Proc Deform UGM KainzA 18052004

Selected Rolling and Smart Material Simulations by Utilizing the Finite Element Package Deform

Dr. Alexander Kainz, Univ.-Prof. Dr. Klaus Zeman

[email protected], [email protected]

Institute for Computer-Aided Methods in Mechanical Engineering

Johannes Kepler University of Linz, Altenbergerstr. 69, A-4040 Linz, Austria

Keywords: Hot Rolling Simulations, Thermo-mechanical coupling, Rigid-viscoplastic and elasto-viscoplastic constitutive laws, Magneto-rheological fluids, Squeeze mode, Parameter identification.

Abstract

By utilizing the commercial finite element package Deform 3D [1], thermo-mechanically coupled massive forming processes can be simulated very efficiently. Within the frame of this simulation study, Deform 3D was applied to several highly praxis-relevant rolling scenarios, including the hot rolling of thick slabs, strips and plates, as well as combined roughing and edging passes to estimate the edging efficiency. Therefore, special emphasis was put on the detailed investigation of 3D-effects, such as lateral flow and strip spread, dog-bone formation (bulging) and back-spread of the dog-bones [3]. Of considerable interest are thermo-mechanically coupled hot rolling simulations with several successive roughing or finishing mill passes. For the two borderline cases of total strain accumulation and total annealing (i.e. full recrystallisation) between two subsequent passes, the deformation behaviour of a metal block undergoing several rolling passes, e.g. seven roughing or nine plate mill passes incl. two 90-degree rotations, was analyzed in detail. Beside the determination of the deformed geometry, the temperature and the effective (logarithmic) strain distribution after each pass were of particular interest, as well as the time dependence of these quantities at selected observer points inside the rolling stock. For plate mill simulations, significant front end bending effects could be observed at the final passes, which can be attributed to a vertical symmetry breaking of either the geometry or the temperature distribution of the ingot. Beside conventional metal forming simulations, novel smart materials like magneto-rheological (MR-) fluids can be treated to some extent by employing the concepts of massive forming processes. By employing Deform 2D in axially-symmetric geometry the massive forming behaviour of magneto-rheological fluids was simulated [6], which enables a deeper insight into the so far poorly understood non-linear squeeze-mode behaviour of MR-fluids. The experimental data obtained by a test rig [6, 7] serve as input for the model calibration and adaptation of rheological and tribological constants.

1. Introduction and Survey

Especially in hot rolling of steel, a precise determination of the occurring strip spread and lateral flow can be considered as key point to produce top quality strip that satisfies the most demanding tolerances. Although the finite element [9] modelling of steady state and transient rigid-viscoplastic as well as elasto-viscoplastic massive forming processes [2, 3] is well established nowadays, the ever increasing demands from customers concerning product quality are the reason for the importance to attain a better understanding of the underlying process details. This is accomplished by employing highly sophisticated and efficient numerical simulation procedures [4, 5]. In the simulation of hot rolling processes, large material deformations and contact phenomena play an essential role. Updated Lagrangian formulations are well suited for problems regarding path dependent material properties and are highly efficient for the determination of the shape of free surfaces and for the simulation of transient aspects, such as the determination of the shape of crop ends in plate rolling. In the present study the commercial finite element package Deform [1] was applied to several highly praxis-relevant thermo-mechanically coupled hot and plate rolling scenarios, which will be outlined in section 2. Special emphasis was put on a detailed investigation of 3D-effects, such as lateral flow and strip spread for flat hot passes, the dog-bone formation (bulging effects) for edging passes and the successive back-spread of the dog-bones for the determination of the edging efficiency at coupled roughing and edging passes. For the quantitative investigation of such 3D effects, finite element methods are essential, as reliable analytical or semi-analytical concepts are scarcely available. For online setup calculations for computer control of operating plants, however, the application of empirical formulae to obtain spread predictions for specified ranges of the relevant influence parameters is widespread. One of the most sophisticated empirical sets of formulae for the prediction of the lateral spread due to the thickness reduction (“natural spread”), and also due to bulging (“dog-bone shape”), was proposed by Shibahara (cf. e.g. [3]). As such parameterized formulae are only reliable for tight ranges of the influence parameters (slab entry thickness and slab aspect ratio, draft, work roll radius, material and frictional parameters…), more sophisticated concepts such as upper bound and FE-methods [1, 9] have to be utilized especially for detailed offline calculations. Further simulation studies in this report comprise coupled block and plate rolling simulations with several consecutive roughing or finishing mill passes, for which the two borderline cases of total strain accumulation and total recrystallisation between two subsequent passes were treated and analysed in detail. In section 3 the phenomenological behaviour of smart materials like magneto-rheological fluids [6 - 8] was investigated numerically by utilizing Deform 2D in axially-symmetrical geometry [6]. It was quite surprising at first glance that the highly non-linear rheological behaviour of such materials can be treated to some extent by utilizing concepts, which were originally developed for massive metal forming processes. It is obvious that more detailed effects, such as the formation and disruption of iron particle chains along the magnetic field lines, cannot be described in full detail within the frame of phenomenological rigid-visco-plastic or elasto-viscoplastic material laws.

