Keywords

1 Introduction

Virtual manufacturing is playing an increasingly important role in optimising manufacturing processes, product quality, and product properties. Flat rolled aluminium products undergo several production steps before reaching their final state, which includes hot rolling, coiling, cold rolling, and heat treatment processes. The development of numerical rolling models for aluminium was subject to numerous studies in the past. Olaogun et al. [1] presented a two-dimensional cold rolling model for AA8015 focussing on the heat transfer in the roll-sheet interface. Bátorfia et al. [2] compared different two-dimensional numerical approaches for identifying coefficients of friction in cold rolling of AA1050. Since two-dimensional models are not applicable in their studies, Gosh et al. [3] and Jiang et al. [4] used three-dimensional models to study edge cracking and strip flatness. However, in those studies, only single pass simulations were conducted. Simon and Falkinger [5] presented a simulation scheme for multi-pass hot rolling of aluminium including a microstructural material model and a comparison of two-dimensional and three-dimensional approaches. A similar approach was presented by Nemetz et al. [6] for multi-pass heavy plate rolling of steel. While single pass and multi-pass models are present in the literature, studies usually focus on either hot rolling or cold rolling. However, a reliable digital representation of the whole production process of flat rolled aluminium products requires modelling of the entire process chain and appropriate interfaces between different manufacturing processes. As the length of the virtual process chain representation increases, tailor-made simplifications become indispensable to answer specific questions within an acceptable effort.

The current work is based on complex numerical three-dimensional models for hot rolling of aluminium alloys that were established in the past [5]. They include elastic deformations of the work rolls and their thermal crowns, backup rolls, and stand deformations. In addition to roll separating forces, temperature, and microstructure evolution, these models provide reliable predictions of the strip profile and edge deformation. Due to the high level of detail, the computational costs of such models are immense, which makes them impractical for advanced numerical studies or multi-pass simulations in an industrial environment. Extending these models to include successive cold rolling passes increases the computational complexity even more. Therefore, the present work establishes a reduced thermo-mechanical cold rolling model using the commercial finite element code LS-DYNA to predict roll separating forces, temperatures, and microstructure evolution. A sensor-based control is introduced to manage strip tensions when the material is coiled. Once verified, this modelling approach is used to simulate a complete industrial pass schedule for an AA6016 aluminium alloy, including hot and cold rolling passes.

2 Modelling Approach

2.1 Modelling of the Cold Rolling Process

As a first step in extending the process chain to cold rolling, models predicting the roll separating forces and microstructural behaviour are beneficial. Using the finite element method, this can be done via two-dimensional models with plane strain conditions [1, 2] or three-dimensional models [3, 4] as shown in the introduction. However, for predicting the specific roll separating forces and the microstructural evolution, the actual width of the strip does not have to be considered. This allows for a reduction of the width dimension as the mentioned variables can be assumed as independent of the width and can therefore be evaluated at the half-width position of the strip. The commercial finite element code LS-DYNA used in this study utilises different solver versions (Shared Memory Parallel and Massively Parallel Processing) for two-dimensional elements and three-dimensional elements. Additionally, different keywords have to be used, e.g., for contact formulations, which can lead to inconsistent results [7]. Therefore, two-dimensional plane strain elements are not suitable for this current research.

Alternatively, a model containing three-dimensional solid elements is introduced in this work. However, it consists of only one row of elements in the width direction. The imposed boundary conditions limit the degrees of freedom of the nodes to translations only in length and thickness direction. Movements in width direction (z-direction, see Fig. 1) are suppressed, leading to plane strain conditions of the elements. The basic setup and boundary conditions are shown in Fig. 1. Both work rolls are modelled, and the symmetry is deliberately not used, since rolling conditions may occur where the strip centre in thickness direction is not horizontally aligned with the roll gap pass line.

Fig. 1
figure 1

Sketch of the basic simulation setup including the boundary conditions (left) and meshing for the reduced model (right): heat transfer coefficients hstrip and hroll, translational strip velocity vstrip, rotational roll velocities ωroll and strip tensions S0 and S1

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To reach a steady state within a short period of computation time, the virtual length of the strip was set to 300 mm. Twelve elements are used in thickness direction with a fixed aspect ratio of 0.5 in length direction (see Fig. 1), to avoid large element distortions during the rolling process. The total number of elements therefore depends on the actual strip thickness and increases with thinner strips. The roll elements are larger in width than the strip elements to avoid line-on-line contact, which may lead to numerical problems.

