Characterization of OT4-1 Alloy by Multi-Dome Forming Test

In this study, the rheological characteristics of a titanium alloy have been obtained by multi-dome bulging test. Free bulging process is an experimental technique that can be used to characterize material in conditions of biaxial tension during superplastic, as well as conventional, hot forming. The constitutive constants are calculated on a base of the information about the bulge geometry, applied pressure, and forming time. A multi-dome forming test allows one to reduce the number of the experiments required for the characterization, since every multi-dome test produces several domes of different size. In this study, a specific die for multi-dome test was used. The die has six holes with different radiuses of 20, 25, 30, 35, 40, and 45 mm. During a test, the specimen is clamped between blank holder and die holder, heated to a specific temperature, and formed by applying constant gas pressure. The experiments were conducted at different temperatures for OT4-1 titanium alloy. The constitutive constants were obtained by processing the experimental data using two different techniques and compared with tensile test results. In order to estimate the influence of friction on the experimental results and to verify obtained material characteristics, finite element (FE) simulation was performed. Finally, the results of FE simulation were compared with the experimental data. The results of the simulation show the advantage of material characterization based on multi dome tests and its interpretation by inverse analysis. The deviations produced by the effect of friction are more significant when the direct approach is applied instead of inverse analysis with a semi analytical model of the bulging process.


Introduction
The characterization of material rheological behavior plays a significant role while the development of forming technologies. It becomes even more important for superplastic forming. A superplasticity effect can be achieved at a narrow range of strain rates which should be provided during the forming process. For some superplastic materials, the elongation at tensile testing could exceed almost 2000% [1]. Low level of flow stress and high strain rate sensitivity of the material are the main features of superplastic deformation. In order to describe material behavior during superplastic deformation, the Backofen constitutive equation [2], using power relation between flow stress σ and strain rate . ε , is commonly used [3,4] where K and m are the material constants. The strain rate sensitivity index m is an important parameter responsible for the stability of plastic flow. A larger m provides better superplasticity. It is considered

Tensile Testing
A tensile test with a stepped strain rate change was performed in order to estimate the temperature and strain rate conditions of the material superplasticity. The samples have a gauge section size F 0 = 6 × 1.55 mm 2 (width = 6 mm and thickness = 1.55 mm) and length l 0 = 17 mm (l 0 = 5.65 √ F 0 ) and were cut parallel to the rolling direction. The specimens were cut from as received OT4-1 sheet and heated for 20 minutes, then deformed in argon atmosphere. The temperature was maintained within an accuracy of ±3 • C. The test was performed in the strain rate range 1 × 10 −5 -5 × 10 −3 s −1 . The specimen geometry before and after the test performed at the target temperature of 840 • C is presented on Figure 1. In order to observe the effect of temperature on the material behavior, additional tests were performed at 790 and 890 • C. The superplastic behavior was characterized by means of uniaxial tensile tests on a Walter Bay LFM-100 test machine (Walter + Bai AG, Löhningen, Switzerland) with a program service Dion-Pro for the control of the traverse motion in real time. The results of the tests performed at different temperatures are presented on Figure 1. It can be seen that the superplastic strain rate range corresponding to the steep part of a stress strain rate curve plotted in logarithmic scale shifts to the right with increasing temperature.

Tensile Testing
A tensile test with a stepped strain rate change was performed in order to estimate the temperature and strain rate conditions of the material superplasticity. The samples have a gauge section size = 6 1.55 mm 2 (width = 6 mm and thickness = 1.55 mm) and length = 17 mm ( = 5.65 ) and were cut parallel to the rolling direction. The specimens were cut from as received OT4-1 sheet and heated for 20 minutes, then deformed in argon atmosphere. The temperature was maintained within an accuracy of ±3°C. The test was performed in the strain rate range 1 10 − 5 10 s −1 . The specimen geometry before and after the test performed at the target temperature of 840 is presented on Figure 1. In order to observe the effect of temperature on the material behavior, additional tests were performed at 790 and 890 °C. The superplastic behavior was characterized by means of uniaxial tensile tests on a Walter Bay LFM-100 test machine (Walter + Bai AG, Löhningen, Switzerland) with a program service Dion-Pro for the control of the traverse motion in real time. The results of the tests performed at different temperatures are presented on Figure 1. It can be seen that the superplastic strain rate range corresponding to the steep part of a stress strain rate curve plotted in logarithmic scale shifts to the right with increasing temperature. The stress strain rate curve was approximated using the Backofen equation in order to calculate the values of constitutive constants characterizing OT4-1 alloy behavior at 840 °C in a strain rate range of 1.5 10 − 2 10 s −1 .