2. Hot Rolling Simulations by utilizing Deform 3D

One of the most important objectives of 3D-hot rolling simulations is the accurate and reliable prediction of the lateral flow and of strip spread, which also enables the determination of relative strip crown changes. The regression of such profile transfer functions, representing the influence of the initial strip crown on the final strip crown, is of particular importance for the process control. It is well known that the shares of the longitudinal and transverse flow depend significantly on the strip exit thickness per pass. For thick strips and slabs, the transverse flow strongly dominates, whereas for thickness values below about 6 mm it can almost be neglected except for the direct neighbourhood of the strip edges. Within a transient region between 6 and 12 mm strip thickness, both the longitudinal and transverse material flow are of comparable order of magnitude.

Fig. 2.1: Combined edging and flat rolling pass: Natural spread and dog-bone formation

Fig. 2.2: Typical Deform-3D simulation model for flat hot passes: rigid work roll, slab, pusher

The final width of the finished hot rolled strip is strongly influenced by the slab aspect ratio (i.e. thickness over width), and by the inhomogeneous plastic deformation during the rolling process. The most common technique to achieve the target strip width in hot rolling is edging, very often with grooved vertical edging rolls, as depicted in Fig. 2.1. During the edging process, a pronounced inhomogeneous plastic “dog-bone” formation at the edges of the slab occurs. Both an increase of the width reduction and a decrease of the initial slab aspect ratio thickness to width 0 0h w lead to an increase of the relative dog bone height. Edging in roughing mills is followed by horizontal rolling. Both the back spread of the dog-bones and the additional “natural spread” due to a thickness reduction are responsible for the final formation of the ratio width over length in the subsequent horizontal passes of the finishing mill. It is obvious that the natural spread (w.r.t. width) increases with a larger reduction during the horizontal pass and with a decreasing ratio of initial width to initial height. To evaluate the efficiency and reliability of Deform-3D [1], and to compare these results with those obtained within the frame of a previous research project [3], the following flat roughing and edging pass simulations were performed. A flat strip of initial width of 800 mm and entry thickness of 20 mm was reduced to a target exit thickness of 12 mm by a rigid work roll with a radius of 350 mm and prescribed angular velocity of 5.7 rad/s. To obtain well pronounced steady state conditions, at least three or four contact lengths have to be simulated, leading to a minimum simulation length of about 250 mm. Due to vertical (i.e. left-right) and horizontal

(i.e. top-bottom) mid-plane symmetry properties, only a quarter of the strip or slab has to be modelled. A typical rolling scenario is depicted in Fig. 2.2, where an additional rigid body serves as a pusher until the bite condition is satisfied and the rolling stock is pulled through the roll gap by the frictional force resulting from the driven work roll with prescribed angular velocity. A rigid-viscoplastic constitutive law including work hardening and rate dependence was assumed. Concerning the tribological properties the Coulomb friction law with a friction coefficient of 0.3µ = was considered. To achieve reliable results in all spatial directions a tetrahedron mesh with 123 000 elements and 32 000 nodes turned out to be essential. This led to a stable time increment of about 0.5-1 ms. and a process simulation time of about 11 hours on a Pentium IV PC with 2.5 GHz CPU. Therefore, when one is primarily interested in the steady state behaviour of such rolling processes, e.g. for systematic parameter studies to obtain regression coefficients for profile transfer functions, there is still a great demand for purely Eulerian or ALE- based finite element packages (cf. e.g. [3]) to simulate such problems with significantly increased efficiency. It should be noted that by using such concepts a reduction of the process calculation time by a factor of 100-1000 seems to be highly realistic.

