Topic Summary of the Dissertation

 1.        The finite element method finds today more and more applications in the simulation, planning and optimization of metal forming processes. This is due to the high demand on product quality and low production costs in the steel industry, as well as the rapid advances in the performance of hard- and software computers and the simultaneously dramatic fall in prices of the hardware. Few studies however, exist which adequately apply this method to complicated shape rolling due to the numerous difficulties involved. The aim of this work is to investigate, based on experimental results of force and power requirements and the local material flow of the angle steel rolling, the performance capability of FEM and the possibilities of its practical application to metal forming processes, especially in shape rolling processes.

 2.        During shape rolling, there exists a complicated distribution of local forming parameters in the roll gap. This results from the different roll diameters and uneven reduction over the section width, as well as the occurrence of inclination of the rolling path against the roll axis. With an increase in the inclination angle, the portion of indirect rolling force grows larger, and as a result, the spread increases. The distribution of normal and tangential stresses at the interface of the workpiece/roll also gets very intricate during rolling. A lateral flow of material exists from where the reduction is larger towards the zone with relative less reduction.

            The empirical calculation of the rolling force and rolling torque of shape rolling can be done according to the dependence of deforming resistance K_wm/K_fm and lever arm coefficient l on the area coefficient A_d/A_m (ratio of deformation zone area to mean cross section area). The area of deformation zone can be determined using various methods, such as the slab model. In the slab model method, the width of a section is at first separated into a number of portions. For each portion a partial area of deformation zone can be calculated just like in flat rolling. The sum of all partial areas gives the area of deformation. The mean temperature of the workpiece after rolling can be decided according to the heat balance of the workpiece and the roll considering the heat generated by mechanical work during the rolling.

            The objective of the roll pass design is to obtain the product of exact form and dimension with as few passes and little roll wear as possible. The methods in use today for the roll pass design of angle steel are those with straight and bent flanges, butterfly method, as well as a special method for continuous mills. In the special method, a workpiece worked from a flat or square groove is at first rolled with a special upper roll to form the angle peak, then edged in the 2nd or 3rd last pass by rolls with H-V arrangement to control the flange length and the end form of the angle steel. The butterfly method possesses the advantage of having small diameters and to a large extent, direct pressure exerted in the groove, which helps to reduce wear and thus redressing losses. It is therefore fairly up-to-date. It has as a disadvantage however, the big barrel length. The method for continuous mills is especially recommended for its special development. As to the roll pass design for channel there are: classical methods using a practically vertical flange, butterfly method, combination of the classical and butterfly methods and the universal method. Today the universal method receives more and more attention, and the combination of the classical and butterfly method is also often employed. The increasing demands on beams with thin, wide and parallel flanges call specially for universal beam mills in the beam production. Without the universal method, it is impossible to produce beams with parallel flanges. A tendency of beam production, especially for those with wide and thin webs, is the usage of beam blank. Concerning this, the style of rolling today is mainly with less reduction of web than of flange, in order to obtain a homogeneous deformation over the cross section. 

            A series of technological questions with which the development of shape rolling is also concerned can be answered through the study of local forming parameters, especially by means of the FEM simulation. Knowledge of the parameter field in the roll gap is an important fundamental for the modern rolling technology. For an optimal design of rolling pass, an exact knowledge of material flow is necessary. The thermomechanical rolling aims at realization of a homogeneous local material flow, a desired temperature distribution and thus, a homogeneous mechanical property. The use of a near-net-shape cast profile can not only shorten the rolling process, but also attain a homogeneous reduction over the cross section of the workpiece and therefore improve the product quality. In the beam production, the beam blank is especially used. In order to produce as many sorts of beams as possible with a certain cast beam blank, the dimension of the beam blank is often adapted either through upsetting in a vertical edging pass or through tension in a special groove, so that various beam blank formats suitable for different mill stands are obtained. During the continuous rolling of section, the tensile stress along the workpiece influences the cross section form, and the tensile stress ranges which permit the rolling within the admissible tolerance zone of the finished dimensions can be determined.

            Local material flow in the roll groove can be determined experimentally through an evaluation of grid distortion or the deformation of pin samples. With each of these methods, workpieces with grid or pin should be produced before rolling. Another possibility is the simulation through the deformation of plastine samples, each of which is made from elements with different colors adjacent to each other.