The work rolls are treated as rigid bodies, however, heat transfer to the strip and the environment is taken into account. The elasto-plastic behaviour of the strip is implemented by a user defined material model according to a modified dislocation density based flow stress model of Kocks-Mecking type [8]. Tensile tests at usual cold rolling temperatures up to 150 °C were performed at the Fraunhofer Institute for Mechanics of Materials (IWM) and used to calibrate the model [9]. The extrapolated flow curves for 100 °C and different strain rates are shown in Fig. 2. Measured temperature dependencies of the Young’s modulus, specific heat capacity, and thermal conductivity according to Table 1 are implemented, since considerable temperature rises may occur during the deformation process. The thermal expansion coefficient and density are set to 2.31·10–5 1/K and 2700 kg/m3, respectively.

Fig. 2
figure 2

Modelled flow stress curves for AA6016 at 100 °C for different strain rates [9]

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Table 1 Thermophysical and mechanical properties used for the rolling simulation
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A surface-to-surface contact with heat transfer between the aluminium strip and the rolls is used, including a Coulomb friction model with a constant coefficient of friction. Based on measured roll separating forces, suitable coefficients of friction between 0.07 and 0.12 are used coinciding with earlier experimental research [10].

In addition to the restrictive boundary conditions, a prescribed motion is imposed on the strip, which depends on strip thickness, reduction, rolling speed, coefficient of friction, and roll diameter. A constant rolling speed is applied on both rolls. Convective heat transfer to the environment is implemented at the top and bottom surface of the strip and at the surfaces of the work rolls according to Fig. 1.

There are further details to consider when running multi-pass simulations. The scheme for transferring the calculated microstructure and deformation history from one pass to the next is adapted from previous work. Python routines are used to automatically setup a finite element model for each pass. After the first pass has finished, a new, undeformed mesh with reduced strip height and fixed initial strip length is generated. The element strains, stresses, user defined history variables, and node temperatures along the strip thickness are then extracted from the centre in rolling direction of the rolled strip and mapped onto the undeformed mesh. This procedure takes place after each rolling pass in order to avoid large element distortions and unnecessary elongation of the strip as the height decreases along the pass schedule. Both contribute to reduced computational costs. To compensate for the reduced virtual strip length in the model, a thermal step is added after each rolling pass. In this step, the rolled strip is cooled by convection considering the real process time of the full-length strip as well as the interpass time in reversing rolling mills. After the multi-pass simulation has finished, further Python routines are used for automatic result extraction and evaluation, minimising postprocessing effort and reducing error-proneness. More detailed information about the Python routines and the simulation schemes is available in [5].

All subsequent rolling simulations are performed using an explicit thermo-mechanical solver, since the process durations and therefore the required timesteps are short. The cooling of the strip in the thermal steps of the multi-pass simulation takes several seconds, wherefore an implicit solver is used to reduce calculation costs.

2.2 Modelling of Sensor-Based Strip Tensions

Strip tensions play an important role in cold rolling as they increase the formability of the rolled material and reduce rolling forces [11]. They are induced by varying strip speeds before and after the roll gap. As the coiling process itself is not considered in this model, the strip tensions are implemented by directly applying a tension load on the cross sections of both strip ends. However, applying the tensions at the start of the simulation, with no contact between the strip and the roll yet, can lead to several problems in the simulation, like deviations in initial strip speed or prevented entry into the rolling gap. Time-dependent activation of strip tension is possible but inconvenient as the starting time must be calculated manually for each rolling configuration and simulation setup. Hence, an event-based control of the strip tensions is introduced in this work by using the *SENSOR keywords implemented in LS-DYNA. Generally, sensors allow for activation and deactivation of boundary conditions and other model elements using element- or node-based criteria [12].

Two sensors are implemented measuring the distance in x-direction between the centre of the roll and the strip’s front and rear end, respectively. Once the sensors exceed a prescribed value dcrit (see Fig. 3), the load boundary conditions switch their status, resulting in a build-up or release of the strip tensions as shown in Fig. 3. Accordingly, the rolling process can be divided into three phases:

Fig. 3
figure 3

Implementation of position-based sensors for applying back tension S0 and front tension S1

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  1. 1.

    Entry into the roll gap and rolling of the front strip without external strip tensions

  2. 2.

    Activation of first sensor switch and both strip back and front tensions, leading to a steady state rolling of the filet

  3. 3.

    Activation of second sensor switch and deactivation of the back tension, whereas front tension remains for rolling of the strip end.

This implementation approach minimises manual effort as the input is independent of the actual rolling parameters and the definition of force-loaded reference points [13] is not required. It is therefore particularly suitable for multi-pass simulations and extended numerical studies.

