Establishment of Cryogenic Constitutive Model for Ti-6Al-2Zr-1Mo-1V Titanium Alloy

doi:10.21203/rs.3.rs-1671760/v1

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Establishment of Cryogenic Constitutive Model for Ti-6Al-2Zr-1Mo-1V Titanium Alloy

https://doi.org/10.21203/rs.3.rs-1671760/v1

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Ti-6Al-2Zr-1Mo-1V (TA15) titanium alloy is widely used in aerospace and other key fields because of its excellent properties such as specific strength, specific stiffness, and corrosion resistance. TA15 titanium alloy has low thermal conductivity and high cutting temperature, which is a typical difficult-to-machine material. The cryogenic machining technology can effectively improve the problems of high cutting temperature and poor machining quality during the machining of titanium alloy. Many researchers have studied the constitutive model of TA15 titanium alloy in order to reveal the machining mechanism, but there has been no research on the cryogenic constitutive model of TA15 titanium alloy. In this paper, a series of tensile tests at low temperature were conducted with TA15 titanium alloy as the research target; The cryogenic constitutive model was established, and the accuracy of the cryogenic constitutive model was verified based on the unequal shear zone model. The test results show that the strength of TA15 titanium alloy increases and the plasticity decrease at low temperatures. the strengthened stage of TA15 titanium alloy was significantly shortened at − 196°C; The maximum error of shear force calculated by orthogonal cutting tests is less than 20% compared with that predicted by cryogenic constitutive model.

TA15 titanium alloy

cryogenic constitutive model

unequal shear zone model

cryogenic mechanical properties

cryogenic machining

The Ti-6Al-2Zr-1Mo-1V (TA15) titanium alloy is a general-purpose high Al-equivalent near-$\alpha$ titanium alloy that belongs to the medium strength titanium alloy family. It is widely utilized in aerospace and other fields because of its high specific strength, creep resistance, corrosion resistance, and outstanding welding properties [1, 2]. Titanium alloy, on the other hand, has low thermal conductivity, a strong chemical activity at high temperatures, and a low elastic modulus. Titanium alloy has high cutting temperatures, a large spring-back of cutting deformation, an easy adhesion characteristic, and poor tool life in the cutting process, making it a typical difficult-to-machine material. [3] As a result, achieving high-quality and efficient titanium alloy machining has become a pressing issue for the industrial industry [4]. Scholars at home and abroad have made a lot of exploration on the high-quality and high-efficiency machining of titanium alloy. It is found that the cryogenic machining with liquid nitrogen as the cooling medium can make the chips in a regular serrated shape, suppress the adiabatic shear band(ASB), and reduce the material microscopic defects [5]. And it plays a positive role in reducing the cutting temperature, cutting force, surface roughness, and tool wear [6–10].

The constitutive model is an equation describing the relationship between stress and strain, strain rate, and temperature during material deformation. Its accuracy is decisive for the analytical prediction of cutting force, cutting temperature, and surface quality in machining, and is an important mean to reveal the machining mechanism. The common constitutive models are Johnson-Cook (J-C) model [11] based on the empirical model, Molinari-Ravichandran (M-R) model [12], Arrhenius model [13], and Bodner-Paton (B-P) model [14] based on the physical model, Rusinek- Klepaczko (R-K) model [15], Zerrilli-Armstrong (Z-A) model [16] and so on. At present, some scholars at home and abroad have conducted a series of studies on the constitutive model of TA15. Hongchao Ji et al. [17] obtained the Johnson-Cook high-temperature thermal deformation constitutive equation of TA15 by compression experimental data. By calculating the constitutive equation, the flow stress was compared with the measured stress-strain curve, and the accuracy of the equation was verified. K.J.Song et al. [18] established an elastic-plastic constitutive model of TA15 titanium alloy welding, which describes the mechanical effects of volume change during diffusional solid phase transformation and transformation induced plasticity (Trp) during diffusionless solid phase transformation. YuanChen et al. [19] studied the effects of cooling rate and deformation on phase transformation through the continuous cooling tests and interrupted tensile tests of TA15 titanium, and came out with a physics-based model coupled with the effect of phase transformation. GZ Quan et al. [20] conducted thermal compression tests on titanium alloy at different temperatures and different strain rates. And based on the established GA-SVR, the continuously three-dimensional relationships among flow stress, temperature, strain, and strain rate were constructed, which improve the simulation accuracy and related research fields where stress-strain data play important roles. Yang Lei et al [21] carried out tensile tests from 750 to 850 A degrees C with an interval of 25 A degrees C and at strain rates of 0.001, 0.01, and 0.1 s(-1) to study the hot tensile behavior of TA15 sheets with bimodal microstructure. A self-consistent model was developed to predict the plastic flow behavior of the TA15 sheets. Li Jiang et al. [22] investigated the effects of strain and temperature on thermal deformation through the exponential-type Zener-Hollomon equation. Isothermal compression was tested at different temperatures and strain rates, and a comprehensive constitutive model involving temperature, strain rate, and flow stress was established and the reliability of the constitutive model was verified. Among various constitutive models, the Johnson-Cook model has been widely and successfully applied to metal impact and nonlinear large deformation problems due to its simple form, clear physical meaning, and easy access to parameters.

