to specify proper order of manipulations. In this work the proprietary RotaStiff software [16] developed in Helsinki University of Technology (TKK) was applied to automate the procedure of these transformations versus specified [uvw] indices of the direction where the [1 0 0] direction of non-rotated tensor should be aligned after rotation, Fig. 2. After selecting target vector [uvw] for new X direction, W (yaw) and H (pitch) angles are determined. The first rotation is made around Z axis by W to get new set of axes (X1,Y1, Z) and after that around Y1 by H to get the next set (X2, Y1, Z2), where new X2 axis now aligned to set [uvw] direction [16]. The procedure could be repeated as many times as needed along new axes to get necessary combination of orientation vectors. For rotation method in Fig. 2, b the rotation cosine matrix Q is
0
1 cos W cos H sin W cos W sin H B C Q ðW; HÞ ¼ @ sin W cos H cos W sin W sin H A: sin H 0 cos H
ð1Þ
For example, by using Fig. 2, b notations, of the required alignments of c- and a2-phases from their original (non-rotated) values could be obtained by the following sequence (rotation angle sign is positive for anticlockwise direction):
9 9 8 8 > = w¼þp=4 > = < h1010i > < h1120i > ! ; h0110i h1100i > > > > ; ; : : h0001i a2 h0001i a2 9 9 9 9 8 8 8 8 h110i h100i > > > > = w¼þp=2 > = = h¼p=5 > = w¼þp=4 < < h112i > < h110i > < ! ! ! : h110i h010i h110i h112i > > > > > > > > ; ; ; ; : : : : h001i c h001i c h111i c h111i c ð2Þ The procedure of rotation described results in identical values if realized through Euler angles or other combination of operations, providing the selection of axes and rotation order are the same. The components of rotated stiffness tensor Crot are calculated from original stiffness C as:
C rot ijkl ¼
3 X 3 X 3 X 3 X
Q pi Q qj Q rk Q sl C ijkl :
ð3Þ
p¼1 q¼1 r¼1 s¼1
When target direction is varied, the angular dependence of elastic properties of single phases could be easily obtained for single phase constituent, but not for an arbitrary lamella structure of
Table 2 Comparison of elastic stiffnesses of c-TiAl at room temperature (GPa). Anisotropy parameters are: A0 = C66/C44, A1 = 2C66/(C11 C12) and A2 = 4C44/(C11 + C33 2C13). Parameter
C11
C33
C12
C13
C44
C66
A0
A1
A2
Theory Ti50Al50 at 0 K [13] Experiment Ti44Al56 [10] This work
190 186 186
185 176 182
105 72 77
90 74 78
120 101 105
50 77 78
0.42 0.76 0.74
1.18 1.35 1.43
2.46 1.89 1.98
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M.M. Gasik / Computational Materials Science 47 (2009) 206–212
Fig. 2. The rotation software screenshot (a) and the rotation scheme adopted in this work (b).
Fig. 3. Polar plots of Young and shear moduli of a2 phase for rotations for pairs (W, H) = (s, 0) shown as solid line and (p/4, s) as broken line for EXX (a), EYY (b), EZZ (c) and GXY (d). Angular dependence of moduli is weak due to hexagonal symmetry of a2 phase [14].
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M.M. Gasik / Computational Materials Science 47 (2009) 206–212
Fig. 4. Polar plots of Young and shear moduli of c phase for rotations for pairs (W, H) = (s, 0) shown as solid line and (p/4, s) as broken line for EXX (a), EYY (b), EZZ (c) and GXY (d). Angular dependence of moduli shows clear cubic symmetry of c phase [14].
