Revised Semiempirical Approach to Predict the Occurrence of Twinning in Titanium Alloys.

A revised semiempirical approach, considering the average values of the valence electron to atom ratio (e/a̅) and a difference in atomic radii of alloying element/s and the base element (Δr̅), is proposed to predict the twin formation in titanium alloys. The revised e/a̅ versus Δr̅ diagram is plotted, considering the reported results of 90 titanium alloys fabricated using various processing methods. A new twin/slip boundary has been plotted and recommended based on the revised e/a̅ versus Δr̅ diagram. The conventional maximum limit reported for the twinning in titanium alloys is e/a̅ = 4.20; however, it has been found that twinning in titanium alloys is possible up to the e/a̅ of 4.30.

Introduction

The last decade has witnessed exponential growth in the development of titanium alloys produced using various thermomechanical processing routes, powder metallurgy, selective laser melting, spark plasma sintering, and many other fabrication techniques for their use in biomedical, aerospace, power plant, and automobile industries.[1,2] A wide range of mechanical, corrosion, and biological properties can be obtained by appropriately tailoring the microstructure of titanium alloys.[3,4] Each alloying element of titanium alloys reacts differently in varying microstructures and mechanical properties.[3,5,6] Therefore, each titanium alloy deforms differently from one another depending on the phases present in the microstructure and their structural parameters.[7,8] Furthermore, the type of deformation mechanism, for example, twin, slip and so forth, plays a vital role in tailoring the mechanical properties of titanium alloys.[9] Twin boundaries present in crystalline alloys improve dislocation storage ability.[10,11] Therefore, alloys that deform by a twin-dominating mechanism demonstrate low yield strength and large elongation/plasticity, whereas alloys that deform by a slip-dominating mechanism demonstrate high yield strength and less elongation/plasticity.[9] In general, twinning helps in achieving significant work hardening in alloys especially used for biomedical and many shape memory applications.[10,12]

Established literature acknowledges that twinning is fundamentally possible in lower-symmetry metals/metal alloys comprising body-centered cubic, hexagonal close-packed, and face-centered cubic structures.[12] Twinning in alloys may occur during deformation, phase transformation, and/or recrystallization by a homogeneous simple shearing of the parent/matrix lattice in alloys with low stacking fault energies.[12] Consequently, deformation twins possess many stacking faults with imperfect structures. Moreover, the chemical composition is one of the influencing factors in the occurrence of twinning in alloys because all the elements react differently in terms of either increasing or decreasing the grain size, the amount of interstitial or substitutional solutes, stacking fault energy, and so forth, and all these parameters also influence the occurrence of twinning.[10,12]

Twinning is a fundamental deformation mode in bcc metastable β titanium alloys.[13] Twinning improves work hardening properties in titanium alloys.[13] The mechanism which provides significant work hardening is known as twin-induced plasticity effect in titanium alloys.[13] Moreover, it has been reported that twining also increases strain in titanium alloys and creates obstacles in gliding dislocations.[13] The deformation mechanisms such as deformation mechanisms including {112}<111> twinning, {332}<113> twinning, and stress-induced α″ martensitic transformation usually remain evident in less stable β titanium alloys which display high work hardening, whereas the slip deformation mechanism remains evident in stable β titanium alloys which display poor ductility.[13] When considering the properties of eutectic alloys, it has been reported that slip systems also dominate in titanium-based (i.e., Ti–Si–Sn, Ti–Fe–Co, Ti–Fe, Ti67.79Fe28.36Sn3.85, etc.),[14,15] aluminum-based (i.e., Al–3.1Ni, Al–17Cu, Al–12.2Si, etc.), copper-based (i.e., Cu–23.1MG, Cu–2.8Zr, Cu–Ge, Cu–26Ag, Cu–19.7B, Cu–CuZrGe, etc.), and nickel-based (i.e., Ni-32.5 atom %, Ni–W, Ni–Mo, Ni–Ni3Si, etc.) eutectic alloys.[15]

