Keywords

1 Introduction

Ring rolling is a bulk forming process used to manufacture seamless rings [1]. It is highly non-linear process with coupled thermo-mechanical effect [2]. Analyzing the ring rolling process using either experimental or analytical method is a material and time-consuming process. However, the use of finite element (FE) simulations to analyze the process is the most convenient tool [3]. Dynamic boundary contact between the rolls and ring is seen in this process and hence the use of dynamic explicit approach is ideal to simulate the ring rolling process [4]. A detailed study of literature shows that there exists a good number of studies done previously pertaining to ring rolling. Seitz et al. [5] transferred the principle of roll bonding to produce seamless radial composite rings using 3D FE model, which combine the advantages of different materials. S. Guenther et al. [6] further extended the work on composite ring rolling process via 3D FE simulations and experiments on steel rings. Cleaver et al. [7] introduced a novel concept for making profiled L shaped rings without ring growth. Seitz et al. [8] used ring rolling process to produce dish shaped rings using 3D FE simulations. Ring rolling of Ti alloys via FE simulations has also been reported in literature. Taek et al. developed a three-dimensional (3D) finite element (FE) model and used processing map to estimate the locations in the rolled Ti-6Al-4V (Ti-64) ring where defects can be expected to develop during the rolling process [9]. Yang et al. developed 3D FE models of ring rolling process to study the effect of size of rectangular-sectioned Ti-64 blanks on strain and temperature distribution [10]. Liang et al. used Response Surface Methodology (RSM) to analyze the effects of instantaneous ring diameter, ring growth velocity and initial ring temperature and on temperature distribution in TA15 alloy ring [11]. Wang et al. analyzed impact of rolling speed, feed rate and initial temperature on volume fraction and grain size of β phase in Ti-64 alloy using 3D FE model [12]. Yeom et al. developed a 3D FE model and simulated strain and temperature and examined microstructure spread in Ti-64 ring [13].

In addition, there exist some studies pertaining to heterogeneity in temperature and equivalent plastic strain (PEEQ) distribution and formation of fishtail defect via FE analysis for different alloys in literature. Li et al. studied the ring rolling of AA 6061 alloy and established that increase in main roll speed resulted in decrease in the heterogeneity of temperature distribution and increase in PEEQ distribution in the ring [14]. Lee and Kim analyzed ring rolling of AISI 1035 steel and found that a reduction in the heterogeneity in the distribution of temperature and PEEQ when the feed rate was increased [15]. Qian and Peng studied the spread of temperature and PEEQ in AISI 5140 steel ring and demonstrated that increase in feed rate and main roll speed resulted in increase in heterogeneity [16]. But a study by Sun et al. on ring rolling of AISI 5140 steel reported a reverse trend of fall in heterogeneity of temperature and PEEQ distribution with increase in feed rate [17]. Therefore, it becomes evident that alteration in material, process parameters, preform dimensions and roll size can change the heterogeneity patterns. In our recent study [18] on the effect of main roll feed (mm/s) and main roll speed (rpm) on heterogeneity of temperature and PEEQ distribution and fishtail defect during rolling of Ti-64 rings, it was ascertained that feed rate is the only significant process parameter impacting heterogeneity of temperature and PEEQ distribution and formation of fishtail defect. A reduction in heterogeneity index and fishtail defect with increase in feed rate was seen. In the present study, a new process parameter, i.e., coefficient of friction (CoF), has been considered apart from the main roll speed and the feed rate and their impact on heterogeneity of temperature and PEEQ distribution and fishtail defect has been looked into. So, an effort has been made in the present study to determine the effects of process parameters (main roll speed and feed rate and coefficient of friction) on heterogeneity of temperature and PEEQ distribution and form defect like fishtail defect. This aim was achieved by conducting FE simulation runs with process parameters decided as per the three parameters two levels, Central Composite Design (CCD) and detailed statistical analysis.

2 Material and Methodology

2.1 Material

Figure 1a and b shows the forged Ti-64 cylindrical billet obtained from Mishra Dhatu Nigam Limited, Hyderabad, India. Doughnut shaped preform shown in Fig. 1c was machined from the forged billet and used as the starting material. Figure 1d shows the ring after rolling with 30% reduction in cross-section.