 3.        Traditional methods used to calculate metal forming parameters are slab method, visioplastic method, slip method, bound theorem, etc. But such methods are mainly limited to plane or symmetrical deformation. They are therefore not adequate for the solution of the shape rolling problems. In the field of shape rolling, the finite element method is being increasingly applied. Not only the global and local parameters, but the processes of microstructure formation and recrystallisation can also be simulated with FEM. FEM finds application especially in the optimization of the shaping processes and the solution of a wide range of technological problems, such as optimal rolling pass design, realization of homogeneity of local material flow for thermomechanical treatment, reduction of groove wear, avoidance of fissure formation and expansion and enlargement of the rolling extent for the metals with low formability. In the modern rolling technology with cast profile, the reasonable form of the cast profile could be found through a FEM simulation.

            The prerequisites for the FEM simulation are, an experimentally assured theory (material law) that can realistically describe the rolling process, detailed knowledge of material data and boundary condition and the computer hardware and software (FEM program).

            The workpiece can be assumed as rigid/plastic, rigid/viscoplastic, elastical-plastical. One of the most important material data is the flow stress dependent on strain, strain rate and temperature. The dependence of E-modulus, Poisson ratio, specific heat, thermal conductivity, coefficient of thermal expansion, etc., on temperature is also important for the simulation of heat rolling process.

            The boundary conditions in the hot rolling are the heat transfer coefficient for workpiece/roll and workpiece/air and the friction coefficient. In terms of friction, the columbic friction and friction factor model are often employed for hot rolling. The friction between the workpiece and roll depends on the material of the contact pair, the rolling condition, the relative velocity and the state of contact surfaces. The influence of the roll (groove) and workpiece geometry on the friction is represented by the distribution of tangential and vertical stresses, in which the friction can be determined according to m = t_ik/s_Nik. A great number of empirical and half empirical formulae have been assumed to describe the friction behavior between the groove and workpiece in various situations. The friction factor model normally gives the upper boundary of the friction. The heat transfer coefficient between workpiece and roll is normally 4500 to 10000 W/m²k, workpiece and cooling water 1000 to 6000  W/m²k, and workpiece and air 40 - 100  W/m²k.

 4.        Two types of FEM analyses are often carried out to simulate heat rolling processes. One of these is the steady-state, Eulerian model by neglecting end effects and using rigid-plastic or viscoplastic behavior. The other is an incremental, updated Lagrangian model used when end effects are being investigated and/or elastoplastic or elastovisoplastic rheology is assumed. Quite a number of investigations have been carried out with various FEM models in the simulation of the rolling process of slab and in stretching grooves. With respect to FEM investigation of shape rolling, a few results have also been published such as the works relating to the rolling of rail, beam and angle steel. Some researcher did the simulation under the application of existing source programs, others tried to, however, directly write their own programs and then analyze the rolling process with the finished program. In a number of investigations some calculated parameters were compared with those of experiment and the FEM was thus examined and optimized. The comparison of local material flow exists, especially in the relatively simple rolling processes such as rolling in stretching grooves. Since the FEM simulation of shape rolling is normally very time consuming, the analysis of the shape rolling was often done through a combination of FEM with other methods (such as slab method, energy method, etc.), in order to cut down the calculation cost. But sometimes such a combination leads to a little sacrifice of the calculation accuracy.

            Most of the published research works for the simulation of shape rolling are inadequate. Many researchers only managed to solve part of the problems of the simulation, or tried to simulate the process with simplified material data and boundary conditions. The dependence of the material data and boundary conditions on the forming parameters such as temperature, strain, strain rate, were not adequately taken into consideration in most of these investigations. Some parameters such as forward and backward slip were often neglected. Only in very few of the investigations were sufficient optimization of boundary conditions and an extensive assessment of the calculation performed, especially with the rolling tests.

 5.        To determine the local material flow and the force, power and temperature during rolling, extensive rolling tests were carried out with grided samples.

            As a pilot test, flat rolling of the samples with outer or inner grids (i.e., surface or cross-section grids) was done to study the grid behavior during rolling and to obtain the rolling parameters and the material flow. During rolling, a so-called "stecker-rolling" was also done by interrupting the roll movement, so that it became possible to establish the form of the workpieces in the roll gap. The pilot experiment verified that, in order to attain an measurable grid, a rolling sample with outer grid should only be rolled for a maximum of three passes.