Fig. 1 with varied volume fraction of phases. To do so, after separate rotation for every a2 and c lamellae towards required compliance, their stiffness components have to be added together according to the micromechanical model [5,7] for C22, C44 and C66 as follows:
C 0kk ¼
a2 C rot; kk
1þ
rot;c S Vc kk rot;a Skk 2 V a2
c C rot; kk
þ
1þ
rot;a2
Skk
V
ð4Þ
a2 rot;c Skk V c
and for all others Cij: c a2 C 0ij ¼ C rot; V a2 þ C rot; Vc ij ij
ð5Þ
where C’ij means specific stiffness component of the composite LRVE (Fig. 1) in compact notation, Vi is volume fraction of the respective phase (Va2 + Vc = 1). The different rule for C22, C44 and C66 is due to selected orientation of lamellae planes in the X–Z plane (Fig. 1). If other axes are chosen, these summation rules have to be respectively changed. Micromechanical mixing rules have such advantages that they could be analytically calculated, do not have fitting parameters, follow dilute estimate asymptotic behaviour and fit in Hashin–Striktman bounds [3,5–8]. Full Poisson matrix is filled from the known relationships [12]:
( rot
S
rot 1
¼ ðC Þ ;
rot ij
m ¼
rot Srot ij =Sjj
if i – j
0
if i ¼ j
Fig. 5. Plots of Young and shear moduli of a2 phase in module space for complete 2p rotations around all axes for EXX (a), EYY (b), EZZ (c) and GXY (d).
ð6Þ
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M.M. Gasik / Computational Materials Science 47 (2009) 206–212
Fig. 5 (continued)
It follows that as expected for anisotropic materials, mij – mji [12]. Both rotated compliance and stiffness matrixes remain symmetric, but in general case they have non-zero off-diagonal ele-
ments for arbitrary selection of [uvw]. Row normalization of compliance and Poisson matrixes is used here as recommended by Tsai [12]. The basic elastic data for single phases after rotation
Fig. 6. Plots of Young and shear moduli of c phase in module space for complete 2p rotations around all axes for EXX (a), EYY (b), EZZ (c) and GXY (d).
M.M. Gasik / Computational Materials Science 47 (2009) 206–212
(2) are shown in Table 1. For example, the Poisson matrix for a lamellar domain of 20% vol. c-TiAl phase with orientation compliance (2) may look as follows:
0
m20% c
0 0:421 0:183 0:028 B 0:416 0 0:133 0:039 B B B 0:127 0:094 0 0:009 ¼B B 0:062 0:086 0:028 0 B B @ 0 0 0 0 0
0
0
0
0 0 0 0 0 0:098
0
1
C C C 0 C C 0 C C C 0:075 A 0
0
3. Results and discussion Examples for a2 (Fig. 3) and c-TiAl (Fig. 4) single phase properties are shown as polar plots for two orientation sets, where s 2 [0; 2p] is running rotation angle (either pitch or yaw), depending on the selection. A better representation of the symmetry could be obtained when data shown in Figs. 3 and 4 are presented in 3D module space, similar to method used in [11], Figs. 5 and 6. Here clear symmetry class of elastic modules could be easily observed (in the col-
211
or version color gradation from blue to red is aligned with the direction of axes in question, i.e. X, Y, Z or XY, respectively). Elastic moduli of LRVE with different composition was calculated and visualized in 3D space. An example for composition with Vc = 0.7 is shown in Fig. 7, where both constituents were rotated first (Fig. 2) as (2) to get their mutual orientation and after that added by (4–5). As one might see, the elastic symmetry of the composite lamellae cannot anymore be clearly assigned to either hexagonal or cubic type. In polycrystals typical for lamella structures [3,4,9] orientations of different subdomains might be considered as statistically random so a quasi-isotropic characterization of elastic properties could be obtained vs. composition – volume fraction Vc in the alloy. Elastic modulus Eii = 1/Sii and shear moduli GXY = 1/S66, GXZ = 1/S55 and GXZ = 1/S44 are shown in Fig. 8 together with additional average (subscript ‘‘ave”) orthotropic approximation [10,14,15], which is based on averaging of bulk and shear moduli of the system, calculated in Voight and Reuss mixture rules. The results of these calculations are compared with other similar simulations [4], although the exact method of such was not explicitly reported in [4]. Note an excellent agreement between the calculations except for pure c-TiAl composition, which for data [4]
Fig. 7. Plots of Young and shear moduli of LRVE composed with 70% vol. of c phase in 3D module space for EXX (a), EYY (b), EZZ (c) and GXY (d).
