archives of civil and mechanical engineering 19 (2019) 1–9
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Original Research Article
Microstructure and mechanical properties of TiC/Ti matrix composites and Ti–48Al–2Cr–2Nb alloy joints brazed with Ti–28Ni eutectic filler alloy Duo Dong a,b,*, Haitao Xu c, Dongdong Zhu b,*, Gang Wang b, Qing He a, Junpin Lin a a
State Key Laboratory for Advanced Metals and Materials, University of Science and Technology Beijing, Beijing 10083, PR China b Key Laboratory of Air-driven Equipment Technology of Zhejiang Province, Quzhou University, Quzhou 324000, PR China c School of Mechanical and Automotive Engineering, Anhui Polytechnic University, Wuhu 241000, PR China
article info
abstract
Article history:
Ti–48Al–2Cr–2Nb and TiC/Ti matrix composites were successfully joined using Ti–28Ni
Received 3 March 2019
eutectic filler metal at different brazing temperatures. Microstructure and mechanical
Accepted 20 July 2019
properties of the joints were systematically studied. The results showed that the joints all
Available online
showed integral interfaces and the microstructures of the joints for 980 8C/15 min were detected as Ti-48Al-2Cr-2Nb/a2-Ti3Al + t3-Al3NiTi2/a2-Ti3Al/Ti(s,s) + d-Ti2Ni + TiC/Ti ma-
Keywords:
trix composites. As brazing temperature increased, continuous Ti2Ni layer diminish and
TMCs
Ti2Ni phase became more discrete. However, a2-Ti3Al rich layer has thickened which
TiAl alloy
deteriorates the properties of the joints. The highest shear strength achieved 469.5 MPa
Brazing
as the joint brazed at 1010 8C for 15 min. The evolution of microstructures brazed with
Microstructure
different temperatures was studied and its relationship with the shear strength was also
Shear strength
revealed in details. © 2019 Published by Elsevier B.V. on behalf of Politechnika Wroclawska.
1.
Introduction
Due to significant weight saving and specific strength than Ni base superalloys, TiAl alloys have been considered as one of the most attractive materials in aerospace and automobile industry [1,2]. The addition of Nb and Cr elements into
TiAl alloys can not only increase the ductility, but also improve high temperature strength and creep resistance above 800 8C [3,4]. In these TiAl alloys, Ti–48Al–2Cr-2Nb has been applied to critical parts in aerospace, such as aircraft turbine engines blades in General Electric's GEnex engine, as a high-temperature as well as weight-saving structural material [5,6].
* Corresponding author at: Quzhou University Kecheng district, Quzhou, Zhejiang Province, China. E-mail addresses:
[email protected] (D. Dong),
[email protected] (D. Zhu). https://doi.org/10.1016/j.acme.2019.07.005 1644-9665/© 2019 Published by Elsevier B.V. on behalf of Politechnika Wroclawska.
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So far, Ti matrix composites have attracted much attention and were successfully applied in aerospace, due to their higher specific strength, relatively lower density and better mechanical properties at elevated temperatures compared with traditional Ti alloys [7,8]. Typically, as TiC particle phase has high modulus, high stiffness, good thermal stability and is totally compatible with Ti matrix, it is considered as one of the best reinforcements of Ti matrix composites [9,10]. Thus it is of great importance to expand the usage of TiAl and TMCs. However, it is quite expensive to deform TiAl alloys and Ti matrix composites into components with large dimensions or complex shapes due to the nature brittleness, poor workability and short deforming region. Thus the application of TiAl alloys and Ti matrix composites is greatly restricted. Take the excellent properties and poor workability of these two materials into account, successful joining of the two materials is thought as an effective way to further extend the applications of TiAl alloys and Ti matrix composites, which is of great importance to the weight saving in aircraft engines. Until now, various methods have been studied to join TiAl alloys and Ti matrix composites. For example, H.S. Ren [11] reported a diffusion bonding of Ti–24Al–15Nb-1Mo alloy and TiAl intermetallics using two Ti based filler metals. The highest shear strength of the joints was about 435 MPa. Y. Wang studied [12] the brazing of Ti2AlNb and TC4 alloys with Ti–Zr–Cu–Ni + Mo composite filler. It was found that Mo particle was beneficial for the homogeneity of the joint microstructure and the formation of b-Ti. The joints showed good mechanical properties at both of room temperature and 6008. Vacuum brazing was also studied to join TiBw/TC4 composite and Ti60 alloy by X.G. Song [13]. The joint strength reached the maximum value of 368.6 MPa. However when brazed at 10608, the formation of coarse lamellar (a+b)-Ti structure would