Thermomechanical processing and manufacturing methods also influence the deformation properties of alloys.[8,16] Cryomilling is a new technology to produce high-strength nanomaterials, in which metallic powders are milled at cryogenic temperature to tailor the microstructure and mechanical behavior of materials.[17,18] The advantages of cryomilling include environmentally friendly nature, less contamination, rapid grain refinement, cost effectiveness, and large-scale production capability of various nanomaterials.[17] Other than additive manufacturing methods, cryomilling could also be crucial for producing titanium alloys to tailor various deformation and strengthening mechanisms.

The atomic radius and electronic parameters including the compositional average values of bond order (Bo̅), the d-orbital energy level (Md̅) and the valence electron to atom ratio (e/a̅) are important parameters in predicting the occurrence of twinning in alloys because these electronic parameters and atomic radii also affect the grain size, the amount of interstitial or substitutional solutes, and the stacking fault energy of alloys.[12,19] Abdel-Hady et al.[20] have proposed the extended phase stability diagram, based on the DV-Xα cluster method suggested by Morinaga et al.[21,22] considering the Bo̅ and Md̅ parameters, which predicts the prospective phases, elastic modulus, and deformation mechanism for titanium alloys. Figure shows the positions of Ti–35Nb–5Ta–7Zr,[23] Ti–35Zr–5Fe–2Mn,[24] Ti–33Zr–3Fe–2Cr,[25] Ti–33Zr–5Fe–2Cr,[25] Ti–33Zr–3Fe–4Cr,[25] Ti–1Fe,[26] and Ti–3Fe[26] on the extended phase stability diagram. It is worth noting that the alloys shown in Figure do not lie in the slip region; however, these alloys demonstrate a slip-dominating deformation mechanism based on the experimental evidence reported in previous studies. This indicates that the DV-Xα cluster method is not completely effective in predicting the proposed deformation mechanism. Hence, there is still a need to develop an improved method for predicting the proposed deformation mechanism in titanium alloys.

Figure 1

Positions of Ti–35Nb–5Ta–7Zr,[23] Ti–35Zr–5Fe–2Mn,[24] Ti–33Zr–3Fe–2Cr,[25] Ti–33Zr–5Fe–2Cr,[25] Ti–33Zr–3Fe–4Cr,[25] Ti–1Fe,[26] and Ti–3Fe[26] on the extended phase stability diagram.[20,27] [Reprinted in part with permission from Ref (20). Copyright 2006 ELSEVIER].

Determination of Parameters

The values of Bo̅ and Md̅ were calculated using eqs and 2, respectively.[27,28]Where x is atomic fraction (atomic weight/atomic mass) of the ith component in the alloy composition and (Bo) and (Md) are the respective values for the ith component.[21]

The values of Δr̅ were determined using eq in which titanium is considered as a base element.[5]where x is the atomic fraction, r is the metallic atomic radius of the ith element, rTi is the atomic radius of titanium, and n is the number of alloying elements. The values of e/a̅ used were estimated using eq .[5]where x is the atomic fraction, e is the number of valence electrons of the ith element, and n is the number of alloying elements.

Results and Discussion

Wang et al.[19] have suggested a semiempirical approach considering the e/a̅ and the compositional average of a difference in atomic radii of the base element and alloying element/s (Δr̅) to understand the deformation mechanisms of β titanium alloys. Figure shows the e/a̅ versus Δr̅ diagram proposed by Wang et al.,[19] which is valid only for solution-treated β titanium alloys and may not be valid for other titanium alloys produced using different processing methods. Investigations on the twin mechanism and the effects of twinning on the mechanical properties have been reported, based on the electron microscopic evidence, in an ample amount to date, which can be used to predict the occurrence of twinning by developing an improved semiempirical approach. Considering these points, the present work tries to propose a revised e/a̅ versus Δr̅ diagram for predicting the occurrence of twinning in titanium alloys considering the various fabrication/processing methods such as casting, solution treating, annealing, ageing, hot rolling, cold rolling, powder metallurgy, and selective laser sintering.

