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Coupled thermo-mechanical finite-element modelling of hot ring rolling process
Journal of Materials Processing Technology 121 (2002) 332–340
Coupled thermo-mechanical finite-element modelling of hot ring rolling process J.L. Songa,*, A.L. Dowsona, M.H. Jacobsa, J. Brooksb, I. Bedenc a
IRC in Materials for High Performance Application and School of Metallurgy and Materials, The University of Birmingham, Edgbaston, Birmingham B15 2TT, UK b DERA, Farnborough, UK c DONCASTERS Blaenavon Ltd., Blaenavon, UK Received 13 October 2000; accepted 2 October 2001
Abstract A general purpose, proprietary finite-element software program, Marc/Mentat, has been used to develop a coupled thermo-mechanical model of the deformation processes occurring during the hot rolling of IN718 rings. Unique features of the model are: (1) it adopts a transient or unsteady state full ring mesh to model the deformation process and shape development; (2) the mandrel and drive rolls are modelled using fully coupled heat-transfer elements as opposed to the more traditional ‘rigid curve’ approach; (3) the model has been validated on an industrial scale using data obtained during the hot ring rolling of planar IN718 rings, and good quantitative agreement has been observed with roll load variations and surface temperature measurements obtained during the rolling operation. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Ring rolling; Finite-element modelling; IN718; Marc/Mentat
1. Introduction Ring rolling is widely used in the production of railway tyres, anti-friction bearing races, flanges of various geometry and rings of different materials and dimensions used in the chemical, aerospace, automotive and nuclear industries. The advantages of ring rolling include short production time, uniform quality, smooth surface, close tolerances and considerable saving in material cost. The principal features of the ring rolling process are illustrated schematically in Fig. 1. Essentially, a typical ring rolling mill incorporates three sets of rolls each with a specific function. First, a set of radial rolls including the main drive roll and mandrel which reduces the radial thickness of the ring. Secondly, a set of axial rolls which restricts axial growth and controls the height of the ring. Finally, the guide rolls maintain the circular shape of the ring and provide support during the rolling operation. The development of finite-element (FE) models for ring rolling is of major commercial and technological interest as they provide a fundamental understanding of the mechanics of the process and an insight into how defects arise in the ring product. They also provide a quick and inexpensive *
Corresponding author.
method of identifying and optimising important process parameters without the need for expensive experimental trials. There have been a number of attempts to model the ring rolling process using both two- and three-dimensional FE techniques [1–6]. This has included recently the introduction of new strategies for the highly non-linear equations involved in modelling the three-dimensional ring rolling process. This is exemplified in the work of Davey and Ward [7], where arbitrary Lagrangian–Eulerian formulation and successive preconditioned conjugate gradient method have been applied, and in the work of Xie et al. [8], who used rigid–viscoplastic dynamic explicit FE method to analyse this process. It is clear that the ring rolling process has been modelled extensively using both two- and three-dimensional FE techniques. The majority of models developed to-date, however, do not take into account the transient nature of the process and in most instances only a small portion of the ring is considered. A second major consideration in the context of the current work is that in most cases, the mandrel is considered to be idle and is configured to move radially against the ring producing the reduction in wall thickness. This is not always the case, particularly, with vertically mounted rolling mills where the radial position of the mandrel is often fixed, and deformation is achieved by
0924-0136/02/$ – see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 1 ) 0 1 1 7 9 - 7
J.L. Song et al. / Journal of Materials Processing Technology 121 (2002) 332–340
333
Fig. 1. Schematic diagram showing the configuration of the workpiece and rolls of a typical ring rolling mill.
the vertical movement of the drive roll towards the mandrel. Finally, in the majority of models developed to-date, the mandrel and drive roll have been modelled using rigid body surfaces maintained at constant temperature. As a consequence, the heat transfer between the workpiece and the rolls has not been adequately considered. This paper attempts to address some of these issues.
