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Microstructural modeling of TiNi alloy high strain rate tension
Available online at www.sciencedirect.com
ScienceDirect Materials Today: Proceedings 4 (2017) 4637–4641
www.materialstoday.com/proceedings
SMA 2016
Microstructural modeling of TiNi alloy high strain rate tension Margarita Evarda, Alexander Motorinb*, Alexander Razova, Aleksandr Volkova Saint Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg, 199034 Russia bITMO University,49 Kronverksky Pr. St. Petersburg, 197101, Russia
a
Abstract Results of the microstructural modeling of high strain rate isothermal straining of an equiatomic TiNi alloy at different structural conditions using the formulae for strain rate dependencies of dislocation yield limit are presented. Simulations were conducted in the temperature range of 20–300°С. It was shown that the calculated values of the phase and dislocation yield stress in the martensitic and austenitic conditions with a good accuracy correspond to the experimental values. The mechanical behavior of the alloy in the entire investigated temperature range was described using only one pair of constants in strain rate sensitivity of dislocation yield limit. © 2017 Elsevier Ltd. All rights reserved. Selection and Peer-review under responsibility of The second conference “Shape memory alloys”. Keywords: TiNi; high strain rate; modeling; microstructural theory; tension.
1. Introduction Changes in the properties of various materials under the influence of high strain rate loading have attracted the interest of scientists and engineers since many years due to the significant practical importance of such research for civilian and special equipment. Alloys with shape memory effect are not an exception, particularly TiNi-based alloys, which have unique properties such as high corrosion resistance and strength, good deformation recovery indicators, high reactive stresses, excellent biocompatibility, and high damping capacity. These alloys have been successfully applied in numerous fields of technology and medicine [1–3]. In addition, shape memory alloys can be
* Corresponding author. Tel.: +7-921-758-1197; fax: +7-812-428-7079. E-mail address: [email protected] 2214-7853 © 2017 Elsevier Ltd. All rights reserved. Selection and Peer-review under responsibility of The second conference “Shape memory alloys”.
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subjected to high strain rate loading during technological operations to improve the functional properties [4] or to manufacture bimetallic materials with shape memory [5–6]. Currently, there several experimental studies on the mechanical behavior of TiNi alloy under high strain rate loading [7–14]; however, the level of theoretical description of these processes is still insufficient. In connection with the foregoing, the aim of this work was to conduct a computer simulation of high strain rate isothermal deformation using the tension of an equiatomic quiatomic TiNi alloy at different structural states using the microstructural model, the basic principles of which are described in the literature [15–17]. Verification of the results was based on the experimental data obtained previously [13, 14]. The high-rate tension at a strain rate of about 103s-1 in the temperature range of 20–300 °С was performed using the Split Hopkinson Pressure Bar technique. 2. Modeling A microstructural model [16, 17] was used for modeling the mechanical behavior of the shape memory alloy. Furthermore, model deformation of each of the 1, 2,…, , …, grains of the polycrystal sample can be considered as a sum of elastic, thermal, phase strains, and micro plastic deformation caused by accommodation of martensite and deformation of twinning. These components were calculated in the way described in [17]. To calculate the athermal plastic deformation, we used the previously described model of plastic deformation for a shape memory alloy [16]. The model assumes that in every grain comprising the polycrystal specific for this alloy, the slip systems become active when the intensity of the shear stress on the slip plane calculated as T( m ,k ) ( 31( m ,k ) ) 2 ( 32( m ,k ) ) 2
riches a critical value:
T( m ,k ) s ( m ,k ) where m is a number of M types of slip planes; k is a number of Km crystallographically equivalent planes belonging to this type; and τ31(m,k) and τ32(m,k) are the shear components of the stress acting on the plane (m ,k). The shear stress components τ31(m,k) and τ32(m,k) can be calculated using the known effective stress applied to the grain using the rotation matrix, which transfers the crystallographic basis of the grain into the crystallographic basis of the (m, k) plane. Components of a shear strain, produced by the tangential stress on the slip plane, are calculated (based on the plane) as 1 2