decrease the joint properties. H.G. Dong [14] studied the joining of g-TiAl and 40Cr steel with friction welding and found that the joint strength was degraded by the formation of martensitic microstructure and TiC phase. J.W. Mao [15] reported that TiB + La2O3 reinforced Ti matrix composites were laser welded and the joints exhibit superior strength but poor ductility because of the existence of a0 martensite and TiBw brittle phase. These joining techniques, brazing excepted, are always carried out at high temperatures above the b-transformation temperature of the base metal. As a result, B2 or b-Ti brittle phase would be formed when Ti or TiAl alloys are exposed at high temperatures during heating process [16,17]. It may deteriorate the mechanical properties greatly. Thus, brazing is taken as the most effective way to join dissimilar materials, attribute to relatively low joining temperature, little influence on the base metal as well as low cost, convenient operating, high precision and excellent joint properties of brazing technique. But to the best of the author's knowledge, the investigation of brazing TiAl alloys to Ti matrix composites has not been reported yet. Z. Mirski [18] brazed Ti–48Al–2Cr-2Nb alloy using different kinds of filler metals and demonstrated that filler metal plays an important role in the joint mechanical quality. It has been widely reported that Ag based [19,20] and Ti based filler [21,22] metal are the main choice for brazing Ti or TiAl alloys for their good wettability on the materials. As both of TiAl alloys and Ti
matrix composites perform great high temperature strength, the brazing filler should possess high molten temperature which is desirable for improving the mechanical stabilities of the joint at high temperature working conditions. However, previous studies [23,24] showed that the continuous brittle intermetallic compound like Ti2Ni or Ti2Cu phase layer was hard to demolish when brazed TiAl alloys with Ti based filler alloy. Fortunately for this issue, we found that Ni element was easy to transport into TiC/Ti alloys through the intergranular gap in our former work [25]. This phenomenon is favorable to homogenize the distribution of Ti2Ni phase layer and reduce the influence of it. In this work, the brazing of TiAl and Ti matrix composites was carried out using Ti–28Ni (wt.%) high-temperature eutectic filler metal at different temperatures. The microstructure and mechanical properties of the joints were investigated. And the effects of brazing temperature on the microstructure and shear strength of joints were studied in details.
2.
Experimental
The nominal composition of Ti–48Al–2Cr–2Nb (at.%) and Ti– 6Al–3Sn–3.5Zr–0.4Mo–0.75Nb–0.35Si (wt.%) + 5 vol% TiC base material used in this study were fabricated by induction skull melting (ISM) in purified argon atmosphere. Electromagnetic stirring was used when the raw materials were melted for a complete homogenization of compositions. Fig. 1 exhibits the interfacial microstructure of the base metals, the results show that Ti–48Al–2Cr–2Nb consists of g-TiAl and a2-Ti3Al, and Ti matrix composite consists of Ti(s,s) and TiC phases. Ti–28Ni brazing filler was fabricated by rapid solidification technique in an arc melting furnace. Fig. 2 displays micromorphology of the Ti–28Ni filler ingot, it can be seen that Ti–28Ni filler is composed of a-Ti and d-Ti2Ni phases. The DSC result in Fig. 2 b) exhibits that the b-transformation and liquidus temperature of Ti–28Ni filler are about 765 8C and 960 8C respectively. So the brazing temperature was determined from 980 8C to 1070 8C in order to make sure of the sufficient diffusion of the filler alloy. Prior to brazing, the substrates were cut into dimensions of 4 4 4 mm and 10 10 4 mm for morphological observation and shear testing. All of the joining surfaces were grounded with SiC papers up to 2000 grit and ultrasonically cleaned in alcohol for 3 min in order to get smooth and unpolluted brazing surfaces. The filler metal was cut into slices of 4 4 0.5 mm and both surfaces were grounded to about 200 mm thick. The assemblies of the joints during brazing are shown as Fig. 3. Ti-28Ni filler was sandwiched by Ti–48Al–2Cr– 2Nb and Ti matrix composite base materials in a graphite mold. A pressure of 0.1 MPa was applied on the joint for the proper contact of materials during brazing process. The heating process of brazing is shown in Fig. 3. The samples were directly heated up to the target brazing temperature (980 8C, 1010 8C, 1040 8C and 1070 8C) at a speed of 10 8C/min. Then the brazing temperature was held for 15 min. Subsequently, the temperature was cooled down to 300 8C in 2 h and finally cooled down spontaneously to room temperature. In the whole process of brazing, the vacuum was kept up to 2.0 103 Pa.