2. Numerical modelling of the ring rolling process 2.1. Applying FE techniques The configuration of the rolling mill used in the development of the FE model is illustrated in Fig. 1. Mounted in the vertical plane, the mill differs slightly from more traditional ring rolling machines in that both the mandrel and the main roll are driven, and the deformation is achieved through the movement of the main roll towards the mandrel. In addition, growth in the axial direction is constrained, i.e. no extension occurs in this direction. On this basis, the process can be approximated by a two-dimensional FE analysis. In order to reduce computation time and simplify the FE analysis, the guide rolls are ignored. This is justified on the basis that the contact stresses between the guide rolls and the ring are small and hence contribution to deformation is insignificant. Similarly, the overall effect on heat loss due to conduction will be small due to the relative small arc of contact. Using a Lagrangian large deformation displacement
technique, the equations governing thermo-mechanical deformation are summarised as [9] @2U @U þ KðTÞU ¼ F; þ DðTÞ @t2 @t @T þ HðTÞT ¼ Q þ QI CðTÞ @t
MðTÞ
where M(T) is the system mass matrix, D(T) the system damping matrix, K(T) the system stiffness matrix, U the nodal displacement vector, @U=@t the velocity vector, @ 2 U=@t2 the acceleration vector, F the force vector, C(T) the heat capacity of the matrix, H(T) the thermal conductivity of the matrix, Q the thermal load vector, QI the internal heat generated due to inelastic deformation, T the nodal temperature vector and @T=@t the time derivative of temperature. Within each small time step (about 104 s), a staggered solution procedure is used in which first a heattransfer analysis occurs and then a stress analysis. This process is repeated until both equations converge before moving onto the next time step. Thermal and mechanical coupling is realised in the first instance by the coefficients in the equation of motion (mass, damping and stiffness) being temperature dependent and secondly by the conversion of inelastic deformation associated with mechanical work into heat which will affect temperature distribution. The two-dimensional mesh used in the analysis, Fig. 2, was established using Mentat. Four-noded, isoparametric, arbitrary quadrilateral full integration elements were used to model the ring blank. The interpolation function was
334
J.L. Song et al. / Journal of Materials Processing Technology 121 (2002) 332–340 Table 1 Temperature dependence of physical properties Temperature Young’s Thermal Specific Thermal (K) modulus expansion heat conductivity (105 MPa) (105 K1) (108 J/kg K) (W/m K) 294 366 477 589 700 811 922 1033 1144 1255 1366
Fig. 2. Initial mesh used in the two-dimensional thermo-mechanical ring rolling FEM model.
assumed to be bilinear, and stiffness was achieved using 4-point Gaussian integration. In order to avoid volumetric locking, a constant dilatation formulation was used. Heat transfer in the main drive roll and mandrel were modelled using fully rigid 4-noded, isoparametric, arbitrary quadrilateral heat-transfer elements, and the heat generated due to plastic work and friction at the interface between the ring and rolls were taken into consideration. An updated Lagrange procedure was used to account for large strain plasticity. The strain, stress, strain rate and temperature were calculated at Gaussian integration points. The conditions at the interfaces between the ring and rolls were described using the Coulomb friction model. A full Newton–Raphson iteration technique was chosen to solve the non-linearity and a sparse direct solver was adopted to solve the simultaneous arithmetic equations. Convergence testing was carried out on the relative displacement. 2.2. Ring rolling parameters and material model The model was validated on an industrial scale using roll load and surface temperature measurements obtained during the hot ring rolling of planar IN718 Ni-based superalloy rings having an initial external diameter of 320 mm and a
2.0 1.96 1.9 1.84 1.78 1.71 1.63 1.54 1.39 1.2 0.99
12.2 1.22 1.33 1.38 1.4 1.44 1.49 1.55 1.58 1.63 1.67
4.3 4.55 4.75 5.0 5.3 5.6 5.9 6.15 6.45 6.7 7.0
11.4 12.6 14.4 16.2 17.9 19.6 21.4 23.2 24.9 26.8 28.7
radial thickness of approximately 45 mm. It is important to note that all of the rolling trials were conducted using a vertically configured rolling mill in which both the mandrel and drive roll were driven and in which feed was applied to the larger driven roll. The rotating speeds of main roll and mandrel used in FE model were based on measurements taken during industrial ring rolling. Relative changes in surface temperature during rolling were monitored using both optical pyrometer and infrared thermal imaging camera linked to a video recorder. Similarly, the surface temperature of the mandrel and drive rolls immediately prior to and after rolling were measured using a hand-held contact pyrometer. Stress, strain, strain rate and temperature relationships used in the FE calculations were derived using a constitutive model developed by Brooks [9]. The main features of this model are summarised in Appendix A. Physical property data (specific heat, thermal conductivity, thermal expansion coefficient and Young’s modulus) and their variation with temperatures were obtained from publications provided by the Nickel Development Institute and the National Physical Laboratory, UK (Table 1).