31a ( m,k ) (m,k )
31 T
1 2
32a ( m,k ) (m,k )
32 T ,
where a ( m,k ) a ( m,k ) (m ,k ) 2 pq pq
is the intensity shear strain rates in the (m, k) plane. The flow stress τs (m,k) using the microstructural approach [16] was considered to consist of the equilibrium value s (m,k) eq, which is the same for all planes of the given m-th type, and the addition caused by deformation hardening τ s(m,k)def τ :
s ( m,k ) s ( m,k ) eq s ( m,k ) def
(1)
Margarita Evard et al./ Materials Today: Proceedings 4 (2017) 4637–4641
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The deformation hardening term τs (m,k)def is assumed to be a linear function of the shear on the plane; therefore, its rate can be calculated as follows:
s ( m,k ) def h( m ) (m,k )
(2)
where h(m) is the hardening modulus for the m-th type of shear planes. To take into account the rate dependence of the flow stress, which was neglected previously [16], a special additional term τ(m,k)rt was added into relation (1); moreover, the power function proposed in a previous study was used [18]:
s ( m,k ) rt PT ( (m,k ) ) where T is the temperature and P and are material constants. On solving equation (2) relative to 𝛤β(m,k) we can determine the rate of plastic deformation produced by the action of a given slip system. The deformation of the entire grain is the sum of the deformations on each of the slip planes in this grain: M
Km
gr P P ( m,k ) m 1 k 1
,
and the macroscopic plastic deformation is obtained by neutralization on all the grains constituting the polycrystal. In this study, a model material with properties similar to that of the equiatomic TiNi alloy was employed [18]. The representative volume under consideration contained 27 grains of a random orientational distribution to exclude of a texture influence on the mechanical properties of the polycrystal model material. For the calculation of the phase and micro plastic strains, the following characteristic temperatures of martensitic transformations were used: Мs = 74°С, Мf = 32°С, As = 74°С, and Af = 98°С. The value of the specific heat of transformation q0 = −150 MJ/m3 was chosen to provide the proper shift of the phase transition temperature, according to the Clausius–Clapeyron relation. The stress of the twinning start was equal to 45 MPa and was supposed to be the same for all the crystallographic variants of martensite. For the calculation of the plastic deformation in accordance with the previous results [16], it was assumed that the dislocation slip in TiNi occurs on two crystallographic planes, namely, {110} and {100}. Both the planes were characterized by their critical shear stress value: for the {100} plane, 110 MPa and for the {110} plane, 175 MPa. The hardening coefficient was calibrated in three ranges depending on the phase composition of the alloy: at deformation temperature of 20–100°С for 1000 MPa, 100–130°С for 1200 MPa, and 130–300°С for 300 MPa. The values of the constants P (7 MPa *с1/4/К) and (0.25) of the strain rate dependence of the flow stress were determined by taking into account all the experimental curves of TiNi alloy mechanical behavior. 3. Results and discussion Using all the parameters of the microstructural approach described above, modeling of the mechanical behavior of equiatomic TiNi alloy during high strain rate isothermal loading at different temperatures was conducted. It was proposed that the temperature field was uniform for the entire sample. Figures 1 and 2 show the calculated stress-strain diagrams for the TiNi alloy samples in martensitic, austenitic, and mixed conditions and the appropriate experimental curves, respectively [13]. A good agreement between the calculated and experimental curves under a high strain rate tension in the single-phase states has been obtained. The experimental and calculated curves in the mixed state and in a state close to the direct martensitic transformation diverge in some areas. This is mainly due to the oscillations, which occur in the experiment, associated with avalanche-type processes of structure adjustment under high-loading. At the same time, the calculated curves are
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Margarita Evard et al./ Materials Today: Proceedings 4 (2017) 4637–4641
near the minima of the first oscillations, which according to our estimates, coincide with the phase yield stress under high strain rate loading [13]. Thus, we can assume that the model describes well the basic parameters of the high strain rate loading, but does not describe avalanche-type of the occurring processes.