Figure 2

Semiempirical approach suggested by Wang et al.[19] considering the e/a̅ (average valence electron to atom ratio) and Δr̅ (atomic radii difference) values to understand the deformation mechanisms for solution-treated β titanium alloys. [Reprinted in part with permission from Wang et al.[19] Copyright 2018 ELSEVIER].

Twin boundaries form as a result of simple shearing that occurs in the parent/base lattice as discussed above. The information on the shearing tendency of the lattice can be obtained based on the modulus of rigidity, which can be determined using the shear stress and shear strain of the parent lattice.[29] It has been reported that the modulus of rigidity increases as the e/a̅ of alloys increases.[29] A relatively high shear stress is required to break the bonds between two atoms and, thereby, for shearing of the lattice when e/a̅ increases and therefore, alloys possessing high e/a̅ values show a slip-dominating mechanism.[19] However, the atomic size misfit effect also plays a crucial role in the occurrence of slip and twin boundaries. The atomic size misfit occurs when alloying element/s have either a higher or lower atomic radius than the base element according to the Hume-Rothery rules.[30] According to the established literature, the values of e/a̅ are used to predict the β phase stability and the formation of intermetallic phases in titanium alloys,[1,25] whereas the values of Δr̅ are used to predict the solid-solution strengthening effects and the deformation mechanisms based on the atomic size misfit phenomenon.[31]Figure depicts the revised e/a̅ versus Δr̅ diagram which has been proposed in the present work.

Figure 3

Revised e/a̅ versus Δr̅ diagram considering the results of titanium alloys reported in previous findings.[13,24−26,32−77]

The regions of slip and twin in the revised e/a̅ versus Δr̅ diagram are found to be different than the regions of slip and twin presented in Figure . In the revised e/a̅ versus Δr̅ diagram, the results of around 90 titanium alloys have been considered for obtaining precise information on the regions of deformation mechanisms. Note that (i) the corresponding findings of these selected titanium alloys demonstrate the evidence of the deformation mechanism (i.e., slip/twin) based on the results of electron microscopy and (ii) the selected titanium alloys not only show a single β phase, but also show the other martensitic (α, α′, α″ ω) and intermetallic phases. Therefore, the revised e/a̅ versus Δr̅ diagram can be used for all the types of titanium alloys, that is, α, α + β, and β alloys.

In the revised e/a̅ versus Δr̅ diagram, the titanium alloys demonstrating (i) twin, (ii) twin and slip, and (iii) twin and/or stress-induced martensite (SIM) are shown in a common twin region to obtain information on the occurrence of twinning in titanium alloys. In Figure , the titanium alloys demonstrating (i) only twin, (ii) twin and slip, and (iii) twin and/or SIM are shown using “half left filled with orange color”, “open symbols with + sign”, and “half down filled with purple color” symbols, respectively. Moreover, the titanium alloys demonstrating only the slip mechanism are separated by a solid-blue line in Figure , and their positions are shown using solid symbols. The titanium alloys displaying the twin mechanism should have less β phase stability, whereas the titanium alloys displaying the slip mechanism should have stable β phase stability.[13]

The values of e/a̅ and Δr̅ for all the selected alloys are summarized in Tables –4 with necessary references. It has been reported in many findings that the slip usually forms in titanium alloys with e/a̅ values greater than 4.20.[19] However, for titanium alloys shown in the slip region of Figure , the values of e/a̅ vary from 4.03 to 4.62, whereas the values of Δr̅ vary from −3.5 × 10–3 to 1.6 × 10–3 nm (Tables –4). This reveals that the slip is possible in titanium alloys with e/a̅ values less than 4.20, which has been reported in alloys such as Ti–5Al–5Mo–5V–3Cr–0.5Fe (wt %),[36] Ti–6Mn–3Mo (wt %),[38] Ti–33Zr–3Fe–2Cr (wt %),[25] Ti–5Al–5Mo–5V–3Cr (wt %),[41] Ti–4Al–7Mo–3V–3Cr (wt %),[41] Ti–1Fe (wt %),[26] and Ti–3Fe (wt %).[26] The slip in these alloys possibly occurs due to atomic size misfit which increases the bonding strength between atoms. As a result, atomic movements become difficult which allows slip deformation instead of twin in these alloys. This phenomenon also suggests the importance of atomic size misfit in terms of Δr̅ in the deformation of titanium alloys.