3. FE results and discussion 3.1. Shape development To initiate the simulation, it was necessary to introduce a preliminary ‘indentation stage’ in order to develop the necessary contact conditions for rolling. During this period (lasting approximately 0.01 s), the drive roll, mandrel and workpiece were maintained stationary while the main drive roll was configured to move vertically towards the mandrel thus imposing an initial pre-load on the ring. At this point, the rolling process was initiated and the friction conditions were sufficiently developed to cause the ring to rotate. The feed rates and roll rotating speeds used in the calculation were specified so as to reproduce the actual conditions measured during the ring rolling trials and output data were generated relating to shape development, roll load, strain,
J.L. Song et al. / Journal of Materials Processing Technology 121 (2002) 332–340
stress and temperature distribution as a function of rolling time. The model was terminated after the main drive roll had traversed a distance of 12 mm which corresponds to a total reduction in thickness of approximately 25–30% and total rolling time is about 20 s. The initial and deformed grids after rolling for 5, 10, 15 and 20 s are shown in Fig. 3(a)–(d). In each figure, the right hand mesh is an enlargement of the area depicted by the circle on the left hand mesh. From Fig. 3, it is clear that the distortion of the grid on both the outer and inner surfaces of the ring was more severe than that at the mid-plane, i.e. high deformation had occurred at both surfaces. A subroutine was written in order to track the time and deformation dependent changes in ring diameter during the ring rolling simulation. In Fig. 4, the diameters determined from the model in both the horizontal and vertical planes and the ratio of these two diameters are plotted as a function of rolling time. These data would tend to suggest that the maximum variation in ring diameter is less than 3 mm with the degree of ovality (Dvertical/Dhorizontal) varying in the range 0.98–1.01, which is consistent with experimental measurement of the final ring profiles.
335
3.2. Strain and stress distributions The total equivalent plastic strain at the inner surface, outer surface and mid-plane are plotted in Fig. 5 as a function of rolling time. Consistent with the grid distortion effects noted in Fig. 3, the strain distribution across the ring section is non-uniform with both the inner and outer surfaces exhibiting higher levels of strain than the mid-plane. This can be related to the small reduction in the radial thickness. Similarly, it should be noted that the strain concentration at the inner surface tends to be higher than that at the outer surface of the ring. This can be attributed to the small diameter of the mandrel and is generally consistent with the finding in other work [4,10]. The second component of stress (the stress in the radial direction) distribution observed within the roll gap is shown in Fig. 6. A significant shear component is observed to develop with the points of maximum stress exhibiting considerable angular displacement on moving from the internal to the external surface of the ring. Similar results have been reported by Mamalis et al. [11] and Lim et al. [6].
Fig. 3. Deformed mesh and initial roll and ring position at rolling time: (a) 5; (b) 10; (c) 15; (d) 20 s.
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Fig. 3. (Continued ).
3.3. Temperature distribution A contour map showing the temperature variations close to the external surface of the ring is represented in Fig. 7. The main point to note is that the temperature is appreciably
lower than the temperature of the interior of the ring, and it is deduced that this is due to considerable heat transfer to the mandrel and drive rolls. This is clearly indicated in Fig. 8 which shows the surface temperature history plots for two rolls during the rolling operation. The start temperatures of
Fig. 4. Horizontal, vertical diameters and ratio of these two diameters changing with rolling time.
J.L. Song et al. / Journal of Materials Processing Technology 121 (2002) 332–340
337
Fig. 5. Nodal strain history at outer surface, mid-plane, and inner surface of the ring during ring rolling process.
Fig. 6. Stress component, syy, contour line plot of the ring blank for a ring rolling time of 10 s.
Fig. 7. Temperature contour line plot near the external surface of the ring blank for a ring rolling time of 10 s.
the main roll and mandrel were 40 and 80 8C, respectively, and from Fig. 8, it is quite evident that these temperatures increased significantly during the rolling reaching 68 and 290 8C, respectively. This clearly highlights the importance of modelling heat transfer between the ring and the rolls during ring rolling simulation. From Fig. 7, it is apparent that the temperature drop at the external surface is significantly greater than that at the internal surface. This is not too surprising considering the large area of contact between the ring and the main roll at the external surface and the large thermal mass of the main roll. In general, the temperature at the surface of the ring is governed by a number of factors, but can be considered to be dictated by the balance between the heat generated as a result of plastic deformation and the rate of heat loss due to conduction, convection and radiation [12]. Although the
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Fig. 8. Surface temperature history of mandrel and main roll during ring rolling process.