Fig. 1. The stress-strain diagrams at high strain rate tension (103s-1) of TiNi alloy at different phase conditions: (а) martensitic at 20°C and (b) mixed at 77°C. 1–modeling and 2–experiment
Fig. 2. The stress-strain diagrams at high strain rate tension (103s-1) of TiNi alloy: (a) austenitic conditions at 110°C and (b) 300°C. 1–modeling and 2–experiment
Fig. 3. The dependence of the yield stresses of TiNi alloy on high strain rate tension test temperature. Experimental values: ■–phase yield stress and ▼–dislocation yield stress. Calculated values: □–phase yield stress and –dislocation yield stress.
Margarita Evard et al./ Materials Today: Proceedings 4 (2017) 4637–4641
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According to the obtained calculation diagrams of high strain rate loading of TiNi alloy at different temperatures, the phase and dislocation yield stress were determined by the tangent method. Figure 3 shows the comparison of the computational and experimental values of the phase and dislocation yield stresses for the macroscopic level. It can be observed that despite some differences between the calculated and experimental diagrams, the calculated values of the phase and dislocation yield stresses in martensitic, austenitic, and mixed conditions are in a good agreement with the experimental values. 4. Conclusion Computer simulation of the mechanical behavior of an equiatomic TiNi alloy under a high strain rate tension was conducted. As a result, a good agreement between the calculated and experimental curves of deformation of the alloy in the martensitic and austenitic state was obtained. The calculated values of the phase and dislocation yield stresses in martensitic, austenitic, and mixed conditions are in good agreement with the experimental values. TiNi alloy deformation behavior during high strain rate loading in all the investigated temperature range was modeled using only one pair of constants (P = 7 MPa* s1/4/К and = 0.25) from the law of the rate sensitivity of dislocation yield strength. Acknowledgements The authors are grateful to the Russian Foundation for Basic Research (grants #15-01-07657 and #16-08-00997). References [1] A. Razov, Phys. Met. Metallogr. 97 (2004) S97–S126. [2] J.M. Jani, M. Leary, A. Subic, M.A. Gibson, Mater. Des. 56 (2014) 1078–1113. [3] I. Khmelevskaya, E. Ryklina, A. Korotitskiy, Mat. Sci. Foundat. 81–82 (2015) 603–637. [4] A. Bragov, A. Danilov, A. Konstantinov, A Lomunov, A. Motorin, A. Razov, Mat. Today: Proc. 2 (2015) S961–S964. [5] S. Belyaev, N. Resnina, V. Borisov, I. Lomakin, V. Rubanik, V. Rubanik, O. Rubanik, Physics Procedia (2010) 52–57. [6] S. Belyaev, N. Resnina, V. Borisov, V. Rubanik, O. Rubanik, Phase Transitions 83 (2010) 276–283. [7] K. Ogawa, J. Phys. IV. Coll.C3 49 (1988) 115–120. [8] Y. Liu, Yu. Li, K. T. Ramesh, J. Van Humbeeck, Scripta Mater. 41 (1999) 89–95. [9] Y. Liu, Yu. Li, Z. Hie, K.T. Ramesh, Phil. Mag. Lett. 82 (2002) 511–517. [10] S. Belyaev, A. Petrov, A. Razov, A. Volkov, Mater. Sci. Eng. 378 (2004) 122–124. [11] S. Nemat-Nasser, J.Y. Choi, Acta Materialia 53 (2005) 449–454. [12] S. Nemat-Nasser, J.Y. Choi, Phil. Mag. 86 (2006) 1173–1187. [13] A. Bragov, A. Danilov, A. Konstantinov, A. Lomunov, A. Motorin, A. Razov, Phys. Met. Metallogr. 116 (2015) 385–392. [14] A. Danilov, A. Razov, Mat. Sci. Foundat. 81–82 (2015) 457–479. [15] M.E. Evard, A.E. Volkov, J. Eng. Marter. Technol. 121 (1999) 102. [16] M. Evard, A. Volkov, Proc. of the Int. Symp.: SME: Fundamentals, Modeling and Industrial Applications (1999) 177–183. [17] M. Evard, N. Markachev, E. Uspenskiy, A. Vikulenkov, A. Volkov, J. Mater. Eng. Perform. 23 (2014) 2719–2726. [18] V. Lihachev, V. Malinin, Strukturno-analiticheskaja teorija prochnosti, SPb: Nauka, 1993. (In Russian).