Table 1

Values of e/a̅ and Δr̅ for Titanium Alloys (Processed Using Varied Processing/Fabrication Techniques) Showing the Slip Mechanisma

alloys	processing method	deformation mechanism	e/a̅	Δr̅ × 10–3 nm	refs.
Ti–15Cr (wt %)	HR and CR	slip	4.28	–2.6554	(32)
Ti–20Cr (wt %)	HR and CR	slip	4.37	–3.5549	(32)
Ti–23Nb–0.7Ta–2Zr–1.2O (atom %)	ST	slip	4.26	–0.9546	(33)
Ti–11Mo (atom %)	annealed	slip	4.22	–0.8822	(34)
Ti–14Mo (atom %)	annealed	slip	4.28	–1.1199	(34)
Ti–17Mo (atom %)	annealed	slip	4.34	–1.3597	(34)
Ti–15Mo–1Fe (wt %)	HR and ST	slip	4.20	–0.8433	(35)
Ti–5Al–5Mo–5V–3Cr–0.5Fe (wt %)	ST	slip	4.08	–1.7660	(36)
Ti–27Nb–7Fe–8Cr (wt %)	cast	slip	4.62	–3.2816	(37)
Ti–27Nb–7Fe–6Cr (wt %)	cast	slip	4.57	–2.8661	(37)
Ti–27Nb–7Fe–4Cr (wt %)	cast	slip	4.53	–2.452	(37)
Ti–35Zr–5Fe–2Mn (wt %)	cast	slip	4.27	1.3826	(24)
Ti–35Zr–5Fe–4Mn (wt %)	cast	slip	4.34	0.9626	(24)
Ti–35Zr–5Fe–6Mn (wt %)	cast	slip	4.40	0.5400	(24)
Ti–35Zr–5Fe–8Mn (wt %)	cast	slip	4.47	0.1147	(24)
Ti–6Mn–3Mo (wt %)	HR and ST	slip	4.19	–1.1925	(38)
Ti–6Mn–4Mo (wt %)	HR and ST	slip	4.20	–1.2397	(38)
Ti–33Zr–3Fe–2Cr (wt %)	cast	slip	4.17	1.6260	(25)
Ti–33Zr–5Fe–2Cr (wt %)	cast	slip	4.25	1.2002	(25)
Ti–33Zr–7Fe–2Cr (wt %)	cast	slip	4.33	0.7714	(25)
Ti–33Zr–3Fe–4Cr (wt %)	cast	slip	4.21	1.2105	(25)
Ti–33Zr–5Fe–4Cr (wt %)	cast	slip	4.29	0.7824	(25)
Ti–33Zr–7Fe–4Cr (wt %)	cast	slip	4.38	0.3514	(25)
Ti–25Nb–1Sn–4Cr (wt %)	cast	slip	4.23	–0.9193	(39)
Ti–25Nb–3Sn–4Cr (wt %)	cast	slip	4.24	–0.8644	(39)
Ti–25Nb–5Sn–4Cr (wt %)	cast	slip	4.24	–0.8079	(39)
Ti–10Mo–3Fe (wt %)	HR and ST	slip	4.21	–0.9931	(40)
Ti–10Mo–5Fe (wt %)	HR and ST	slip	4.29	–1.3780	(40)
Ti–5Al–5Mo–5V–3Cr (wt %)	HR and ST	slip	4.06	–1.6761	(41)
Ti–4Al–7Mo–3V–3Cr (wt %)	HR and ST	slip	4.08	–1.4667	(41)
Ti–1Fe (wt %)	HR and annealed	slip	4.03	–0.1803	(26)
Ti–3Fe (wt %)	HR and annealed	slip	4.10	–0.5426	(26)
Ti–10Fe (wt %)	HR and annealed	slip	4.35	–1.826	(42)
Ti–20Mo (wt %)	HR	slip	4.22	–0.8872	(43)