contact stresses between the ring and the rolls are much higher at the internal surface and, as a consequence, the potential for heat loss due to conduction is much greater, heat losses due to convection and radiation are considerably reduced, and the small size of the mandrel means that the temperature at the internal surface is retained. Fig. 9 compares the measured and predicted surface temperatures during rolling. In the initial stages of the rolling operation, since both the main roll and mandrel are in a relatively cold condition, there is a large temperature drop on passing through the roll gap. This is not directly apparent from the experimental data. However, as rolling progresses, both the main roll and mandrel increase in temperature (as shown in Fig. 8), and the temperature of the ring decreases such that during the final stages of rolling the measured and predicted temperatures converge. The
discrepancies between the measured and predicted temperatures during the initial stages of the rolling operation can be explained in part by the fact that the predicted data corresponds to points within the roll gap immediately in contact with the mandrel and drive rolls. Temperatures obtained experimentally correspond to a point on the external surface of the ring following exit from the roll gap. At this point, some recovery in temperature would be expected due to the transfer of internal heat towards the external surface. This conclusion is clearly supported by the through-thickness temperature contours plotted in Fig. 7. 3.4. Rolling load Fig. 10 compares the measured and predicted loads during rolling with the latter fluctuating with the element contacts
Fig. 9. Comparison between measured and predicted surface temperatures.
J.L. Song et al. / Journal of Materials Processing Technology 121 (2002) 332–340
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Fig. 10. Comparison between predicted and measured loads.
and releases in the roll gap. Apart from the initial stages of rolling, the results show reasonable agreement. The reason for the difference at the initial stage of rolling can be explained as follows. In the actual rolling process, the rolling bite was built up gradually, so the rolling load is steadily increased. However, in the FE modelling, the rolling conditions were set up within the first 0.01 s, so the load quickly reached steady state.
4. Conclusions A coupled two-dimensional elastic–plastic thermal– mechanical model has been developed to simulate the hot ring rolling process. A major feature of the model is that the complete ring and work roll assembly are considered and, consequently, the model is capable of generating quantitative information regarding the ring shape, temperature, stress and strain distributions. Heat transfer between the ring and the work rolls has been modelled using fully coupled heat-transfer elements and important information has been generated. The calculated rolling load and surface temperature have been compared with experimental measurements and showed good agreement.
Particular thanks are due to the engineering and production staff at Doncasters, Blaenavon, for their support during the ring rolling and model validation trials.
Appendix A. Constitutive equations for IN718 derived by Brooks The temperature compensated strain rate can be expressed using the Zener–Holloman parameter such that Qa Z ¼ e_ exp RT Following from this, the flow stress at steady state can be related to Z by a simple power law lss ¼ l0 Z q The microstructure can be described by a parameter, l, which varies towards steady state exponentially with strain l ¼ lss þ ð1 lss Þ expðaeÞ An intermediate parameter K can be defined as K ¼ kln The target stress is then approached exponentially as a function of strain
Acknowledgements Funding from the UK Engineering and Physical Sciences Research Council (EPSRC), Innovative Manufacturing Initiative (IMI)—Aerospace Programme is gratefully acknowledged. Thanks are also extended to Prof. M.H. Loretto (Director, IRC) and support staff within the IRC for the provision of facilities and technical support to the programme. Finally, the authors would like to acknowledge the financial and technical support provided by Doncasters plc, Rolls Royce plc and Spray Forming Developments Ltd.
s ¼ si K expðbeÞ where k is a scaling constant, l0 a scaling constant related to the steady state stress, n the structure–stress exponent (commonly unity so that structure behaves like stress), m the rate sensitivity of stress, q the rate sensitivity of structure, a an exponential damping constant relating strain and structure, b an exponential damping constant relating strain and stress, Qa the activation energy, R the gas constant, e_ the strain rate (per second), T the temperature (K), Z the Zener– Holloman parameter, lss the steady state structure (defined
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as (steady state stress)/(initial stress)), e the strain, s the stress, initially zero, and l the microstructural parameter, initially unity. References [1] D.Y. Yang, K.H. Kim, Int. J. Mech. Sci. 30 (8) (1988) 571–580. [2] S.G. Xu, J.C. Lian, J.B. Hawkyard, Int. J. Mech. Sci. 33 (5) (1991) 393–401. [3] Y.K. Hu, W.K. Liu, Int. J. Numer. Meth. Eng. 33 (1992) 1217–1236. [4] N. Kim, S. Machida, S. Kobayashi, Int. J. Mach. Tools Manuf. 30 (4) (1990) 569–577. [5] Z.M. Hu, I. Pillinger, P. Hartley, S. McKenzie, P.J. Spence, J. Mater. Process. Technol. 45 (1994) 143–148.