However, the above constitutive models of TA15 titanium alloy are only applicable to normal temperature and high-temperature conditions, and there is no relevant research on the constitutive model at low temperatures. In this paper, the cryogenic mechanical properties of TA15 titanium alloy were obtained by a series of cryogenic tensile tests. Based on the Johnson-Cook constitutive model of TA15 titanium alloy, a modified model considering low-temperature cooling was established. Combined with the orthogonal cutting test, the reliability of the modified constitutive model is verified by the shear force prediction model based on the unequal shear zone model.

In this section, TA15 titanium alloy was taken as the research object, and the mechanical properties of TA15 titanium alloy at low temperature were obtained through a series of cryogenic tensile tests. The temperature term of J-C constitutive model of TA15 titanium alloy was modified at low temperatures, and the cryogenic constitutive equation of TA15 titanium alloy was obtained.

2.1 Test materials

The tensile test material was Ti-6AL-2Zr-1Mo-1V titanium alloy, which was treated by solution and aging. The mass fraction is shown in Table 2.1。

Table 2.1

Chemical composition of TA15 titanium alloy
Element	Al	N	H	Fe	C	Mo	V	Ti
Content(%)	6.55	0.009	0.01	0.2	0.08	0.94	0.82	Bal.

2.2 Cryogenic tensile test

The titanium alloy specimens had a nominal diameter of 5 mm, a nominal length of 27 mm (Fig. 2.1), and a testing rate of 2 mm/min (nominal strain rate of approximately 1.333e-4s^− 1). The tensile tests were performed on the electronic universal testing machine (MTS SANS CMT5000, USA) at 25°C, 0°C, -50°C, -100°C, -150°C, and − 196°C. The temperature test system was KEITHLEY 2000 MULTIMETER, and the CryoLab low-temperature system was used to maintain the temperature inside the cryogenic box. The tensile test platform is shown in Fig. 2.2.

The engineering stress-strain curve can be obtained from the load and displacement data in the tensile process, and the engineering stress S and engineering strain e are defined as

$$S=\frac{P}{{{A_0}}}$$

$$e=\frac{{l - {l_0}}}{{{l_0}}}$$

Where is the engineering stress, is the applied axial load, ${A_0}$ is the initial cross-sectional area of the specimen, is the current gauge length of the specimen, and ${l_0}$ is the initial gauge length of the specimen.

Through the above calculation, the engineering stress-strain curves of TA15 titanium alloy in the tensile process at test temperatures can be obtained, as shown in Fig. 2.3. It can be seen from the figure that TA15 titanium alloy has no obvious yield platform at all temperatures. However, with the decrease in temperature, the strength of TA15 titanium alloy increases and the fracture strain decreases, indicating that the plasticity of TA15 decreases with the decrease in temperature.