            The main test was an angle steel rolling in a special roll pass system developed through a combination of the butterfly method and the design method for continuous mill. A new rolling pass schedule was used in which the original workpiece with a cross-section of 48x48mm was rolled until the final form of angle steel 40x40x9mm was achieved, so that it was possible to mill the grid on all sides of the surfaces and cross-sections of the samples before each pass as well as obtain the measurable deformed grids. Before the main experiment, a pre-experiment was done to produce samples K6, K5, ... K2, which were rolled in the groove sequence K6, K5, ...K2 for one to five passes from the original form. Inner grids were milled with 4x4x0.3mm for the sample K6, 3x3x0.3 mm for the K5, K4, K3 and 2x2x0.3mm for the K2. Each of the samples for the inner grids was at first sawn into two halves, milled with grid and then welded together again. Outer grids were made only for samples with original form and the sample K4, all with 4x4x0.3 mm. Since the surface form before the other passes is very complicated, it was therefore very difficult to mill the outer grids on other samples. The samples with the outer grids were prepared for so-called "stecker-rolling" in each pass, to establish the shape of the deformation zone. During rolling the workpiece was fetched from the furnace at about 950°C and put into the roll gap at 900°C. Every workpiece with inner grid was rolled for one to six passes at the lowest possible speed of 0.15m/s, so that the results could be comparable with those of so-called "stecker-rolling". After the rolling of every pass, the workpiece was cooled in the water bath.

 6.        The rolling force, moment, power and temperature were first evaluated after rolling. In addition to photographs for each useable sample, the evaluation of the deformed grids included chiefly the calculation and graphical presentation of so-called partial upsetting, partial spread and partial elongation over the cross section and surfaces according to the coordinates of the grid points before and after deformation. The partial upsetting, partial spread and partial elongation are of special importance for the roll pass design. The measurement of the coordinates was performed in a modern Universal Precision Measure Center (UPMC). The data processing and diagram figuring were done with a personal computer. The photos of various grids and the diagrams of the partial upsetting, partial spread and partial elongation were used for comparison with FEM simulation.

 7.        The FEM simulation of the processes of rolling, especially angle steel rolling (supported by DFG, the German Research Foundation) was carried out with the FEM code MARC.

            For a simulation of the rolling process with code MARC, an input file with special format should normally be written before hand. The input file consists of the data for mesh generation, material data and boundary conditions, rolling parameters, parameters relating to the output (out-file, post file and restart file), special control parameters for FEM calculation (such as convergence errors for temperature and stress or displacement, time step for each increment), etc.. The input data can be directly fed with a pre- and postprocessor MENTAT or, to reduce the size of the input file and make it easier for reading and further working, through a combined application of the MENTAT and the editor vi. To feed a complicated description of an input data, e.g. a mathematical formula, a user subroutine can be used. In the code MARC, a series of existing user subroutines are available.

            The rolls and workpiece were taken into account separately as rigid and elastic-plastic. At first the cross section of a groove was fed. The groove (more precisely, the surface of the groove) can be obtained through rotation of the cross section, the input of the workpiece followed by the input of the cross section and the expansion of the cross section in the length direction. At the beginning of the rolling, a push movement had to be given against the workpiece nodes towards the roll gap. After the workpiece has contacted with the rolls with about four nodes in the length direction, it should automatically run into the roll gap without further push action. The program then begins to run after connecting the input file (and user subroutines, if they are also used) with the main Program of code MARC.

            Through the isothermal analysis of flat rolling and comparison with the experiment, a three dimensional FEM model of flat rolling was built. Each of the six passes of complicated angle rolling were then simulated. On this basis the simulation of the angle steel rolling processes with coupled thermo-mechanical model, a much more intricate calculation, was carried out.

 8.        The flow stress of the high strength constructional steel used in the investigation was measured at various strains, strain rates and temperatures through a torsion test. A FEM simulation was done to determine the real temperature during the torsion test. With the results of the simulation, a mathematical model was developed to predict the temperature rise resulting from the plastic work. The measured flow stress was then formulated as a state variable dependent on the strain, the strain rate and the temperature taking into consideration the temperature increment due to plastic work on the torsion samples. The temperature dependence of E-modulus of the steel was measured between 20°C and 1000°C. Other material data, such as specific heat, thermal conductivity, coefficient of thermal expansion and mass density were selected according to literature and fed into the program in terms of temperature dependence.