[6] T. Lim, I. Pillingger, P. Hartley, J. Mater. Process. Technol. 80–81 (1998) 199–205. [7] K. Davey, M.J. Ward, An efficient solution method for finite element ring rolling simulation, Int. J. Numer. Meth. Eng. 47 (2000) 1997– 2018. [8] C. Xie, X. Dong, S. Li, S. Huang, Rigid–viscoplastic dynamic explicit FEA of the ring rolling process, Int. J. Mach. Tools Manuf. 40 (2000) 81–93. [9] J.W. Brooks, Modelling data for inconel alloy 718, Doncasters Internal Report, 1997. [10] S.G. Xu, K.J. Weinmann, D.Y. Yang, J.C. Lian, Trans. ASME 19 (1997) 542–549. [11] A.G. Mamalis, J.B. Hawkyard, W. Johnson, Spread and flow pattern in ring rolling, Int. J. Mech. Sci. 18 (1976) 11–16. [12] Z.M. Hu, I. Pillinger, P. Hartley, J. Mater. Process. Technol. 45 (1994) 143–148.
Coupled thermo-mechanical finite-element modelling of hot ring rolling process J.L. Songa,*, A.L. Dowsona, M.H. Jacobsa, J. Brooksb, I. Bedenc a
IRC in Materials for High Performance Application and School of Metallurgy and Materials, The University of Birmingham, Edgbaston, Birmingham B15 2TT, UK b DERA, Farnborough, UK c DONCASTERS Blaenavon Ltd., Blaenavon, UK Received 13 October 2000; accepted 2 October 2001
Abstract A general purpose, proprietary finite-element software program, Marc/Mentat, has been used to develop a coupled thermo-mechanical model of the deformation processes occurring during the hot rolling of IN718 rings. Unique features of the model are: (1) it adopts a transient or unsteady state full ring mesh to model the deformation process and shape development; (2) the mandrel and drive rolls are modelled using fully coupled heat-transfer elements as opposed to the more traditional ‘rigid curve’ approach; (3) the model has been validated on an industrial scale using data obtained during the hot ring rolling of planar IN718 rings, and good quantitative agreement has been observed with roll load variations and surface temperature measurements obtained during the rolling operation. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Ring rolling; Finite-element modelling; IN718; Marc/Mentat
1. Introduction Ring rolling is widely used in the production of railway tyres, anti-friction bearing races, flanges of various geometry and rings of different materials and dimensions used in the chemical, aerospace, automotive and nuclear industries. The advantages of ring rolling include short production time, uniform quality, smooth surface, close tolerances and considerable saving in material cost. The principal features of the ring rolling process are illustrated schematically in Fig. 1. Essentially, a typical ring rolling mill incorporates three sets of rolls each with a specific function. First, a set of radial rolls including the main drive roll and mandrel which reduces the radial thickness of the ring. Secondly, a set of axial rolls which restricts axial growth and controls the height of the ring. Finally, the guide rolls maintain the circular shape of the ring and provide support during the rolling operation. The development of finite-element (FE) models for ring rolling is of major commercial and technological interest as they provide a fundamental understanding of the mechanics of the process and an insight into how defects arise in the ring product. They also provide a quick and inexpensive *
Corresponding author.
method of identifying and optimising important process parameters without the need for expensive experimental trials. There have been a number of attempts to model the ring rolling process using both two- and three-dimensional FE techniques [1–6]. This has included recently the introduction of new strategies for the highly non-linear equations involved in modelling the three-dimensional ring rolling process. This is exemplified in the work of Davey and Ward [7], where arbitrary Lagrangian–Eulerian formulation and successive preconditioned conjugate gradient method have been applied, and in the work of Xie et al. [8], who used rigid–viscoplastic dynamic explicit FE method to analyse this process. It is clear that the ring rolling process has been modelled extensively using both two- and three-dimensional FE techniques. The majority of models developed to-date, however, do not take into account the transient nature of the process and in most instances only a small portion of the ring is considered. A second major consideration in the context of the current work is that in most cases, the mandrel is considered to be idle and is configured to move radially against the ring producing the reduction in wall thickness. This is not always the case, particularly, with vertically mounted rolling mills where the radial position of the mandrel is often fixed, and deformation is achieved by
0924-0136/02/$ – see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 1 ) 0 1 1 7 9 - 7
J.L. Song et al. / Journal of Materials Processing Technology 121 (2002) 332–340
333
Fig. 1. Schematic diagram showing the configuration of the workpiece and rolls of a typical ring rolling mill.
the vertical movement of the drive roll towards the mandrel. Finally, in the majority of models developed to-date, the mandrel and drive roll have been modelled using rigid body surfaces maintained at constant temperature. As a consequence, the heat transfer between the workpiece and the rolls has not been adequately considered. This paper attempts to address some of these issues.