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M.M. Gasik / Computational Materials Science 47 (2009) 206–212
Fig. 8. Elastic modulus E (a) and shear modulus G (b) for composite LRVE vs. volume fraction of c phase calculated in this work (lines) and in [4] as symbols.
and to extend evaluation for the whole specimen scale, simplifying numerical analysis. Although data for room temperature moduli were shown, substitution of proper data for high-temperature with exact known composition of the phases gives a possibility to re-run calculations and to obtain similar dependencies for other conditions. The generic algorithm used also is able to simplify calculations for similar lamellar systems of other composition providing the data for single phases and orientation are available.
Acknowledgements Support from EU and European Space Agency FP6 project ‘‘Intermetallic materials processing in relation to Earth and space solidification” (IMPRESS), contract No. NMP3-CT-2004-500635, is gratefully acknowledged. Fig. 9. Poisson ratios for composite LRVE vs. volume fraction of c phase calculated in this work (lines) and in [4] as symbols.
exhibits deviation from the general trend in some of moduli. There also some differences in Poisson ratios, Fig. 9. Unfortunately original data from [4] are not available so it was not possible to check their method of calculation. Calculated moduli in different directions could be assigned to specific lamella orientations in separate grains if microstructure of the specimen in known. The sum of grains i.e. the specimen bulk modulus could be obtained using standard micromechanical procedures described elsewhere [3,5–8], so anisotropy of the specimen at meso- and macroscale could be evaluated. If a rough estimation is sufficient, Hill’s values (orthotropic in respect to LRVE) could be used to mimic behaviour of single lamellar LRVE in numerical calculations because these values could be easily tabulated or approximated for fast retrieval during FEM processing. 4. Summary The micromechanical method was shown to be able to calculate elastic properties of lamella LRVE counting for phase anisotropy
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
[10] [11] [12] [13] [14] [15] [16]
J.Y. Jung, J.K. Park, C.H. Chun, Intermetallics 7 (1999) 1033–1041. T. Tetsui, Intermetallics 10 (2002) 239–245. S.M. Schlögl, F.D. Fischer, Comput. Mater. Sci. 7 (1996) 34–39. R.A. Brockmann, Int. J. Plast. 19 (2003) 1749–1772. M.M. Gasik, Comp. Mater. Sci. 13 (1998) 42–55. V. Romanova, R. Balokhonov, E. Soppa, S. Schmauder, Comput. Mater. Sci. 39 (2007) 274–281. S. Ueda, M.M. Gasik, J. Therm. Stresses 23 (2000) 395–409. M. Gasik, B. Zhang, Comput. Mater. Sci. 18 (2000) 93–101. C. Zambaldi, F. Roters, T. Bieler, D. Raabe, Micromechanical modeling of neargamma titanium aluminides, in: Presented at World Titanium Congress, Japan, 2007. Y. He, R.B. Schwarz, A. Migliori, S.H. Wang, J. Mater. Res. 10 (1995) 1187–1195. Y. He, R.B. Schwarz, T. Darling, M. Hundley, S.H. Whang, Z.M. Wang, Mater. Sci. Eng. A239–240 (1997) 157–163. S.W. Tsai, Composites Design, 4th ed., Think Composites, Dayton, 1988. C.L. Fu, M.H. Yoo, Philos. Mag. Lett. 62 (1990) 159–165. H.K.D.H. Bhadeshia, Worked Examples in Geometry of Crystals, 2nd ed., The Inst. of Metals, London, UK, 2001. D. Balzara, R.B. Von Dreele, K. Bennett, H. Ledbetter, J. Appl. Phys. 84 (1998) 4822–4833. T. Hirsi, M. Gasik, RotaStiff 1.0 software, Helsinki University of Technology – TKK, Espoo, Finland, 2006.
