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plastic relaxation of internal stress in giant-magnetostriction particle reinforced metal-matrix composite
Materials Science and Engineering A 387–389 (2004) 900–904
Diffusional/plastic relaxation of internal stress in giant-magnetostriction particle reinforced metal-matrix composite Eiichi Sato∗ , Atsushi Yamaguchi, Koichi Kitazono, Kazuhiko Kuribayashi Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, 3-1-1 Yoshinodai, Sagamihara, Kanagawa 229-8510, Japan Received 25 August 2003; received in revised form 25 November 2003
Abstract A giant-magnetostriction particle reinforced tin-matrix composite, Sn/Terfenol-D, was fabricated, and the relaxation behavior of internal stress caused by magnetostriction of the particles was directly observed as an average strain change of the composite with and without an external stress. The diffusional relaxation under zero external stress was not observed. During steady-state creep, the internal stress by magnetostriction suppressed dislocation movement and removal of the magnetostriction activated the dislocation movement. This implies that the plastic relaxation of internal stress was observed directly from the average strain change. © 2004 Published by Elsevier B.V. Keywords: Diffusional relaxation; Plastic relaxation; Metal matrix composite; Giant-magnetostriction particle; Micromechanics analysis; Dislocation movement
1. Introduction Lately, the industrial world demands materials which can be used under higher temperature and larger load condition for improving performance and fuel consumption of heat engines. One of high temperature strengthening methods is introducing second phase into the matrix, and the understanding of this strengthening mechanism is important for further material development. High temperature mechanical behavior of metal matrix composites strongly depends on relaxation mechanisms which relax the inhomogeneous internal stress field, which is generated by the difference in elastic and creep properties between the second phase and matrix. The internal stress is considered to be relaxed by interfacial diffusion of the matrix atoms (diffusional relaxation) or inhomogeneous creep deformation of the matrix (plastic relaxation). As an experimental proof of the diffusional relaxation, the diffusional relaxation peak of internal friction has been reported in Al–Si alloy [1]. The internal friction peak appears when a period of external vibration agrees with the diffusional relaxation time. As an experimental proof of the plastic relaxation, the plastically accommodated power law ∗
Corresponding author. Tel.: +81 42 759 8263; fax: +81 42 759 8461. E-mail address: [email protected] (E. Sato).
0921-5093/$ – see front matter © 2004 Published by Elsevier B.V. doi:10.1016/j.msea.2003.12.077
creep and diffusionally accommodated linear creep have been reported in Ti/TiB(w) composite [2]. In the former, the internal stress is relaxed by inhomogeneous creep deformation of the matrix caused by active dislocation movement over the second phase, and in the latter, the internal stress is relaxed by interfacial diffusion of dislocation core atoms stacked in front of the second phase. However, nobody has observed the diffusional relaxation directly as the time-dependant profile of the macroscopic strain change. Recently, Mori et al. pointed out that the plastic relaxation cannot exist theoretically in the creep condition [3]. We already pointed out [4] a mathematical mistake of Mori et al., but a clear experimental proof of plastic relaxation is still required. In the present paper, we conducted an experiment where macroscopic strain change in the composite is directly measured by strain gauge for high temperature during diffusional and plastic relaxation processes.
2. Diffusional relaxation under zero external stress Consider a giant-magnetostriction particle reinforced metal-matrix composite kept at a high temperature and in a magnetic field. An average strain is generated in the composite by the internal stress caused by magnetostriction of the particles. The internal stress is, then, gradually relaxed
E. Sato et al. / Materials Science and Engineering A 387–389 (2004) 900–904
901
H H
Matrix(Sn)
Terfenol-D
strain
H =
H
⊥H
time Fig. 1. Schematic diagram of diffusional relaxation of internal stress caused by magnetostriction in a particle under zero external stress.