2. Numerical modelling of the ring rolling process 2.1. Applying FE techniques The configuration of the rolling mill used in the development of the FE model is illustrated in Fig. 1. Mounted in the vertical plane, the mill differs slightly from more traditional ring rolling machines in that both the mandrel and the main roll are driven, and the deformation is achieved through the movement of the main roll towards the mandrel. In addition, growth in the axial direction is constrained, i.e. no extension occurs in this direction. On this basis, the process can be approximated by a two-dimensional FE analysis. In order to reduce computation time and simplify the FE analysis, the guide rolls are ignored. This is justified on the basis that the contact stresses between the guide rolls and the ring are small and hence contribution to deformation is insignificant. Similarly, the overall effect on heat loss due to conduction will be small due to the relative small arc of contact. Using a Lagrangian large deformation displacement
technique, the equations governing thermo-mechanical deformation are summarised as [9] @2U @U þ KðTÞU ¼ F; þ DðTÞ @t2 @t @T þ HðTÞT ¼ Q þ QI CðTÞ @t
MðTÞ
where M(T) is the system mass matrix, D(T) the system damping matrix, K(T) the system stiffness matrix, U the nodal displacement vector, @U=@t the velocity vector, @ 2 U=@t2 the acceleration vector, F the force vector, C(T) the heat capacity of the matrix, H(T) the thermal conductivity of the matrix, Q the thermal load vector, QI the internal heat generated due to inelastic deformation, T the nodal temperature vector and @T=@t the time derivative of temperature. Within each small time step (about 104 s), a staggered solution procedure is used in which first a heattransfer analysis occurs and then a stress analysis. This process is repeated until both equations converge before moving onto the next time step. Thermal and mechanical coupling is realised in the first instance by the coefficients in the equation of motion (mass, damping and stiffness) being temperature dependent and secondly by the conversion of inelastic deformation associated with mechanical work into heat which will affect temperature distribution. The two-dimensional mesh used in the analysis, Fig. 2, was established using Mentat. Four-noded, isoparametric, arbitrary quadrilateral full integration elements were used to model the ring blank. The interpolation function was
334
J.L. Song et al. / Journal of Materials Processing Technology 121 (2002) 332–340 Table 1 Temperature dependence of physical properties Temperature Young’s Thermal Specific Thermal (K) modulus expansion heat conductivity (105 MPa) (105 K1) (108 J/kg K) (W/m K) 294 366 477 589 700 811 922 1033 1144 1255 1366
Fig. 2. Initial mesh used in the two-dimensional thermo-mechanical ring rolling FEM model.
assumed to be bilinear, and stiffness was achieved using 4-point Gaussian integration. In order to avoid volumetric locking, a constant dilatation formulation was used. Heat transfer in the main drive roll and mandrel were modelled using fully rigid 4-noded, isoparametric, arbitrary quadrilateral heat-transfer elements, and the heat generated due to plastic work and friction at the interface between the ring and rolls were taken into consideration. An updated Lagrange procedure was used to account for large strain plasticity. The strain, stress, strain rate and temperature were calculated at Gaussian integration points. The conditions at the interfaces between the ring and rolls were described using the Coulomb friction model. A full Newton–Raphson iteration technique was chosen to solve the non-linearity and a sparse direct solver was adopted to solve the simultaneous arithmetic equations. Convergence testing was carried out on the relative displacement. 2.2. Ring rolling parameters and material model The model was validated on an industrial scale using roll load and surface temperature measurements obtained during the hot ring rolling of planar IN718 Ni-based superalloy rings having an initial external diameter of 320 mm and a
2.0 1.96 1.9 1.84 1.78 1.71 1.63 1.54 1.39 1.2 0.99
12.2 1.22 1.33 1.38 1.4 1.44 1.49 1.55 1.58 1.63 1.67
4.3 4.55 4.75 5.0 5.3 5.6 5.9 6.15 6.45 6.7 7.0
11.4 12.6 14.4 16.2 17.9 19.6 21.4 23.2 24.9 26.8 28.7
radial thickness of approximately 45 mm. It is important to note that all of the rolling trials were conducted using a vertically configured rolling mill in which both the mandrel and drive roll were driven and in which feed was applied to the larger driven roll. The rotating speeds of main roll and mandrel used in FE model were based on measurements taken during industrial ring rolling. Relative changes in surface temperature during rolling were monitored using both optical pyrometer and infrared thermal imaging camera linked to a video recorder. Similarly, the surface temperature of the mandrel and drive rolls immediately prior to and after rolling were measured using a hand-held contact pyrometer. Stress, strain, strain rate and temperature relationships used in the FE calculations were derived using a constitutive model developed by Brooks [9]. The main features of this model are summarised in Appendix A. Physical property data (specific heat, thermal conductivity, thermal expansion coefficient and Young’s modulus) and their variation with temperatures were obtained from publications provided by the Nickel Development Institute and the National Physical Laboratory, UK (Table 1).
