Thermomechanical simulation of a rolling process with an FEM approach

Journal of Materials Processing Technology 92±93 (1999) 494±501

Thermo-mechanical simulation of a rolling process with an FEM approach L.M. Galantuccia,1, L. Tricaricob,* a

Dipartimento di Progettazione e Produzione Industriale, Politecnico di Bari, viale Japigia 182 Bari, Italy b Dipartimento di Progettazione e Produzione Industriale, Politecnico di Bari, Bari, Italy

Abstract The authors propose a model for the study of a hot rolling process, and the approach is based on thermo-mechanical analysis using the Finite-Element Method (FEM). The model can be used to speed up and improve the design and evaluation of the roughing and ®nishing phases in plate and sheet production. It is able to calculate the temperature distribution in the roll and the plate, the stress and strain ®elds, throughout a transient analysis done starting from the ®rst phases of the process. The main hypotheses adopted in the formulation are: the elasto-plastic behaviour of the material; and rolling under plane-deformation conditions. The main variables that characterise the rolling process, such as the geometry of the plate and the roll, the loads and the boundary conditions (radius of the rolls, rolling speed, initial and ®nal thickness, initial temperatures of the plate and the roll), have been expressed in a parametric form, this approach giving a good ¯exibility to the model. During the simulation, an iterative procedure enables the calculation and updating of the load conditions, such as the heat produced by friction on the plate±roll contact arc, and that caused by plastic deformation. The congruence of the results has been evaluated using experimental and theoretical data available in the literature. # 1999 Elsevier Science S.A. All rights reserved. Keywords: Rolling; Finite-Element Method (FEM); Simulation; Thermal; Mechanical

1. Introduction During hot rolling, the material is deformed plastically at temperatures higher than the recrystallization temperature, passing under a sequence of rolling mill stands. Each stand has a gap, de®ned by the distance between the surface of the rolls, the ®nal width of the plate being achieved through the progressive reduction of the gap of the stands in tandem rolling [1]. During recent years, steel makers have perceived an increasing interest in the market towards ¯at laminates with high performance in terms of mechanical characteristics and reliability; moreover, the hot rolling practice tends to obtain higher quality production with lower production costs by means of the optimisation of the manufacturing process. Acting on the chemical composition of the steel, and above all, on some of the parameters that set the rolling process (temperatures of the plate, fast cooling, rolling speed, per*Corresponding author. Tel.: +39-85460778; fax: +39-85460788 E-mail addresses: [email protected] (L. Tricarico); [email protected] (L.M. Galantucci) 1 Tel.: +39-85460764; fax: +39-85460788

centages of reduction in each individual stand), it is possible to control and drive the microstructure formation in the steel, increasing the more suitable characteristics. Rolling in control in the austenitic ®eld of niobium micro-alloyed steels [2], it is possible, for example, to enhance the strength properties of the steel and to drop the ductile-to-brittle transition temperature; rolling, instead, in the ferritic phase of steels that could develop a two-phase structure (ferritic± martensitic), it is possible to improve the cold formability characteristics of the material, or the drawability [3]. Various approaches have been proposed to simulate the material ¯ow during the process and to predict the material ®nal structure, and the latter is tied to the strain, the stresses, the deformation speed, and also the thermal cycle. The Finite-Element Method (FEM) is an effective tool for this purpose; the deformations and the thermal exchanges during rolling are in fact complex, and the approach with the classical theories can give only some of the information needed to design and control the process. Many models developed using the FEM have been proposed [4±10], differing with respect to: (i) the type of analysis (transient, steady-state); (ii) the type of formulation (incremental or solid, variational or ¯ow); (iii) the solution technique

0924-0136/99/$ ± see front matter # 1999 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 9 9 ) 0 0 2 4 2 - 3

L.M. Galantucci, L. Tricarico / Journal of Materials Processing Technology 92±93 (1999) 494±501