Fig. 2.3: Mises stress distribution inside the roll-gap near the strip edge

Fig. 2.4: Lateral velocity field inside the roll-gap (local strip necking effect on roll-gap entry side)

Of particular interest are the stationary field distributions. In Fig. 2.3 the Mises-stress distribution is shown, which tends to zero very fast outside the roll-gap. The lateral velocity field (i.e. in width direction) near the strip edge is depicted in Fig. 2.4. Although strip rolling is primarily a “forming under pressure” process technology, however locally, due to the prevailing tensile stresses on the roll gap entry side, well pronounced strip necking effects can be observed there. The accumulated lateral displacement field downstream outside the roll gap, however, does not exhibit any necking effects, as expected. For the simulation of the edging pass under consideration, a rigid cylindrical work roll without grooves was assumed for simplicity. A slab of 200 mm thickness and 1200 mm initial width was reduced in one single edging pass to a final target width of 1100 mm by a rigid (vertical) edging roll with a radius of 550 mm. Compared to the flat pass described above, the tetrahedron mesh could be chosen much coarser (about 68 000 elements and 16 000 nodes) on account of the more advantageous aspect ratio (width over thickness) of the rolling stock. Therefore, the calculation time took merely about 1.8 CPU-hours, again on a Pentium IV 2.5 GHz CPU. The well pronounced dog-bone formation is depicted in Fig. 2.5, where the vertical bulging effect of about 23 mm corresponds satisfactorily with the expectations from

empirical formulae. The corresponding Mises-stress distribution, as represented in Fig. 2.6, confirms the empirical observation that the plastic forming zone is restricted primarily to the edge zones of the slab.

Fig. 2.5: Vertical displacement field with well-pronounced dog-bone formation (bulging)

Fig. 2.6: The corresponding Mises-stress distribution near the (vertical) contact surface

Beside the treatment of flat roughing and edging passes, as described above, the thermo-mechanically coupled massive forming behaviour of a metal cube with an initial edge length of 200 mm and with a homogenous temperature distribution of 1200°Centigrade (immediately after leaving the reheating furnace) was investigated. The metal block undergoes seven consecutive reversing passes at a rougher and is reduced to a target final thickness of 34 mm according to an underlying pass schedule. Again, due to vertical and horizontal symmetry properties, only a quarter of the block had to be simulated. For the block of metal a rigid-viscoplastic material behaviour was assumed, as elastic effects can totally be neglected for such high temperatures. The contact between the cylindrical rigid work roll and the block of metal was modelled according to the Coulomb friction law with a friction coefficient of

0.5µ = . Concerning radiation an emissivity coefficient of 0.8ε = was chosen to be realistic. The massive forming behaviour was investigated for the two borderline cases of total strain accumulation (accumulated work hardening for successive passes) and total annealing (full recrystallisation between consecutive passes), as these two extreme cases yield upper and lower bounds for the resulting aspect ratios of the deformed geometry. Numerical investiga-tions confirmed that for the case of full recrystallisation, lateral flow and spread effects become more pronounced compared to scenarios with almost total strain accumulation.

Fig. 2.7: The Mises-stress distribution inside the rolling stock during pass 5 (71 s process time)

Fig. 2.8: The temperature distribution inside the ¼-block after pass 4 (62 s process time)

The resulting stress, strain-, strain-rate and temperature distributions inside the rolling stock are of special interest. In Fig. 2.7 the Mises-stress distribution inside the quarter of the block during pass 5 is represented inside the roll gap at the contact surface and at the vertical free surface. The temperature distribution after pass 4 is depicted in Fig. 2.8 with both the vertical and horizontal mid-symmetry plane being visible. As expected, the core temperature maintains its initial value, whereas the free edges are already cooled down below 900°C. These spatial distributions were analyzed in more detail along specified paths. Besides, the time dependence of these quantities at selected observer points MP (mid-point) and SNP (near the contact surface vertically above the mid-point) were determined. In Fig. 2.9 the time dependence of the effective (i.e. logarithmic) strain is depicted for the case of total strain accumulation. The effective strain at the location SNP near the contact surface is significantly higher compared to the mid-point values MP, which is due to the significant shear effects that are induced near the block surface by the work rolls. The time dependence of the temperature values at these selected locations MP and SNP are represented in Fig. 2.10. It is obvious that surface near regions are cooled down much faster than the mid-point of the block.