Multi-Dome Forming Testing
The multi-dome forming tests were performed on the same material machined to rectangular sheet blanks with the sides equal to 320 and 370 mm. The initial thickness of every specimen was 1 mm.
The tests were carried out at different constant pressures ( ) and forming times ( , ). For each pressure value, three tests were performed with different forming times. The values of pressure and times are presented in Table 2.  The stress strain rate curve was approximated using the Backofen equation in order to calculate the values of constitutive constants characterizing OT4-1 alloy behavior at 840 • C in a strain rate range of 1.5 × 10 −5 -2 × 10 −3 s −1 .

Multi-Dome Forming Testing
The multi-dome forming tests were performed on the same material machined to rectangular sheet blanks with the sides equal to 320 and 370 mm. The initial thickness of every specimen was 1 mm. The tests were carried out at different constant pressures (P i ) and forming times (t i,j ). For each pressure value, three tests were performed with different forming times. The values of pressure and times are presented in Table 2. During testing, the formed sheet was clamped between the blank holder with six forming holes and the die holder securing the specimen. The experiments were carried out on the FSP 60T forming machine produced by An Aries Alliance Company (ACB) (Nantes, France). Figure 2a shows the scheme of the mold with forming holes of different diameters: 20, 25, 30, 35, 40, and 45 mm. The photograph of the specimen formed at P 1 = 0.3 MPa, during t 1 = 1200 s is presented on Figure 2b. During testing, the formed sheet was clamped between the blank holder with six forming holes and the die holder securing the specimen. The experiments were carried out on the FSP 60T forming machine produced by An Aries Alliance Company (ACB) (Nantes, France). Figure 2a

Mathematical Model and Characterization Technique
The scheme of the free bulging test is presented on Figure 3. A sheet specimen of initial thickness is deformed by pressure in to a cylindrical die with an aperture radius and entry radius , forming a dome. The free part of the dome is assumed to be a spherical surface with radius at an instant moment of time ( ).
is the height of the dome and is the current thickness of the specimen at the dome apex.

Mathematical Model and Characterization Technique
The scheme of the free bulging test is presented on Figure 3. A sheet specimen of initial thickness s 0 is deformed by pressure P in to a cylindrical die with an aperture radius R 0 and entry radius ρ 0 , forming a dome. The free part of the dome is assumed to be a spherical surface with radius ρ at an instant moment of time (t). H is the height of the dome and s is the current thickness of the specimen at the dome apex. During testing, the formed sheet was clamped between the blank holder with six forming holes and the die holder securing the specimen. The experiments were carried out on the FSP 60T forming machine produced by An Aries Alliance Company (ACB) (Nantes, France). Figure

Mathematical Model and Characterization Technique
The scheme of the free bulging test is presented on Figure 3. A sheet specimen of initial thickness is deformed by pressure in to a cylindrical die with an aperture radius and entry radius , forming a dome. The free part of the dome is assumed to be a spherical surface with radius at an instant moment of time ( ).
is the height of the dome and is the current thickness of the specimen at the dome apex.  The mathematical model describing the forming process is based on following assumptions:

•
The material is isotropic • Elastic strains are negligible • At any given time, the metal sheet is shaped as a part of the sphere • The sheet is rigidly clamped Considering the stress equilibrium of a small element in the dome apex, one can express the value of equivalent stress (σ ) as The curvature radius ρ can be expressed as a function of height using the simple geometrical formula Equivalent strain (ε ) at the dome apex can be expressed as By applying of Equation (2) to the results of multi-dome forming test, one can calculate the value of stress corresponding to each bulge at the final moment of forming. The strain rate corresponding to each stress can be estimated as the value of ε calculated by (4) and divided by forming The obtained pairs of stress strain rates then can be approximated by Backofen equation. This simple technique was used for material characterization in [3,8]. The drawback is that the Equation (5) gives a very approximate estimation of the strain rate. The strain rate varies significantly during the test and the results based on Equation (5) could become a source of errors.
An alternative technique is based on the mathematical model proposed in [15], allowing one to predict the evolution of a dome height. Using this model, the constitutive constants are calculated by inverse analysis, minimizing the error function constructed as where N is a number of domes obtained by all experiments, t i and H i are the forming time and the measured height value of the i-th dome; H i (t) is the predicted evolution of the height of the i-th dome.
To construct the prediction of a dome height, the stress rate should be expressed as the derivation of the effective strain.
where the value of apex thickness is considered to be a function of dome height [20] Combining the Equations (2), (7), and (8) with Equation (1), one can construct the differential equation for dome height evolution The parameter B i can be calculated based on the experimental results as where s i is the experimental thickness at the dome apex.

Material Characterization
The processing of multi-dome forming results was performed using two interpretation techniques. According to the simple direct technique described in Section 3, five pairs of effective stress σ and effective strain rate . ε values for each of nine tests were calculated using Equations (2) and (5). The obtained 45 (σ , . ε ) points were fitted by the power equation shown on Figure 4 and the constitutive constants were found as: K = 444, m = 0.394.
Comparing these results with the tensile test data which were approximated by Backofen equation with the constants K = 1471 and m = 0.577, significant deviations can be observed. The differences between the constitutive constants obtained using different experimental techniques were noticed in many studies [6][7][8][9]. The results of free bulging tests are considered to be more convenient for simulation of SPF while they are obtained in conditions of biaxial stress tension which are close to the ones realized in production.
Inverse analysis of the multi dome forming results were performed in two steps. At first, for every dome apex point (H i , s i ) on every specimen, the coefficient B i was calculated by Equation (10). The objective function F err was constructed according to Equation (6) using numerical solution of Equation (9) for the evaluation of H i (t). The Backofen constants were found to be those corresponding to the minimum of F err at the values of K = 494 and m = 0.375.
The comparison of constitutive data obtained by different techniques is presented on Figure 4. The results of inverse analysis are plotted by the blue solid line. The (σ , . ε ) points obtained by the direct method are illustrated by square red markers. The power fitting of these points is plotted by the red dashed line. The green markers and the green dotted line illustrate the results obtained by the tensile test. It can be seen that, in the given strain rate, the multi-dome forming tests produce higher stresses and lower strain rate sensitivity than the tensile ones.
The parameter can be calculated based on the experimental results as where is the experimental thickness at the dome apex.

Material Characterization
The processing of multi-dome forming results was performed using two interpretation techniques. According to the simple direct technique described in Section 3, five pairs of effective stress and effective strain rate values for each of nine tests were calculated using Equations (2) and (5). The obtained 45 ( , ) points were fitted by the power equation shown on Figure 4 and the constitutive constants were found as: = 444, = 0.394.
Comparing these results with the tensile test data which were approximated by Backofen equation with the constants = 1471 and = 0.577 , significant deviations can be observed.
The differences between the constitutive constants obtained using different experimental techniques were noticed in many studies [6][7][8][9]. The results of free bulging tests are considered to be more convenient for simulation of SPF while they are obtained in conditions of biaxial stress tension which are close to the ones realized in production. Inverse analysis of the multi dome forming results were performed in two steps. At first, for every dome apex point ( , ) on every specimen, the coefficient was calculated by Equation (10). The objective function was constructed according to Equation (6) using numerical solution of Equation (9) for the evaluation of . The Backofen constants were found to be those corresponding to the minimum of at the values of = 494 and = 0.375.
The comparison of constitutive data obtained by different techniques is presented on Figure 4. The results of inverse analysis are plotted by the blue solid line. The ( , ) points obtained by the direct method are illustrated by square red markers. The power fitting of these points is plotted by the red dashed line. The green markers and the green dotted line illustrate the results obtained by the tensile test. It can be seen that, in the given strain rate, the multi-dome forming tests produce higher stresses and lower strain rate sensitivity than the tensile ones.

