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Transformation-induced plasticity of expandable tubulars materials
Materials Science and Engineering A 438–440 (2006) 459–463
Transformation-induced plasticity of expandable tubulars materials Xu Ruiping a,∗ , Liu Jie b , Zhang Yuxin a , Guo Ruisong a , Liu Wenxi a a
School of Materials Science and Engineering, Tianjin University, Tianjin 300072, China b Shanghai Tianhe Shape Memory Limited Corporation, Shanghai 201615, China
Received 8 May 2005; received in revised form 18 January 2006; accepted 28 February 2006
Abstract A new concept on strengthening-plasticity is presented in this paper, along with the design rules of the alloys with high expandable performance. The plasticity of expandable alloys induced by the ␥→ martensitic transformation, can be obviously improved. In order to enhance transformationinduced plasticity, alloys with low stacking-fault energy and high starting temperature of martensite Ms() should be looked for. Therefore, a formula for the calculation of Ms() of Fe-based alloys is also given in this paper in order to facilitate the design of expandable tubular materials. © 2006 Published by Elsevier B.V. Keywords: Strengthening-plasticity; Transformation-induced plasticity; Fe-based alloys; martensite
1. Introduction Expandable tubular technology is an innovative technology in oil industry and has been viewed as a revolutionary well construction in the 21st century. Many papers or articles previously published have introduced its concept and applications in drilling. However, there are only few papers to introduce how to choose and design expandable tubular materials at present. This paper will present the design rules of expandable tubular materials and give a new concept, strengthening-plasticity, to express its comprehensive properties. 2. Strengthening-plasticity of materials An idea of solid expandable tubular [1] (SET) was first put forward by the Royal Dutch Shell Technology Ventures, Inc. in 1992. Its principle [2,3] is to put the tubular into an oil well and make them expanded and deformed by pulling a specially designed grounder or pushing an expansion cone or mandrel. The main purpose of using expandable tubular is to diminish drilling dimension and widen oil-extracting pathway. SETs have been successfully installed in oil and gas fields since November 1999 [4], in a variety of environments in wells on land, offshore and in deepwater to solve a range of drilling and completion challenges. Practical field applications have shown that expand∗
Corresponding author. Tel.: +86 22 27890485; fax: +86 22 27404471. E-mail address: [email protected] (X. Ruiping).
0921-5093/$ – see front matter © 2006 Published by Elsevier B.V. doi:10.1016/j.msea.2006.02.173
able tubular technology can reduce the overall cost of drilling greatly and shorten the construction period. Expandable tubular technology can be applied in a series of engineering technology, including petroleum drilling, fixing, completing and repairing wells. So expandable tubular technology has been viewed as a revolution in petroleum well tubular. Many petroleum companies at abroad and home have begun to develop and research this technology in recent years. However, there are many difficulties in researching the materials of expandable tubular. Because expandable tubular are often installed in oceans or deepwater and have to bear great outside force, its materials should have high strength. Besides, the materials should also have excellent plasticity in order to meet the requirements of large deformation during the expansion process. So a new concept, which is strengthening-plasticity expressed by a parameter k, will be introduced in this paper to show its comprehensive properties. The parameter k, equals to the product of tensile strength σ b and elongation rate δ, which are both measured under annealing condition: k = σb δ
(1)
If the unit of σ b is MPa and δ is %, the unit of k is MPa%. A three-dimensional coordinate system can be used to show the relations among k, σ b and δ. The value of k can be got from the upright axis of the plane of σ b –δ, as shown in Fig. 1. From this figure, we can get the shape of this three-dimensional coordinate system looks like “hills” from top to bottom and the projection of the hills’ contour lines gives hyperbolas. The authors of this
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X. Ruiping et al. / Materials Science and Engineering A 438–440 (2006) 459–463
Fig. 1. The coordinates of strengthening-plasticity k.
Fig. 3. The mechanism for the stress induced ␥– transformation.