However, to accurately describe the dynamic mechanical properties of materials and obtain accurate constitutive equations of materials, it is necessary to convert engineering stress and strain into true stress and strain data. True stress is defined as:

$$\sigma =\frac{P}{A}$$

Where is the current cross-sectional area of the specimen;

True strain is defined as.

$$d\varepsilon =\frac{{dl}}{l}$$

Namely.

$$\varepsilon =\int\limits_{{{l_0}}}^{l} {\frac{{dl}}{l}} =\ln (\frac{l}{{{l_0}}})$$

The relationship between the real strain and the engineering strain is

$$\varepsilon =\ln (1+e)$$

The relationship between real and engineering stresses can be obtained from the incompressible theory of plasticity mechanics:

$$A*l={A_0}*{l_0}$$

$$\sigma =S*(1+e)$$

However, the use of this transformation formula should meet the incompressible conditions. In the uniaxial tensile test, the deformation concentration of the specimen after necking occurs near the necking area, and the elongation detected in the experiment is the entire gauge length section. Therefore, the relationship between the true stress and the engineering stress after necking cannot be obtained through this relationship.

Through the above relationship, the true stress and strain during the tensile process of the specimen from the yield stage to the tensile strength stage can be obtained. The true stress-strain curves at various temperatures are shown in Fig. 2.4. According to the stress-strain curve, the yield strength and tensile strength at each temperature can be obtained in Table 2.2.

Table 2.2

Yield strength and tensile strength at different temperatures.
Test temperature (℃)	Yield strength (MPa)	Tensile strength (MPa)
25	608.2	658.6
0	700.8	734.8
-50	741.7	808.6
-100	889.9	930.1
-150	989.9	1041.9
-196	1021.5	1176.7

2.3 Johnson-Cook constitutive model parameter identification

J-C constitutive model is a material constitutive model proposed by Johnson and Cook for large strain, high strain rate, and high temperature of materials, which can comprehensively describe the strain hardening, strain rate hardening, and temperature softening effect of materials. Since the cutting process involves large strain, high strain rate, and, high temperature, the model is often used to describe the mechanical behavior of materials in the cutting process. The function model is as follows:

$$\sigma =(A+B*{\varepsilon ^n})*[1+C*\ln (\frac{{\mathop \varepsilon \limits^{.} }}{{\mathop {{\varepsilon _0}}\limits^{.} }})]*[1 - {(\frac{{T - {T_0}}}{{{T_m} - {T_0}}})^m}]$$

Where $\sigma$ is the flow stress, is the yield stress of the material at the reference temperature, and is the strain reinforcement factor. is the work hardening index, is the strain rate strengthening factor, $\mathop \varepsilon \limits^{.}$ is the equivalent plastic strain rate, $\mathop {{\varepsilon _0}}\limits^{.}$ is the reference equivalent plastic strain rate, is the temperature softening factor, is the test temperature, ${T_0}$ is the reference temperature, and ${T_m}$ is the material melting point.

To obtain the relevant parameters of the J-C constitutive model, the constitutive equation is first simplified to $\sigma =(A+B*{\varepsilon ^n})$, and the parameters , and are fitted through the tensile tests at reference strain rate and reference temperature; then the equation was simplified to$\sigma =A*[1+C*\ln (\frac{{\mathop \varepsilon \limits^{.} }}{{{\varepsilon _0}}})]$, and the strain rate hardening coefficients is fitted by Hopkinson compression bar tests at different strain rates (high strain rate) under reference temperature; finally, the constitutive equation is simplified to$\sigma =A*[1 - {(\frac{{T - {T_0}}}{{{T_m} - {T_0}}})^m}]$, and the temperature softening coefficient is fitted by tensile tests at different temperatures under reference strain rate. Hongchao Ji [17] et al. modified the strain hardening and temperature softening terms of the J-C model of TA15 titanium alloy at high temperature and high strain rate through experiments, and obtained the following model:

$$\sigma =(279.579 - 122.153\varepsilon - 14.103{\varepsilon ^2})[1+0.152\ln (\frac{{\mathop \varepsilon \limits^{.} }}{{\mathop {{\varepsilon _0}}\limits^{.} }})][ - 0.011+0.001\ln (\frac{{\mathop \varepsilon \limits^{.} }}{{\mathop {{\varepsilon _0}}\limits^{.} }})(\frac{{T - {T_0}}}{{{T_m} - {T_0}}})]$$

By analyzing the real stress-strain curve, it can’t be effectively fitted because the strengthening stage of TA15 titanium alloy is too short at-196°C. Therefore, − 150°C was selected as the reference temperature to fit the strain hardening term of the TA15 titanium alloy constitutive model, and A = 989.3, B = 910.9, n = 0.75466 were obtained. The fitting curve is shown in Fig. 2.5.