Finite Element Verification
The constitutive constants obtained by different methods were verified by finite element (FE) simulation of the multi-dome forming process. Three-dimensional simulation was carried out by MSC. Marc software (Newport Beach, CA, USA). A finite element mesh was generated using four-node regular plane elements. The deformable sheet of an initial thickness of 1 mm was divided into 195,520 elements. The simulations were made for all the pressures used in the experiments. The pressure and temperature were considered to be constant in every simulation. The die was simulated as a rigid undeformable body. Three different pairs of constitutive constants obtained by the techniques described above were used for simulation of material properties. The results of FE simulation of multi-dome forming at 0.3 MPa during 3600 s are presented on Figure 5. The color field illustrates thickness distribution within the specimen.

Finite Element Verification
The constitutive constants obtained by different methods were verified by finite element (FE) simulation of the multi-dome forming process. Three-dimensional simulation was carried out by MSC. Marc software (Newport Beach, CA, USA). A finite element mesh was generated using four-node regular plane elements. The deformable sheet of an initial thickness of 1 mm was divided into 195,520 elements. The simulations were made for all the pressures used in the experiments. The pressure and temperature were considered to be constant in every simulation. The die was simulated as a rigid undeformable body. Three different pairs of constitutive constants obtained by the techniques described above were used for simulation of material properties. The results of FE simulation of multi-dome forming at 0.3 MPa during 3600 s are presented on Figure 5. The color field illustrates thickness distribution within the specimen.
The comparison of the reults of FE simulation with the experimental data is presented in Table 3 and Figure 6. Table 3 contains the values of mean deviation between the measured height and FE predictions made for different pressures. The curves plotted on Figure 6 illustrate the dependence of relative error averaged by all forming pressures and times on the radius of the bulge. It can be seen that inverse analysis allows one to obtain much better results than other techniques. The larger deviations from measured values are observed in simulations made in which constitutive equations are obtained by tensile testing.   The comparison of the reults of FE simulation with the experimental data is presented in Table 3 and Figure 6. Table 3 contains the values of mean deviation between the measured height and FE predictions made for different pressures. The curves plotted on Figure 6 illustrate the dependence of relative error averaged by all forming pressures and times on the radius of the bulge. It can be seen that inverse analysis allows one to obtain much better results than other techniques. The larger deviations from measured values are observed in simulations made in which constitutive equations are obtained by tensile testing.

Effect of Friction
In order to estimate the influence of friction on the accuracy of the results obtained by interpretation of multi-dome forming tests an additional series of calculations with different values of friction factor was performed. Analyzing the results of finite element simulation, one can estimate how the effect of friction affects the results of multi-dome bulging test. Considering a single dome with maximum aperture radius ( = 45 mm) at a fixed moment of time and pressure ( = 4000 s, = 0.3 MPa) it is possible to notice the monotonic decrease of height and thickness at the dome apex with increasing friction as it is plotted on Figure 7.
The values of height and thickness of the specimen at each bulge calculated by FEM were used as the initial data both for direct and inverse techniques. The constitutive constants obtained by this method were compared with the ones which were used in the simulations. This information allows one to estimate how large the difference could be between the results of interpretation of multi-dome forming test and the real material constants, and how this difference is affected by friction and interpretation technique. The results of comparison are illustrated on Figure 8.

Effect of Friction
In order to estimate the influence of friction on the accuracy of the results obtained by interpretation of multi-dome forming tests an additional series of calculations with different values of friction factor was performed. FE simulations were carried out with the following friction coefficient values: 0.1, 0.3, 0.5, 0.7, and 0.9. The calculations were proceed with material characteristics obtained from inverse analysis (K = 494, m = 0.375). The values of pressure and forming time were set at P 1 = 0.3 MPa, t 1 = 4000 s; P 2 = 0.5 MPa, t 2 = 1500 s; P 3 = 0.7 MPa, t 3 = 700 s.
Analyzing the results of finite element simulation, one can estimate how the effect of friction affects the results of multi-dome bulging test. Considering a single dome with maximum aperture radius (R 0 = 45 mm) at a fixed moment of time and pressure (t = 4000 s, P = 0.3 MPa) it is possible to notice the monotonic decrease of height and thickness at the dome apex with increasing friction as it is plotted on Figure 7.