Fig. 2.9: Time dependence of the plastic equiva-lent (logarithmic) strain distribution at two selec-ted observer locations MP (mid-point) and SNP (near the upper contact surface)

Fig. 2.10: Time dependence of the temperature distribution at the same observer positions MP and SNP

Fig. 2.11: Plate rolling pass 3, where the block of steel is reduced from 136.6 mm to a final (target) thickness of 103.8 mm.

Fig. 2.12: The Mises-stress distribution inside the roll gap during pass 5, where the plate is rolled from 75 mm to 62.4 mm.

To conclude this section, another rolling case, namely a thermo-mechanically coupled plate rolling simulation with nine consecutive passes and two 90-degree rotations of the plate after

1000

1050

1100

1150

1200

1250

0 30 60 90 120Time [s.]

Tem

pera

ture

[°C

]

Location MP

Location SNP

0

0.5

1

1.5

2

2.5

3

0 30 60 90Time [s.]

Eff

ecti

ve S

trai

n

Location MP

Location SNP

passes 4 and 8 is presented. According to the underlying pass-schedule the block of metal is reduced from an initial thickness of 205 mm to a final target plate thickness of 21.9 mm by a rigid cylindrical work roll with a diameter of 1000 mm. Due to the high initial temperature of 1200°C, a rigid-viscoplastic material behaviour was assumed. Special emphasis was put on the formation of the deformed geometry at the crop ends after each pass. Therefore, the auxiliary pusher concept for the generation of a contact surface between the work roll and the plate was replaced by a kinematic velocity constraint. This modelling concept ensures the shape conservation of the crop end far outside the roll gap during the entry of the plate into the roll gap, i.e. during the first 5-15 ms. real process time of each plate rolling pass. In Fig. 2.13 and Fig. 2.14 a significant front end bending effect occurring after plate pass No. 8 is depicted, which can be attributed to a vertical symmetry breaking of either the geometry (mesh) or the plate temperature distribution. Of particular interest is the accurate determina-tion of the time dependence of the total rolling force and torque during each pass, the behaviour of which is respectively represented in Fig. 2.15 and Fig. 2.16 for the plate rolling pass No. 9. As expected, the rolling force increases significantly during the rolling of the crop ends, which are significantly colder and therefore harder deformable.

Fig. 2.13: Significant front end bending effect after plate rolling pass 8.

Fig. 2.14: The rolling scenario immediately before plate rolling pass 9, with the plate being rotated by an angle of 90° after pass 8.

Fig. 2.15: The total rolling force (z-load) vs. time exhibits a well pronounced steady state and increases at the crop ends.

Fig. 2.16: The corresponding rolling torque vs. time for the plate rolling pass 9 with reduction from 31 mm to 21.9 mm.

3. Investigation of the Squeeze-Mode of Magneto-Rheological (MR) Fluids

Magneto-rheological (MR) fluids represent a class of smart materials, the rheological properties of which can be controlled by the application of an external magnetic field. The essential characteristic behaviour of such suspensions of micron-sized ferromagnetic particles (e.g. iron) in a non-magnetic carrier fluid (e.g. oil including special additives and antiwear agents) is the rapid and reversible transition from the state of a Newtonian-like fluid to the behaviour of a stiff semi-solid by applying a magnetic field of the order of about 0.1-0.4 Tesla. This feature, called the magneto-rheological (MR)-effect [6-8], can be understood from the fact that, in the presence of a magnetic field, the particles form chain-like structures aligned in field direction. Therefore, the columnar microstructure drastically increases its resistance to an applied shear strain. This behaviour has inspired the development and design of new products and applications, such as semi-active vibration dampers, brakes and clutches.

