A three-dimensional steady-state finite element analysis of square die extrusion by using automatic mesh generation
Introduction
Extrusion is the process of squeezing a heated billet to flow through a die orifice of desired section shape. Extrusion has been applied to the manufacture of round or complicated sections, and has continuously widened its area of application since the extruded products can be manufactured with high productivity, good precision and surface finish. Among the industrial extrusion processes, square die (or, flat faced die) extrusion has been widely used for extrusion of profiled light metal products such as aluminum or copper alloys. Square die extrusion is usually carried out at an elevated temperature without lubrication. The design of extrusion dies is still an art rather than a science with increasing complexity of shape and thinness of section. Therefore, most of the die design is still dependent on personal judgement, intuition and experience. In order to manufacture defect-free products and to achieve proper die design and process control, a systematic analysis of extrusion is required. In recent years, more detailed information for the die design has been obtained from numerical methods of analysis. The finite element method has been widely used for an effective analysis of the extrusion process.
Iwata et al. [1] analyzed hydrostatic extrusion by the finite element method. Subsequently, finite element analyses on plane or axisymmetric problems were performed by many researchers [2], [3], [4], [5], [6], [7]. Three-dimensional analyses of extrusion problems have been carried out recently. Böer and Webster [8] analyzed the drawing of a square section using optimization technique. Yang and the author [9], [10] analyzed the three-dimensional extrusion of arbitrarily shaped sections for a cold and steady-state process. Mori et al. [11] performed steady-state calculation on the curvature of the extruded products. Lam et al. [12], [13] analyzed extrusion with flow guide for a steady-state isothermal process. Mooi et al. [14] and Yang et al. [15] analyzed extrusion for a non-steady-state problem using updated Lagrangian or arbitrary Lagrangian-Eulerian method. The finite element methods for non-steady-state (Lagrangian description) have simple governing equation, but a large number of remeshing is needed. The steady-state method (Eulerian description) is convenient for problems which involve abrupt deformation such as square die extrusion, because remeshing is not required and it requires relatively short computation time. Convective effect cannot be considered in simulating square die extrusion with steady-state description. However, the convective effect can be ignored since the extrusion speed is very slow in industrial practice. Although non-steady-state stages exist at the beginning and the end of the process, deformation in the steady-state is dominant. In addition, extrusion with a constant exit temperature is of practical interest for achieving a uniform product quality or for making the most efficient use of the maximum speed. Therefore, the steady-state method is sufficient for the simulation of square die extrusion. The steady-state method for the analysis of square die extrusion, however, was generally restricted to only isothermal or cold state.
The meshing can be simplified by using a shifting technique because the section shape of square die extrusion is constant along extrusion direction. And, the conventional looping methods [16], [17] are employed for generation of two-dimensional meshes before shifting.
The objective of this study is to develop a steady-state finite element method and automatic mesh generation technique for hot extrusion through square dies, and to provide theoretical basis for an optimal die design and process control for the extrusion technology. In the present investigation, both the analyses of deformation and temperature are treated as steady-state and decoupled model. The analysis of temperature distribution includes heat transfer. Convection link element is adopted for heat transfer analysis between the billet and the container, and also between the billet and the die. Computations are carried out for hot extrusion of square section, L-shaped section and clamp shaped section. Distributions of temperature, effective strain rate, velocity and mean stress are discussed for effective design of an extrusion die.
Section snippets
Theory
For a steady-state three-dimensional control volume V, equilibrium demands the following relations:where, denote volumetric strain rate.
The constitutive equation is given as follows:where, σ′ij, , and denote deviatoric stress tensor, effective strain rate, flow stress and strain rate tensor, respectively.
Let δvi be an arbitrary variation of the velocity field compatible with the boundary conditions. The variational equation is then
Results and discussion
Hot extrusion through square dies has been widely used in manufacturing Al alloy sections. Computations are carried out for the hot extrusion of square section, L-shaped section and clamp shaped section of Al alloy. These examples are collected from an extrusion company. In order to validate the developed finite element simulation program, the computational results for square section are compared with the results by Yang et al. [15], [16] under the same process condition. The billet and the die
Conclusion
Rigid-viscoplastic finite element analyses for three-dimensional hot extrusion of sections through square dies by using automatic mesh generation were performed providing a theoretical basis for the optimal die design and process control. In the present investigation, steady-state assumption is used for both the analyses of deformation and temperature. The analysis of temperature distribution includes heat transfer, and carried out by decoupling from the analysis of deformation. A convection
Acknowledgements
This work was supported by the Korea Science and Engineering Foundation (KOSEF) through the Machine Tool Research Center at Changwon National University.
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