by interfacial diffusion of the matrix atoms around the particles, and the average strain also reduces in an exponential manner as shown in Fig. 1 as time proceeds. This average strain change should be measured directly using strain gauge for high temperature use. 3. Diffusional and plastic relaxation during steady-state creep Consider the case where the composite shows steady-state creep by applying a uniaxial compressive load. The repulsive stress field between the stacked dislocations is increased by the internal stress caused by the magnetostriction of the particles. If this steady-state creep is achieved by the interfacial diffusion of the stacked dislocation core atoms, the diffusion of dislocation core atoms would be enhanced by the increased repulsive stress between dislocations, and the creep strain rate would increase as shown in Fig. 2. On the other hand, if the steady-state creep is achieved by plastically accommodated creep where dislocations climb over the particle, the dislocation movement would be disturbed by the average back stress, i.e. the internal stress caused by the magnetostriction of the particle, and the creep strain rate would decrease. After magnetostriction is removed, the stacked dislocations would rapidly get over the particle, and the creep strain rate would increase as shown in Fig. 3.
Fig. 2. Schematic diagram of diffusional relaxation of internal stress caused by magnetostriction in a particle during steady-state creep under an external stress.
[6], the initial strain of the composite immediately after the generation of magnetostriction can be described as 15µI (1 − vM )f εT , (1 − f)µM (7 − 5vM ) +µI {8 − 10vM + f(7 − 5vM )} 1 = − ε0H . 2
ε0H =
ε0⊥H
(1)
4. Quantitative analysis by micromechanics Diffusional relaxation around a spherical particle has been analyzed by Okabe et al. [1]. Using Eshelby’s equivalent inclusion theory [5] and Mori–Tanaka’s mean field theory
Fig. 3. Schematic diagram of plastic relaxation of internal stress caused by magnetostriction in a particle during steady-state creep under an external stress.
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E. Sato et al. / Materials Science and Engineering A 387–389 (2004) 900–904
Equation (1) implies that apparent Poisson’s ratio of the composite is 0.5. Here, ε0H and ε0⊥H are the initial strains in a parallel and perpendicular directions to the magnetic field, respectively, µI , vI , µM , and vM are the shear modulus and Poisson’s ratios of the particle and matrix, respectively, and f is the volume fraction of the particles. The magnetostriction of the particles parallel to the magnetic field is expressed by εT and the magnetostriction perpendicular is (−1/2)εT . Considering the rate equation and the Gibbs’s free energy change in the relaxation process, the diffusional relaxation time can be calculated as
Strains parallel and perpendicular to the magnetic field were measured by strain gauges for high temperature use up to 300 ◦ C attached on the specimen surface and a data logger with resolution of 1 × 10−6 . The compressive load was applied with a balance made of stainless steel.
(1 − f){µM (1 − f)(7 − 5vM ) +µI (8 − 10vM + f(7 − 5vM ))} r3 kT τD = , 30 DI hΩ f 2 µI µM (7 − 5vM )
Fig. 5 shows the result of the diffusional relaxation test under zero external stress at 180 ◦ C. It is observed that the strains in two directions change abruptly by applying the magnetic field of 0.8 T (indicated by no. 1 in Fig. 5), then negligible changes in the strains are observed for the following period of 10 h with the static magnetic field (indicated by no. 2 in Fig. 5). Though monotonous creep behavior with small strain rate is observed during the period, it is considered as the drift of strain gauges in the high temperature condition. It is noted that unlike the gauge drift, the strains parallel and perpendicular to the magnetic field are in the opposite directions to each other. The initial strains after applying magnetic field are obtained as εH = 97 × 10−6 and ε⊥H = −50 × 10−6 in the parallel and perpendicular directions to the magnetic field. Similarly to the behavior at 180 ◦ C, at room temperature, the initial strains after applying the magnetic field of 0.8 T were obtained as εH = 148 × 10−6 and ε⊥H = −70 × 10−6 .
(2)
where DI is the interfacial diffusional coefficient of matrix atom, h the interface thickness, r the average particle radius of spherical particle, and Ω the matrix atomic volume.