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Fig. 1 – Backscattered images of base materials. a) Ti-48Al-2Cr-2Nb alloy; b) Ti matrix composites.
Fig. 2 – Backscattered image and DSC analysis results of Ti-28Ni filler. a) Backscattered image of Ti-28Ni filler; b) DSC image of Ti-28Ni filler.
Fig. 3 – Assemblies of brazing samples and thermal cycling curve of brazing process. a) assemblies of brazing samples; b) thermal cycling curve.
After brazing, the microstructure and chemical composition of the joints were characterized using scanning electronic microscope (SEM) equipped with an energy dispersive spectrometer (EDS). Precipitation phases in the joints were identified by X-ray diffractometer (XRD) with Cu Ka at a scanning speed of 28/min. The XRD analysis was carried out on the fracture surfaces of the specimens after shear testing. The mechanical properties of the joints brazed at diverse temperatures were measured by shear testing with an electronic universal testing machine. At least five shear test samples were used to obtain average shear strength. After that, the
fracture surfaces were investigated by SEM to study the fracture mode and fracture location of the joints.
3.
Results and discussions
3.1. Microstructure of the Ti–48Al–2Cr–2Nb/Ti–28Ni/Ti matrix composites joint Due to the typical interfacial microstructure of the joint brazed at 9808, the results for brazing at 9808 were presented. Fig. 4
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Fig. 4 – Microstructure of the joint brazed at 980 8C for 15 min. a) microstructure of the whole joint; b) microstructure of zone I; c) microstructure of zone II; d) microstructure of the joint/Ti matrix composites interface.
shows the backscattered images of Ti–48Al–2Cr–2Nb/Ti–28Ni/ Ti–28Ni/Ti matrix composites joint brazed at 980 8C for 15 min. It can be seen that the width of the joint was about 219 mm. An increase about 20% of the joint width was observed in comparison with the original Ti–28Ni thickness of 200 mm. It proves that elemental diffusion has happened between molten brazing filler and the base materials. The entire interfacial of the joint is divided into two zones (zone I and zone II as shown in Fig. 4a). The microstructures of each zone are displayed in Fig. 4b–d. The high magnification of zone I shows a continuous reaction zone adjacent to Ti–48Al–2Cr– 2Nb which is about 7.6 mm wide. As Fig. 4b shows, zone I is divided into two layers, marked as I1 and I2. I1 is about 4.2 mm formed by gray and white phases while I2 layer is a continuous layer about 3.4 mm wide with monotonous gray contrast. The phases in zone I are marked as A, B and C. Similarly, zone II is a diffusion zone of about 212 mm wide between zone II and TiC/ Ti matrix composites. The phases in zone II are marked as D, E, F, G, H and J. It is worth noting that, a continuous layer with large white block phase was formed in this zone, and its width is about 60 mm. Some micro-cracks were observed in the higher magnified feature of this layer in Fig. 4c, indicating that this layer might be the weak point of the brazed joint. Phase D, in gray contrast, is uniformly distributed. Phase E is black and only exists in the region between the continuous white phase layer and Ti matrix composite base metal. With regard to phase F, it becomes more and more discrete in the direction from the brazing seam to Ti matrix composite. It makes the boundary between the brazing seam and the Ti matrix composites obscure. To study the diffusion process in the joint during brazing, the elemental distributions of the joint were measured by EDS.