Since the J-C constitutive model is mainly used to describe the dynamic mechanical properties of the material at high temperature, the temperature softening coefficient fitted at low temperature cannot accurately reflect the relationship between material properties and temperature. Therefore, the temperature term of the J-C constitutive model is modified. The modified temperature term is$\sigma =A*[1 - {m_1}{(\frac{{T - {T_0}}}{{{T_m} - {T_0}}})^{{m_2}}}]$, and the fitting results are ${m_1}$= 3.8772, ${m_2}$ = 0.9862. The results show that the modified J-C constitutive temperature term can better characterize the influence of temperature on the mechanical properties of TA15 titanium alloy. The fitting curve is shown in Fig. 2.6.

For the strain rate strengthening coefficient of the constitutive model, the strain rate term of the J-C model of TA15 titanium alloy obtained by Hongchao Ji [17] et al. can be selected. where the strain rate strengthening coefficient = 0.152. Finally, the J-C constitutive model is sorted out, and the cryogenic constitutive model of TA15 titanium alloy is obtained as follows:

$$\sigma =(989.3+910.9{\varepsilon ^{0.75466}})[1+0.152\ln (\frac{{\mathop \varepsilon \limits^{.} }}{{\mathop {{\varepsilon _0}}\limits^{.} }})][1 - 3.872{(\frac{{T - {T_0}}}{{{T_m} - {T_0}}})^{0.9862}}]$$

where the reference strain rate$\mathop {{\varepsilon _0}}\limits^{.}$ = 0.0013s^− 1, the reference temperature ${T_0}$= 123K, and the TA15 melting point temperature ${T_m}$= 1800K.

3.1 Orthogonal cutting test

Orthogonal cutting refers to the cutting when the blade inclination angle of the main cutting edge of the tool is ${\lambda }_{s}=0$. At this time, the main cutting edge and the cutting speed direction are the right angles, so it is also called the right angle cutting. This method is one of the most effective ways to study the basic cutting theory. The orthogonal cutting test is carried out on the CD6140A horizontal lathe-produced Dalian Machine Tool Group. The experimental setup is shown in Fig. 3.1. The test workpiece is a TA15 titanium alloy rod, and the outer diameter of the workpiece is 100 mm. The cutting tool is DNMG-150608-MM1125 cemented carbide coated insert provided by Sandvik. The cutting rank angle is 5$^\circ$, the cutting clearance angle is 7$^\circ$, and the nose radius is 0.8mm. Liquid nitrogen is provided by China Tianhai DPL-175 liquid nitrogen tank. The main cutting force and feed force during the cutting process are obtained by the YDCB-III05 three-dimensional piezoelectric quartz dynamometer developed by the Dalian University of Technology. The cutting temperature is acquired by a T610 thermal imager designed by FLIR Systems (USA). The specific cutting parameters and test results are shown in Table 3.1.

Table 3.1

Cutting parameters and measurement results of TA15 titanium alloy orthogonal test
Serial number	Cutting speed (m/min)	Feeding (mm/r)	Depth of cut (mm)	Main cutting force (N)	Feeding force (N)	Temperature (℃)
1	50	0.2	0.4	164	64	140
2	70	0.2	0.4	137	46	160
3	90	0.2	0.4	170	117	164
4	110	0.2	0.4	152	93	171
5	50	0.25	0.4	194	64	94
6	70	0.25	0.4	179	57	113
7	90	0.25	0.4	187	68	127
8	110	0.25	0.4	184	72	158
9	50	0.3	0.4	270	116	105
10	70	0.3	0.4	236	85	114
11	90	0.3	0.4	217	78	155
12	110	0.3	0.4	251	75	165

3.2 Validation of the cryogenic constitutive model

To verify the validity of the constitutive model, the sheer force prediction model is established based on the unequal shear zone model. The constitutive equation of TA15 titanium alloy, the cutting force, and cutting temperature measured in the orthogonal cutting test are used to verify from the perspective of shear force. The basic line is shown in Fig. 3.2.