5. Experimental procedure Tin was selected as the matrix of low melting temperature (232 ◦ C) and ETREMA TERFENOL-D (Tb0.27 Dy0.73 Fe1.9 ) was selected as the particles with giant magnetostriction. Sn/Terfenol-D (20 vol.%) composite was fabricated as follows. Tin powders were mixed with Terfenol-D powders by a ball mill, and were uniaxially pressed at room temperature. The mean particle size of Terfenol-D was estimated as 30 m by optical microscopy. Phase identification was done by the X-ray diffraction, indicating that only the peaks of TbFe2 and DyFe2 , the constituents of Terfenol-D, and those of Sn were observed. Relative density of the specimen was 99.9% by the Archimedes method. Macroscopic strain change of the Sn/Terfenol-D composite by the magnetostriction and by the relaxation process were measured as follows (Fig. 4). Magnetic field of 0.8 T was generated by an electromagnet. The specimen of 8 mm × 7 mm × 6 mm was heated by a microheater up to 180 ◦ C, and the temperature of the specimen surface was controlled within ±1 ◦ C.
6. Results 6.1. Relaxation test under zero external stress
6.2. Relaxation test during steady-state creep Fig. 6 shows the result of the relaxation test during steady-state creep at 180 ◦ C under a compressive stress of 8.4 MPa. Similarly to the behavior under zero external stress, the abrupt changes in strain by applying the magnetic field are observed (indicated by no. 1 in Fig. 6). On the other hand, obviously different from that, decreases
Sn-20vol%Terfenol-D(P) 100 × 10
SUS316
Micro Heater
strain
Electromagnet
=
ε
-6
2
1
97×10-6 H
T=180
Magnetic field
ε⊥ H
Specimen weight Strain gauge
H=0 T
Thermocouple
4.5
Fig. 4. Experimental apparatus.
-50×10-6 H=0.8 T
6
7.5
t/104s
H=0 T
9
10.5
12
Fig. 5. Strain change profile in the relaxation test under zero external stress at 180 ◦ C.
E. Sato et al. / Materials Science and Engineering A 387–389 (2004) 900–904
903
considering that vacancies cannot be generated nor disappear by a small matrix stress caused by the magnetostriction. On the other hand, during steady-state creep under an external load, relaxation behavior shown in Fig. 3 based on plastically accommodated creep was observed in the strain profile in Fig. 6. First, creep rates decrease by the magnetostriction (no. 2), and second, after removal of the magnetostriction, the strain rate is increased and then gradually decreased to the original creep rate (no. 3). The average internal stress in the matrix by the magnetostriction σijM is calculated as 2.3 MPa using the equation: ∗
σijM = −fCM ijkl (Sklmn − Iklmn )εmn ,
Fig. 6. Strain change profile in the relaxation test during steady-state creep under external stress of 8.4 MPa at 180 ◦ C.
in strain rate are observed (indicated by no. 2 in Fig. 6). The steady-state creep rates, ε˙ H = 6.9 × 10−9 s−1 and ε˙ ⊥H = −1.4 × 10−9 s−1 , decrease to ε˙ H = 3.2 × 10−9 s−1 and ε˙ ⊥H = −5.6 × 10−10 s−1 . In addition, immediately after the removal of the magnetostriction, large increases in strain rate are observed (indicated by no. 3 in Fig. 6). These increased strain rates decrease to the original strain rates, ε˙ H = 6.2 × 10−9 s−1 and ε˙ ⊥H = −1.2 × 10−9 s−1 , after the period of 2 h of nil magnetic field.