The corresponding results were shown in Fig. 5 and Table 1. According to our previous study [25], the relative error of the element C is less than 6%. As can be seen in Fig. 5, zone I is rich in Ti, Al and Ni elements. It demonstrated that diffusion of Ni element from the molten filler into solid Ti–48Al–2Cr–2Nb substrate happened during brazing. Zone II was occupied by Ti and Ni element with small regions of C enrichment. Ni and Ti element diffused into the Ti matrix composites through the intergranular gap [25], and it is faster than other diffusive mode. However, the low brazing temperature of 980 8C was not favorable to the sufficient diffusion of Ni element. Thus Ni dominated the continuous white phase layer in zone as seen in Fig. 5d. Based on the EDS analysis of spot A and B in Table 1, the continuous reaction layer I is mainly comprised of a2-Ti3Al and a few t3-Al3NiTi2 phases. The phases can also be found in X.G. Song's study [26] when joining high Nb containing TiAl using TiNi-Nb filler alloy. The EDS results of phase C indicated that it was composed of 70.49% Ti and 27.16% Al which could be identified as a2-Ti3Al. During brazing process, elemental diffusion motivated by concentration gradients took place at the liquid/solid interface. Ti and Ni element would diffuse into base materials and Al to the brazing seam. Then a2-Ti3Al was formed firstly. Simultaneously Ni element was expelled to Ti– 48Al–2Cr–2Nb base metal interface and zone II, due to the low solubility of Ni element in a2-Ti3Al phase [27]. So a continuous a2-Ti3Al reaction layer I2 was formed in the joint. The formation of a2-Ti3Al was at high temperature, once the a2Ti3Al phase was formed, further elemental diffusion between the liquid filler and Ti–48Al–2Cr–2Nb base metal was obstructed. However, Al element continued diffusing from Ti–48Al–2Cr–2Nb into zone I during the entire brazing process.
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Fig. 5 – EDS mapping analysis of joint brazed at 980 8C for 15 min. a) Microstructure of the joints; b) element distribution of Ti; c) element distribution of Al; d) element distribution of Ni; e) element distribution of C.
Table 1 – Chemical compositions of spots in Fig. 4. Spots
A B C D E F G H J
Compositions (at.%)
Possible phase
Ti
Al
Ni
Nb
Cr
C
31.64 63.54 70.49 69.81 51.04 87.91 91.24 49.50 51.41
43.60 31.38 27.16 8.77 3.71 5.30 4.45 5.77 2.64
20.13 2.67 0.56 11.21 32.02 1.75 1.25 28.42 0.42
1.33 1.37 1.19 – – – – – –
3.29 1.25 0.60 – – – – – –
– – – 10.21 13.22 5.04 3.06 16.31 45.53
Then t3-Al3NiTi2 was formed due to a relatively high content of Ni and Al element in layer I1. The EDS results of phase D shows that it contains 69.81% Ti and 11.21% Ni which is supposed to be a mix of Ti2Ni and Ti(s,s). Phase E and H indicated an elemental molar ratio of about Ti:Ni = 2:1. According to Ti-Ni binary phase diagram, it can be determined as d-Ti2Ni phase formed by reaction: Ti + Ni!Ti2Ni at 984 8C. Phase F and G was mainly composed of Ti element with a friction of 88.6%, it should be Ti(s,s) formed as the equation of b-Ti!Ti(s,s). Previous study [38] showed that the maximum solubility of Ni in b-Ti and a-Ti is 12% and 0.5%, respectively. During the formation of Ti(s,s), Ni would be expelled to the boundary of Ti (s,s), then Ti2Ni phase would be formed by b-Ti!a-Ti + Ti2Ni. Combining with the EDS data and the morphology of phase J, it is determined to be TiC particles. The TiC particles can only be found in the area between the continuous d-Ti2Ni layer and Ti matrix composite. It is mainly because the diffusion of C element into the brazing seam was hampered by the continuous Ti2Ni layer. To further identify the phases analyzed above, XRD was performed on the joint and the result is shown in Fig. 6. Based on the EDS and XRD analysis, the microstructure of the joint brazed at 980 8C for 15 min is
Al3NiTi2 Ti3Al Ti3Al Ti2Ni +Ti(s,s) Ti2Ni Ti(s,s) Ti(s,s) Ti2Ni TiC
Ti–48Al–2Cr–2Nb/a2-Ti3Al + t3-Al3NiTi2/a2-Ti3Al/Ti(s,s)+ Ti2Ni + TiC/Ti matrix composites.
d-
3.2. Effects of brazing temperatures on the interfacial microstructure of Ti–48Al–2Cr–2Nb/Ti–28Ni/Ti matrix composites brazed joints Fig. 7 illustrates the back-scattered images of joints brazed at different temperatures for 15 min. The width of each zone brazed under different parameters is shown in Table 2. It can be seen that the widths of each zone in the brazing joint increased as brazing temperature increased. Although the width of both layer I1 and layer I2 were enlarged, layer I1 grew up much faster than layer I2. According to the microstructure analysis, t3-Al3NiTi2 can only be found I1 layer. So it can be concluded that more filler element of Ti and Ni diffused into layer I1 as the brazing temperature increases. It made the a2Ti3Al layer become thicker and more t3-Al3NiTi2 was formed in layer I1. It can also be found the size of t3-Al3NiTi2 phase became smaller and the distribution became more discrete as the brazing temperature increased. It was caused by the relatively low diffusive rate of element Ni compared with other
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Table 2 – Width of each zone under different brazing parameters. Brazing temperature
980 8C 1010 8C 1040 8C 1070 8C
Width of different zones (mm) I1
I2
Whole joint
4.2 12.1 14.3 25.1
3.4 5.1 6.3 12.5
219 222 293 319
Fig. 6 – XRD patterns of the phases in the joint brazed at 980 8C for 15 min.