(updated Lagrangian, Eulerian); (iv) the constitutive law hypothesised for the material behaviour (elasto-plastic, elasto-viscoplastic; rigid±plastic, viscoplastic); (v) the type of discretization (2-D in the case of plane deformation, 3-D in form rolling); and (vi) the type of analysis (mechanical, thermal, uncoupled or coupled thermo-mechanical). 2. Formulation of the simulation model In the formulation of a rolling process model, it is necessary to consider the interaction of the thermal and mechanical phenomena (coupled analysis). The constitutive law for the behaviour of the material is in fact tied to the rolling temperature, and at the same time, the friction and the work of plastic deformation generate an increase in the temperature of the rolled plate. The procedure for an FEM analysis could be formulated in an indirect, or a direct way. In the indirect or uncoupled approach [5], the mechanical phenomenon and the thermal phenomenon are correlated in sequence, applying the results of the thermal model as the boundary condition for the mechanical model [6]; this approach is suitable for unidirectional coupling because it offers a better ¯exibility in the formulation of both models. In the direct solution approach, the distribution of temperature is obtained simultaneously with the ®eld of velocity [7±9]. The model proposed in this work follows this approach; it couples the vectors of structural load {F} and thermal load {Q}, according to the matrix equation [11] 0 0 fu_ g 0 fug fF g K 0 C t T_ 0 K t fT g fQ g where {T} is the vector of the temperatures, {Q} is the vector of the total heat ¯ow (given by the sum of the contributions due to the convection, to the surface loads and to the heat generated internally), {u} is the vector of the displacements, [K] is the stiffness matrix and [Kt] and [Ct] are the total conductivity and the speci®c heat matrix, respectively. 3. Simulation model The simulation model makes a transient analysis in plane deformation, taking into account: (i) the roll±plate contact; (ii) the heat transfer due to the conduction and convection in the rolls and the plate; and (iii) the heat caused by plastic deformation and that generated by friction. A starting procedure settles the geometry, the materials and the boundary conditions as functions of the main geometric parameters (radius R of the rolls, initial size hi and ®nal size hf of the plate) and the process parameters (distribution of the initial temperatures, rolling speed v, roll speed vr). In the following paragraphs have been synthesised the choices made in the implementation of the model.

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3.1. Mesh generation The de®nition of the element mesh foresees the discretization of the roll and the plate (restricted to only one of the two zones in contact with the roll, for obvious reasons of symmetry), and the schematisation of the contact between the two materials. Elements having two degrees of structural freedom (the displacements along the x and y directions of the nodes) and one thermal degree of freedom (the temperature in the node) have been used. They allow the thermomechanical coupled analysis in two dimensions, under conditions of large deformations and with material nonlinearities. The length of the plate, proportional to the radius of the roll, is chosen to enable the investigation of the ®rst phase of the rolling process (around 908 of roll rotation). The mesh of the rolling cylinder has been re®ned in the external zone to take into account the higher structural and thermal stress gradients; the mesh used for the plate is tied to the roll mesh and also to the aspect-ratio limit of the elements. The undeformed initial geometry is highlighted in Fig. 1(a) and in more detail in Fig. 1(b). The contact at the roll±plate interface has been schematised with 2-D elements; they have been positioned in the potential contact zones, called, respectively, the contact surface and the target surface. The contact happens when a node of the contact surface (node of contact) penetrates the target surface, passing beyond the external circumference of the roll. The contact is described specifying the normal contact stiffness KN, the coef®cient of friction at the roll±plate interface and the thermal contact conductance COND; in particular, KN is the penalty stiffness that acts in the normal direction on the target surface and enforces the displacement compatibility by limiting the penetration of the target base. The thermal contact conductance COND allows consideration of the imperfect contact and the temperature discontinuity across the contact interface. 3.2. Definition of the materials The thermo-physical and mechanical characteristics of the roll and plate materials, such as the thermal conductivity, the speci®c heat and the curve of the ¯ow stress of the material, are given as functions of the temperature.

Fig. 1. Mesh adopted for the roll and the plate.