Fig. 2. The comparison of strengthening-plasticity among various types of steels.
paper have made many surface and field experiments on expandable tubular materials and compare the strengthening-plasticity of different kinds of steels. The result is as shown in Fig. 2. After many experiments, a kind of expandable alloys with excellent expandable performance was obtained at last, whose properties lies in the hatched zone in Fig. 2. This figure shows that the k of LSX80 researched by Enventure Global Technology L.L.C., is 30,000 MPa%, while the k of our expandable tubular materials reaches 60,000 MPa%, nearly two times that of Enventure Global Technology L.L.C. The measured mechanical properties of our researched expandable alloys are that σ b is 990 MPa, σ s 365 MPa, δ 61%, ψ 57% and work-hardened index n is 0.40, respectively. The k of high strengthening-toughness steels and high plasticity steels with low carbon seen from this figure, are both below 20,000 MPa%. 3. Design for high strengthening-plasticity steels and transformation-induced plasticity The great trouble in designing expandable tubular materials is that the two properties of plasticity and strength are always contradictory, that is to say, the plasticity is low while the strength is high, such as the high strengthening-toughness
steels. In turn, the high plasticity steels with low carbon have good plasticity, its strength is very low. In order to improve the strength and plasticity of expandable tubular materials at the same time, the mechanism of transformation-induced plasticity was used when we designed some compositions of Fe-based expandable alloys. This kind of alloy has been proven to have high strengthening-plasticity and outstanding expandable performance. The mechanism of transformation-induced plasticity is induced by the transformation from parent phase austenite ␥ to martensite when the materials are subjected to outside force. Partial plastic deformation has gone with phase transformation. Meanwhile, the double phase structure produced during the phase transformation can improve strain-hardening rate and prevent early necking phenomenon from happening. Therefore, transformation-induced plasticity can improve the strengthening-plasticity obviously. Besides, when designing expandable tubular material, it should also have high work hardening rate. The reason is to make the materials yield easily at the early period of expansion and improve their strength quickly at the later period, which can save a lot of expandable energies. The value of work hardening rate n discussed in this paper is more than 0.25, which can meet stable plastic deformation condition dσ/dε > σ [5,6]. So there are three characteristics for expandable tubular materials, i.e. high strength and high plasticity and high work hardening rate. It is important to know it when designing expandable tubular materials. The composition of expandable alloys should lie in austenite zone according to Schaefflers figure [7]. So the addition of alloy elements should be strictly controlled to ensure that the structure of alloys is a single austenite phase. Only if the structure of alloys is an austenite, does the mechanism of transformation-induced plasticity function when expandable alloys deform under the action of stress. The mechanism of ␥→ martensite phase transformation is as shown in Fig. 3. The top and bottom plane of triangular prism show the place of close-packed atoms and the height of triangular prism is a planar distance between close-packed atoms. The arrowheads in this figure show the
X. Ruiping et al. / Materials Science and Engineering A 438–440 (2006) 459–463
461
moving direction of Shockley partial dislocations. The crystal structure of the parent phase γ is face-centered cubic (fcc) and that of martensite is a hcp structure. The fcc can be looked as layer to layer stacking along (1 1 1)␥ close-packed and its stacking way is ABCABC· · ·. The hcp can be looked as layer to layer stacking along (0 0 0 1) coherency, its stacking way is ABABAB· · ·. When materials are bearing outside force, a/6[1 1 2] Shockley partial dislocations [8] will occur in fcc and the atoms of every other (1 1 1)␥ move the distance by a/6[1 1 2] along the same direction. As a result, a shear angle of 19◦ 28 is formed. Meanwhile, the stacking arrangement of atoms will change from ABCABC· · · to ABABAB· · ·. Consequently, fcc crystal structure changes to hcp structure and martensite [9] is formed. During the ␥→ martensite phase transformation, plastic deformation also occurs, which can decrease resistance of plastic deformation. Therefore, transformation-induced plasticity improves the plasticity greatly. 4. The formula of Ms() for Fe-based alloys Hsu [10] suggested that martensite forms and grows up by the mechanism of staking faults nucleation for Fe-based alloys, which is a kind of alloys with low stacking faults energy. The stacking faults energy plays an important role in the process of ␥→ martensite phase transformation. The lower the stacking faults energy, the easier martensite phase transformation happens. Accordingly, the starting temperature of martensite transformation Ms() increases. Ms() can be recognized as an index to evaluate the inclination of martensite phase transformation. So the determination of Ms() is very important and necessary to facility to design the expandable tubulars materials. But it is difficult to measure Ms() by experiments because the resistance, specific capacity and magnetism between parent phase ␥ and martensite are very close and the heat effects of the phase transformation are small. For the above reasons, a method of calculating Ms() of Fe-based alloys is given in this paper. The micro-mechanism of martensite phase transformation has been investigated by surface relief [11–14] and atomic force microscopy (AFM) [15] showing that martensite has two morphologies. One is stress-induced martensite, the other is thermal induced martensite. Fig. 3 also shows the mechanism of stress-induced martensite, one of whose character is to form 19◦ 28 -shear angle. As temperature drops, thermally induced martensite will form. It can be concluded from Figs. 3 and 4 that the mechanism of these two martensite is different. One obvious distinction is that the moving orientation of stacking faults of thermally induced martensite is not on a line but a left–right interval style. Though its arrangement of atoms are ABABAB· · ·, its shear angle is 0◦ . And its Shockley partial dislocations expressed with arrowhead in Fig. 4 move to the left at first then to the right and hold still at last. The moving of its Shockley partial dislocations looks like the crossed arrangements of sawtooth of sawblade. Based on the above-mentioned model, the formula of Ms() for Fe-based alloy is deduced from the mechanism of thermal induced martensite. Every nucleus embryo of thermal induced martensite expressed with the length of arrowhead in the middle of Fig. 4,
Fig. 4. The mechanism for the thermal induced ␥– transformation.