elements. As the brazing temperature increases to 1010 8C, the width of zone II shows little change. However, the continuous d-Ti2Ni layer disappeared. At higher brazing temperatures, the fluidity of the filler increased, which made more liquid filler squeezed out of the joint. On the other hand, the elemental exchange of Ti and Ni was highly accelerated at the filler/Ti matrix composites interface at higher brazing temperatures. Due to the disappearance of the continuous d-Ti2Ni layer, more C element diffused into the joints, and TiC particles can be found in the whole area of zone II. Despite the continuous dTi2Ni layer disappeared, large amount of d-Ti2Ni phase can also be found in the brazing seam. Further increase the brazing temperature to 1040 8C and 1070 8C, the brazing seam became much wider and the volume fraction of d-Ti2Ni
Fig. 8 – Shear strength of joints brazed at different temperatures.
decreased. Due to the nature brittleness of d-Ti2Ni phase, the disappearance of continuous d-Ti2Ni layer is benefit to the mechanical properties. Although the morphology of the joints changed greatly as brazing temperature increased, no new phase was formed.
Fig. 7 – Interfacial microstructure of the joints brazed at different temperatures. a) e) 980 8C; b) f) 1010 8C; c) g) 1040 8C; d) h) 1070 8C.
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According to the analysis above, the mechanism of the brazing process can be concluded as follows. Firstly, as the brazing temperature rose to the liquidus temperature of the Ti28Ni filler metal, the filler metals spread out and wetted the base materials. During brazing, some liquid filler was squeezed out of the joint by the pressure applied on the joint. Reactions of diffusion and dissolution took place at the interface between the filler and base materials. Ti and Ni element would diffuse to the Ti–48Al–2Cr–2Nb and Ti matrix composite substrates. During the brazing process, a layer rich in Ti, Al and Ni was formed at the interface between Ti–48Al–2Cr–2Nb and the filler metal. During cooling process, the a2-Ti3Al layer was
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formed at the interface. At the same time, Ni element was expelled out from a2-Ti3Al due to the low solid solubility of Ni element in a2-Ti3Al phase. Further elemental exchange between the substrate and the filler became harder due to the a2-Ti3Al layer which formed at high temperature. As Al element continued diffusing from Ti–48Al–2Cr-2Nb into the joint, the excess Ni and Al together with Ti element reacted to form t3-Al3NiTi2 phase. The t3-Al3NiTi2 phase and the a2-Ti3Al together formed layer I1. As for zone, a lot of Ni element stayed at the middle of the joint because of the insufficient elemental diffusion at relatively low brazing temperature of 980 8C. As a result, a continuous d-Ti2Ni layer was formed in the joint
Fig. 9 – Fracture path and fracture morphology of joints brazed at different temperatures. a) b) 980; c) d) 1010 8C; e) f) 1040; g) h) 1070 8C.
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through the reaction of Ti + Ni!d-Ti2Ni. The residual filler metal would transform to a-Ti. When the brazing temperature increased, the fluidity of the filler was enhanced, more liquid filler was squeezed out of the joint. Higher brazing temperature also intensified the diffusion of the element, leading to the growth of each zone and the disappearance of the continuous d-Ti2Ni layer. Higher temperature also made the distribution of t3-Al3NiTi2 and d-Ti2Ni phase more homogeneous. It is worth noting that, the b-transformation temperature of Ti matrix composites used in this experiment is about 1080 8C [28]. Further increasing the brazing temperature, b-Ti would be formed, which may deteriorate the properties of the base metal. So the brazing experiments at higher temperatures were not studied.
3.3.