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The behaviour of the working material has been hypothesised as being of the elasto-plastic type, with bilinear isotropic hardening. For temperatures lower than the recrystallization temperature, the yield stress and the slope of linear hardening have been considered as functions of the rolling temperature; for higher temperatures, the stress± strain curve has been hypothesised independent of the deformation, whilst the ¯ow stress is obtained as CT "_

mT

The strain rate "_ is calculated with an average value in the zone of deformation as " # v hi "_ p ln hf Rhi ÿ hf where the constants C(T) and m(T) depend on the material and the temperature. 3.3. Loads and boundary conditions The mechanical constraints allow only the rotation of the roll around its centre and the translation of the plate along its axis. Moreover, elastic axial elements with high stiffness have been used to constraint the roll in its centre (Fig. 1(a)), whilst an elastic axial element with negligible stiffness has been used to constraint the plate in the origin of the reference system (Fig. 1(b)). The symmetry has been imposed with thermal constraints for the nodes on the x axis of the plate, considering this surface as adiabatic. The loading conditions have been applied in different load steps. As concerns mechanical loads, in the ®rst step, an axial pressure is applied on the plate to allow the entrance in the stand and to avoid sliding between the contact and target surfaces; in the following steps, only roll rotation has been assigned to the roll. The thermal phenomenon is driven by the initial conditions of the roll and the plate (the temperatures assigned on the nodes of the plate as a function of their distance from the core), and by the thermal ¯ow due to the convection on the external surfaces, to the plastic deformation and to the friction on the interface. The heat ¯ow exchanged from the roll and the plate is q hT ÿ Ta , where h is the convective coef®cient, Ta the room temperature and T the material temperature. The value of h has been diversi®ed for the roll to take into account the in¯uence of the higher cooling in the zone of the spray jet (as schematised in Fig. 1(a), along an arc of 458). _ is a The heat generated through plastic deformation W fraction of the power needed for the deformation; in fact, for hot metalworking also, it has been veri®ed experimentally that 90±95% of the power of deformation is transformed into heat, whilst the residual fraction causes distortions in the crystalline lattice, and dislocations and alterations on the _ is calculated by integratboundary of grains. The power W ing the power of deformation for the unit of volume w_ on the

control volume V: Z _ W w_ dV V

where w_ is w_ ij "_ ij Splitting the deviatoric and the hydrostatic components of the stress, considering the condition of incompressibility of the material and the criterion of Von Mises, we obtain r 1 2 "_ ij "_ ij w_ p30 2 where 0 is the yield stress. An algorithm has been developed for the automatic calculation of the heat generated during the deformation; it has been used inside each load step of the solution, calculating the value in each element of the plate. The equivalent stress eqv , the total stain "tot , the total strain rate "_ tot (the latter from the "tot of the preceding load step and from the step time of the actual load step) are calculated in the nodes of the element. The heat produced through plastic deformation is considered as the thermal load for unit volume of the element. The heat produced through friction is Q pS jvr ÿ vj where is the friction coef®cient, p and S are the pressure and the contact area, respectively, and vr ÿ v is the roll± plate relative speed. Due to complexity, an average value of the relative speed and then an average value of the heat generated in the nodes that come into contact on the interface have been assumed. Compared to the theoretical results, this solution was found to be fully satisfying, also because the heat generated by friction has only a slight in¯uence on the temperature obtained. An implemented algorithm calculates the mean value of the heat due to friction and converts it in a thermal input (in terms of heat per unit of surface) on the nodes of the plate located in the roll±plate interface zone.

4. Results and discussion The FEM model described has been utilised to simulate a rolling process for which the thermal pro®les on the surface and in the core of the plate were available [9]. The roll has a radius of 120 mm and a roll speed of 3.8 rpm; the plate, starting from an initial size of 19 mm, has a thickness reduction of 21.3%. The initial temperatures are that of 8508C in the core, and 8308C on the surface. The steel considered is a low-carbon one (0.1% C, 0.55% Mn, 0.1% Cr). Fig. 2 shows the plate deformation and the equivalent stresses in the roll and the plate. In this case, the material


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Fig. 2. Deformation and equivalent stresses (N/m2) in the plate during the entrance phase and after 3 s (vr 3.8 rpm).

behaviour has been hypothesised, considering the temperature for hot rolling, and that the material is insensitive to hardening. The ±" curves have been obtained interpolating the constitutive equations at 8008C and 10008C CT800 120 MPa; mT800 0:10; CT1000 110 MPa; mT1000 0:12. Fig. 3 shows the isothermal curves in the plate after 3 s from the beginning of the process (approximately 708 of roll rotation). The thermal gradient caused by conduction towards the roll is highlighted: it gives lower temperatures in the contact zone. Outside the rolling stand, where there is only the convective heat exchange with the surrounding environment, this gradient disappears because the conduction from the core compensates for the convection on the surface: this is more evident when comparing the time±temperature curves for the nodes in a cross-section of the plate.

