Modeling of Wire Rolling
INTRODUCTION
Shape rolling is a metalworking process where typically blooms or billets reheated in a walking beam furnace is passed through rolling stands that successively reduces and reshapes the cross-section. The rolling process is usually classified as roughing and finishing. Rough rolling breaks the as-cast microstructure and reduces its cross-section quickly at a temperature of about 1200°C. Finish rolling where the reductions are progressively lowered in-order to assure the final product with close dimensional tolerances follows this. In the case of wire rolling, the finish rolling is performed in a block where the wire speeds can reach up to tens of meters per second. Designing the roll-pass has traditionally been more of art than science. Owing to the recent advances in steel grades and our increased understanding about metalworking, thermo-mechanically controlled rolling became the state of the art. Simulation of the rolling process is more and more used in roll-pass design lately (1-10). This article describes the architecture of a software platform for optimization of shape rolling process. The platform consists of an advanced microstructure based material model, generalized plane strain, large deformation finite element solver and optimization routines for multi-objective optimization of the process, product, and productivity.
MODELING APPROACHES
The empirical models for roll-force, roll-torque, etc. are fast and simple and have been in use for more than 70 years (11). These approximate models are based on mean values of stress, strain, force, etc. and are functions based on calibrating various rolling conditions like temperature, rolling speed, material properties, surface quality, etc. with respect to experiments or even plant trials (12-15). Though very beneficial for scheduling type tasks, these models lack the knowledge of what is happening within the material. Numerical continuum models based on Finite Element Method (FEM) in 2 and 3 dimensions (2D & 3D) has proven to be very useful to model metalworking processes. This modelling approach involves the assemblage of a range of technologies like Computer Aided Design (CAD), large strain-deformation-rotation formulation, contact formulation involving friction, rate, and temperature dependent elasto-plastic deformation, thermo-mechanical-microstructure dependent plastic flow, parallel processing capabilities, etc. Advancements in all of these technologies in the previous decades made it possible to perform detailed modelling of complex metalworking processes (16-27).
CONTINUUM MODELS IN ROLLING
Rao & Kumar used finite-element method based computer program in a simulation of rolling for the first time in 1977 (17). They used a transient plane strain elasto-plastic formulation to simulate cold rolling. Since then, a lot of researchers have worked on this topic (18-24) focusing on the design of the rolling process. Osio (25) pioneered the use of FEM to perform a sensitivity analysis of the shape rolling process. Microstructural behaviour was also included in rolling simulations since decades ago (26). Multi-physics problem: The problem of thermo-mechanical-microstructural coupling is solved by a staggered approach as shown in the figure below. The thermal field solved by implicit time integration computes the heat losses by conduction, convection, and radiation. The non-linear thermo-physical properties of the material demand an iterative solution for the thermal field. Based on the computed temperature, the microstructure model computes the phase composition and its evolution. This, along with the temperature is given as input to the mechanical domain.
Mechanical and physical properties are heavily dependent on temperature and phase composition. During deformation, part of the plastic deformation energy is converted to heat. Adiabatic heating, latent heat evolution and volume change during phase evolution are also included here. Since the inertia forces are not significant, the mechanical problem can be simplified as quasi-static, which is, solved implicitly using an iterative scheme.
Large Deformations:
The updated Lagrangian formulation of mesh moving with the materials pioneered by McMeeking and Rice (16) is most suited for computation of large strain-deformation-rotation elasto-plastic problems. As opposed to the Eulerian formulation (fixed mesh), the main advantage of using this technology is the ease of tracing free surfaces and taking the deformation history that enables the accurate computation of residual stress. Shape rolling involves the motion of free surfaces of the workpiece and tracing the surfaces of the roll and the workpiece material which makes the formulation befitting. 'Updated' refers to the fact that the variables are related to the last calculated (updated) configuration.
Generalized Plane Strain:
2D Plane strain formulation provides the exact solution to a 3D cylinder, which has its ends, constrained by frictionless walls. While modelling shape rolling, where much of the deformation is normal to the plane, this approach is not appropriate. Generalized plane strain formulation overcomes this deficiency by including one additional degree of freedom normal to the plane. It also assumes that a plane normal to the axis always remain normal. From the figure below, Vz Out; the velocity of the material at the roll gap can be calculated by assuming constant volume during plastic deformation and knowing Vz In; the in-going velocity of material (27).
Contact Modeling:
The rolls are modelled as rigid and driven by a prescribed displacement. In order to simulate the rotary motion of the roll, the vertical motion of a point in the roll that comes in contact with the work-piece is computed based on the angular velocity (see figure below).
In the simulation of shape rolling, modelling of contact and friction between roll and workpiece is paramount. The Lagrangian multiplier method and penalty method are the most common contact algorithms available in the literature. However, a simpler algorithm called ‘direction constraint method’ proposed by S. Riljak (27) is used here. This algorithm works by constraining the nodes in contact using the normal and tangential unit vectors of the polygon sides defining the tool. The roll-separating-force and roll-torque are computed based on the nodal contact forces which in turn consist of normal and friction components described by Coulomb's law. In order to include the friction force acting in the normal direction of the cross-section plane, a correction factor is introduced based on the relative sliding velocity between the workpiece and the roll.