Mechanical properties of the brazed joints
The shear strengths of joints brazed at different temperatures are shown in Fig. 8. It can be seen that the shear strengths of the brazed joints all achieved high level, which are higher than 400 MPa. This level is higher than that in the previous works [29,30] which were roughly below 350 MPa. At 980 8C, the shear strength achieved 429.4 MPa, and the shear strength increased to 469.5 MPa when the brazing temperature increased to 1010 8C. However when further increased the brazing temperature to 1040 8C and 1070 8C, the joint properties decreased to 450.2 MPa and 420.5 MPa respectively. In order to illustrate the reason why the shear strength of the brazed joints changed with increasing of the brazing temperature, the morphology in the fracture of joints after shear testing were also studied. Fig. 9a, c, e, g shows the fracture path of the brazing seam at different brazing temperatures, corresponding fracture morphology were shown in Fig. 9b, d, f, h. It can be seen the fracture propagated along the continuous d-Ti2Ni layer in zone when brazed at 980 8C. Previous study [31] shows that d-Ti2Ni and a2-Ti3Al are brittle phases and their existence is detrimental to the joining properties, especially the d-Ti2Ni layer. The micro cracks observed in the continuous d-Ti2Ni layer in zone II would be fracture resources during shear testing. When the brazing temperature was elevated to 1010 8C, the continuous d-Ti2Ni layer in zone II disappeared, which is beneficial to improve the mechanical properties. Zone I which is composed of a2-Ti3Al phase and t3-Al3NiTi2 phase and the continuous d-Ti2Ni layer in zone II, are intermetallic compound (IMC) layers. The existence of IMC layer has great influence on the bonding strength. During brazing, the formation of IMC indicates a good reaction and a solid bonding between the filler metal and the substrate. However, due to the nature brittleness of the IMC, further increasing of IMC layer would result in higher stress and deteriorate the joint strength. Consequently, when brazed at 980 8C, the fracture was located in the continuous d-Ti2Ni layer and changed to zone I as the brazing temperature increased. It should be noted that, although both layer I1 and layer I2 consisted a lot of a2-Ti3Al, the formation and even distribution of t3-Al3NiTi2 phase enhanced the properties of layer I1, so rare fractures located in layer I1. It can be seen in Fig. 9b, d, f, h that all the samples displayed small river patterns and cleavage facets, implying that the
samples all fractured in quasi-cleavage fracture mode. In conclusion, the formation of continuous a2-Ti3Al and d-Ti2Ni layers was both harmful to the properties of the joint. Elevated brazing temperature was good for the disappearance of the dTi2Ni layer, but it also improved the growth of the a2-Ti3Al interlayer. Thus the shear strength rose when the brazing temperature increased from 980 8C to 1010 8C, and then decreased as the brazing temperature was elevated to 1040 8C and 1070 8C.
4.
Conclusions
(1) Sound joints of Ti–48Al–2Cr–2Nb/Ti–28Ni/Ti matrix composites were obtained by vacuum brazing at different brazing temperatures. The typical interfacial microstructure of the joint brazed at 980 8C for 15 min was Ti-48Al2Cr-2Nb/a2-Ti3Al + t3-Al3NiTi2/a2-Ti3Al/Ti(s,s) + d-Ti2Ni + TiC/Ti matrix composite. (2) The continuous d-Ti2Ni layer diminished at 1010 8C and became more and more discrete when the temperature was elevated to 1040 8C and 1070 8C. But with the increase of brazing temperature a2-Ti3Al rich layer (zone) became thicker. (3) The shear strength of the joints rose at the brazing temperature from 980 8C to 1010 8C, and decreased at higher brazing temperatures. The joints fractured at the continuous d-Ti2Ni layer at 980 8C. When d-Ti2Ni layer disappeared the crack broke into the enlarged a2-Ti3Al layer. All the samples fractured in quasi-cleavage fracture mode.
Funding This work was supported by the National Natural Science Foundation of China (Grant No. 51801112, 51501100, 51704001), the Zhejiang Provincial Natural Science Foundation of China (Grant No. Y18E010014) (Grant No. Y18E010003).
Conflict of Interest The authors declare no conflicts of interest.
Ethical Statement This manuscript is our original work and has not been published nor has it been submitted to any other journal. All authors have checked the manuscript and approved this submission.
Acknowledgement This work was supported by the National Natural Science Foundation of China (Grant No. 51801112, 51501100, 51704001),
archives of civil and mechanical engineering 19 (2019) 1–9
the Zhejiang Provincial Natural Science Foundation of China (Grant No. Y18E010014) (Grant No. Y18E010003).
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