Fig. 1
figure 1

Dimensions and photographs of forged Ti-64 cylindrical billet captured in two different views a side and b top. Photographs of the c preform and d the final ring

Full size image

2.2 Lab Scale Ring Rolling Facility

The preform was initially soaked in the furnace at 950 °C for 1.5 h. Soaking helps in attaining uniform temperature in the preform cross-section and improves formability [19, 20]. Then the preform was rolled in the lab scale ring rolling facility, when the preform touched 880 °C. The main roll undergoes both radial and linear motion, as shown in Fig. 2. The reduction of cross-section of ring in the deformation process results in circumferential extrusion and increase in ring diameter. The guide roll and mandrel are freely mounted and undriven. Guide roll has the role to maintain circularity of the ring and stabilize the rolling process. Rolling parameters and dimensions of different rollers, preform and ring are provided in Table 1. Table 1 shows that two rings rolled with two different feed rates of 1 and 2.5 mm/s by keeping the other parameters constant. More details of the rolling experiments can be found elsewhere [18].

Fig. 2
figure 2

Lab scale ring rolling facility showing the rotational and translational motion of the main roll

Full size image
Table 1 Parameters used in lab scale ring rolling experiment
Full size table

FLIR T620 thermal imaging camera was used to monitor the temperature in the zone of deformation while the preform was rolled. A cross-section reduction of 30% was achieved in both the rings. After rolling, the rings were left to be cooled in air.

2.3 Development of Ring Rolling FE Model

Figure 3a shows the developed 3D coupled thermo-mechanical FE model for the lab scale ring rolling machine using ABAQUS/Explicit software. C3D8RT, the eight-node thermally coupled brick element was used to mesh the ring. Adaptive remeshing was applied to improve the quality of simulation results. Guide roll, mandrel, and main roll are considered analytically rigid. Reduced integration and hourglass control has been implemented. In order to reduce the computational time, mass scaling factor of 1000 was applied to the FE models [21]. Mesh size of 1.85 was chosen, resulting in 56,241 elements, using the mesh sensitivity analysis provided in Fig. 3b. Contact pairs have been provided with coefficient of friction of 0.3 [11]. Convection and radiation are considered on all free surfaces. Taylor-Quinney coefficient is assumed to be 0.9 [22]. Thermo-physical properties of Ti-64 alloy are given in Table 2. The values of specific heat, conductivity, density, and Young’s modulus were taken from literature [23,24,25,26].

Fig. 3
figure 3

a FE model of lab scale ring rolling set-up developed using ABAQUS/Explicit platform b mesh sensitivity analysis

Full size image
Table 2 Thermo-physical properties of Ti-64 alloy
Full size table

Figure 4 shows the comparison of the experimental results with the FE models developed. Figure 4a and b respectively compares the height of the ring at ID (inner diameter), OD (outer diameter), and center in the final cross sections of the simulated and experimental rings for LFR condition. Similarly, Fig. 4c and d, respectively, shows the final cross-sections of the simulated and experimental rings for HFR condition. Maximum relative error of 13.60% was observed at OD of LFR ring while predicting the height. A comparision of predicted and experimental temperature profile is given in Fig. 4e. Maximum relative error of 3% was observed in temperature prediction in LFR ring. Figure 4f shows the rate of outer diameter growth in both LFR and HFR ring. Maximum relative error of 2.8% in growth predictions was seen in LFR rings.

Fig. 4
figure 4

Cross-sections of a, c simulated and b, d experimental rings for the LFR and the HFR conditions, respectively, e temperature variation in the deformation zone in simulated (sim) and experimental (exp) rings for the LFR and the HFR conditions and f outer diameter growth rate for LFR and HFR rings observed in simulation and experiment

Full size image

In order to develop the constitutive equation, Ti-64 cylindrical specimens (15 mm height and 10 mm diameter) were compressed resulting in 60% reduction in height attaining strain level of 0.91 in a strain rate and temperature range of 0.001–10 s−1 and 750–950 °C, respectively. The obtained flow curves were adiabatically corrected and subsequently hyperbolic sine equation was developed, which was used as a constitutive equation to run simulations. Details of development of the constitutive equation can be found elsewhere [27]. The general form of hyperbolic sine equation is given in Eq. (1).