2.3 Three-Dimensional Cold Rolling Model Considering Strip Width

To verify if the reduced modelling approach is valid for prediction of roll separating forces, temperatures, and material microstructure, as well as to quantify the reduction in computational cost, a three-dimensional model considering the half-width of the strip is used as shown in Fig. 4. Only the upper work roll and the half-thickness of the strip are modelled resulting in a quarter model to keep the computation time reasonable. The identical physical boundary and contact conditions are used for both models. However, translational motion in width direction is only restricted in the mid of the strip where the x–y symmetry plane is located. Additionally, convective heat transfer is applied also at the lateral surface of the strip.

Fig. 4
figure 4

Three-dimensional quarter model for the cold rolling process with x–y and x–z symmetry planes

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3 Model Verification and Validation

In the context of model development, model verification relates to reviews that are restricted to models exclusively, without using measurements from the model’s original. Model verification thus includes checking the plausibility of its own results, i.e., comparing the results of the model against other results of the same model (model-internal verification), as well as checking the plausibility of the model’s results against those of other models (model-external verification). In contrast, model validation relates to reviews that comprise the comparison of model results against measurements from suitable originals of the model. Model validation thus includes checking the validity of model results against measured data from some of the model’s originals [14].

The first step is to verify the features of the presented model, namely the sensor-based tension control and the width reduction. The modelling approach is then used to simulate a complete industrial pass schedule for an AA6016 aluminium alloy including hot rolling and cold rolling.

3.1 Verification of Sensor-Based Strip Tensions

For the verification of the sensor-based strip tension control and its impact on the roll separating force, a single cold rolling pass with typical rolling parameters for an AA6016 aluminium alloy is used. The back and front tensions are set to 15 MPa and 25 MPa, respectively.

As the external strip tensions are not applied throughout the rolling process, their influence is visible on the transient roll separating force, as shown in Fig. 5. With no external loads applied, the force reaches its maximum at the start of the rolling process. After the first sensor reaches its critical value and both front and back tensions are applied, the roll separating force drops by 16% and the steady state begins. At the end of the process, the second sensor switch is activated, and the force increases again, but to a lower level than at the beginning, as the front tension remains.

Fig. 5
figure 5

Normalised roll separating force during a cold rolling pass with sensor-based strip tension control

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The averaged stress in rolling direction over the cross section at the centre of the strip is shown in Fig. 6. The stress equals the applied back tension when the evaluated element row has not reached the roll gap yet. Due to the deformation in the roll gap, a compressive stress peak occurs for a limited time until the longitudinal stress reaches the applied front tension after the roll gap. These results verify the sensor-based strip tension approach.

Fig. 6
figure 6

Averaged longitudinal stress at the centre of the strip during a cold rolling pass with sensor-based strip tension control

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3.2 Comparison of Reduced and Quarter Cold Rolling Model

To verify the effectiveness of the reduced modelling approach in predicting roll separating forces, temperatures, and microstructural behaviour, and to quantify the computational savings, three single pass simulations were performed using both the reduced model as well as the quarter model introduced in Sect. 2.3. The chosen rolling parameters are based on the actual industrial production. The initial strip thickness, starts at around 10 mm, and the pass reduction decreases from pass to pass, while the rolling speeds increase. This allows for model verification for different sets of rolling parameters used in industrial productions. Strip tensions are kept nearly constant for each pass. Identical numerical parameters and hardware settings were used to ensure comparability. Normalised roll separating forces, number of nodes, and resulting calculation times are shown in Fig. 7. The roll forces were normalised by the resulting roll separating force of the quarter model in the first pass. With a maximum deviation of 0.5% in the third pass, the roll separating forces can be considered identical in both models. As the initial strip thickness reduces from pass one to three, the total number of nodes increases drastically. The computation times of the three-dimensional quarter model exceed those of the reduced model by a factor of 8 to 10.

Fig. 7
figure 7

Comparison of normalised roll separating forces, number of nodes, and computation time for three single pass simulations using the reduced and the three-dimensional quarter model

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To evaluate the comparability of the material behaviour, Fig. 8 shows the transient temperature, strain, and dislocation density of an element at the surface and an element at the x–z symmetry plane for one exemplary pass. The temperature curves were normalised by the maximum occurring value at the surface of the quarter model. The evaluated elements are located at the centre of the strip in x-direction and lay in the x–y symmetry plane for the quarter model (see Fig. 4). All three material properties match exactly between both modelling approaches. In summary, the reduced model is suitable for the evaluation of width-independent variables, leading to the same kind and quality of information at a fraction of the computational costs of the quarter model.