Fig. 3.1: MR-modes of operation [9]: a) Valve (Flow) Mode, b) Direct Shear Mode, c) Squeeze-Mode

The MR-Fluid offers three modes of operation, whereby only two of them are used at present in the vast majority of commercial applications. The direct shear mode is characterized by the relative motion of two magnetic poles separated by the MR-Fluid generating shear forces. In the valve-mode (or flow mode), the magneto-rheological effect is used to throttle the flow through passages. The resulting pressure difference is used for hydrostatic force generation. In the third mode of operation, the so called squeeze mode, the MR-fluid is squeezed out of a narrowing gap, thereby building up high pressures, which can be used to develop small devices capable of carrying high loads. However, due to a lack of understanding of the highly non-linear complicated material behaviour, commercial applications of the squeeze mode are up to now restricted to small amplitude vibration damping. To obtain a thorough understanding of the underlying material behaviour of the MR-fluid in the squeeze mode, a test rig for the exploration of the MR-fluid behaviour was developed [6-7]. The purpose of the test rig is the acquisition of measurement data from experiments. While the basic task of squeezing the MR-fluid out of a narrowing gap can easily be accomplished, the measurement of the distributions of the hydrostatic pressure and of the shear stresses at the boundary of the MR-fluid turns out to be very challenging. To keep the geometry of the gap filled by the MR-fluid as simple as possible, a cylindrical container is equipped with a coil for the generation of the magnetic field. The rotationally symmetric squeezing gap filled by the MR-fluid is formed by the bottom of the container and a piston that dives into the MR-fluid. The container is attached to a servo-hydraulic drive, which enables a controlled translatory motion in the vertical direction, while the piston is attached to a shaft driven by an electrical servo motor for controlled rotary motion. Due to the high degree of symmetry of the set-up, both the evaluation and interpretation of integral measurement categories, such as the

reaction forces and torques (e.g. integrated normal pressure on the gap boundary) are simplified. The measurement is performed by a system of strain gauges, which also allows for the detection of the point of attack of the equivalent vertical force corresponding to a potentially asymmetric behaviour of the MR-fluid filled gap, e.g. due to an inhomogeneous distribution of magnetic particles in the carrier fluid. The data obtained by the test rig serve as input for the development and analysis of adequate macroscopic material models for the behaviour of MR-fluids in the squeeze mode. In the present numerical study this mode is investigated by utilizing the commercial finite element software package Deform [1], which has proven to be best suited for the simulation of massive forming processes, where high plastic equivalent strain and strain rate values occur and successive mesh updating procedures are essential.

Fig. 3.2: Experimental test-rig for the analysis of the squeeze mode (cf. [6, 7])

Fig. 3.3: Numerical simulation model (axially-symmetrical Deform 2D modelling approach)

To gain a well-founded understanding of the phenomenological behaviour of such MR-materials, systematic numerical studies with different assumptions and approximations on the underlying constitutive laws have to be performed, until the measured results are recovered satisfactorily. One of the simplest approaches is based on the well known Bingham-equation, where the shear yield stress is controlled by the applied magnetic field and the fluid shear stresses increase linearly with the fluid shear rates. In the present study we assumed an isotropic rigid-viscoplastic constitutive law with an (almost) linear strain-rate dependence. A systematic parameter identification for a test case, where the gap is reduced from 4 to 2 mm within a real process time of 2 sec., led to an effective yield stress of 2.7 kPa. A comparison of the measured vertical load versus time curve with the simulation result led to a very satisfactory coincidence except for the first 10 msec. This deviation results from elastic effects and was analysed in more detail for a test case, where the gap of initial thickness of 4 mm is merely reduced by 0.2 mm according to a sinusoidal movement with time periods in the range of 0.1 sec. up to 10 sec. The resulting characteristic hysteresis curves in the vertical load versus vertical displacement diagram were compared and calibrated against the measurements. The deviations of calculated from measured hysteresis loops are due to the formation and disruption of iron particle chains along the magnetic field lines. It is obvious that such effects cannot be described correctly within the frame of elasto-viscoplastic constitutive laws and massive forming concepts.