Note that hot-rolling, cold-rolling and solution treating are abbreviated as HR, CR, and ST, respectively.

Table 4

Values of e/a̅ and Δr̅ for Titanium Alloys (Processed Using Varied Processing/Fabrication Techniques) Showing the Twin and/or SIM Mechanismsa

alloys	processing method	deformation mechanism	e/a̅	Δr̅ × 10–3 nm	refs.
Ti–6Cr–4Mo–2Al–2Sn–1Zr (wt %)	ST	twin and SIM	4.12	–1.2575	(60)
Ti–8.5Cr–1.5Sn (wt %)	HR	twin and SIM	4.16	–1.4663	(61)
Ti–7.5Mo (wt %)	HR and ST	twin and SIM	4.08	–0.3110	(62)
Ti–2Al–9.2Mo–2Fe (wt %)	ST	twin and SIM	4.13	–0.8992	(63)
Ti–5.3Mo–6.5Sn–10.2Nb–10Zr (wt %)	cast	twin and SIM	4.13	0.7244	(64)
Ti–25Nb–10Ta–5Zr (wt %)	ST	twin and SIM	4.20	0.2378	(65)
Ti–29Nb–13Ta–4.6Zr (wt %)	ST	twin and SIM	4.25	0.1752	(65)
Ti–35Nb–10Ta–5Zr (wt %)	ST	twin and SIM	4.28	0.1832	(65)
Ti–26Nb (atom %)	ST	twin and SIM	4.26	–0.2596	(66)
Ti–18Nb–8Zr (atom %)	ST	twin and SIM	4.18	0.8622	(66)
Ti–16Nb–10Zr (atom %)	ST	twin and SIM	4.16	1.1442	(66)
Ti–12Mo (wt %)	ST and CR	twin and SIM	4.13	–0.5096	(67)
Ti–25Ta–25Nb (wt %)	CT and ST	twin and SIM	4.28	–0.2805	(68)
Ti–35Nb–2Ta–3Zr (wt %)	SLS	twin and SIM	4.23	0.0238	(69)
Ti–4Mo (wt %)	HR and ST	twin and SIM	4.04	–0.1629	(70)
Ti–3Al–8Mo–7V–3Cr (wt %)	HR and ST	twin and SIM	4.15	–1.9580	(41)
Ti–10V–4Cr–1Al (wt %)	CR and ST	twin and SIM	4.32	–2.4176	(71)
Ti–36Nb–2Ta–3Zr (wt %)	annealed	twin and SIM	4.24	0.0174	(13)
Ti–10V–3Fe–3Al (wt %)	HR and ST	twin and SIM	4.14	–1.9493	(72)
Ti–27.96Nb–11.97Ta–5.02Zr (wt %)	CR and ST	twin and SIM	4.23	0.2216	(73)
Ti–34Nb (atom %)	HR and ST	twin and SIM	4.34	–0.3400	(74)
Ti–25Nb–0.7Ta–2Zr (atom %)	CR	twin and SIM	4.26	0.0039	(75)
Ti–25Nb–10Ta–5Zr (wt %)	powder metallurgy and HT	twin or SIM	4.20	0.2378	(52)

Note that hot-rolling, cold-rolling, solution treating, selective laser sintering, heat treatment, and stress-induced martensite are abbreviated as HR, CR, ST, SLS, HT, and SIM, respectively.

