Extrusion of a cylindrical metal bar with frictional heat generation

The bar has an initial radius of 100 mm and is 300 mm long. Figure 1 shows half of the cross-section of the bar, modeled with first-order axisymmetric elements (CAX4T and CAX4RT elements in Abaqus/Standard and CAX4RT elements in Abaqus/Explicit).

In the primary analysis in both Abaqus/Standard and Abaqus/Explicit, heat transfer between the deformable bar and the rigid die is not considered, although frictional heating is included. A fully coupled temperature-displacement analysis is performed with the die kept at a constant temperature. In addition, an adiabatic analysis is presented using Abaqus/Standard without accounting for frictional heat generation. The node-to-surface (default) and surface-to-surface contact pair formulations as well as general contact in Abaqus/Standard are presented. In the case of Abaqus/Explicit, penalty and kinematic contact formulations are used in the definition of contact interactions.

Various techniques are used to model the rigid die. In Abaqus/Standard the die is modeled with CAX4T elements made into an isothermal rigid body using an isothermal rigid body and with an analytical rigid surface. In Abaqus/Explicit the die is modeled with an analytical rigid surface and discrete rigid elements (RAX2). The fillet radius is set to 0.075 for models using an analytical rigid surface to smoothen the die surface.

The Abaqus/Explicit simulations are also performed with Arbitrary Lagrangian-Eulerian (ALE) adaptive meshing and enhanced hourglass control.

The material model is chosen to reflect the response of a typical commercial purity aluminum alloy. The material is assumed to harden isotropically. The dependence of the flow stress on the temperature is included, but strain rate dependence is ignored. Instead, representative material data at a strain rate of 0.1 sec⁻¹ are selected to characterize the flow strength.

The interface is assumed to have no conductive properties. Coulomb friction is assumed for the mechanical behavior, with a friction coefficient of 0.1. Gap heat generation is used to specify the fraction, $f_g$, of total heat generated by frictional dissipation that is transferred to the two bodies in contact. Half of this heat is conducted into the workpiece, and the other half is conducted into the die. Furthermore, 90% of the nonrecoverable work due to plasticity is assumed to heat the work material.

In the first step, the bar is moved to a position where contact is established and slipping of the workpiece against the die begins. In the second step, the bar is extruded through the die to realize the extrusion process. This is accomplished by prescribing displacements to the nodes at the top of the bar. In the third step, contact is deactivated in preparation for the cool down portion of the simulation. In Abaqus/Standard this is performed in a single step: the bar is allowed to cool down using film conditions, and deformation is driven by thermal contraction during the fourth step.

Volume proportional damping is applied to two of the analyses that are considered. In one case the automatic stabilization scheme with a constant damping factor is used. A nondefault damping density is chosen so that a converged and accurate solution is obtained. In another case the adaptive automatic stabilization scheme with a default damping density is used. In this case the damping factor is automatically adjusted based on the convergence history.

In Abaqus/Explicit the cool down simulation is broken into two steps: the first introduces viscous pressure to damp out dynamic effects and, thus, allow the bar to reach static equilibrium quickly; the balance of the cool down simulation is performed in a fifth step. The relief of residual stresses through creep is not analyzed in this example.

In Abaqus/Explicit mass scaling is used to reduce the computational cost of the analysis; nondefault hourglass control is used to control the hourglassing in the model. The default integral viscoelastic approach to hourglass control generally works best for problems where sudden dynamic loading occurs; a stiffness-based hourglass control is recommended for problems where the response is quasi-static. A combination of stiffness and viscous hourglass control is used in this problem.

For purposes of comparison a second problem is also analyzed, in which the first two steps of the previous analysis are repeated in a static analysis with the adiabatic heat generation capability. The adiabatic analysis neglects heat conduction in the bar. Frictional heat generation must also be ignored in this case. This problem is analyzed only in Abaqus/Standard.