7. Discussion The abrupt change in the macroscopic strain caused by the magnetostriction is compared to micromechanics analysis. The magnetostriction of polycrystalline Terfenol-D under 0.8 T is estimated as 1020×10−6 at room temperature and 560 × 10−6 at 180 ◦ C from the data of single crystal [7]. Using the elastic modulus at room temperature and the estimated magnetostrictions at two temperatures, Eq. (1) predicts that the abrupt change in the macroscopic strains are εH = 178 × 10−6 and ε⊥H = −89 × 10−6 at room temperature and εH = 98 × 10−6 and ε⊥H = −49 × 10−6 at 180 ◦ C. Considering ambiguity in material parameters of Terfenol-D, the magnitude of the strain change and apparent Poisson’s ratio are consistent between those by the measurement and theoretical estimations. As time proceeds for 10 h after applying the magnetic field under zero external stress, relaxation behavior shown in Fig. 1 was not observed in the strain profile in Fig. 4. The relaxation time for interfacial diffusion is calculated as 11.3 h at 180 ◦ C by Eq. (2) using the same pre-exponential factor and 0.65 of the activation energy of the lattice diffusion of Sn [8]. Since nil relaxation behavior was observed for almost the same period as the estimated relaxation time, we concluded that the diffusional relaxation was not activated in this condition. The result of no relaxation might be explained
(3)
M is the elastic modulus of the matrix, S where Cijkl klmn the Eshelby’s tensor, Iklmn the identity tensor, and ε∗mn the eigen strain in the equivalent inclusion. In the plastically accommodated creep, macroscopic creep rate is expressed by the power law equation [4]:
ε˙ = Aσ n ,
(4)
where A is a constant and n is the stress exponent. If the internal stress of −2.3 MPa is superimposed to the external stress of 8.4 MPa, the macroscopic strain rate decreases from 6.9 × 10−9 to 6.9 × 10−9 × ((8.4–2.3)/8.4)2.6 s−1 = 3.0 × 10−9 s−1 . The measured strain rate under the magnetic field, 3.2 × 10−9 s−1 , agrees well with this theoretical prediction. Therefore, we can conclude that the average internal stress is considered to disturb dislocation movement and also that the creep of this composite is plastically accommodated. At the end, during steady-state creep, strain rate increased immediately after magnetostriction disappeared. It is considered that when magnetostriction disappeared, the stacked dislocations became free, and the dislocations rapidly got over the particles.
8. Conclusion Giant-magnetostriction particle reinforced tin-matrix composite, Sn/Terfenol-D, was fabricated, and the relaxation behavior of internal stress caused by magnetostriction of the particle was directly observed as an average strain change of the composite with and without external stress: (1) The magnitude of the strain change caused by the magnetostriction of particles in the both directions and apparent Poisson’s ratio were consistent with the theoretical prediction. (2) The diffusional relaxation under zero external stress was not observed. (3) The internal stress by magnetostriction suppressed dislocation movement of the creep deformation, and removal of the magnetostriction activated the dislocation movement. It implies that the plastic relaxation of
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E. Sato et al. / Materials Science and Engineering A 387–389 (2004) 900–904
internal stress was observed directly from the average strain change. Acknowledgements The authors express their thanks to Mr. Kenichi Ushijima, former graduated student, Tokai University for his preliminary study on the relaxation behavior of Pb/Terfenol-D composites. This study was financially supported by Grants-in-Aids for Scientific Research from the Japan Society for the Promotion of Science.
References [1] M. Okabe, T. Mori, T. Mura, Phil. Mag. A 44 (1981) 1. [2] K. Kawabata, E. Sato, K. Kuribayashi, Acta Mater. 51 (2003) 1909. [3] T. Mori, J. Huang, M. Taya, Acta Mater. 45 (1997) 429. [4] E. Sato, T. Ookawara, K. Kuribayashi, S. Kodama, Acta Mater. 46 (1998) 4153. [5] J.D. Eshelby, Proc. R. Soc. A 252 (1959) 561. [6] T. Mori, K. Tanaka, Acta Metall. 21 (1973) 571. [7] D.C. Jiles, J.B. Thoelke, Phys. Status Solid. A 147 (1995) 535. [8] C. Coston, N.H. Nachtrieb, J. Phys. Chem. 68 (1964) 2219.