Remeshing and Data Mapping:
The biggest drawback of using the updated Lagrangian formulation where the mesh is attached to the material is that deformation can introduce severe mesh distortions that result in degeneration of results. In order to avoid this, new mesh is progressively generated from the distorted ones and the old state variables are mapped onto the new mesh. The methodology used for solving this problem is described in a previous post available here.
Multi-pass Rolling:
Multi-pass shape rolling is threading wire/rod through a sequence of roll pairs to progressively reduce the cross-section area. In the case considered here, (see figure below) 8 pairs of oval and round rolls are arranged at 90° angle to each other in a finishing block. Owing to the symmetry, only a quarter of the model is included here. In this model, it is assumed that the edge that initiates contact with the wire remains in the plane perpendicular to its axis. The initial position of the roll-edge is shown in the figure below. The wire (cross-section in 2D case) moves through each of the roll-pairs, gets deformed progressively by the crosswise motion of the contact edge.
Material Modeling:
The deformation mechanisms during the thermo-mechanical processing of alloys are complex as it involves, different phases and microstructural mechanisms like recovery, recrystallization, precipitation, etc. In order to perform simulations of the hot-rolling process, the material behaviour needs to be modelled numerically. This includes modelling of Thermo-Mechanical, Thermo-Physical and Phase Kinetics behaviour of the material (28-30).
Modeling the Plastic Flow:
The parameters for a flow stress model are identified by matching the flow strength measurement with computed flow stress. The experimental results from a compression or tension test machine include stress, strain, and temperature as a function of time denoted by σ , ε and T respectively. The model for flow stress can be written as σ y= F ( ε , T ,ε̊ , p (T , ε )) . The parameters of the model, p is calibrated by matching the measurements with the model. This post explains how calibration is performed. The required measurements are performed in a Gleeble thermo-mechanical simulator. In addition to this, a series of validation tests are also performed in order to validate the material model independent of the process.
Modeling the Physical Properties:
The thermo-physical properties of the material like thermal conductivity, elastic modulus, specific heat capacity, Poisson's ratio, expansion coefficient etc. are also required in the computation. Owing to the heavy temperature dependence of these physical properties, either temperature-dependent functions or curves are given as input to the solver (28-30).
Modeling the Phase Evolution:
The phase evolution models to be used are based on Kirkaldy and Venugopalan (31) and Li et al. (32), which are used in the automotive industry during the last decades with great success predicting the final properties of hot-stamped components. The model employed here enable prediction of the various phase evolution as well as accounting for the transformation induced plasticity and the uneven temperature shrinkage (33).
3D Modeling:
Full 3D Finite-element simulation of shape rolling using solid elements provides the most generic solution to the problem by making the least number of assumptions. Since the mid-90s, 3D FEM has been used in the simulation of rolling and roll-pass design (22). In this study, MSC Marc software was used to perform 3D simulations of rolling (see cover image). The rolls were assumed to be rigid (ignoring roll-deflection) and modelled using analytic surfaces. The length of the wire is chosen by trial and error so that the section of interest has negligible end-effects. The wire is given an initial displacement so that it starts threading between the rolls and thereafter driven only by friction. The friction coefficient between wire and rolls is calibrated based on the wire velocity measured from the experiment. It is well established that the 3D simulation of rolling using solid elements is very much resource intensive. The purpose of performing this experiment is to provide a reference case for comparison with the first pass of the multi-pass 2D generalized plane strain case described earlier. Here, the maximum temperature is slightly above 1000°C with a peak equivalent plastic strain of 60%.
RESULTS AND DISCUSSIONS
The cross-section of the wire during the 8-pass finish rolling is shown in the figure below. The alternate oval and round shapes of the tool is optimized to produce a final round cross-section.
Figure below shows how the temperature, equivalent plastic strain and peak strain-rate vary during each pass at the centre of the cross-section (axis of the wire). The temperature at the centre rises from 950°C to 1375°C. The strain at the centre reaches close to 480% whereas the peak strain rate is above 7000s-1. One of the main challenges involved in the finishing stages of wire rolling is the local melting of the wire. This can lead to porosity and in worst-case breakage.
However, in this case, the simulation results show that this problem is avoided. Comparison between 2D and 3D simulations ensures that the roll-force, temperature and peak strain lies within 5% deviation. S. Riljak (27) performed a systematic comparison between 2D and 3D cases of shape rolling between different shapes like flat, square, round, oval, H, diamond, etc. and found that except for the diamond to square (15%), the deviation was less than 10%. Temperature [°C] evolution during high-speed multi-pass shape rolling of stainless steel wire in a finishing block is given in the animation below.
Equivalent plastic strain evolution during high-speed multi-pass shape rolling of stainless steel wire in a finishing block is given in the animation below.
ACKNOWLEDGEMENTS
This work was performed partly within the project OptiRoll: A tool for optimization of rolling of long products (32078) funded by VINNOVA. The FE code for single pass shape rolling originally written by S. Riljak is adapted here for multi-pass rolling simulations.
REFERENCES
Download the PDF version for references.