Fig. 4 shows the thermal cycles in different nodes of the initial cross-section of the plate mesh (Fig. 3): the nodes in the core (Cnt) and on the intermediate planes under the surface (Pl_1, Pl_2, Pl_3 and Pl_4). In the deformation zone can be noted different thermomechanical phenomena (conduction in the roll, heat generated by friction and by plastic deformation, and homogenisation due to the conduction of the working material); in particular, higher conduction can be noted on the nodes closer to the roll±plate interface (Pl_1, Pl_2), whilst plastic deformation causes a moderate increase in the temperature in the core. The amplitude of the zone of in¯uence of the roll contact on the isotherms of the plate is, of course, related to the rolling speed as the thermal phenomenon depends on time; Figs. 3 and 4 underline, however, that for the case examined,

Fig. 3. Temperature distribution in the plate after 3 s (vr 3.8 rpm).

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Fig. 6. Comparison of the thermal cycles on the surface. Fig. 4. Thermal cycles in the initial section with a total roll rotation of 908 (4 s).

this region is contained inside the length of the simulated rolling arc (R/2). Contrarily, the greater gradients of the more signi®cant variable of the process, such as the equivalent stress (Fig. 2, in which one can observe high stresses in the worked zone), and the plastic deformation along the y direction (Fig. 5) are circumscribed in the roll±plate contact zone. These considerations still remain valid even if the rolling speed is varied. 4.1. Influence of roll rotation speed The simulation results obtained with a double roll rotation speed (7.6 rpm) highlight that the material ¯ow stress increases due to the increase in the average strain rate "_ in the deformation zone. The isotherms underline, instead, that the plate temperature increases when the rolling speed increases; this is due to the heat generated during plastic deformation and due to the decrease in the heat exchange between the roll and the plate, related to the shorter contact time. 4.2. Comparison with experimental results The comparison between the thermal cycles obtained with the simulation and the experimental results reported in the

literature reference [9] is discussed. Obviously, this comparison has been done in a critical manner, assuming a particular number of material parameters because some of the experimental conditions are not known. Fig. 6 refers to the experimental condition on the same material (roll radius 120 mm, initial temperature 8258C, initial thickness 38 mm, percentage thickness reduction 20%), there being a very good agreement between the temperatures predicted by the simulation and those obtained by experimentation. 4.3. Influence of roll cooling The results described previously have been obtained considering a steady temperature of the roll (1008C); the analysis of the rolling process considering only a 908 roll rotation does not allow examination in a complete manner of the effect of the sprinklers on the thermal cycles. To investigate the mutual interaction between the heat exchanged in the deformation zone and that lost by convection in the cooling zone of the roll, a new FEM model has been implemented; compared to the previous model, it has a coarser grid mesh on the roll and a greater length of the plate; and it allows the analysis, as shown in Fig. 7, of a complete rotation of the roll. Fig. 8 shows the isotherms in the roll after 3.261 s from the beginning of the process. The simulation has been carried out using a roll speed of 3.8 rpm and an initial roll

Fig. 5. Strains along the y direction after 3 s (vr 3.8 rpm).


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Fig. 7. Mesh for the analysis of a complete roll rotation.

temperature of 1008C. These results give a complete and detailed view of the temperature distribution in the plate and the roll after a roll rotation of 708. Fig. 9 shows the thermal cycles during the ®rst 10 s of the rolling process for four nodes of the roll mesh (in the R_1 and R_4 sections of Fig. 7). The temperature of the roll surface in the R_4 section, after a fast increase due to the contact with the plate, decreases due to the convective heat exchange with the surrounding environment and due to the thermal diffusion towards the internal part; the temperature increase in the internal node is, instead, smoother. The action of the sprinklers after a roll rotation of 2208 causes remarkable cooling (nodes in the R_1 section); in the following 908 roll rotation, the temperature increases for thermal diffusion, tending to the initial temperature. These considerations are still valid for the thermal cycles of the R_2 and R_3 sections, which differ from the previous cycles only because the phase related to the roll rotation speed is shifted in time. The results obtained are in agreement with the behaviour of the experiments reported in [10]; for what concerns, instead, the temperature variations Tcooling and Tcontact due to the passing of the roll in the zone cooled with the