$$\dot{\varepsilon }=\mathrm{\rm A}{\left[{\text{sinh}}\alpha \sigma \right]}^{n} \cdot {\text{exp}}\left(-\frac{Q}{RT}\right)$$
(1)

where n, α, and A are constants while Q is the activation energy. Equation (1) lacks the strain term. Hence, in order to integrate the strain term, polynomial fit of all the constants (n, α, A and Q) with respect to strain is plotted in Fig. 5. The polynomial equations mentioned are then applied in simulation through a user-defined material subroutine (VUMAT).

Fig. 5
figure 5

Plot representing the data as a function of strain with polynomial fit for determining the constants of Arrhenius equation

Full size image

2.4 Development of Simulation Matrix

Main roll feed, main roll speed, temperature, and coefficient of friction (μ) are the important process parameters for ring rolling process. In this study, a three-factor (main roll feed (mm/s), main roll speed (rpm) and coefficient of friction (CoF or μ)) and two-level Central Composite Design (CCD) were constituted and are shown in Fig. 6. CCD is a fractional factorial design and is integral part of Response Surface Methodology (RSM). Table 3 shows the minimum and maximum values of process parameters considered for analysis in this study. A range of values chosen for the main roll feed (mm/s) is to ensure that there is no adiabatic damage. The entire functional range of main roll speed (rpm) in the ring rolling set-up is chosen for the study. In case of coefficient of friction a range of 0.1–0.3 was chosen for analysis initially. But, a low coefficient of friction value of 0.1 failed to achieve rolling, as can be seen in Fig. 7. In the process of ring rolling, it is essential that the ring is drawn into the gap between the main roll and mandrel [28]. This process results in gradual reduction in thickness and expansion of the ring. But, in case the friction is less, the drawn-in of the ring to the deformation zone between the main roll and mandrel won’t be possible resulting in non-rotation of the ring and compression as shown in Fig. 7. Hence, coefficient of friction range had to be changed to 0.2–0.3. The aim was to see if reduced friction does make any impact. The total deformation level of about 30% was maintained for all the simulations run.

Fig. 6
figure 6

Central composite design (CCD) showing the scheme of simulations run

Full size image
Table 3 Factors and levels of the CCD
Full size table
Fig. 7
figure 7

FE Simulation depicting failure to achieve ring rolling operation at low coefficient of friction (μ = 0.1)

Full size image

It was found out that there was a noticeable difference between the feed rate that the rolling mill was set to achieve (prescribed feed rate) and the feed rate that it could actually achieve (effective feed rate) during rolling. Hence, the main roll feed range of 0.44–1.76 mm/s was used for simulation, as given in Table 3. Similar difference in prescribed and effective feed rates was also reported in literature [29].

3 Recorded Responses

20 simulations were run following CCD shown in Fig. 6. PEEQ and temperature distribution and fishtail coefficient are recorded in Table 4.

Table 4 Simulation runs as per CCD parameters and responses
Full size table

3.1 Heterogeneity in PEEQ and Temperature Distribution

Heterogeneity studies of PEEQ and temperature distribution are important for bulk forming processes like ring rolling. The preform in FE simulation was divided into 56,241 elements, and the PEEQ and temperature in each element were measured after rolling process. Coefficient of variation (CoV) was used in the present study as a heterogeneity index to analyze equivalent plastic strain and temperature distribution. It is defined as the ratio of standard deviation and mean [18]. CoV is unit-less and higher is its values suggesting higher is the inhomogeneity in PEEQ and temperature distribution. The formula used to determine the mean (\(\upmu )\) and standard deviation (\(\upsigma )\) are given in Eqs. (2) and (3), respectively, for PEEQ. Equation (4) is used to calculate CoV, as presented below.

$$\upmu = \sum\limits_{{\text{p}}=1}^{{\text{M}}}{{\text{PEEQ}}}_{{\text{p}}}/{\text{M}}$$
(2)
$$\upsigma = \sqrt{\frac{{{\sum}_{{\text{p}}=1}^{{\text{M}}}\left({{\text{PEEQ}}}_{{\text{p}}}-\upmu \right)}^{2}}{{\text{M}}-1}}$$
(3)
$${\text{CoV}}= \frac{\upsigma }{\upmu }$$
(4)

where, M is the total number of elements into which the ring has been divided. PEEQp is the equivalent plastic strain of each element. Similarly, CoV was determined to study the heterogeneity in temperature distribution. CoV calculation for temperature analysis was done in Kelvin scale. Figures 8 and 9, respectively, show the distribution of PEEQ and temperature in the ring cross-section, run as per CCD, after attaining 30% deformation.