Diffusional/plastic relaxation of internal stress in giant-magnetostriction particle reinforced metal-matrix composite Eiichi Sato∗ , Atsushi Yamaguchi, Koichi Kitazono, Kazuhiko Kuribayashi Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, 3-1-1 Yoshinodai, Sagamihara, Kanagawa 229-8510, Japan Received 25 August 2003; received in revised form 25 November 2003
Abstract A giant-magnetostriction particle reinforced tin-matrix composite, Sn/Terfenol-D, was fabricated, and the relaxation behavior of internal stress caused by magnetostriction of the particles was directly observed as an average strain change of the composite with and without an external stress. The diffusional relaxation under zero external stress was not observed. During steady-state creep, the internal stress by magnetostriction suppressed dislocation movement and removal of the magnetostriction activated the dislocation movement. This implies that the plastic relaxation of internal stress was observed directly from the average strain change. © 2004 Published by Elsevier B.V. Keywords: Diffusional relaxation; Plastic relaxation; Metal matrix composite; Giant-magnetostriction particle; Micromechanics analysis; Dislocation movement
1. Introduction Lately, the industrial world demands materials which can be used under higher temperature and larger load condition for improving performance and fuel consumption of heat engines. One of high temperature strengthening methods is introducing second phase into the matrix, and the understanding of this strengthening mechanism is important for further material development. High temperature mechanical behavior of metal matrix composites strongly depends on relaxation mechanisms which relax the inhomogeneous internal stress field, which is generated by the difference in elastic and creep properties between the second phase and matrix. The internal stress is considered to be relaxed by interfacial diffusion of the matrix atoms (diffusional relaxation) or inhomogeneous creep deformation of the matrix (plastic relaxation). As an experimental proof of the diffusional relaxation, the diffusional relaxation peak of internal friction has been reported in Al–Si alloy [1]. The internal friction peak appears when a period of external vibration agrees with the diffusional relaxation time. As an experimental proof of the plastic relaxation, the plastically accommodated power law ∗
Corresponding author. Tel.: +81 42 759 8263; fax: +81 42 759 8461. E-mail address: [email protected] (E. Sato).
0921-5093/$ – see front matter © 2004 Published by Elsevier B.V. doi:10.1016/j.msea.2003.12.077
creep and diffusionally accommodated linear creep have been reported in Ti/TiB(w) composite [2]. In the former, the internal stress is relaxed by inhomogeneous creep deformation of the matrix caused by active dislocation movement over the second phase, and in the latter, the internal stress is relaxed by interfacial diffusion of dislocation core atoms stacked in front of the second phase. However, nobody has observed the diffusional relaxation directly as the time-dependant profile of the macroscopic strain change. Recently, Mori et al. pointed out that the plastic relaxation cannot exist theoretically in the creep condition [3]. We already pointed out [4] a mathematical mistake of Mori et al., but a clear experimental proof of plastic relaxation is still required. In the present paper, we conducted an experiment where macroscopic strain change in the composite is directly measured by strain gauge for high temperature during diffusional and plastic relaxation processes.
2. Diffusional relaxation under zero external stress Consider a giant-magnetostriction particle reinforced metal-matrix composite kept at a high temperature and in a magnetic field. An average strain is generated in the composite by the internal stress caused by magnetostriction of the particles. The internal stress is, then, gradually relaxed
E. Sato et al. / Materials Science and Engineering A 387–389 (2004) 900–904
901
H H
Matrix(Sn)
Terfenol-D
strain
H =
H
⊥H
time Fig. 1. Schematic diagram of diffusional relaxation of internal stress caused by magnetostriction in a particle under zero external stress.