Fig. 8
figure 8

Transient normalised temperature, strain, and dislocation density of surface and half-thickness elements for the reduced model and the three-dimensional quarter model

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3.3 Comparison with Industrial Pass Schedules

The reduced model is used to simulate an industrial pass schedule for an AA6016 aluminium alloy. The 600 mm thick cast slab is reduced to a 1 mm thin strip by 21 hot rolling passes followed by three cold rolling passes, which start at a strip thickness of around 10 mm. Strip tensions are applied during the last five passes in which the material is coiled. The coiling process itself is not modelled in this work. For the cold rolling passes, the actual initial strip temperatures for each pass vary due to different downtimes between successive passes and are implemented based on measurements. Between the first and second cold rolling pass the coiled strip is annealed to reduce the work hardening and to increase formability. As the heat treatment is not modelled, the material history is not transferred between these passes. Instead, a new mesh with initial material model variables for cold rolling is used as a simplification. For the contact zone, a speed-dependent friction coefficient in the range between 0.4 and 0.075 is used. For hot rolling, roll separating forces are almost independent on the coefficient of friction. Therefore, a nearly constant coefficient of friction is chosen there. Its value is determined from the bite condition, i.e., from the requirement that the entry of the strip into the roll gap is guaranteed for every set of rolling parameters. For cold rolling, coefficients of friction based on rolling speed are provided by the Level 2 system of the cold rolling mill and fitted under the constraint, that the coefficient of friction is strictly inversely proportional to the rolling speed. The hot rolling simulation includes the same thermal and mechanical boundary conditions as the cold rolling model described in Fig. 1. More information about the hot rolling model can be found in earlier research in [5].

Figure 9 shows the roll separating forces along the entire process chain, normalised by the measured value of the first pass. Thus, the information about the development of the roll separating force along the pass schedule is given. The exit temperatures of the strip after leaving the roll gap are illustrated in Fig. 10. The temperature measurements should be taken with caution as different measurement methods are used for the plate and coil passes and experience has shown that measurement errors may occur. As measured values are not available for each pass, the calculated exit temperatures from the Level 2 system, which is used to design the rolling setup in an industrial environment, are added. Therefore, the Level 2 temperature of the first pass is used to normalise the temperature values in Fig. 10.

Fig. 9
figure 9

Normalised roll separating forces along the entire process chain of hot and cold rolling for an AA6016 aluminium alloy strip

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Fig. 10
figure 10

Normalised strip exit temperatures along the entire process chain of hot and cold rolling for an AA6016 aluminium alloy strip

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With an average error below 5% in predicting the roll separating forces, the model agrees well with measured data from the rolling mills. Comparing the strip exit temperatures with the Level 2 system data, results in an average error of less than 5% as well. The greatest deviations in prediction of the force occur at the 20th pass and the 24th pass.

At the 20th pass, the roll separating force is underestimated by almost 12%. By evaluating the strip exit temperature in Fig. 10, this can be correlated with an overestimation of the material temperature. As the yield stress decreases at higher temperatures, this results in lower roll separating forces. Inaccurate variables in the thermal contact definitions due to increased coolant supply may be responsible for the increased temperature deviations during both hot rolling coil passes. The largest error of 13% occurs in the last cold rolling pass. Two reasons for this deviation seem conceivable: First, the coefficient of friction is held constant for the last two passes, since the same rolling speed is used. As more complex friction models are not yet implemented, this assumption may be inadmissible. Additionally, the maximum strain rate during the last cold rolling pass is significantly higher compared to previous passes, exceeding the maximum strain rate used for the calibration of the material model significantly. This may also contribute to the inaccuracy of the roll separating force’s prediction in this pass.

4 Conclusion and Outlook

The present work introduces a reduced three-dimensional model for rolling of aluminium alloys. Its aim is to reduce the computational effort with increasing length of the virtual process chain, by introducing tailor-made simplifications. An approach for sensor-based strip tensions for coiled material stocks is implemented and verified.

The results confirm that the reduced model concept is suitable for predicting width-independent variables like specific roll separating forces, temperatures, and the microstructural behaviour of the strip with acceptable accuracy compared to a conventional three-dimensional quarter model. Computational costs are reduced drastically compared to the quarter model considering the strip`s half-width. Using the established model to simulate an entire industrial pass schedule of hot and cold rolling operations shows good agreement with measured roll separating forces.

Future work will focus on implementing more complex friction models based on rolling parameters. Additionally, the process chain will be extended by modelling the coiling and decoiling processes as well as heat treatments during or after rolling processes.