Fig. 8
figure 8

FE modeling results showing the PEEQ spread after attaining 30% deformation using CCD parameters

Full size image
Fig. 9
figure 9

FE modeling results showing the temperature spread after attaining 30% deformation using CCD parameters

Full size image

3.2 Fishtail Formation

Fishtail is one of the critical form defects encountered in the ring rolling industry where irregular height in the cross-section of the formed ring results because of unrestricted material flow in the axial direction [30], as can be seen in Fig. 10. Fishtail defect is quantified by fishtail coefficient calculated using Eq. (5) [18]. The maximum height of the ring cross-section is subtracted from the minimum height of the ring cross-section and the difference is divided by the preform height. Fishtail coefficient varies between 0 and 1 as it is a normalized value. But, for implementation of the ANOVA technique, an unrestricted range of (−∞, ∞) is required. Hence, for further analysis, logit transformation with log base 10 function has been used, primarily to pull out the ends of the distribution to an unrestricted range of (−∞, ∞), given by Eq. (6).

$$ {\text{Fishtail}}\,{\text{coefficient}}\,\left( {\text{x}} \right) = \frac{{{\text{Max}}.\,{\text{height}}\,{\text{of}}\,{\text{ring}}\,{\text{cross}}\,{\text{section}} - {\text{Min}}.\,{\text{height}}\,{\text{of}}\,{\text{ring}}\,{\text{cross}}\,{\text{section}}}}{{{\text{Height}}\,{\text{of}}\,{\text{the}}\,{\text{undeformed}}\,{\text{preform}}}} $$
(5)
$${\text{y}}={\text{logit}}\left({\text{x}}\right)={\text{log}}\left(\frac{{\text{x}}}{1-{\text{x}}}\right)$$
(6)

4 Regression Equation and Analysis of Variance (ANOVA)

FE simulations yielded CoV (PEEQ), CoV (temperature), and logit transformation of fishtail coefficient as responses. These responses can be denoted as function (fi) of main roll speed (r), main roll feed (v), and coefficient of friction (μ) [31], as shown below:

$$ {\text{CoV}}\,({\text{PEEQ}})\, = \,{\text{f}}_{{1}} \left( {{\text{r}},{\text{v}},\mu } \right) $$
(7)
$$ {\text{CoV}}\,({\text{Temperature}})\, = \,{\text{f}}_{{2}} \left( {{\text{r}},{\text{v}},\mu } \right) $$
(8)
$$ {\text{logit}}\,{\text{Fishtail}}\,{\text{Coefficient}}\, = \,{\text{f}}_{{3}} \left( {{\text{r}},{\text{v}},\mu } \right) $$
(9)

The polynomial equation can be stated as:

$$\begin{aligned} {\text{f}}_{{\text{i}}} \left( {{\text{r}},{\text{v}},\mu } \right) & \, = \,{\text{a}}_{0} + {\text{a}}_{1} {\text{r}} + {\text{a}}_{2} {\text{v}} + {\text{a}}_{3} \mu + {\text{a}}_{{12}} {\text{rv}} + \\ & \quad {\text{a}}_{{13}} {\text{r}}\mu + {\text{a}}_{{23}} {\text{v}}\mu + {\text{a}}_{{11}} {\text{r}}^{2} + {\text{a}}_{{22}} {\text{v}}^{2} + {\text{a}}_{{33}} \mu ^{2} + \varepsilon \\ \end{aligned} $$
(10)

where a0 is a constant and is estimated as average of the responses and a1, a2, a3, a12, a13, a23, a11, a22, and a33 are the regression coefficients for the linear, interaction, and squared terms in Eq. (10). ε is the error term [32].