by interfacial diffusion of the matrix atoms around the particles, and the average strain also reduces in an exponential manner as shown in Fig. 1 as time proceeds. This average strain change should be measured directly using strain gauge for high temperature use. 3. Diffusional and plastic relaxation during steady-state creep Consider the case where the composite shows steady-state creep by applying a uniaxial compressive load. The repulsive stress field between the stacked dislocations is increased by the internal stress caused by the magnetostriction of the particles. If this steady-state creep is achieved by the interfacial diffusion of the stacked dislocation core atoms, the diffusion of dislocation core atoms would be enhanced by the increased repulsive stress between dislocations, and the creep strain rate would increase as shown in Fig. 2. On the other hand, if the steady-state creep is achieved by plastically accommodated creep where dislocations climb over the particle, the dislocation movement would be disturbed by the average back stress, i.e. the internal stress caused by the magnetostriction of the particle, and the creep strain rate would decrease. After magnetostriction is removed, the stacked dislocations would rapidly get over the particle, and the creep strain rate would increase as shown in Fig. 3.
Fig. 2. Schematic diagram of diffusional relaxation of internal stress caused by magnetostriction in a particle during steady-state creep under an external stress.
[6], the initial strain of the composite immediately after the generation of magnetostriction can be described as 15µI (1 − vM )f εT , (1 − f)µM (7 − 5vM ) +µI {8 − 10vM + f(7 − 5vM )} 1 = − ε0H . 2
ε0H =
ε0⊥H
(1)
4. Quantitative analysis by micromechanics Diffusional relaxation around a spherical particle has been analyzed by Okabe et al. [1]. Using Eshelby’s equivalent inclusion theory [5] and Mori–Tanaka’s mean field theory
Fig. 3. Schematic diagram of plastic relaxation of internal stress caused by magnetostriction in a particle during steady-state creep under an external stress.
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E. Sato et al. / Materials Science and Engineering A 387–389 (2004) 900–904
Equation (1) implies that apparent Poisson’s ratio of the composite is 0.5. Here, ε0H and ε0⊥H are the initial strains in a parallel and perpendicular directions to the magnetic field, respectively, µI , vI , µM , and vM are the shear modulus and Poisson’s ratios of the particle and matrix, respectively, and f is the volume fraction of the particles. The magnetostriction of the particles parallel to the magnetic field is expressed by εT and the magnetostriction perpendicular is (−1/2)εT . Considering the rate equation and the Gibbs’s free energy change in the relaxation process, the diffusional relaxation time can be calculated as
Strains parallel and perpendicular to the magnetic field were measured by strain gauges for high temperature use up to 300 ◦ C attached on the specimen surface and a data logger with resolution of 1 × 10−6 . The compressive load was applied with a balance made of stainless steel.
(1 − f){µM (1 − f)(7 − 5vM ) +µI (8 − 10vM + f(7 − 5vM ))} r3 kT τD = , 30 DI hΩ f 2 µI µM (7 − 5vM )
Fig. 5 shows the result of the diffusional relaxation test under zero external stress at 180 ◦ C. It is observed that the strains in two directions change abruptly by applying the magnetic field of 0.8 T (indicated by no. 1 in Fig. 5), then negligible changes in the strains are observed for the following period of 10 h with the static magnetic field (indicated by no. 2 in Fig. 5). Though monotonous creep behavior with small strain rate is observed during the period, it is considered as the drift of strain gauges in the high temperature condition. It is noted that unlike the gauge drift, the strains parallel and perpendicular to the magnetic field are in the opposite directions to each other. The initial strains after applying magnetic field are obtained as εH = 97 × 10−6 and ε⊥H = −50 × 10−6 in the parallel and perpendicular directions to the magnetic field. Similarly to the behavior at 180 ◦ C, at room temperature, the initial strains after applying the magnetic field of 0.8 T were obtained as εH = 148 × 10−6 and ε⊥H = −70 × 10−6 .
(2)
where DI is the interfacial diffusional coefficient of matrix atom, h the interface thickness, r the average particle radius of spherical particle, and Ω the matrix atomic volume.