Based on the single shear plane model of MERCHANT [23], the plastic shear theory of OXLEY [24], and the unequal shear zone theory of ASTAKHOVVP et al. [25], the expressions of stress, strain, and strain rate of the unequal shear zone are given in this section. In the metal cutting theory, the traditional shear zone model usually uses a straight line to replace the plastic deformation shear boundary line. It is considered that a shear zone is a single plane without thickness, as shown in Fig. 3.3.

3.2.1 Calculation of shear force ${F_s}$ on actual shear plane by measured cutting force

Combined with the geometric relation of the single shear plane model and the main cutting force and feed force obtained in the orthogonal cutting test, the friction and positive pressure on the rake face of the tool can be obtained by the following relation.

$${F_u}={F_c}\sin \gamma +{F_f}\cos \gamma$$

$${F_v}={F_c}\cos \gamma - {F_f}\sin \gamma$$

Where $\gamma$ is the rank angle of the tool.

According to the definition of friction coefficient, the relationship between friction angle $\beta$ and measuring forces obtained by the orthogonal cutting test is as follows:

$$\beta =\arctan (\frac{{{F_u}}}{{{F_v}}})$$

Shear angle is the angle between the cutting speed and the main shear plane, which reflects the cutting deformation. According to the minimum energy theory of MERCHANT [23], the shear angle $\phi$ in the cutting model is:

$$\phi =\frac{\pi }{4}+\frac{{\gamma - \beta }}{2}$$

Combined with the geometric relationship of single the shear plane model and the main cutting force ${F_c}$ and feed force ${F_f}$ obtained in the orthogonal cutting test, the main shear plane shear force ${F_s}$ can be obtained by the following relationship:

$${F_s}={F_c}\cos \phi - {F_f}\sin \phi$$

3.2.2 Equivalent strain and equivalent strain rate obtained by cutting parameters

In the orthogonal cutting unequal shear zone model proposed by OXLEY [24], the middle line AB represents the main shear plane, as shown in Fig. 3.4. The model points out that in orthogonal cutting, the strain rate field $\mathop {{\varepsilon _{AB}}}\limits^{.}$ at the main shear surface AB is:

$$\mathop {{\varepsilon _{AB}}}\limits^{.} =\frac{{(\eta +1)V\cos \gamma }}{{h\cos (\phi - \gamma )}}$$

Where $\eta$ is the uncertainty factor, which describes the non-conformity of the tangential velocity in the shear zone, and its value is related to the material properties and cutting state; SHI B et al. [26] showed that the strain rate distribution in the shear zone can be well described when $\eta$= 4; is the cutting speed; is the thickness of the shear zone, and according to the geometric relationship:

$$h=\frac{{t{}_{c}\cos (\phi - \gamma )}}{{\sin \phi }}$$

Where ${t_c}$ is the cutting thickness.

The shear stress ${\varepsilon _{AB}}$ at the main shear plane AB is

$${\varepsilon _{AB}}=\mathop {{\varepsilon _{AB}}}\limits^{.} \frac{{\alpha h}}{{(\eta +1)V\sin \phi }}$$

Where $\alpha$ indicates that the distance from the main shear plane AB to the initial shear line CD is times the thickness of the whole shear zone. According to the geometric relationship, it is obtained that:

$$\alpha =\frac{{\cos \phi \cos (\phi - \gamma )}}{{\cos \gamma }}$$

According to the Mises criterion, the equivalent strain $\overline {\varepsilon }$ and the equivalent strain rate $\overline {{\mathop \varepsilon \limits^{.} }}$ at the main shear zone AB is calculated by Formulas 21 and 22 respectively:

$$\overline {\varepsilon } =\frac{\varepsilon }{{\sqrt 3 }}$$

$$\overline {{\mathop \varepsilon \limits^{.} }} =\frac{{\mathop \varepsilon \limits^{.} }}{{\sqrt 3 }}$$