3. FE results and discussion 3.1. Shape development To initiate the simulation, it was necessary to introduce a preliminary ‘indentation stage’ in order to develop the necessary contact conditions for rolling. During this period (lasting approximately 0.01 s), the drive roll, mandrel and workpiece were maintained stationary while the main drive roll was configured to move vertically towards the mandrel thus imposing an initial pre-load on the ring. At this point, the rolling process was initiated and the friction conditions were sufficiently developed to cause the ring to rotate. The feed rates and roll rotating speeds used in the calculation were specified so as to reproduce the actual conditions measured during the ring rolling trials and output data were generated relating to shape development, roll load, strain,
J.L. Song et al. / Journal of Materials Processing Technology 121 (2002) 332–340
stress and temperature distribution as a function of rolling time. The model was terminated after the main drive roll had traversed a distance of 12 mm which corresponds to a total reduction in thickness of approximately 25–30% and total rolling time is about 20 s. The initial and deformed grids after rolling for 5, 10, 15 and 20 s are shown in Fig. 3(a)–(d). In each figure, the right hand mesh is an enlargement of the area depicted by the circle on the left hand mesh. From Fig. 3, it is clear that the distortion of the grid on both the outer and inner surfaces of the ring was more severe than that at the mid-plane, i.e. high deformation had occurred at both surfaces. A subroutine was written in order to track the time and deformation dependent changes in ring diameter during the ring rolling simulation. In Fig. 4, the diameters determined from the model in both the horizontal and vertical planes and the ratio of these two diameters are plotted as a function of rolling time. These data would tend to suggest that the maximum variation in ring diameter is less than 3 mm with the degree of ovality (Dvertical/Dhorizontal) varying in the range 0.98–1.01, which is consistent with experimental measurement of the final ring profiles.
335
3.2. Strain and stress distributions The total equivalent plastic strain at the inner surface, outer surface and mid-plane are plotted in Fig. 5 as a function of rolling time. Consistent with the grid distortion effects noted in Fig. 3, the strain distribution across the ring section is non-uniform with both the inner and outer surfaces exhibiting higher levels of strain than the mid-plane. This can be related to the small reduction in the radial thickness. Similarly, it should be noted that the strain concentration at the inner surface tends to be higher than that at the outer surface of the ring. This can be attributed to the small diameter of the mandrel and is generally consistent with the finding in other work [4,10]. The second component of stress (the stress in the radial direction) distribution observed within the roll gap is shown in Fig. 6. A significant shear component is observed to develop with the points of maximum stress exhibiting considerable angular displacement on moving from the internal to the external surface of the ring. Similar results have been reported by Mamalis et al. [11] and Lim et al. [6].
Fig. 3. Deformed mesh and initial roll and ring position at rolling time: (a) 5; (b) 10; (c) 15; (d) 20 s.
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Fig. 3. (Continued ).
3.3. Temperature distribution A contour map showing the temperature variations close to the external surface of the ring is represented in Fig. 7. The main point to note is that the temperature is appreciably
lower than the temperature of the interior of the ring, and it is deduced that this is due to considerable heat transfer to the mandrel and drive rolls. This is clearly indicated in Fig. 8 which shows the surface temperature history plots for two rolls during the rolling operation. The start temperatures of
Fig. 4. Horizontal, vertical diameters and ratio of these two diameters changing with rolling time.
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Fig. 5. Nodal strain history at outer surface, mid-plane, and inner surface of the ring during ring rolling process.
Fig. 6. Stress component, syy, contour line plot of the ring blank for a ring rolling time of 10 s.