5. Experimental procedure Tin was selected as the matrix of low melting temperature (232 ◦ C) and ETREMA TERFENOL-D (Tb0.27 Dy0.73 Fe1.9 ) was selected as the particles with giant magnetostriction. Sn/Terfenol-D (20 vol.%) composite was fabricated as follows. Tin powders were mixed with Terfenol-D powders by a ball mill, and were uniaxially pressed at room temperature. The mean particle size of Terfenol-D was estimated as 30 m by optical microscopy. Phase identification was done by the X-ray diffraction, indicating that only the peaks of TbFe2 and DyFe2 , the constituents of Terfenol-D, and those of Sn were observed. Relative density of the specimen was 99.9% by the Archimedes method. Macroscopic strain change of the Sn/Terfenol-D composite by the magnetostriction and by the relaxation process were measured as follows (Fig. 4). Magnetic field of 0.8 T was generated by an electromagnet. The specimen of 8 mm × 7 mm × 6 mm was heated by a microheater up to 180 ◦ C, and the temperature of the specimen surface was controlled within ±1 ◦ C.
6. Results 6.1. Relaxation test under zero external stress
6.2. Relaxation test during steady-state creep Fig. 6 shows the result of the relaxation test during steady-state creep at 180 ◦ C under a compressive stress of 8.4 MPa. Similarly to the behavior under zero external stress, the abrupt changes in strain by applying the magnetic field are observed (indicated by no. 1 in Fig. 6). On the other hand, obviously different from that, decreases
Sn-20vol%Terfenol-D(P) 100 × 10
SUS316
Micro Heater
strain
Electromagnet
=
ε
-6
2
1
97×10-6 H
T=180
Magnetic field
ε⊥ H
Specimen weight Strain gauge
H=0 T
Thermocouple
4.5
Fig. 4. Experimental apparatus.
-50×10-6 H=0.8 T
6
7.5
t/104s
H=0 T
9
10.5
12
Fig. 5. Strain change profile in the relaxation test under zero external stress at 180 ◦ C.
E. Sato et al. / Materials Science and Engineering A 387–389 (2004) 900–904
903
considering that vacancies cannot be generated nor disappear by a small matrix stress caused by the magnetostriction. On the other hand, during steady-state creep under an external load, relaxation behavior shown in Fig. 3 based on plastically accommodated creep was observed in the strain profile in Fig. 6. First, creep rates decrease by the magnetostriction (no. 2), and second, after removal of the magnetostriction, the strain rate is increased and then gradually decreased to the original creep rate (no. 3). The average internal stress in the matrix by the magnetostriction σijM is calculated as 2.3 MPa using the equation: ∗
σijM = −fCM ijkl (Sklmn − Iklmn )εmn ,
Fig. 6. Strain change profile in the relaxation test during steady-state creep under external stress of 8.4 MPa at 180 ◦ C.
in strain rate are observed (indicated by no. 2 in Fig. 6). The steady-state creep rates, ε˙ H = 6.9 × 10−9 s−1 and ε˙ ⊥H = −1.4 × 10−9 s−1 , decrease to ε˙ H = 3.2 × 10−9 s−1 and ε˙ ⊥H = −5.6 × 10−10 s−1 . In addition, immediately after the removal of the magnetostriction, large increases in strain rate are observed (indicated by no. 3 in Fig. 6). These increased strain rates decrease to the original strain rates, ε˙ H = 6.2 × 10−9 s−1 and ε˙ ⊥H = −1.2 × 10−9 s−1 , after the period of 2 h of nil magnetic field.