Fig. 10
figure 10

Photographs of preform and sectioned ring with fishtail defect

Full size image

4.1 PEEQ Heterogeneity Analysis

ANOVA test results for PEEQ are presented in Table 5. It shows the various sources of variation and ascertains the p-value for the linear, squared, and interaction terms. Statistically significant terms have p-value less than 0.05. Main roll speed, main roll feed, and square of main roll feed are found to be significant based on the p-values with a contribution of 4.14%, 84.15%, and 6.8%, respectively.

Table 5 ANOVA test results for PEEQ heterogeneity (R2 = 0.9909)
Full size table

The empirical relationship developed for PEEQ using the coefficients is presented below:

$$ \begin{aligned} {\text{CoV}}\,({\text{PEEQ}}) & \, = \,0.{732}\, + \,0.00{\text{61r}} - 0.{\text{842v}}\, + \,{2}.{65}\mu - 0.0000{\text{29r}}^{{2}} \\ & \quad + \,0.{\text{2666v}}^{{2}} {-}{5}.{46}\mu^{{2}} - 0.000{\text{31Rv}}\, + \,0.00{\text{51r}}\mu - \, 0.{\text{227f}}\mu \\ \end{aligned} $$
(11)

The final empirical relationship using coefficients from significant terms in Table 5 is presented below:

$$ {\text{CoV}}\,({\text{PEEQ}})\, = \,0.{732}\, + \,0.00{\text{61r}} - 0.{\text{842v}}\, + \,0.{\text{2666v}}^{{2}} $$
(12)

The R2 value was found to be 0.9909, which explains that 99.09% of statistical variation can be explained by the chosen parameters.

4.2 Temperature Heterogeneity Analysis

The empirical relationship representing the impact of factors and interactions is formulated using coefficients as given below:

$$ \begin{aligned} 10000 \times {\text{CoV}}\,({\text{temperature}}) & \, = \,201.3 - 0.44{\text{r}}{-}95.9{\text{v}}\, + \,193\mu \, + \,0.0037{\text{r}}^{2} \\ & \quad + \,20.32{\text{v}}^{2} {-}448\mu^{2} - 0.037{\text{rv}}\, + \,0.13{\text{r}}\mu \, + \,15.5{\text{v}}\mu \\ \end{aligned} $$
(13)

The significant factors provided in Table 6 contributing to heterogeneity in temperature distribution are the main roll speed, the main roll feed, and square of the main roll feed with contribution of 0.27%, 95.90%, and 2%, respectively.

Table 6 ANOVA test results for temperature heterogeneity (R2 = 0.9953)
Full size table

The final empirical relationship involving significant term is given below:

$$ 10000 \times {\text{CoV}}\,({\text{temperature}})\, = \,201.3 - 0.44{\text{r}} - 95.9{\text{v}}\, + \,20.32{\text{v}}^{2} $$
(14)

The R2 value was found to be 0.9953.

4.3 Fishtail Coefficient Analysis

ANOVA results for the fishtail defect are listed in Table 7. The significant terms are found to be the main roll feed rate, square of main roll feed rate, and square of friction factor with contribution of 64.19%, 6.37%, and 4.5%, respectively.

Table 7 ANOVA test results for fishtail defect formation (R2 = 0.9101)
Full size table

The empirical relationship developed for fishtail coefficient using the coefficients is presented below:

$$\begin{aligned} {\text{Logit}}\,{\text{transformation}}\,{\text{of}}\,{\text{fishtail}}\,{\text{coefficient}} & \, = \,{\text{1}}.{\text{126}}\, + \,0.0{\text{247r}} - 0.{\text{566v}} - {\text{14}}.{\text{29}}\mu \\ & \quad - \,0.000{\text{6}}0{\text{2r}}^{{\text{2}}} + \,0.{\text{168}}0{\text{v}}^{{\text{2}}} \, + \,{\text{24}}.{\text{6}}\mu ^{{\text{2}}} \\ & \quad + \,0.00{\text{385rv}} - 0.0{\text{65r}}\mu - 0.{\text{5}}0{\text{1v}}\mu \\ \end{aligned}$$
(15)

The final empirical relationship using significant term from Table 7 is given below:

$$\begin{aligned} {\text{Logit}}\,{\text{transformation}}\,{\text{of}}\,{\text{fishtail}}\,{\text{coefficient}} & \, = \,{\text{1}}.{\text{126}} - 0.{\text{566v}}\, \\ & \quad + \,0.{\text{168}}0{\text{v}}^{{\text{2}}} \, + \,{\text{24}}.{\text{6}}\mu ^{{\text{2}}} \\ \end{aligned}$$
(16)

The R2 value was of 0.9101.