is composed of six layers of atom plane [16,17]. Only one stacking fault interface exists between one nucleus embryo and parent phase, so the interface energy σ of per area equals to stacking faults energy F. Besides, the chemical energy Gchem is three times stacking faults energy because every nucleus embryo includes three stacking faults. Other differences energy of crystal lattice and the phase resistance energy are expressed by Eresist . At the temperature of Ms() , the free energy of phase transformation equals to zero: nρ(Gchem + σ) + Eresist = 0
(2)
where n is the thickness of stacking faults, ρ the atomic density of per area, σ is interface energy of per area. From the above equation, the following equation can be deduced: Eresist = Gchem + σ = 4F nρ
(3)
It is supposed that the stacking faults energy show a linear relation with absolute temperature in Ref. [15]. F = F0 + k T
(4)
where F0 is the stacking faults energy at T = 0 and k is a proportionality coefficient. When alloy A and alloy B are at their corresponding Ms() temperatures, i.e. TA = Ms()A , TB = Ms()B , their stacking faults energy are expressed with the following equations, respectively: FA = F0A + kMs()A
(5)
FA = F0B + kMs()B
(6)
F on the right side of Eq. (3) can be substituted by FA and FB , respectively because the change of Eresist is not too large and nρ is almost a constant. The left side of Eq. (3) is a constant, so the
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X. Ruiping et al. / Materials Science and Engineering A 438–440 (2006) 459–463
Table 1 The Ms() values of nine kinds of Fe-based alloysa No.
Mn
Si
Cr
Ni
Change of stacking faults energy, F0
Calculated values
Measured values
Error
1 2 3 4 5 6 7 8 9
31.5 27.6 20.4 19.9 16.25 16.0 13.0 14.0 0
6 6.1 5.0 5.1 5.0 5.1 4.7 6.9 0
– 5.0 8.0 10.1 9.2 11.6 11.4 9.0 18
– – 5.0 5.0 4.0 4.9 6.8 5.3 8
−109 −101 −75 −73.4 −69 −66.6 −57.1 −74.8 0
283 278.4 263.3 262.4 259.9 258.5 253 263 220
293 283 261.2 255 273 267 244 255 221
+10 +4.6 −2.1 −7.4 +13.1 +8.5 −9 −8 +1
a
The numbers indicate the weight percentage of alloy elements and the balance is Fe.
following equations can be easily obtained: F0A + k Ms()A = F0B + k Ms()B 1 F0 k
Ms()B = Ms()A −
5. Conclusions (7) (8)
where F0 equals to the total change of stacking faults energy resulting from the compositional change of every alloy element. F0 =
n
xi fi
(9)
i=1
If 0Cr18Ni8 stainless steel is regarded as a standard alloy whose martensite phase transformation starting temperature is 220 K, we can get the following formula: n
Ms() = 220 −
1 xi fi k
(10)
i=1
where xi is the change of atom percentage compared to the standard alloy, i is the number of alloy elements. fi is the effect of alloy elements of per atom percentage on the stacking faults energy. Table 1 shows the weight percentage of nine kinds of Febased alloys and their Ms() values calculated with the above formula. Table 2 exhibits the effect of several alloy elements on the stacking faults energy. We can conclude from Table 1 that the error is not too big and it is convenient to get the values of Ms() for Fe-based alloys. Table 2 The effect of alloy elements on the stacking faults energy [18,19] Alloying element
fi (erg/cm2 /%)
Cr Ni Si Co Cu Nb Mo Mna
+0.5 +1.4 −3.4 −0.55 +3.6 +3.2 +0.1 −1.68
a
Mn is a supplementary value near to the value in Ref. [20].