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Elastic properties of lamellar Ti–Al alloys Michael M. Gasik * Helsinki University of Technology - TKK, P.O. Box 6200, FIN-02015 TKK, Finland
a r t i c l e
i n f o
Article history: Received 9 June 2009 Accepted 23 July 2009 Available online 27 August 2009 Keywords: Elastic properties Anisotropy Ti–Al Rotation Micromechanical modelling
a b s t r a c t Ti–Al intermetallic alloys have a great potential for high-temperature lightweight applications. They also have rather complicated microstructure and sophisticated behaviour under thermal–mechanical loading. In many cases, Ti–Al component optimization is sufficient when using effective elastic properties of these alloys. Computation of such effective properties is difficult due to various orientations of the phases micro-constituents. Here a simple micromechanical model is applied to calculate effective stiffness and other elastic parameters of Ti–Al alloys as function of composition, orientation and temperature. The dependence of these properties is computed and possibility of simplified, quasi-isotropic or orthotropic properties application is discussed. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Modern aero-engines and their key components are expected to respond to extreme requirements concerning reliability, minimum weight, high performance, cost-efficiency and long durability with reduced noise and pollutant emissions. There is growing interest in new solutions like light intermetallic materials due to their low density and enhanced temperature capability. Their recent major drawback is brittleness especially at low temperatures [1]. Since in the early stages of development of TiAl alloys the application targets have been set for low-temperature usage, such as low-pressure turbine blades for jet engines, and exhaust valves for passenger vehicle engines [2]. Gamma-based two-phase titanium aluminides induces four different types of microstructures, i.e. near gamma, duplex, near lamellar, and fully lamellar [1,2]. Each of these microstructures exhibits its unique mechanical properties, which greatly depend not only on its solidification structures but also on the respective heat treatment. The presence of peritectic phase at an interdendritic region of primary a phase is mainly responsible for the development of near gamma or duplex microstructures [1]. The mechanical properties of these structures are greatly influenced by a dispersion of interdendritic c-phase [1,2]. The cast c-based (a2 + c) two-phase titanium aluminide alloys are the best potential candidates for engineering materials for high-temperatures applications. These alloys normally have a lamellar type microstructure composed of the c-TiAl phase (tetragonal L10) and a2-phase (non-stoichiometric Ti3Al, hexagonal D019) [3]. Due to crystallographic orientations, six major alignments of these two * Tel.: +358 9 4512769; fax +358 9 4512799. E-mail address: [email protected].fi. 0927-0256/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2009.07.009
phases are possible [3,4]. Due to variety of microstructural features and different scale of the constituents, design of components requires a proper engineering approximation of effective elastic properties on macro- and mesoscale. In this work, evaluation of elastic modulus and Poisson ratios was performed for different simulated microstructures with the aim of calculation of quasi-homogeneous properties, which might be used in further 3D FEM computations.
2. Modelling approach In the present work a simplified micromechanical model was used which is based on the compliance between strains under different virtual loading schemes [3,5,6]. Such models are being efficiently used in composites and multi-phase materials, including anisotropic ones like FGM [5,7,8], where apparent or averaged properties of the material are needed and detailed computation on the lattice or microscopic level is not feasible. It is known [6] that 2D models cannot generally represent 3D behaviour of heterogeneous composites and thus explicit 3D analysis is needed. Earlier attempts of applying explicit micromechanical models to TiAl-type alloys were mainly based on crystal plasticity (a single domain analysis) based on Schmid’s law [3,9] which has to be solved with numerical methods. Orientation variation of elastic properties was either experimentally measured using single crystals of c-TiAl [10] or combined with other numerical methods [4]. Explicit calculations for c-TiAl were made for single parameter such as Young modulus (1/S1111) for some symmetry directions only [10] and for whole set of the rotation in 3D [11]. Such analytical analysis for arbitrary ‘‘composite” lamellar structure was not performed.