ScienceDirect Materials Today: Proceedings 4 (2017) 4637–4641
www.materialstoday.com/proceedings
SMA 2016
Microstructural modeling of TiNi alloy high strain rate tension Margarita Evarda, Alexander Motorinb*, Alexander Razova, Aleksandr Volkova Saint Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg, 199034 Russia bITMO University,49 Kronverksky Pr. St. Petersburg, 197101, Russia
a
Abstract Results of the microstructural modeling of high strain rate isothermal straining of an equiatomic TiNi alloy at different structural conditions using the formulae for strain rate dependencies of dislocation yield limit are presented. Simulations were conducted in the temperature range of 20–300°С. It was shown that the calculated values of the phase and dislocation yield stress in the martensitic and austenitic conditions with a good accuracy correspond to the experimental values. The mechanical behavior of the alloy in the entire investigated temperature range was described using only one pair of constants in strain rate sensitivity of dislocation yield limit. © 2017 Elsevier Ltd. All rights reserved. Selection and Peer-review under responsibility of The second conference “Shape memory alloys”. Keywords: TiNi; high strain rate; modeling; microstructural theory; tension.
1. Introduction Changes in the properties of various materials under the influence of high strain rate loading have attracted the interest of scientists and engineers since many years due to the significant practical importance of such research for civilian and special equipment. Alloys with shape memory effect are not an exception, particularly TiNi-based alloys, which have unique properties such as high corrosion resistance and strength, good deformation recovery indicators, high reactive stresses, excellent biocompatibility, and high damping capacity. These alloys have been successfully applied in numerous fields of technology and medicine [1–3]. In addition, shape memory alloys can be
* Corresponding author. Tel.: +7-921-758-1197; fax: +7-812-428-7079. E-mail address: [email protected] 2214-7853 © 2017 Elsevier Ltd. All rights reserved. Selection and Peer-review under responsibility of The second conference “Shape memory alloys”.
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Margarita Evard et al./ Materials Today: Proceedings 4 (2017) 4637–4641
subjected to high strain rate loading during technological operations to improve the functional properties [4] or to manufacture bimetallic materials with shape memory [5–6]. Currently, there several experimental studies on the mechanical behavior of TiNi alloy under high strain rate loading [7–14]; however, the level of theoretical description of these processes is still insufficient. In connection with the foregoing, the aim of this work was to conduct a computer simulation of high strain rate isothermal deformation using the tension of an equiatomic quiatomic TiNi alloy at different structural states using the microstructural model, the basic principles of which are described in the literature [15–17]. Verification of the results was based on the experimental data obtained previously [13, 14]. The high-rate tension at a strain rate of about 103s-1 in the temperature range of 20–300 °С was performed using the Split Hopkinson Pressure Bar technique. 2. Modeling A microstructural model [16, 17] was used for modeling the mechanical behavior of the shape memory alloy. Furthermore, model deformation of each of the 1, 2,…, , …, grains of the polycrystal sample can be considered as a sum of elastic, thermal, phase strains, and micro plastic deformation caused by accommodation of martensite and deformation of twinning. These components were calculated in the way described in [17]. To calculate the athermal plastic deformation, we used the previously described model of plastic deformation for a shape memory alloy [16]. The model assumes that in every grain comprising the polycrystal specific for this alloy, the slip systems become active when the intensity of the shear stress on the slip plane calculated as T( m ,k ) ( 31( m ,k ) ) 2 ( 32( m ,k ) ) 2
riches a critical value:
T( m ,k ) s ( m ,k ) where m is a number of M types of slip planes; k is a number of Km crystallographically equivalent planes belonging to this type; and τ31(m,k) and τ32(m,k) are the shear components of the stress acting on the plane (m ,k). The shear stress components τ31(m,k) and τ32(m,k) can be calculated using the known effective stress applied to the grain using the rotation matrix, which transfers the crystallographic basis of the grain into the crystallographic basis of the (m, k) plane. Components of a shear strain, produced by the tangential stress on the slip plane, are calculated (based on the plane) as 1 2