(i) The product of tensile strength and elongation rate can be employed as an index to evaluate the strengtheningplasticity of materials. The concept of strengtheningplasticity can reflect the comprehensive properties of expandable alloys. (ii) Expandable tubular materials should have three characteristics: high plasticity, high strength and high work hardening rate. (iii) The property of strengthening plasticity can be improved by transformation-induced plasticity. We should find the alloys with low stacking faults energy and high martensite starting temperature Ms() when designing expandable tubular materials. (iv) The formula of Ms() of Fe-based alloys is as following: n
Ms()
1 = 220 − xi fi k i=1
References [1] A. Filippov, et al., Proceedings of the SPE Annual Technical Conference, Houston, October 3–6, 1999. [2] C.L. Lohoefer, et al., in: K. Waddell, et al. (Eds.), Enventure Global Technology, Expandable Liner Hanger Provides Cost-Effective Alternative Solution, Proceedings of the IADC/SPE Drilling Conference, New Orleans, February, 2000. [3] R.C. Haut, Proceedings of the Deep Offshore Technology International Conference, Stavanger, October 19–21, 1999. [4] K.D. Kenneth, Shell Deepwater Development, Inc., D.B. Campo, Proceedings of the SPE/IADC Drilling Conference, Amsterdam, Netherlands, February 27–March 1, 2001. [5] H.J. Klemola, M.A. Nieminen, Metal. Trans. A 5 (1974) 18. [6] H.J. Klemola, M.A. Nieminen, Metal. Trans. A 7 (1976) 1752. [7] A.L. Schaeffler, Met. Prog. 11 (1969) 680. [8] A. Sato, E. Chishima, K. Soma, T. Mori, Acta Met. 30 (1982) 1177. [9] E. Gartsein, A. Rarinkin, Acta Metal. 27 (1979) 1053. [10] T.Y. Hsu Xu Zuyao, Martensite Transformation and Martensite, Science Press, 1999. [11] J.W. Christian, The Theory of Phase Transformations in Metals and Alloys, Pergamon Press, Oxford, 1965, p. 802. [12] Z. Nishiyama, Martensitic Transformation, Academic Press, New York, 1978, p. 48. [13] S. Tanckeuchi, T. Honma, J. Jpn. Inst. Met. 19 (1955) 652.
X. Ruiping et al. / Materials Science and Engineering A 438–440 (2006) 459–463 [14] P. Gaunt, J.W. Christian, Acta Metal. 7 (1955) 529. [15] Wang Defa, Study on Martensitic Transformation and Shape Memory Mechanism of FeMnSi alloys, Doctor Dissertation of Tianjin University, 2000. [16] G.B. Olson, M. Cohen, Metal. Trans. A 7 (1976) 1897. [17] G.B. Olson, M. Cohen, Metal. Trans. A 7 (1976) 1915.
463
[18] D. Dulicu, J. Nutting, Iron Steel Inst., SR 86 (1964) 140. [19] X. Jimei, Metal Problem of Stainless Steels, Metallurgical Industry Press, 1983. [20] Dong Zhizhong, Study on Corrosion Resistance Fe-based Shape Memory Alloy and Martensitic Transformation, Doctor Dissertation of Tianjin University, 2001.