sprinklers, and that in the roll±plate contact zone, respectively, the simulation results show a strong sensitivity to the conductance values of these zones. In particular, the results shown in Fig. 8 have been obtained using the conductance value COND estimated with the experimental data reported in [9]. The convective coef®cient in the sprinkler zone was, instead, equal to 4800 W/m2 K [9,10]. 4.4. Thermal cycles Finally, the following part shows the thermal cycles on the plate. Fig. 10(a) and (b) highlight, respectively, the temperature of points on the surface and in the core, the examined section being those represented in Fig. 7. The analysis of the thermal cycle con®rms the remarks made describing the results of the previous model (Fig. 4). The sequence of the thermal cycles suggests, instead, a steady-state for what concerns the effects of the heat exchanged at the roll±plate interface and that generated during plastic deformation. This sequence, moreover, underlines: (i) the temperature homogenisation inside the plate (decrease in the temperature

Fig. 8. Isotherms in the roll after 3.261 s (vr 3.8 rpm).

500


Fig. 9. Thermal cycles at different points of the roll (Fig. 7).

Fig. 11. Thermal cycles at different points located in section S_1.

on the thermal and structural behaviour of the roll and the plate, providing good agreement with the thermal cycles reported in the experimental experiences available in the literature. The thermal diffusion at the roll±plate interface and the convection in the sprinkler zone have major effects on the simulation results. The model has been developed using a parametric approach, not only for the geometrical and process parameters, such as the rolling speed, the roll radius, and the initial and ®nal thickness of the plate, but also for the thermo-physical and structural parameters. The former are tied to the plate temperature, whilst the latter are related also to the average strain rate in the deformation zone. This approach gives a good ¯exibility in simulating single rolling stands on a tandem rolling line. This model can, therefore, be considered an important design tool, especially for rolling in control, where it is extremely important to follow the thermal evolution of the material along all of the tandem rolling line. Fig. 10. Thermal cycles at different cross-sections of: (a) the core; and (b) the surface plate.

on the surface and increase in the core) due to thermal diffusion; (ii) the decrease in the temperature due to the convective heat exchange between the plate surface and the surrounding environment; and (iii) the higher temperature decrease for the roll±plate interaction. Considering the thermal cycles, such as those of section S_1 (Fig. 11), the overall effect can be evaluated by means of the thermal gradients: 0.38C/s for the plate at the entrance and in the exit zone; and 308C/s for that close to the deformation zone. 5. Conclusions The FEM simulation model proposed for the thermomechanical analysis of the ®rst phases of the hot rolling process is able to estimate the effect of process parameters

Acknowledgements The authors want to express their thanks to Prof. Ing. Attilio Alto for his ®rm and careful guidance during all of the development of this research. This work has been funded by Italian CNR 93.00451.CT07 and MURST 40%. References [1] B. Avitzur, Metal Forming: Processes and Analysis, McGraw-Hill, New York, 1987. [2] E. Anelli, A. Cerquitella, M. Pontremoli, A. de Vito, L. Masi, Nastri laminati a caldo per applicazioni speciali, La Metallurgia Italiana, 1986, 303±310. [3] E. Anelli, Acciai Bifasici ad altissima trafilabilitaÁ. Indagine sull'attuale stato di sviluppo e sulle prospettive di impiego, Ricerca Monoaziendale Nuova Italsider, Rome, 1988. [4] S. Kobayashi, Metal Forming and The Finite Element Method, Oxford University Press, Oxford, 1989. [5] O.C. Zienkiewicz, R.L. Taylor, The Finite Element Method, McGraw-Hill, New York, 1989.

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rolling by a 3D finite element method, ASME J. Eng. Industry 114 (1992) 336±344. [10] A.A. Tseng, A.S. Gunderia, D.S. Ni, Thermal behavior of roll and strip in steel rolling with non-conventional roll design, in: R.K. Shah, H.Md. Roshan, V.M.K. Sastri, K.A. Padmanabhan (Eds.), Thermomechanical Aspects of Manufacturing and Materials Processing, An advanced study institute book, 1992. Hemisphere Publishing Corporation, New York. [11] ANSYS user's manual, revision 5.0, Swanson Analysis System, Inc., 1992.

Thermomechanical simulation of a rolling process with an FEM approach

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