31a ( m,k ) (m,k )
31 T
1 2
32a ( m,k ) (m,k )
32 T ,
where a ( m,k ) a ( m,k ) (m ,k ) 2 pq pq
is the intensity shear strain rates in the (m, k) plane. The flow stress τs (m,k) using the microstructural approach [16] was considered to consist of the equilibrium value s (m,k) eq, which is the same for all planes of the given m-th type, and the addition caused by deformation hardening τ s(m,k)def τ :
s ( m,k ) s ( m,k ) eq s ( m,k ) def
(1)
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The deformation hardening term τs (m,k)def is assumed to be a linear function of the shear on the plane; therefore, its rate can be calculated as follows:
s ( m,k ) def h( m ) (m,k )
(2)
where h(m) is the hardening modulus for the m-th type of shear planes. To take into account the rate dependence of the flow stress, which was neglected previously [16], a special additional term τ(m,k)rt was added into relation (1); moreover, the power function proposed in a previous study was used [18]:
s ( m,k ) rt PT ( (m,k ) ) where T is the temperature and P and are material constants. On solving equation (2) relative to 𝛤β(m,k) we can determine the rate of plastic deformation produced by the action of a given slip system. The deformation of the entire grain is the sum of the deformations on each of the slip planes in this grain: M
Km
gr P P ( m,k ) m 1 k 1
,
and the macroscopic plastic deformation is obtained by neutralization on all the grains constituting the polycrystal. In this study, a model material with properties similar to that of the equiatomic TiNi alloy was employed [18]. The representative volume under consideration contained 27 grains of a random orientational distribution to exclude of a texture influence on the mechanical properties of the polycrystal model material. For the calculation of the phase and micro plastic strains, the following characteristic temperatures of martensitic transformations were used: Мs = 74°С, Мf = 32°С, As = 74°С, and Af = 98°С. The value of the specific heat of transformation q0 = −150 MJ/m3 was chosen to provide the proper shift of the phase transition temperature, according to the Clausius–Clapeyron relation. The stress of the twinning start was equal to 45 MPa and was supposed to be the same for all the crystallographic variants of martensite. For the calculation of the plastic deformation in accordance with the previous results [16], it was assumed that the dislocation slip in TiNi occurs on two crystallographic planes, namely, {110} and {100}. Both the planes were characterized by their critical shear stress value: for the {100} plane, 110 MPa and for the {110} plane, 175 MPa. The hardening coefficient was calibrated in three ranges depending on the phase composition of the alloy: at deformation temperature of 20–100°С for 1000 MPa, 100–130°С for 1200 MPa, and 130–300°С for 300 MPa. The values of the constants P (7 MPa *с1/4/К) and (0.25) of the strain rate dependence of the flow stress were determined by taking into account all the experimental curves of TiNi alloy mechanical behavior. 3. Results and discussion Using all the parameters of the microstructural approach described above, modeling of the mechanical behavior of equiatomic TiNi alloy during high strain rate isothermal loading at different temperatures was conducted. It was proposed that the temperature field was uniform for the entire sample. Figures 1 and 2 show the calculated stress-strain diagrams for the TiNi alloy samples in martensitic, austenitic, and mixed conditions and the appropriate experimental curves, respectively [13]. A good agreement between the calculated and experimental curves under a high strain rate tension in the single-phase states has been obtained. The experimental and calculated curves in the mixed state and in a state close to the direct martensitic transformation diverge in some areas. This is mainly due to the oscillations, which occur in the experiment, associated with avalanche-type processes of structure adjustment under high-loading. At the same time, the calculated curves are
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near the minima of the first oscillations, which according to our estimates, coincide with the phase yield stress under high strain rate loading [13]. Thus, we can assume that the model describes well the basic parameters of the high strain rate loading, but does not describe avalanche-type of the occurring processes.