Transformation-induced plasticity of expandable tubulars materials Xu Ruiping a,∗ , Liu Jie b , Zhang Yuxin a , Guo Ruisong a , Liu Wenxi a a
School of Materials Science and Engineering, Tianjin University, Tianjin 300072, China b Shanghai Tianhe Shape Memory Limited Corporation, Shanghai 201615, China
Received 8 May 2005; received in revised form 18 January 2006; accepted 28 February 2006
Abstract A new concept on strengthening-plasticity is presented in this paper, along with the design rules of the alloys with high expandable performance. The plasticity of expandable alloys induced by the ␥→ martensitic transformation, can be obviously improved. In order to enhance transformationinduced plasticity, alloys with low stacking-fault energy and high starting temperature of martensite Ms() should be looked for. Therefore, a formula for the calculation of Ms() of Fe-based alloys is also given in this paper in order to facilitate the design of expandable tubular materials. © 2006 Published by Elsevier B.V. Keywords: Strengthening-plasticity; Transformation-induced plasticity; Fe-based alloys; martensite
1. Introduction Expandable tubular technology is an innovative technology in oil industry and has been viewed as a revolutionary well construction in the 21st century. Many papers or articles previously published have introduced its concept and applications in drilling. However, there are only few papers to introduce how to choose and design expandable tubular materials at present. This paper will present the design rules of expandable tubular materials and give a new concept, strengthening-plasticity, to express its comprehensive properties. 2. Strengthening-plasticity of materials An idea of solid expandable tubular [1] (SET) was first put forward by the Royal Dutch Shell Technology Ventures, Inc. in 1992. Its principle [2,3] is to put the tubular into an oil well and make them expanded and deformed by pulling a specially designed grounder or pushing an expansion cone or mandrel. The main purpose of using expandable tubular is to diminish drilling dimension and widen oil-extracting pathway. SETs have been successfully installed in oil and gas fields since November 1999 [4], in a variety of environments in wells on land, offshore and in deepwater to solve a range of drilling and completion challenges. Practical field applications have shown that expand∗
Corresponding author. Tel.: +86 22 27890485; fax: +86 22 27404471. E-mail address: [email protected] (X. Ruiping).
0921-5093/$ – see front matter © 2006 Published by Elsevier B.V. doi:10.1016/j.msea.2006.02.173
able tubular technology can reduce the overall cost of drilling greatly and shorten the construction period. Expandable tubular technology can be applied in a series of engineering technology, including petroleum drilling, fixing, completing and repairing wells. So expandable tubular technology has been viewed as a revolution in petroleum well tubular. Many petroleum companies at abroad and home have begun to develop and research this technology in recent years. However, there are many difficulties in researching the materials of expandable tubular. Because expandable tubular are often installed in oceans or deepwater and have to bear great outside force, its materials should have high strength. Besides, the materials should also have excellent plasticity in order to meet the requirements of large deformation during the expansion process. So a new concept, which is strengthening-plasticity expressed by a parameter k, will be introduced in this paper to show its comprehensive properties. The parameter k, equals to the product of tensile strength σ b and elongation rate δ, which are both measured under annealing condition: k = σb δ
(1)
If the unit of σ b is MPa and δ is %, the unit of k is MPa%. A three-dimensional coordinate system can be used to show the relations among k, σ b and δ. The value of k can be got from the upright axis of the plane of σ b –δ, as shown in Fig. 1. From this figure, we can get the shape of this three-dimensional coordinate system looks like “hills” from top to bottom and the projection of the hills’ contour lines gives hyperbolas. The authors of this
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X. Ruiping et al. / Materials Science and Engineering A 438–440 (2006) 459–463
Fig. 1. The coordinates of strengthening-plasticity k.
Fig. 3. The mechanism for the stress induced ␥– transformation.
Fig. 2. The comparison of strengthening-plasticity among various types of steels.
paper have made many surface and field experiments on expandable tubular materials and compare the strengthening-plasticity of different kinds of steels. The result is as shown in Fig. 2. After many experiments, a kind of expandable alloys with excellent expandable performance was obtained at last, whose properties lies in the hatched zone in Fig. 2. This figure shows that the k of LSX80 researched by Enventure Global Technology L.L.C., is 30,000 MPa%, while the k of our expandable tubular materials reaches 60,000 MPa%, nearly two times that of Enventure Global Technology L.L.C. The measured mechanical properties of our researched expandable alloys are that σ b is 990 MPa, σ s 365 MPa, δ 61%, ψ 57% and work-hardened index n is 0.40, respectively. The k of high strengthening-toughness steels and high plasticity steels with low carbon seen from this figure, are both below 20,000 MPa%. 3. Design for high strengthening-plasticity steels and transformation-induced plasticity The great trouble in designing expandable tubular materials is that the two properties of plasticity and strength are always contradictory, that is to say, the plasticity is low while the strength is high, such as the high strengthening-toughness