207
M.M. Gasik / Computational Materials Science 47 (2009) 206–212
Fig. 1. Lamellae layout in the LRVE used for modelling in this work.
Table 1 Starting elastic properties of single phases used from [4] as non-rotated and used after orientation rotation in this work. Phase
c-TiAl
Condition
Non-rotated
a2 Rotated
Non-rotated
Rotated
Elastic modulus, GPa 140 Exx 140 Eyy 135.4 Ezz
173.5 197.5 221.2
125 125 190.9
125.0 125.0 189.3
Shear modulus, GPa Gxy 78 105 Gyz Gzx 105
80.2 60.0 64.9
43 62 62
43 62 62
0.284 0.298 0.298
0.454 0.101 0.101
0.454 0.101 0.101
Poisson’s ratio
mxy myz mzx
0.284 0.2977 0.2977
In this work the focus is on providing engineering solution for predicting elastic properties of lamellar alloys in the whole applicable range of composition, rotation angles and temperatures. For lamellar structures the model could be simplified to transversely orthotropic material [12]. Here Y-direction is adopted as through the thickness of lamellae in the LRVE – local representative volume element (Fig. 1). Dark and light phases represent in the case of duplex TiAl alloys c-TiAl and a2-Ti3Al, respectively. The original (non-rotated) values of the elastic moduli and Poisson ratios are shown in Table 1 for room temperature for separate phases [4]. Based on these values, respective non-rotated compliance tensor could be calculated using standard procedures [12]. Data of Table 2 show that calculated stiffness values are well cor-
related with experimental data [10,13] (He et al. [10] have discussed the reason for differences between room temperature and the phases 0 K data for single crystals c-TiAl). The co-orientation of in the LRVE is specified by relationships: h111ic h0001ia2 , h110i c h1120ia2 [3,4,14,15]. Assembling of the effective compliance matrix does not generally possess a problem, but in this case the crystallographic orientation should be taken into account since both phases (LRVE constituents) are anisotropic. For the assembly of LRVE both lamellae should be property rotated [14,15]. Doing proper rotation of the compliance tensor in 3D space requires some care to specify proper order of manipulations. In this work the proprietary RotaStiff software [16] developed in Helsinki University of Technology (TKK) was applied to automate the procedure of these transformations versus specified [uvw] indices of the direction where the [1 0 0] direction of non-rotated tensor should be aligned after rotation, Fig. 2. After selecting target vector [uvw] for new X direction, W (yaw) and H (pitch) angles are determined. The first rotation is made around Z axis by W to get new set of axes (X1,Y1, Z) and after that around Y1 by H to get the next set (X2, Y1, Z2), where new X2 axis now aligned to set [uvw] direction [16]. The procedure could be repeated as many times as needed along new axes to get necessary combination of orientation vectors. For rotation method in Fig. 2, b the rotation cosine matrix Q is
0
1 cos W cos H sin W cos W sin H B C Q ðW; HÞ ¼ @ sin W cos H cos W sin W sin H A: sin H 0 cos H
ð1Þ
For example, by using Fig. 2, b notations, of the required alignments of c- and a2-phases from their original (non-rotated) values could be obtained by the following sequence (rotation angle sign is positive for anticlockwise direction):
9 9 8 8 > = w¼þp=4 > = < h1010i > < h1120i > ! ; h0110i h1100i > > > > ; ; : : h0001i a2 h0001i a2 9 9 9 9 8 8 8 8 h110i h100i > > > > = w¼þp=2 > = = h¼p=5 > = w¼þp=4 < < h112i > < h110i > < ! ! ! : h110i h010i h110i h112i > > > > > > > > ; ; ; ; : : : : h001i c h001i c h111i c h111i c ð2Þ The procedure of rotation described results in identical values if realized through Euler angles or other combination of operations, providing the selection of axes and rotation order are the same. The components of rotated stiffness tensor Crot are calculated from original stiffness C as:
C rot ijkl ¼
3 X 3 X 3 X 3 X
Q pi Q qj Q rk Q sl C ijkl :
ð3Þ
p¼1 q¼1 r¼1 s¼1
When target direction is varied, the angular dependence of elastic properties of single phases could be easily obtained for single phase constituent, but not for an arbitrary lamella structure of
Table 2 Comparison of elastic stiffnesses of c-TiAl at room temperature (GPa). Anisotropy parameters are: A0 = C66/C44, A1 = 2C66/(C11 C12) and A2 = 4C44/(C11 + C33 2C13). Parameter
C11
C33
C12
C13
C44
C66
A0
A1
A2
Theory Ti50Al50 at 0 K [13] Experiment Ti44Al56 [10] This work
190 186 186
185 176 182
105 72 77
90 74 78
120 101 105
50 77 78
0.42 0.76 0.74
1.18 1.35 1.43
2.46 1.89 1.98
208
M.M. Gasik / Computational Materials Science 47 (2009) 206–212
Fig. 2. The rotation software screenshot (a) and the rotation scheme adopted in this work (b).