Fig. 1. The stress-strain diagrams at high strain rate tension (103s-1) of TiNi alloy at different phase conditions: (а) martensitic at 20°C and (b) mixed at 77°C. 1–modeling and 2–experiment
Fig. 2. The stress-strain diagrams at high strain rate tension (103s-1) of TiNi alloy: (a) austenitic conditions at 110°C and (b) 300°C. 1–modeling and 2–experiment
Fig. 3. The dependence of the yield stresses of TiNi alloy on high strain rate tension test temperature. Experimental values: ■–phase yield stress and ▼–dislocation yield stress. Calculated values: □–phase yield stress and –dislocation yield stress.
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According to the obtained calculation diagrams of high strain rate loading of TiNi alloy at different temperatures, the phase and dislocation yield stress were determined by the tangent method. Figure 3 shows the comparison of the computational and experimental values of the phase and dislocation yield stresses for the macroscopic level. It can be observed that despite some differences between the calculated and experimental diagrams, the calculated values of the phase and dislocation yield stresses in martensitic, austenitic, and mixed conditions are in a good agreement with the experimental values. 4. Conclusion Computer simulation of the mechanical behavior of an equiatomic TiNi alloy under a high strain rate tension was conducted. As a result, a good agreement between the calculated and experimental curves of deformation of the alloy in the martensitic and austenitic state was obtained. The calculated values of the phase and dislocation yield stresses in martensitic, austenitic, and mixed conditions are in good agreement with the experimental values. TiNi alloy deformation behavior during high strain rate loading in all the investigated temperature range was modeled using only one pair of constants (P = 7 MPa* s1/4/К and = 0.25) from the law of the rate sensitivity of dislocation yield strength. Acknowledgements The authors are grateful to the Russian Foundation for Basic Research (grants #15-01-07657 and #16-08-00997). References [1] A. Razov, Phys. Met. Metallogr. 97 (2004) S97–S126. [2] J.M. Jani, M. Leary, A. Subic, M.A. Gibson, Mater. Des. 56 (2014) 1078–1113. [3] I. Khmelevskaya, E. Ryklina, A. Korotitskiy, Mat. Sci. Foundat. 81–82 (2015) 603–637. [4] A. Bragov, A. Danilov, A. Konstantinov, A Lomunov, A. Motorin, A. Razov, Mat. Today: Proc. 2 (2015) S961–S964. [5] S. Belyaev, N. Resnina, V. Borisov, I. Lomakin, V. Rubanik, V. Rubanik, O. Rubanik, Physics Procedia (2010) 52–57. [6] S. Belyaev, N. Resnina, V. Borisov, V. Rubanik, O. Rubanik, Phase Transitions 83 (2010) 276–283. [7] K. Ogawa, J. Phys. IV. Coll.C3 49 (1988) 115–120. [8] Y. Liu, Yu. Li, K. T. Ramesh, J. Van Humbeeck, Scripta Mater. 41 (1999) 89–95. [9] Y. Liu, Yu. Li, Z. Hie, K.T. Ramesh, Phil. Mag. Lett. 82 (2002) 511–517. [10] S. Belyaev, A. Petrov, A. Razov, A. Volkov, Mater. Sci. Eng. 378 (2004) 122–124. [11] S. Nemat-Nasser, J.Y. Choi, Acta Materialia 53 (2005) 449–454. [12] S. Nemat-Nasser, J.Y. Choi, Phil. Mag. 86 (2006) 1173–1187. [13] A. Bragov, A. Danilov, A. Konstantinov, A. Lomunov, A. Motorin, A. Razov, Phys. Met. Metallogr. 116 (2015) 385–392. [14] A. Danilov, A. Razov, Mat. Sci. Foundat. 81–82 (2015) 457–479. [15] M.E. Evard, A.E. Volkov, J. Eng. Marter. Technol. 121 (1999) 102. [16] M. Evard, A. Volkov, Proc. of the Int. Symp.: SME: Fundamentals, Modeling and Industrial Applications (1999) 177–183. [17] M. Evard, N. Markachev, E. Uspenskiy, A. Vikulenkov, A. Volkov, J. Mater. Eng. Perform. 23 (2014) 2719–2726. [18] V. Lihachev, V. Malinin, Strukturno-analiticheskaja teorija prochnosti, SPb: Nauka, 1993. (In Russian).