Hence, it can be clearly stated that the main roll feed is the most significant factor contributing to heterogeneity of PEEQ and temperature and fishtail formation. The second-order polynomial fit of responses (CoV (PEEQ), CoV (Temperature) and logit fishtail) with respect to process parameters are presented in Fig. 11. The results echo the findings of ANOVA in Tables 5, 6 and 7 and affirm that the main roll feed is the most significant factor. There is a negative correlation and quadratic relationship between responses and main roll feed rate.

Fig. 11
figure 11

a, d, g CoV (PEEQ), b, e, h CoV (temperature) and c, f, i fishtail coefficient variation with main roll speed, main roll feed, and coefficient of friction

Full size image

4.4 Interpretation of DoE Optimization

Relationship, given in Eq. (17), between feed per revolution (Δhi) and process parameters like main roll speed (n1) and main roll feed (υ) was proposed Hawkyard et al. [33]:

$$\Delta {{\text{h}}}_{{\text{i}}}= \frac{2\mathrm{\pi \upsilon }{{\text{R}}}_{{\text{i}}}}{{{\text{n}}}_{1}{{\text{R}}}_{1}}$$
(17)

where the terms Ri and R1 are the outer radius of the deforming ring and radius of the driver roll, respectively. Equation (17) shows that feed per revolution is directly proportional to main roll feed rate (υ) and inversely proportional to main roll speed (n1). Higher feed rate makes penetration of plastic deformation easier into the radial thickness, thus creating a more uniform strain distribution.

In Fig. 11, it can be seen that CoV (PEEQ) reduces with increase in feed rate. Increase in feed rate leads to greater penetration in the ring cross-section resulting in greater uniformity in strain distribution and reduced CoV (PEEQ). Increase in feed rate also results in reduced CoV (temperature) as adiabatic heating increases during high feed rate deformation increasing the temperature [34], resulting in internal temperature rise. In case of high feed rate, the desired deformation is attained in less time leading to lesser temperature loss through the surfaces. Hence, a uniform temperature distribution can be resulted in case of high feed rate deformation. Another reason for more uniform deformation in case of high feed rate deformation can be attributed to lower material deformation resistance at higher temperature resulting more uniform deformation [4]. Both the main roll speed (rpm) and coefficient of fraction play little role as suggested both in ANOVA and Fig. 11. It has been reported in literature, a more uniform deformation results in reduction in fishtail coefficient [35], and hence, higher feed rate resulted in reduced fishtail defect.

5 Conclusion

In the present study, the effects of process parameters such as main roll speed, main roll feed and coefficient of friction on heterogeneity of strain (PEEQ) and temperature distribution and form defect like fishtail defect were looked into. A series of 20 3D FE simulations of ring rolling process was run based on a three-parameter two-level Central Composite Design (CCD). Coefficient of Variation (CoV) was used as a heterogeneity index. The extent of Fishtail defect is captured via fishtail coefficient. The main conclusions are as follows:

  • FE simulations show that reduced coefficient of friction (μ = 0.1) resulted in failure to achieve the rolling process.

  • Feed rate has been established to be the most significant factor affecting the severity of fishtail defect and heterogeneity in temperature and equivalent plastic strain (PEEQ) distribution.

  • The effects of other terms like main roll speed, square of feed, and square of coefficient of friction have been found to be minimal.

  • ANOVA study shows that CoV (PEEQ), CoV (temperature), and logit fishtail have a quadratic relationship with feed rate (mm/s).

  • The responses (CoV (PEEQ), CoV (temperature), and logit of fishtail) shared a negative correlation with feed rate.