Fig. 3. Polar plots of Young and shear moduli of a2 phase for rotations for pairs (W, H) = (s, 0) shown as solid line and (p/4, s) as broken line for EXX (a), EYY (b), EZZ (c) and GXY (d). Angular dependence of moduli is weak due to hexagonal symmetry of a2 phase [14].
209
M.M. Gasik / Computational Materials Science 47 (2009) 206–212
Fig. 4. Polar plots of Young and shear moduli of c phase for rotations for pairs (W, H) = (s, 0) shown as solid line and (p/4, s) as broken line for EXX (a), EYY (b), EZZ (c) and GXY (d). Angular dependence of moduli shows clear cubic symmetry of c phase [14].
Fig. 1 with varied volume fraction of phases. To do so, after separate rotation for every a2 and c lamellae towards required compliance, their stiffness components have to be added together according to the micromechanical model [5,7] for C22, C44 and C66 as follows:
C 0kk ¼
a2 C rot; kk
1þ
rot;c S Vc kk rot;a Skk 2 V a2
c C rot; kk
þ
1þ
rot;a2
Skk
V
ð4Þ
a2 rot;c Skk V c
and for all others Cij: c a2 C 0ij ¼ C rot; V a2 þ C rot; Vc ij ij
ð5Þ
where C’ij means specific stiffness component of the composite LRVE (Fig. 1) in compact notation, Vi is volume fraction of the respective phase (Va2 + Vc = 1). The different rule for C22, C44 and C66 is due to selected orientation of lamellae planes in the X–Z plane (Fig. 1). If other axes are chosen, these summation rules have to be respectively changed. Micromechanical mixing rules have such advantages that they could be analytically calculated, do not have fitting parameters, follow dilute estimate asymptotic behaviour and fit in Hashin–Striktman bounds [3,5–8]. Full Poisson matrix is filled from the known relationships [12]:
( rot
S
rot 1
¼ ðC Þ ;
rot ij
m ¼
rot Srot ij =Sjj
if i – j
0
if i ¼ j
Fig. 5. Plots of Young and shear moduli of a2 phase in module space for complete 2p rotations around all axes for EXX (a), EYY (b), EZZ (c) and GXY (d).
ð6Þ
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Fig. 5 (continued)
It follows that as expected for anisotropic materials, mij – mji [12]. Both rotated compliance and stiffness matrixes remain symmetric, but in general case they have non-zero off-diagonal ele-
ments for arbitrary selection of [uvw]. Row normalization of compliance and Poisson matrixes is used here as recommended by Tsai [12]. The basic elastic data for single phases after rotation
Fig. 6. Plots of Young and shear moduli of c phase in module space for complete 2p rotations around all axes for EXX (a), EYY (b), EZZ (c) and GXY (d).