Fig. 3.4: Velocity field inside the MR-Fluid after step 200 (2 s. process time)

Fig. 3.5: Strain-rate distribution inside the MR-Fluid after step 200 (2 s process time)

Fig. 3.6: Comparison of the calculated (solid line) and the measured (dashed line) vertical load vs. time curve

Fig. 3.7: The spatial distribution of the normal pressure along the surface of the MR-Fluid

4. Conclusions and Outlook:

In the present simulation study the commercial finite element package Deform was applied to several praxis-relevant rolling scenarios, including the hot rolling of thick slabs, strips and plates as well as combined flat and edging passes. The thermo-mechanically coupled rolling simulations were accomplished by utilizing the impressive capabilities of Deform 3D. Special emphasis was put on the investigation of 3D-effects, such as lateral flow and strip spread, dog-bone formation (bulging) and back-spread. Further development steps comprise sys-tematic parameter studies of coupled rougher and edging passes, which enables the regression of profile transfer and of dog-bone shape functions, and especially of parameterized spread formulae due to natural spread (i.e. the thickness reduction of the strip) and due to bulging (i.e. back spread of the dog-bones). For plate rolling simulations, significant front end bending effects could be observed, which can be attributed to a vertical symmetry breaking of either the geometry or the temperature distribution of the plate.

It was quite surprising at first glance that the highly non-linear rheological behaviour of smart materials like magneto-rheological fluids can be treated to some extent by utilizing concepts, which were originally developed for massive metal forming processes. However, the descrip-tion of more detailed material behaviour aspects, such as the formation and disruption of iron particle chains along the magnetic field lines, cannot be performed correctly within the frame of elasto-viscoplastic constitutive laws.

Acknowledgements:

The financial support of the main part of this work by the “Linz Center of Competence in Mechatronics” within the framework of the Kplus program of the Austrian government is gratefully honoured. The company SFTC (Scientific Forming Technologies Corporation) is acknowledged for the provision of the Finite-element software package Deform. Special tribute should be paid to Dr. Bernhard Manhartsgruber, Dr. Edwin Till, Dr. Erik Parteder, Dr. Bodo Heise, Dr. Josef Synka, Dr. Andreas Obereder and DI. Erwin Reiss for valuable contributions.

List of References:

[1] Deform 2D V8.0, Deform 3D V5.0, Deform Tools V4.1, Trademark of SFTC (Scientific Forming Technologies), 5038 Reed Road, Columbus, Ohio, USA.

[2] A. Kainz and G. Finstermann, "A New Eulerian Lagrangian Hybrid Finite Element Method for the Numerical Simulation of Stationary Rolling Processes", in J.H.Benyon et al., Modelling of Metal Rolling Processes 3, IOM Conference Papers, Church House, London UK, 1999, pp. 104-113.

[3] A. Kainz and A. Obereder, “Finite Element Simulations of the Visco-Plastic Strip Behaviour for Steady State Flat Hot Rolling Processes”, Proceedings of the Fifth World Congress on Computational Mechanics WCCM V, July 7-12, 2002, Vienna, Austria, Editors: Mang, H.A.; Rammerstorfer, F.G.; Eberhardsteiner, J., Publisher: Vienna University of Technology, Austria, ISBN 3-9501554-0-6.

[4] J. Synka and A. Kainz, Extension of the Concept of Material Objectivity to Mixed Eulerian-Lagrangian Reference Systems, Acta Mechanica, Vol. 166, No. 1-4, Springer-Verlag Wien, pp.13-25, 12, 2003.

[5] J. Synka and A. Kainz, "A Novel Mixed Eulerian-Lagrangian Finite-Element Method for Steady-State Hot Rolling Processes", International Journal of Mechanical Sciences, Vol. 45 (12), pp. 2043-2060, Elsevier, 2003.

[6] B. Manhartsgruber, A. Kainz, R. Scheidl, Computer Aided Design and Optimization of a Test-Rig for Magneto-Rheological Fluid Behaviour", in: Conference Proceedings for the First International Conference on Computational Methods in Fluid Power Technology, Stecki, Jacek S (Editor), Fluid Power Net Publications, ISBN 0-9578574-1-1, November 26-28, 2003, Melbourne, Australia.

[7] B. Winkler, B. Manhartsgruber, R. Scheidl, A Visco-Elasto-Plastic Material Law for Magneto-Rheological Liquids, 2nd International Workshop on Computer Software for Design, Analysis and Control of Fluid Power Systems, September 5-7, 2001, Ostrava-Malenovice, Czech Republic, pp. 175-182.

[8] R.E. Rosensweig, Ferrohydrodynamics, Cambridge University Press, Cambridge, London, New York, 1985.

[9] O.C. Zienkiewicz and R.L. Taylor, “The Finite Element Method”, Vol. 1-3, Butterworth, Heinemann, 5th edition, 2000.