Fig. 7. Temperature contour line plot near the external surface of the ring blank for a ring rolling time of 10 s.
the main roll and mandrel were 40 and 80 8C, respectively, and from Fig. 8, it is quite evident that these temperatures increased significantly during the rolling reaching 68 and 290 8C, respectively. This clearly highlights the importance of modelling heat transfer between the ring and the rolls during ring rolling simulation. From Fig. 7, it is apparent that the temperature drop at the external surface is significantly greater than that at the internal surface. This is not too surprising considering the large area of contact between the ring and the main roll at the external surface and the large thermal mass of the main roll. In general, the temperature at the surface of the ring is governed by a number of factors, but can be considered to be dictated by the balance between the heat generated as a result of plastic deformation and the rate of heat loss due to conduction, convection and radiation [12]. Although the
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Fig. 8. Surface temperature history of mandrel and main roll during ring rolling process.
contact stresses between the ring and the rolls are much higher at the internal surface and, as a consequence, the potential for heat loss due to conduction is much greater, heat losses due to convection and radiation are considerably reduced, and the small size of the mandrel means that the temperature at the internal surface is retained. Fig. 9 compares the measured and predicted surface temperatures during rolling. In the initial stages of the rolling operation, since both the main roll and mandrel are in a relatively cold condition, there is a large temperature drop on passing through the roll gap. This is not directly apparent from the experimental data. However, as rolling progresses, both the main roll and mandrel increase in temperature (as shown in Fig. 8), and the temperature of the ring decreases such that during the final stages of rolling the measured and predicted temperatures converge. The
discrepancies between the measured and predicted temperatures during the initial stages of the rolling operation can be explained in part by the fact that the predicted data corresponds to points within the roll gap immediately in contact with the mandrel and drive rolls. Temperatures obtained experimentally correspond to a point on the external surface of the ring following exit from the roll gap. At this point, some recovery in temperature would be expected due to the transfer of internal heat towards the external surface. This conclusion is clearly supported by the through-thickness temperature contours plotted in Fig. 7. 3.4. Rolling load Fig. 10 compares the measured and predicted loads during rolling with the latter fluctuating with the element contacts
Fig. 9. Comparison between measured and predicted surface temperatures.
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Fig. 10. Comparison between predicted and measured loads.
and releases in the roll gap. Apart from the initial stages of rolling, the results show reasonable agreement. The reason for the difference at the initial stage of rolling can be explained as follows. In the actual rolling process, the rolling bite was built up gradually, so the rolling load is steadily increased. However, in the FE modelling, the rolling conditions were set up within the first 0.01 s, so the load quickly reached steady state.
4. Conclusions A coupled two-dimensional elastic–plastic thermal– mechanical model has been developed to simulate the hot ring rolling process. A major feature of the model is that the complete ring and work roll assembly are considered and, consequently, the model is capable of generating quantitative information regarding the ring shape, temperature, stress and strain distributions. Heat transfer between the ring and the work rolls has been modelled using fully coupled heat-transfer elements and important information has been generated. The calculated rolling load and surface temperature have been compared with experimental measurements and showed good agreement.
Particular thanks are due to the engineering and production staff at Doncasters, Blaenavon, for their support during the ring rolling and model validation trials.
Appendix A. Constitutive equations for IN718 derived by Brooks The temperature compensated strain rate can be expressed using the Zener–Holloman parameter such that Qa Z ¼ e_ exp RT Following from this, the flow stress at steady state can be related to Z by a simple power law lss ¼ l0 Z q The microstructure can be described by a parameter, l, which varies towards steady state exponentially with strain l ¼ lss þ ð1 lss Þ expðaeÞ An intermediate parameter K can be defined as K ¼ kln The target stress is then approached exponentially as a function of strain
Acknowledgements Funding from the UK Engineering and Physical Sciences Research Council (EPSRC), Innovative Manufacturing Initiative (IMI)—Aerospace Programme is gratefully acknowledged. Thanks are also extended to Prof. M.H. Loretto (Director, IRC) and support staff within the IRC for the provision of facilities and technical support to the programme. Finally, the authors would like to acknowledge the financial and technical support provided by Doncasters plc, Rolls Royce plc and Spray Forming Developments Ltd.
s ¼ si K expðbeÞ where k is a scaling constant, l0 a scaling constant related to the steady state stress, n the structure–stress exponent (commonly unity so that structure behaves like stress), m the rate sensitivity of stress, q the rate sensitivity of structure, a an exponential damping constant relating strain and structure, b an exponential damping constant relating strain and stress, Qa the activation energy, R the gas constant, e_ the strain rate (per second), T the temperature (K), Z the Zener– Holloman parameter, lss the steady state structure (defined
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