Upsetting a Billet: upsetBillet
Prepared by Philip Cardiff and Ivan Batistić
Tutorial Aims
- Demonstrate a large-strain elastoplastic analysis with rough solid contact in an axisymmetric model.
- Compare the predicted force-displacement response and deformation pattern with finite-element results.
Case Overview
This case considers the upsetting of a cylindrical billet between two parallel rough dies. The benchmark has been used to assess numerical methods for metal forming simulations [1, 2, 3].
The billet has an initial diameter of \(20\) mm and an initial height of \(30\) mm. It is upset by \(60\%\), corresponding to a total die displacement of \(18\) mm. Because of symmetry, only the upper half of the billet is modelled, so the rigid die in this half model is displaced by \(9\) mm. The problem geometry and loading are shown in Figure 1.
Gravitational and transient effects are neglected. The loading is applied over 2000 quasi-static time increments. Rough contact between the die and billet is modelled using the solidContact boundary condition with a Coulomb friction coefficient of \(0.5\). Taylor [1] showed that this high friction coefficient approximates the rough surface condition.
The case is represented as a 2-D axisymmetric model, see Figure 2. The default tutorial mesh consists of 168 cells. The updated Lagrangian solid model is selected in constant/solidProperties using nonLinearGeometryUpdatedLagrangian.
Figure 1 - Problem geometry [4]
Figure 2 - Axisymmetric mesh [4]
The billet is modelled using the neoHookeanElasticMisesPlastic mechanical law. The mechanical properties are given in Table 1. The tabulated hardening curve in constant/plasticStrainVsYieldStress corresponds to a hardening modulus of \(300\) MPa.
Table 1 - Mechanical properties
| Parameter | Symbol | Value |
|---|---|---|
| Young's modulus | \(E\) | \(200\) GPa |
| Poisson's ratio | \(\nu\) | \(0.3\) |
| Initial yield stress | \(\sigma_Y\) | \(700\) MPa |
| Initial density | \(\rho\) | \(7833\) kg/m\(^3\) |
| Hardening modulus | \(\kappa\) | \(300\) MPa |
Expected Results
Figure 3 shows the predicted force-displacement response for different mesh densities reported by Cardiff et al. [4]. The Lagrangian finite-element result from Taylor [1] is included for comparison. The finite-volume predictions quickly approach a mesh-independent solution and are close to the finite-element result.
Figure 3 - Force-displacement response for different mesh densities [4]
The predicted deformed shape of the billet, equivalent plastic strain distribution, and comparison with Abaqus finite-element results are available in Cardiff et al. [4].
The solidForces function object is added in system/controlDict. It records the reaction force on the symmPlane patch, which can be used to plot the force-displacement response shown in Figure 3.
Video 1 - Upsetting a billet animation
Running the Case
The tutorial case is located at solids4foam/tutorials/solids/elastoplasticity/upsetBillet. The case can be run using the included Allrun script, i.e. > ./Allrun.
The Allrun script first checks the case format using solids4Foam::convertCaseFormat. It then creates the blockMeshDict file from system/blockMeshDict.cylinder.m4 using the m4 scripting language and runs blockMesh to create the mesh. For foam-extend, the axis edges are manually removed using collapseEdges. Finally, the case is run using the solids4Foam solver.
This case is currently skipped by Allrun when using OpenFOAM.org.
References
[1] L. M. Taylor, "A finite element analysis for large deformation metal forming problems involving contact and friction," The Institute for Computational Engineering and Sciences, The University of Texas at Austin, ICES Report 81-15, 1981.
[2] Dassault Systémes Simulia Corp. "Abaqus 6.11 documentation", 2012. Accessed August 2015.