7. Discussion The abrupt change in the macroscopic strain caused by the magnetostriction is compared to micromechanics analysis. The magnetostriction of polycrystalline Terfenol-D under 0.8 T is estimated as 1020×10−6 at room temperature and 560 × 10−6 at 180 ◦ C from the data of single crystal [7]. Using the elastic modulus at room temperature and the estimated magnetostrictions at two temperatures, Eq. (1) predicts that the abrupt change in the macroscopic strains are εH = 178 × 10−6 and ε⊥H = −89 × 10−6 at room temperature and εH = 98 × 10−6 and ε⊥H = −49 × 10−6 at 180 ◦ C. Considering ambiguity in material parameters of Terfenol-D, the magnitude of the strain change and apparent Poisson’s ratio are consistent between those by the measurement and theoretical estimations. As time proceeds for 10 h after applying the magnetic field under zero external stress, relaxation behavior shown in Fig. 1 was not observed in the strain profile in Fig. 4. The relaxation time for interfacial diffusion is calculated as 11.3 h at 180 ◦ C by Eq. (2) using the same pre-exponential factor and 0.65 of the activation energy of the lattice diffusion of Sn [8]. Since nil relaxation behavior was observed for almost the same period as the estimated relaxation time, we concluded that the diffusional relaxation was not activated in this condition. The result of no relaxation might be explained
(3)
M is the elastic modulus of the matrix, S where Cijkl klmn the Eshelby’s tensor, Iklmn the identity tensor, and ε∗mn the eigen strain in the equivalent inclusion. In the plastically accommodated creep, macroscopic creep rate is expressed by the power law equation [4]:
ε˙ = Aσ n ,
(4)
where A is a constant and n is the stress exponent. If the internal stress of −2.3 MPa is superimposed to the external stress of 8.4 MPa, the macroscopic strain rate decreases from 6.9 × 10−9 to 6.9 × 10−9 × ((8.4–2.3)/8.4)2.6 s−1 = 3.0 × 10−9 s−1 . The measured strain rate under the magnetic field, 3.2 × 10−9 s−1 , agrees well with this theoretical prediction. Therefore, we can conclude that the average internal stress is considered to disturb dislocation movement and also that the creep of this composite is plastically accommodated. At the end, during steady-state creep, strain rate increased immediately after magnetostriction disappeared. It is considered that when magnetostriction disappeared, the stacked dislocations became free, and the dislocations rapidly got over the particles.
8. Conclusion Giant-magnetostriction particle reinforced tin-matrix composite, Sn/Terfenol-D, was fabricated, and the relaxation behavior of internal stress caused by magnetostriction of the particle was directly observed as an average strain change of the composite with and without external stress: (1) The magnitude of the strain change caused by the magnetostriction of particles in the both directions and apparent Poisson’s ratio were consistent with the theoretical prediction. (2) The diffusional relaxation under zero external stress was not observed. (3) The internal stress by magnetostriction suppressed dislocation movement of the creep deformation, and removal of the magnetostriction activated the dislocation movement. It implies that the plastic relaxation of
904
E. Sato et al. / Materials Science and Engineering A 387–389 (2004) 900–904
internal stress was observed directly from the average strain change. Acknowledgements The authors express their thanks to Mr. Kenichi Ushijima, former graduated student, Tokai University for his preliminary study on the relaxation behavior of Pb/Terfenol-D composites. This study was financially supported by Grants-in-Aids for Scientific Research from the Japan Society for the Promotion of Science.
References [1] M. Okabe, T. Mori, T. Mura, Phil. Mag. A 44 (1981) 1. [2] K. Kawabata, E. Sato, K. Kuribayashi, Acta Mater. 51 (2003) 1909. [3] T. Mori, J. Huang, M. Taya, Acta Mater. 45 (1997) 429. [4] E. Sato, T. Ookawara, K. Kuribayashi, S. Kodama, Acta Mater. 46 (1998) 4153. [5] J.D. Eshelby, Proc. R. Soc. A 252 (1959) 561. [6] T. Mori, K. Tanaka, Acta Metall. 21 (1973) 571. [7] D.C. Jiles, J.B. Thoelke, Phys. Status Solid. A 147 (1995) 535. [8] C. Coston, N.H. Nachtrieb, J. Phys. Chem. 68 (1964) 2219.