Table 2

Values of e/a̅ and Δr̅ for Titanium Alloys (Processed Using Varied Processing/Fabrication Techniques) Showing the Twin Mechanisma

alloys	processing method	deformation mechanism	e/a̅	Δr̅ × 10–3 nm	refs.
Ti–24Nb–2Hf (atom %)	ST	twin	4.24	–0.0011	(44)
Ti–24Nb–4Hf (atom %)	ST	twin	4.24	0.2394	(44)
Ti–26Nb–2Hf (atom %)	ST	twin	4.26	–0.0221	(44)
Ti–26Nb–4Hf (atom %)	ST	twin	4.25	0.2115	(44)
Ti–15Mo (wt %)	HR and ST	twin	4.16	–0.6473	(45)
Ti–32Zr–30Nb (wt %)	CR and ST	twin	4.22	2.8876	(46)
Ti–10Mo (wt %)	HR and ST	twin	4.11	–0.4202	(40)
Ti–10Mo–1Fe (wt %)	HR and ST	twin	4.14	–0.6106	(40)
Ti–25Nb–3Zr–3Mo–2Sn (wt %)	HR and ST	twin	4.19	0.0169	(47)
Ti–5Fe (wt %)	HR and annealed	twin	4.17	–0.9064	(42)
Ti–20V (wt %)	HR and ST	twin	4.19	–2.4731	(48)
Ti–10Mo–1Fe (wt %)	HR and ST	twin	4.14	–0.6106	(49)
Ti–16Nb (at %)	HR and ST	twin	4.16	–0.1601	(50)
Ti–14Mo (wt %)	HR	twin	4.15	–0.6009	(43)
Ti–13Al (at %)	ST and annealed	twin	3.87	–0.5346	(51)
Ti–10Al (at %)	ST and annealed	twin	3.90	–0.4004	(51)
Ti–7Al (at %)	ST and annealed	twin	3.93	–0.2753	(51)
Ti–3Al (at %)	ST and annealed	twin	3.97	–0.1191	(51)
Ti–20Nb–10Ta–5Zr (wt %)	powder metallurgy and HT	twin	4.16	0.2626	(52)

Note that hot-rolling, cold-rolling, solution treating, and heat treatment are abbreviated as HR, CR, ST, and HT, respectively.

Furthermore, for titanium alloys shown in the twin region of Figure , the values of e/a̅ vary from 3.87 to 4.34 and the values of Δr̅ vary from −2.7 × 10–3 to 2.8 × 10–3 nm (Table ). Out of all the titanium alloys demonstrating twinning in Figure , 81.5% of titanium alloys possess the e/a̅ and Δr̅ values that range from 4.10 to 4.30 and −1 × 10–3 to 1 × 10–3 nm, respectively. This indicates that the chances of twinning in titanium alloys remain high for the e/a̅ values from 4.10 to 4.30 and the Δr̅ values from −1 × 10–3 to 1 × 10–3 nm. Many findings have concluded that the possibility of twinning in titanium alloys remains high for e/a̅ values not exceeding 4.20.[19] Consequently, the e/a̅ value of 4.20 is believed to be the maximum limit for the occurrence of twinning because Coulombic forces between two positive ions increase as e/a̅ increases, and thereby, the shearing of the lattice becomes difficult. However, the shearing of the lattice also depends on the atomic size misfit other than e/a̅. Hence, if titanium alloys possess e/a̅ values greater than 4.20, but possess the Δr̅ close to 0 nm in the twin region shown in Figure , then twinning is possible in these alloys. This phenomenon can be seen in many titanium alloys demonstrating the twin mechanism despite comprising e/a̅ values greater than 4.20 (Figure ).