steels. In turn, the high plasticity steels with low carbon have good plasticity, its strength is very low. In order to improve the strength and plasticity of expandable tubular materials at the same time, the mechanism of transformation-induced plasticity was used when we designed some compositions of Fe-based expandable alloys. This kind of alloy has been proven to have high strengthening-plasticity and outstanding expandable performance. The mechanism of transformation-induced plasticity is induced by the transformation from parent phase austenite ␥ to martensite when the materials are subjected to outside force. Partial plastic deformation has gone with phase transformation. Meanwhile, the double phase structure produced during the phase transformation can improve strain-hardening rate and prevent early necking phenomenon from happening. Therefore, transformation-induced plasticity can improve the strengthening-plasticity obviously. Besides, when designing expandable tubular material, it should also have high work hardening rate. The reason is to make the materials yield easily at the early period of expansion and improve their strength quickly at the later period, which can save a lot of expandable energies. The value of work hardening rate n discussed in this paper is more than 0.25, which can meet stable plastic deformation condition dσ/dε > σ [5,6]. So there are three characteristics for expandable tubular materials, i.e. high strength and high plasticity and high work hardening rate. It is important to know it when designing expandable tubular materials. The composition of expandable alloys should lie in austenite zone according to Schaefflers figure [7]. So the addition of alloy elements should be strictly controlled to ensure that the structure of alloys is a single austenite phase. Only if the structure of alloys is an austenite, does the mechanism of transformation-induced plasticity function when expandable alloys deform under the action of stress. The mechanism of ␥→ martensite phase transformation is as shown in Fig. 3. The top and bottom plane of triangular prism show the place of close-packed atoms and the height of triangular prism is a planar distance between close-packed atoms. The arrowheads in this figure show the
X. Ruiping et al. / Materials Science and Engineering A 438–440 (2006) 459–463
461
moving direction of Shockley partial dislocations. The crystal structure of the parent phase γ is face-centered cubic (fcc) and that of martensite is a hcp structure. The fcc can be looked as layer to layer stacking along (1 1 1)␥ close-packed and its stacking way is ABCABC· · ·. The hcp can be looked as layer to layer stacking along (0 0 0 1) coherency, its stacking way is ABABAB· · ·. When materials are bearing outside force, a/6[1 1 2] Shockley partial dislocations [8] will occur in fcc and the atoms of every other (1 1 1)␥ move the distance by a/6[1 1 2] along the same direction. As a result, a shear angle of 19◦ 28 is formed. Meanwhile, the stacking arrangement of atoms will change from ABCABC· · · to ABABAB· · ·. Consequently, fcc crystal structure changes to hcp structure and martensite [9] is formed. During the ␥→ martensite phase transformation, plastic deformation also occurs, which can decrease resistance of plastic deformation. Therefore, transformation-induced plasticity improves the plasticity greatly. 4. The formula of Ms() for Fe-based alloys Hsu [10] suggested that martensite forms and grows up by the mechanism of staking faults nucleation for Fe-based alloys, which is a kind of alloys with low stacking faults energy. The stacking faults energy plays an important role in the process of ␥→ martensite phase transformation. The lower the stacking faults energy, the easier martensite phase transformation happens. Accordingly, the starting temperature of martensite transformation Ms() increases. Ms() can be recognized as an index to evaluate the inclination of martensite phase transformation. So the determination of Ms() is very important and necessary to facility to design the expandable tubulars materials. But it is difficult to measure Ms() by experiments because the resistance, specific capacity and magnetism between parent phase ␥ and martensite are very close and the heat effects of the phase transformation are small. For the above reasons, a method of calculating Ms() of Fe-based alloys is given in this paper. The micro-mechanism of martensite phase transformation has been investigated by surface relief [11–14] and atomic force microscopy (AFM) [15] showing that martensite has two morphologies. One is stress-induced martensite, the other is thermal induced martensite. Fig. 3 also shows the mechanism of stress-induced martensite, one of whose character is to form 19◦ 28 -shear angle. As temperature drops, thermally induced martensite will form. It can be concluded from Figs. 3 and 4 that the mechanism of these two martensite is different. One obvious distinction is that the moving orientation of stacking faults of thermally induced martensite is not on a line but a left–right interval style. Though its arrangement of atoms are ABABAB· · ·, its shear angle is 0◦ . And its Shockley partial dislocations expressed with arrowhead in Fig. 4 move to the left at first then to the right and hold still at last. The moving of its Shockley partial dislocations looks like the crossed arrangements of sawtooth of sawblade. Based on the above-mentioned model, the formula of Ms() for Fe-based alloy is deduced from the mechanism of thermal induced martensite. Every nucleus embryo of thermal induced martensite expressed with the length of arrowhead in the middle of Fig. 4,
Fig. 4. The mechanism for the thermal induced ␥– transformation.