M.M. Gasik / Computational Materials Science 47 (2009) 206–212
(2) are shown in Table 1. For example, the Poisson matrix for a lamellar domain of 20% vol. c-TiAl phase with orientation compliance (2) may look as follows:
0
m20% c
0 0:421 0:183 0:028 B 0:416 0 0:133 0:039 B B B 0:127 0:094 0 0:009 ¼B B 0:062 0:086 0:028 0 B B @ 0 0 0 0 0
0
0
0
0 0 0 0 0 0:098
0
1
C C C 0 C C 0 C C C 0:075 A 0
0
3. Results and discussion Examples for a2 (Fig. 3) and c-TiAl (Fig. 4) single phase properties are shown as polar plots for two orientation sets, where s 2 [0; 2p] is running rotation angle (either pitch or yaw), depending on the selection. A better representation of the symmetry could be obtained when data shown in Figs. 3 and 4 are presented in 3D module space, similar to method used in [11], Figs. 5 and 6. Here clear symmetry class of elastic modules could be easily observed (in the col-
211
or version color gradation from blue to red is aligned with the direction of axes in question, i.e. X, Y, Z or XY, respectively). Elastic moduli of LRVE with different composition was calculated and visualized in 3D space. An example for composition with Vc = 0.7 is shown in Fig. 7, where both constituents were rotated first (Fig. 2) as (2) to get their mutual orientation and after that added by (4–5). As one might see, the elastic symmetry of the composite lamellae cannot anymore be clearly assigned to either hexagonal or cubic type. In polycrystals typical for lamella structures [3,4,9] orientations of different subdomains might be considered as statistically random so a quasi-isotropic characterization of elastic properties could be obtained vs. composition – volume fraction Vc in the alloy. Elastic modulus Eii = 1/Sii and shear moduli GXY = 1/S66, GXZ = 1/S55 and GXZ = 1/S44 are shown in Fig. 8 together with additional average (subscript ‘‘ave”) orthotropic approximation [10,14,15], which is based on averaging of bulk and shear moduli of the system, calculated in Voight and Reuss mixture rules. The results of these calculations are compared with other similar simulations [4], although the exact method of such was not explicitly reported in [4]. Note an excellent agreement between the calculations except for pure c-TiAl composition, which for data [4]
Fig. 7. Plots of Young and shear moduli of LRVE composed with 70% vol. of c phase in 3D module space for EXX (a), EYY (b), EZZ (c) and GXY (d).
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M.M. Gasik / Computational Materials Science 47 (2009) 206–212
Fig. 8. Elastic modulus E (a) and shear modulus G (b) for composite LRVE vs. volume fraction of c phase calculated in this work (lines) and in [4] as symbols.
and to extend evaluation for the whole specimen scale, simplifying numerical analysis. Although data for room temperature moduli were shown, substitution of proper data for high-temperature with exact known composition of the phases gives a possibility to re-run calculations and to obtain similar dependencies for other conditions. The generic algorithm used also is able to simplify calculations for similar lamellar systems of other composition providing the data for single phases and orientation are available.
Acknowledgements Support from EU and European Space Agency FP6 project ‘‘Intermetallic materials processing in relation to Earth and space solidification” (IMPRESS), contract No. NMP3-CT-2004-500635, is gratefully acknowledged. Fig. 9. Poisson ratios for composite LRVE vs. volume fraction of c phase calculated in this work (lines) and in [4] as symbols.
exhibits deviation from the general trend in some of moduli. There also some differences in Poisson ratios, Fig. 9. Unfortunately original data from [4] are not available so it was not possible to check their method of calculation. Calculated moduli in different directions could be assigned to specific lamella orientations in separate grains if microstructure of the specimen in known. The sum of grains i.e. the specimen bulk modulus could be obtained using standard micromechanical procedures described elsewhere [3,5–8], so anisotropy of the specimen at meso- and macroscale could be evaluated. If a rough estimation is sufficient, Hill’s values (orthotropic in respect to LRVE) could be used to mimic behaviour of single lamellar LRVE in numerical calculations because these values could be easily tabulated or approximated for fast retrieval during FEM processing. 4. Summary The micromechanical method was shown to be able to calculate elastic properties of lamella LRVE counting for phase anisotropy
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