Effect of Friction
In order to estimate the influence of friction on the accuracy of the results obtained by interpretation of multi-dome forming tests an additional series of calculations with different values of friction factor was performed.  Figure 7.
The values of height and thickness of the specimen at each bulge calculated by FEM were used as the initial data both for direct and inverse techniques. The constitutive constants obtained by this method were compared with the ones which were used in the simulations. This information allows one to estimate how large the difference could be between the results of interpretation of multi-dome forming test and the real material constants, and how this difference is affected by friction and interpretation technique. The results of comparison are illustrated on Figure 8.  The values of height and thickness of the specimen at each bulge calculated by FEM were used as the initial data both for direct and inverse techniques. The constitutive constants obtained by this method were compared with the ones which were used in the simulations. This information allows one to estimate how large the difference could be between the results of interpretation of multi-dome forming test and the real material constants, and how this difference is affected by friction and interpretation technique. The results of comparison are illustrated on Figure 8. The red quadric markers on Figure 8 correspond to the parameters , obtained by application of Equations (2) and (5) to the results of FE simulations and their approximation by Backofen power law according to the direct method. The blue triangle markers correspond to the results of inverse analysis and the black marker corresponds to the initial parameters. It can be seen that the values of strain rate sensitivity ( ), obtained by inverse analysis are very close to the referenced one. The values of calculated by direct method are generally higher than the referenced one at 0.01-0.015, which can be treated as a neglectable error in most cases. The deviations of are more significant and reach 20% for the direct method and 6% for the inverse one. The effect of friction on the deviations between the calculated constitutive constants and the reference constants is more significant when the direct approach is applied.

Conclusions
In this study, the multi-dome forming process is studied numerically and experimentally. Backofen constitutive constants OT4-1 titanium alloy were evaluated by multi-dome forming tests using both direct and inverse techniques. The results were compared with the data obtained by tensile testing and verified by finite element simulation. The deviations produced by the effect of friction on the experimental results were estimated for both direct and inverse methods.
The results of experimental data processing point to a difference of material characteristics obtained from tensile tests and free bulging experiments. For the OT4-1 alloy at 840 ºC, the stress values calculated from multidome tests are higher than those obtained from tensile tests for the same strain rates. It was shown that the inverse analysis based on a semi analytical model of free bulging process allows one to perform more accurate interpretation of the results of a multi-dome forming test than the direct approach. At the same time, using the results of tensile testing with stepped changing of strain rate may lead the appearance of large errors in the simulation of SPF processes. The red quadric markers on Figure 8 correspond to the parameters K, m obtained by application of Equations (2) and (5) to the results of FE simulations and their approximation by Backofen power law according to the direct method. The blue triangle markers correspond to the results of inverse analysis and the black marker corresponds to the initial parameters. It can be seen that the values of strain rate sensitivity (m), obtained by inverse analysis are very close to the referenced one. The values of m calculated by direct method are generally higher than the referenced one at 0.01-0.015, which can be treated as a neglectable error in most cases. The deviations of K are more significant and reach 20% for the direct method and 6% for the inverse one. The effect of friction on the deviations between the calculated constitutive constants and the reference constants is more significant when the direct approach is applied.

Conclusions
In this study, the multi-dome forming process is studied numerically and experimentally. Backofen constitutive constants OT4-1 titanium alloy were evaluated by multi-dome forming tests using both direct and inverse techniques. The results were compared with the data obtained by tensile testing and verified by finite element simulation. The deviations produced by the effect of friction on the experimental results were estimated for both direct and inverse methods.
The results of experimental data processing point to a difference of material characteristics obtained from tensile tests and free bulging experiments. For the OT4-1 alloy at 840 ºC, the stress values calculated from multidome tests are higher than those obtained from tensile tests for the same strain rates. It was shown that the inverse analysis based on a semi analytical model of free bulging process allows one to perform more accurate interpretation of the results of a multi-dome forming test than the direct approach. At the same time, using the results of tensile testing with stepped changing of strain rate may lead the appearance of large errors in the simulation of SPF processes.