Figure shows that although titanium alloys, i.e., Ti–10V–4Cr–1Al (wt %)[71] and Ti–34Nb (atom %),[74] lie in the twin region, these alloys possess e/a̅ values greater than 4.30. This suggests that despite comprising e/a̅ values greater than 4.30, these alloys still possess low stacking fault energy to trigger the formation of twin boundaries. It is also worth noting that Ti–10V–4Cr–1Al (wt %) shows twinning at the e/a̅and Δr̅ values of 4.32 and −2.4 × 10–3 nm, respectively, whereas Ti–5Al–5Mo–5V–3Cr–0.5Fe (wt %),[36] Ti–5Al–5Mo–5V–3Cr (wt %),[41] and Ti–4Al–7Mo–3V–3Cr (wt %)[41] show the slip by contrast at the e/a̅and Δr̅ values of 4.08 and −1.8 × 10–3 nm, 4.06 and −1.7 × 10–3 nm, and 4.08 and −1.5 × 10–3 nm. These alloys show contrasting results than the conventional limits of e/a̅ and Δr̅ for twinning in titanium alloys (e/a̅ < 4.20 and Δr̅ close to 0 nm) because of using different processing parameters/techniques. In Ti–10V–4Cr–1Al (wt %), the twinning-induced plasticity and transformation-induced plasticity effects have been observed following the martensitic transformation of the orthorhombic α″ phase. Therefore, Ti–10V–4Cr–1Al (wt %) displays exceptional mechanical properties at an e/a̅of 4.32 including a yield strength of 420 MPa, an ultimate tensile strength of 1200 MPa, and a uniform elongation of 35%.[71] Similar kinds of results have also been reported in other research.[78]

Conclusions

The revised e/a̅ versus Δr̅ diagram has been proposed to predict the occurrence of twinning in titanium alloys considering the previously reported results of titanium alloys fabricated using various processing methods. The revised e/a̅ versus Δr̅ diagram suggests that the slip is also possible below the e/a̅ value of 4.10 and the twin is also possible above the e/a̅ value of 4.20. Furthermore, the parameter Δr̅ is also important in predicting the occurrence of twinning other than e/a̅. It has also been observed that the chances of the twin formation in titanium alloys remain high for e/a̅ values between 4.10 and 4.30 and Δr̅ values between −1 × 10–3 and 1 × 10–3 nm. Therefore, the maximum limit for the occurrence of twinning is found to be up to the e/a̅ value of 4.30 in this work and should not be up to the e/a̅ value of 4.20. The revised e/a̅ versus Δr̅ diagram is helpful to predict the deformation mechanism when designing new titanium alloys for industry applications.

Table 3

Values of e/a̅ and Δr̅ for Titanium Alloys (Processed Using Varied Processing/Fabrication Techniques) Showing the Twin and Slip Mechanismsa

alloys	processing method	deformation mechanism	e/a̅	Δr̅ × 10–3 nm	refs.
Ti–12Mo (wt %)	ST	twin and slip	4.13	–0.5096	(53)
Ti–40Nb (wt %)	cold rolled and aged	twin and slip	4.26	–0.2557	(54)
Ti–15Mo–5Zr (wt %)	CR and ST	twin and slip	4.17	–0.2859	(55)
Ti–24Nb–4Zr–8Sn (wt %)		twin and slip	4.15	0.4670	(76)
Ti–10Mo–1Fe (wt %)	HR	twin and slip	4.14	–0.6106	(56)
Ti–10Mo–3Fe (wt %)	HR	twin and slip	4.21	–0.9931	(56)
Ti–3Al–5Mo–7V–3Cr (wt %)	HR and ST	twin and slip	4.12	–1.8076	(41)
Ti–20Mo (wt %)	ST	twin and slip	4.22	–0.8872	(57)
Ti–4.4Ta–1.9Nb (wt %)	ST	twin and slip	4.02	–0.0224	(58)
Ti–23Nb–3Zr–2Ta (wt %)	cast	twin and slip	4.14	0.0941	(59)
Ti–7Mo–3Cr (wt %)	CR and ST	twin and slip	4.13	–0.8354	(77)

Note that hot-rolling, cold-rolling, and solution treating are abbreviated as HR, CR, and ST, respectively.