3.2.3 Calculation of shear force using J - C constitutive model

The equivalent stress can be calculated by the cutting temperature in the orthogonal test that can be measured by the thermal imager, and the equivalent strain and equivalent strain rate obtained above are introduced into the J-C constitutive equation.

According to the Mises criterion, the shear stress of the main shear surface can be obtained.

$$\sigma =\sqrt 3 {\tau _{AB}}$$

According to the orthogonal cutting theory, the sheer force $F_{s}^{'}$ predicted by the constitutive model can be obtained

$$F_{s}^{'}={\tau _{AB}}\frac{{{t_w}{t_c}}}{{\sin \phi }}$$

Where ${t_w}$ is the cutting width.

3.3 Comparison of prediction results

The cutting force obtained by the orthogonal cutting experiment is converted to the shear force by a series of conversions, and then compared with the sheer force calculated by the constitutive model, the calculation results and experimental results are shown in Fig. 3.5. Figure 3.5 (a) for each cutting parameter test and model to get the sheer force contrast diagram, from the diagram can be seen with the increase of feed, the shear force is also increasing. The effect of cutting speed on the shear force is not obvious. Figure 3.5 (b) is the sheer force prediction error diagram for each cutting parameter. It can be seen from the diagram that the minimum error between the predicted value and the experimental value of the shear force is 1.27%, and the maximum error is 18.42%. It shows that the constitutive model established in this paper can accurately describe the material deformation in TA15 actual cryogenic machining, and the model has high reliability.

In this paper, the cryogenic constitutive model of TA15 titanium alloy was obtained through cryogenic tensile tests, and the accuracy of the model was verified by orthogonal cutting tests. The following conclusions were obtained:

The cryogenic mechanical properties of TA15 were obtained from the cryogenic tensile test of TA15 titanium alloy. After analysis, with the decrease in temperature, the strength of TA15 increases, and the plasticity decreases. At − 196 °C, the strengthened stage of TA15 during the tensile process was significantly shortened.
Based on the real stress-strain data obtained and the Johnson-Cook constitutive equation of TA15 titanium alloy, the temperature softening term was corrected, and the cryogenic constitutive model of TA15 titanium alloy was established.
According to the obtained Johnson-Cook constitutive model of TA15 titanium alloy, the shear force was predicted by the cutting theory based on the unequal shear zone, and compared with the sheer force obtained from the test results. The minimum error between the predicted results and the test results was 1.27 %, and the maximum error was 18.42 %, which verified the reliability of the cryogenic constitutive model.

Funding

This research is funded in part by the National Key R&D Program of China (2019YFB2005400) and the National Natural Science Foundation of China (No. U20B2033)

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Availability of data and material

Not applicable.

Code availability

Not applicable.

Ethical approval

The authors oblige the rules regarding the ethic in publication.

Consent to participate

All authors consent to participate.

Consent for publication

All authors consent to publish.

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Establishment of Cryogenic Constitutive Model for Ti-6Al-2Zr-1Mo-1V Titanium Alloy

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Abstract

Figures

1 Introduction

2 Establishment Of Cryogenic Constitutive Model For Ta15 Titanium Alloy

2.1 Test materials

2.2 Cryogenic tensile test

2.3 Johnson-Cook constitutive model parameter identification

3 Validation Of The Constitutive Model

3.1 Orthogonal cutting test

3.2 Validation of the cryogenic constitutive model

3.2.1 Calculation of shear force \({F_s}\) on actual shear plane by measured cutting force

3.2.2 Equivalent strain and equivalent strain rate obtained by cutting parameters

3.2.3 Calculation of shear force using J - C constitutive model

3.3 Comparison of prediction results

4 Conclusion

Declarations

References

Status:

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