Rolling of thick plates

Hot rolling is a basic manufacturing technique used to transform preformed shapes into a form suitable for further processing. Rolling processes can be divided into different categories, depending on the complexity of metal flow and on the geometry of the rolled product. Finite element computations are used increasingly to analyze the elongation and spread of the material during rolling (Kobayashi, 1989). Although the forming process is often carried out at low roll speed, this example shows that a considerable amount of engineering information can be obtained by using the explicit dynamics procedure in Abaqus/Explicit to model the process.

The rolling process is first investigated using plane strain computations. These results are used to choose the modeling parameters associated with the more computationally expensive three-dimensional analysis.

Since rolling is normally performed at relatively low speeds, it is natural to assume that static analysis is the proper modeling approach. Typical rolling speeds (surface speed of the roller) are on the order of 1 m/sec. At these speeds inertia effects are not significant, so the response—except for rate effects in the material behavior—is quasi-static. Representative rolling geometries generally require three-dimensional modeling, resulting in very large models, and include nonlinear material behavior and discontinuous effects—contact and friction. Because the problem size is large and the discontinuous effects dominate the solution, the explicit dynamics approach is often less expensive computationally and more reliable than an implicit quasi-static solution technique.

The computer time involved in running a simulation using explicit time integration with a given mesh is directly proportional to the time period of the event. This is because numerical stability considerations restrict the time increment to

where the minimum is taken over all elements in the mesh, $L^{el}$ is a characteristic length associated with an element, $\rho$ is the density of the material in the element, and $\lambda$ and $\mu$ are the effective Lamé's constants for the material in the element. Since this condition effectively means that the time increment can be no larger than the time required to propagate a stress wave across an element, the computer time involved in running a quasi-static analysis can be very large. The cost of the simulation is directly proportional to the number of time increments required, $n=T/\mathrm{\Delta }t$ if $\mathrm{\Delta }t$ remains constant, where T is the time period of the event being simulated. ($\mathrm{\Delta }t$ will not remain constant in general, since element distortion will change $L^{el}$ and nonlinear material response will change the effective Lamé constants and density; but the assumption is acceptable for the purposes of this discussion.) Thus,

To reduce n, we can speed up the simulation compared to the time of the actual process; that is, we can artificially reduce the time period of the event, T. This will introduce two possible errors. If the simulation speed is increased too much, the inertia forces will be larger and will change the predicted response (in an extreme case the problem will exhibit wave propagation response). The only way to avoid this error is to find a speedup that is not too large. The other error is that some aspects of the problem other than inertia forces—for example, material behavior—may also be rate dependent. This implies that we cannot change the actual time period of the event being modeled. But we can see a simple equivalent—artificially increasing the material density, $\rho$, by a factor $f^2$ reduces n to $n/f$, just as decreasing T to $T/f$ does. This concept, which is called “mass scaling,” reduces the ratio of the event time to the time for wave propagation across an element while leaving the event time fixed, thus allowing treatment of rate-dependent material and other behaviors while having exactly the same effect on inertia forces as speeding up the time of simulation. Mass scaling is attractive because it allows us to treat rate-dependent quasi-static problems efficiently. But we cannot take it too far or we allow the inertia forces to dominate and, thus, change the solution. This example illustrates the use of mass scaling and shows how far we can take it for a practical case.

The trial and error method works well for most generic quasi-static problems; however, for rolling processes Abaqus/Explicit can set the mass scaling factor automatically based on the rolling geometry and mesh properties. An acceptable value for the stable time increment is calculated, and the appropriate mass scaling factor is applied on an element-by-element basis. The value of the stable time increment is based on the average element length in the rolling direction, the average velocity of the product through the rollers, and the number of nodes in the cross-section of the mesh.