is composed of six layers of atom plane [16,17]. Only one stacking fault interface exists between one nucleus embryo and parent phase, so the interface energy σ of per area equals to stacking faults energy F. Besides, the chemical energy Gchem is three times stacking faults energy because every nucleus embryo includes three stacking faults. Other differences energy of crystal lattice and the phase resistance energy are expressed by Eresist . At the temperature of Ms() , the free energy of phase transformation equals to zero: nρ(Gchem + σ) + Eresist = 0
(2)
where n is the thickness of stacking faults, ρ the atomic density of per area, σ is interface energy of per area. From the above equation, the following equation can be deduced: Eresist = Gchem + σ = 4F nρ
(3)
It is supposed that the stacking faults energy show a linear relation with absolute temperature in Ref. [15]. F = F0 + k T
(4)
where F0 is the stacking faults energy at T = 0 and k is a proportionality coefficient. When alloy A and alloy B are at their corresponding Ms() temperatures, i.e. TA = Ms()A , TB = Ms()B , their stacking faults energy are expressed with the following equations, respectively: FA = F0A + kMs()A
(5)
FA = F0B + kMs()B
(6)
F on the right side of Eq. (3) can be substituted by FA and FB , respectively because the change of Eresist is not too large and nρ is almost a constant. The left side of Eq. (3) is a constant, so the
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X. Ruiping et al. / Materials Science and Engineering A 438–440 (2006) 459–463
Table 1 The Ms() values of nine kinds of Fe-based alloysa No.
Mn
Si
Cr
Ni
Change of stacking faults energy, F0
Calculated values
Measured values
Error
1 2 3 4 5 6 7 8 9
31.5 27.6 20.4 19.9 16.25 16.0 13.0 14.0 0
6 6.1 5.0 5.1 5.0 5.1 4.7 6.9 0
– 5.0 8.0 10.1 9.2 11.6 11.4 9.0 18
– – 5.0 5.0 4.0 4.9 6.8 5.3 8
−109 −101 −75 −73.4 −69 −66.6 −57.1 −74.8 0
283 278.4 263.3 262.4 259.9 258.5 253 263 220
293 283 261.2 255 273 267 244 255 221
+10 +4.6 −2.1 −7.4 +13.1 +8.5 −9 −8 +1
a
The numbers indicate the weight percentage of alloy elements and the balance is Fe.
following equations can be easily obtained: F0A + k Ms()A = F0B + k Ms()B 1 F0 k
Ms()B = Ms()A −
5. Conclusions (7) (8)
where F0 equals to the total change of stacking faults energy resulting from the compositional change of every alloy element. F0 =
n
xi fi
(9)
i=1
If 0Cr18Ni8 stainless steel is regarded as a standard alloy whose martensite phase transformation starting temperature is 220 K, we can get the following formula: n
Ms() = 220 −
1 xi fi k
(10)
i=1
where xi is the change of atom percentage compared to the standard alloy, i is the number of alloy elements. fi is the effect of alloy elements of per atom percentage on the stacking faults energy. Table 1 shows the weight percentage of nine kinds of Febased alloys and their Ms() values calculated with the above formula. Table 2 exhibits the effect of several alloy elements on the stacking faults energy. We can conclude from Table 1 that the error is not too big and it is convenient to get the values of Ms() for Fe-based alloys. Table 2 The effect of alloy elements on the stacking faults energy [18,19] Alloying element
fi (erg/cm2 /%)
Cr Ni Si Co Cu Nb Mo Mna
+0.5 +1.4 −3.4 −0.55 +3.6 +3.2 +0.1 −1.68
a
Mn is a supplementary value near to the value in Ref. [20].
(i) The product of tensile strength and elongation rate can be employed as an index to evaluate the strengtheningplasticity of materials. The concept of strengtheningplasticity can reflect the comprehensive properties of expandable alloys. (ii) Expandable tubular materials should have three characteristics: high plasticity, high strength and high work hardening rate. (iii) The property of strengthening plasticity can be improved by transformation-induced plasticity. We should find the alloys with low stacking faults energy and high martensite starting temperature Ms() when designing expandable tubular materials. (iv) The formula of Ms() of Fe-based alloys is as following: n
Ms()
1 = 220 − xi fi k i=1
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