In a finite element analysis of angle steel production process with six passes, the Updated Lagrangian approach was applied and good correlation between predicted and measured results was reached [96].

Coupled analysis problems

Rolling process, especially hot rolling, involves both mechanical and thermal behaviors. Most mechanical and physical properties and boundary conditions are temperature related. The heat flow and stress analysis cannot be
analyzed separately. The analysis requires a thermally coupled approach.

The sources of the coupling effect are [133]:

- heat generated by plastic work
- heat generated through friction energy in the interface workpiece/roll
- thermal contact between the (hot) workpiece and the (cold) roll
- influence of the temperature on the material properties and boundary condition (e.g. friction)
- convective heat transport due to high velocities during the rolling

Material Formulation

In the material to be rolled, both stress analysis and heat transfer must be allowed. As to the material behaviors, besides elastic-plastic or elastic-viscoplastic formulation, the rigid-plastic or rigid-viscoplastic model is also acceptable, and maybe better, in most cases since plastic strains usually outweigh elastic strains. With the elastic-plastic finite-element formulation, time steps should be shorter due to the nature of elastic-plastic constitutive equations.

Roll Formulation

In the roll modeling, three levels of approximations can be applied:

- rigid with constant temperature
- rigid with heat transfer in the roll
- deformable with heat transfer

In most hot rolling processes, it is practical to define rolls as rigid. Using deformable rolls will tremendously increase the computational time since great number of elements are required to define a roll, as a solid cylinder, or even with groove on the surface. However, in some situation, roll deformation needs to be considered, such as cold rolling of thin strip in which roll flattening has to be considered.

Automatic Contact

Contact is a primary problem in rolling and other forming processes, and the contact problem is more critical in the rolling process. An FEM program for rolling process should have facilities for both mechanical contact and thermal contact.

Some early version of finite element programs used gap or interface element to handle the contact problem. The user had to define such gap element in the contact problems. In the current technological level, contact problem should be handled automatically. An FEM program should have the following abilities:

- to handle both rigid-deformable as well as deformable-deformable contact in 2D and 3D
- to handle both continuum elements as well as shell elements
- to specify bodies as well as possible body movements in a flexible manner
- to handle thermo-mechanically coupled contact phenomena
- to handle dynamic contacts using implicit and explicit solution techniques
- to fully address friction during the contact

Various strategies have been developed to handle contact problem. Two examples are as follows:

- For each boundary node it is checked whether a boundary segment/patch of a surface is crossed. If so, automatically a set of local transformations is applied to the node, which is in contact with the surface. Prescribed boundary conditions are applied such that the node can only slide along the surface. If a node is in contact with a deformable surface automatically a mufti-point constraint (tying) is created such that the contacting node remains on the edge or patch of the segment to be contacted [133].
- It is checked whether tensile forces occur at nodes which are in contact: if so the node will be enabled to separate from the contacted segment through the removal of boundary conditions or tying constraints[133].

For thermal contact, following steps, among others, can be done[133]:

- Specifying convective heat transfer between a body and the surrounding air (including radiation) or between the body and another body.
- Checking if a node is in contact with another surface. If it is, a set of convective boundary conditions based on a film coefficient and a sink temperature are added to both surfaces. If the surface to be contacted is a rigid surface with no heat transfer, the temperature of the rigid surface is used as the sink temperature.

Friction Formulation

Friction is the most important driving force in rolling process, so a finite element program should be able to handle friction problem correctly. Various friction types should be available, such as shear and Coulomb friction types. In addition, a user should be allowed to define his own friction model as a function of a lot of variables, such as the normal force at the point of contact, the relative sliding velocity between the contacting bodies and the local temperature, etc.

In contact problems, the occurrence of so-called "neutral lines" may easily result in numerical difficulties. Smooth approximation should be made. For example, MARC program address the problem with an approximation as showed in Fig. 1, in which f_{t} is the friction force, v_{r} is the relative sliding velocity between the contacting bodies, and C is a constant to be chosen by the user.

Figure 1: Smoothing function used for friction [133]

Solver techniques

The request to analyze larger and larger 3D problems have led to looking for new solver technology. For instance, a 3D slab rolling problem with 40 elements in rolling direction, 10 in height and 10 in width direction, requires already 4000 brick elements if the roll is taken to be rigid. A deformable roll could easily double the number of elements. The use of direct solvers not only require long computation times, but also large amounts of disk space. We have recently obtained some experience with an iterative element by element solver and a sparse conjugate gradient solver, which may both be combined with a preconditioner. This new technology in solving sets of equations shows that more realistic models can be analyzed at reasonable costs. A typical example is outlined in the next section [133].

Establishment of Rolling

Suppose a workpiece with 40 elements in the rolling direction is pushed into the roll gap to undergo deformation. The rolling stage before the workpiece fills the whole deformation zone may be called the stage of rolling establishment. This stage costs about 20-30% or more of the total increments. This computational cost is simply to establish the rolling condition, because the results in stage are not for practical use. A lot of work can be done to reduce this part of computational cost, for example:

- Analyze the stage of rolling establishment in only one step, with different approach, such as Eulerian approach
- Use coarse mesh during stage of roll establishment for those elements outside the roll gap. When they are drawn near the roll gap, remesh them with finer mesh.
- Develop new solver technology. Equations for those nodes beyond the roll gap may not need to be solved at all! The only meaningful changes for them are their coordinates. Temperature change in this short period can be neglected.

References

[52] Shiro Kobayashi, Soo-Ik Oh, Taylan Altan: Metal forming and the finite-element method. 1989. ISBN 0-19-504402-9.

[96] Bingji Li: Compared Experimental and Theoretical Investigations of Forming Technical Parameters in Shape Rolling with Example of the Hot Rolling of Angle Steels. TU Bergakademie Freiberg, Freiberg, Germany, 1996. ISBN 3-86012-029-8

[133] A.W.A. Konter and A. Bout: Finite Element Techniques in the Analysis of Rolling Processes. Marc Analysis Research Corp. - Europe.