            In addition to the friction, the heat transfer in the interfaces workpiece/roll and workpiece/air were specially examined as boundary conditions. The friction behavior was studied with respect to the surface temperature and flow stress of the workpiece, the distribution of vertical and tangential stresses, as well as, the surface state, the geometry and the relative speed of the workpiece and the roll. The friction coefficient of each pass was then also optimized through variation together with comparison with the rolling experiment. As a result of the studies, the friction coefficients for all the passes was chosen as m = 0.35 - 0.6 corresponding with the columbic friction law, and m = 0.30 - 0.55 where the friction factor model was employed. The real friction behavior could be assessed between the two friction laws. The heat transfer coefficient in the interface workpiece/air was determined both through calculations according to the heat flux of convection and radiation and the cooling test with samples K4, K2 and K1. The heat transfer coefficient for the interface workpiece/roll was firstly determined through special rolling tests, then optimized through the comparison of the simulated and measured surface temperatures of the workpiece at the exit.

 9.        To increase the calculation accuracy, various measures were taken. In addition to the most exactly possible input of the material data and boundary conditions, the outer contours of the workpieces and the grooves and all the rolling parameters were fed at a high accuracy corresponding to the rolling test. A correction of the groove form according to the rolled workpiece was carried out to remove the error caused by groove wear (» 1.0 mm). Before the rolling simulation of each pass, an FEM analysis of the cooling process was done to determine the temperature profile of the cross-section of the workpiece. Here, each of the samples was cooled down in the air from an original temperature of 950°C to a surface temperature of the sample at 900°C. These initial temperature distributions were then fed into the input files.

            For an economical simulation with FEM, mesh techniques for a complete and the most time economical calculations were specially studied, in addition to the investigation of the relationship between the mesh of the workpiece and the roll, as well as the control parameters (convergence errors, time steps etc.). The relative large grid was used for the calculation. The convergence errors of stress and temperature were set as 15% and 10°C, respectively. Due to large efforts, the calculation of a rolling pass took only 15 - 20 hours in the workstation HP APOLLO 720, although half of the geometry had to be computed (The isothermal calculation of a rolling pass needed about 10 hours).

 10.       The directly calculated parameters are rolling force, rolling torque, elastic and plastic strain, elastic and plastic strain rate, plastic work, shape of deformation zone, stress and temperature distribution, as well as the workpiece geometry during and after rolling. The also calculated partial upsetting, partial spread and partial elongation were fed as input into the MARC main program through a user subroutine. A graphical presentation and evaluation of the analyzed parameter fields followed with the pre- and postprocessor MENTAT. The calculated rolling force and rolling torques could be read directly in the out-file. The shape of the deformation zone could be determined either in agreement with the tendency of the strain rate in the direction of reduction or according to the working zone of the external force upon the workpiece nodes. It is also possible to obtain the deformation of the front and rear ends as well as the residual stress distributions of the rolled product.

 11.       After studies of the relative movement of the workpiece and rolls, a special upsetting model for shape rolling simulation was developed through a functional combination of the slab method and the FEM. The workpiece was represented as a three dimensional slab element pair. Each slab element was then divided into a number of FEM elements (36 elements for the first pass of the angle steel rolling, for example). The upper or lower roll was defined through input of the groove cross section and their rotation expansion around the corresponding roll axes for 3 - 5° (relating to the length of the slab elements). The slab method is for the determination of the elongation of the workpiece and the relative velocities between workpiece and rolls in rolling direction, while the FEM is for the calculation of other parameters. The distribution of the elongation along the arc of contact has an important influence on the accuracy of the simulation. The backward and forward slip don't play a great role. With this model, all parameters over and above the cross section, such as strain, stress, temperature and strain rate, as well as spreading and reduction can be calculated. The parameter distribution over the length of the deformation zone can also be obtained in accordance with the parameter course over the upsetting time. With this simplified model, the simulation of an angle rolling pass was performed in only 5 to 10 minutes with sufficient accuracy.

 12.       To examine the validity and accuracy of the FEM simulation and also to optimize the FEM model, various paramaters or parameter fields of the simulation were compared with corresponding results of the rolling tests.

            The comparison of the outer contours for each pass shows an excellent agreement. Except in the formation of angle steel peak in the third and finishing passes, where the errors of workpiece heights go up to 8%, the calculated height, width, and flange thickness of rolling stock etc. normally leave an error below 3%, even < 1%.

            A comparison of grid distortion on the surface und cross-section of the workpiece also shows a good correlation. But such a comparison has a prerequisite - the grid before a rolling pass must be comparable. So for most of the passes, the examination of the calculated local material flow was done mainly through a comparison of partial upsetting, partial spread and partial elongation of FEM analysis with those of rolling tests. By reading the parameter fields of the analysis and the rolling tests over a cross section or another face, the validity of the simulation could be directly checked at a qualitative level. A further quantitative comparison of these followed, especially for the pass by pass, through an evaluation and graphical presentation of the relative change in stock height and width over the stock thickness and flange length. The comparison achieved a good correlation. The possible reasons for an error can mainly be traced back to the rolling tests. During the tests, the rolling condition (roll gap, rolling temperature etc.) for the rolling by workpiece production and the main rolling experiment could not be made exactly the same, though a great effort at adaptation was made. The welding together of the two halves of the workpiece with the inner grids could be done only around the edge of the cross-section, so that during rolling, there actually exists a relatively "free" elongation on the cross-section.

            The relative change in height and width over the flange length showed a descent for the 1st and 2nd passes and an upward course for the other passes except the upsetting pass (4th pass). This happened because of the distribution of reduction. In the first and second passes, a relatively less reduction exists in the middle of the samples. On the other hand, the reduction in the middle of samples in the 3rd, 5th and 6th pass is larger than in other parts of the stock side.

            The rolling forces and rolling torques of FEM-analyses were also compared with those of rolling tests. A difference of about -10% gave a relative good agreement for most of the passes. Relatively large errors of  about -20% in the finishing pass were mainly due to inaccuracy in the temperature at entry. Before the rolling process, the legs cooled down very quickly in the air since they are very thin. On  the other hand, because of the large lengths (» 800 mm) of the workpieces before the last pass and a low rolling speed, the back part of the workpiece actually went into the roll gap with a quite low temperature. But for the FEM-simulation, the lengths of the workpieces used were about 100 mm. Therefore, the mean temperature of the simulation was in reality higher than that of the test.

 13.       The transferability of the finished FEM model of angle steel rolling was also investigated. This was carried out on the basis of the fact that all the rolling processes are to some extent similar to each other and that the only difference is in the cross sections of the workpieces and grooves (rolls). Through the new input of the cross section of the workpiece and grooves (rolls), together with an adaptation of the FEM control parameters, the rolling process of other flat or long products could also be simulated.

            With respect to this, the H-beam rolling with cast profile was at first analyzed, taking the example of rolling of IPE140. The pass sequence consists of three universal and two edging passes in a continuous mill. The rolling processes of three universal passes in a continuous mill were simulated. Rolling parameters for the simulation corresponds to those of a practical rolling test: initial temperature with 1000 - 1015°C, and rolling speed with 4 - 6 m/s. Usually the vertical rolls are not driven in a universal rolling mill, so during the simulation, the angular velocity of the vertical rolls was so optimized that their drivetorque was against zero. This was done by using code MARC through variation together with the employment of the restart file. The FEM models were improved through a comparison with actual rolling. The calculation time for each pass lasted about 10 hours in the same workstation.

            A further example is the study of the cast-rolling process of a thin slab with a liquid core. At first, the cooling process of the cross-section from liquid state of 1394°C to a state with a thickness of solid shell of 10mm, 15mm, 20mm, 25mm, respectively, was studied by means of two-dimensional finite element simulation in which the liquid-solid interface was determined. Then the three-dimensional simulation of workpieces with corresponding thicknesses of the solid shell was carried out, in which the workpieces were simplified to hollow bodies. Meshing of the cross section with each wall thickness was studied. A comparison of the analytical results with those of experiments described in the literature followed for each calculation.

 14.       Several research works of special industrial interest could be carried out with the continued study of the above investigations. An examination of the structure formation combining the above simulation with microstructure models is very important for the industrial application of the thermo-mechanical treatment of the shape rolling. The simulation study of the whole cast-rolling process of thin wide slab (e.g. width up to 1800 mm) with a liquid core, considering the real cooling process, could be very helpful in the solution of various practical problems and the further development of the technology. In shape rolling with a near net shape cast profile, a reasonable form of the cast profile could be determined with the help of the FEM simulation, so as to obtain a possibly good homogeneous deformation and microstructure and the most possible minimal resistance of material flow. It is economical to produce various sections (angle steel, Channel, etc.) taking the cast-rolled thin slab as raw material since less passes than those necessary for the traditional shape rolling are needed. It is possible and very practical to process an executable program package for the automatic simulation of various rolling processes with an input of merely the boundary conditions and the data of materials, rolls (grooves) and rolling processes. In addition, it is of special importance to carry out a further investigation of the simplified model through a combination of the slab method and FEM and the study of the distribution of the elongation over the length of deformation zone for various rolling processes.

(This summary was written in the Spring 1995, as I was still in Germany)