An upper bound solution of ECAE process with outer curved corner
Introduction
Equal channel angular extrusion (ECAE) or pressing is a deformation technique to deform materials through a die with two channels being equal in cross-section and intersecting at a certain angle [1]. During the ECAE process, the grain refinement occurs together with significant strain hardening resulting in remarkable enhancement of strength in many engineering materials [2], [3], [4], [5], [6], [7], [8], [9]. However, referring to the literature it will be appreciated that in contrast to numerous publications on structure characterization of ECAE products, few papers have been published on the mechanics of ECAE. Further investigations are of interest to comprehensively assess the effect of process parameters on the mechanics of ECAE.
Regarding the previous works on ECAE [10], [11], [12], [13], [14], [15], not enough attempts have been made on the upper bound analysis of the process because none of them has considered the effects of all parameters such as friction, die angle and the outer curved corner angle all together, on the mechanics of the process. In fact, the process has been analyzed just considering one or two parameters of the process. In other words, considering the previous publications on ECAE using the upper bound method, only the effects of certain parameters on extrusion pressure have been assessed and none of them has considered the effects of all the process parameters simultaneously. For example in one of the first analytical approaches developed by Segal [4], the effects of friction coefficient and outer curved corner angle were not considered. Lee [14] analyzed the stresses and strains in channel angular deformation (CAD), in which two channels are not equal in cross-section. He considered ECAE as a special form of CAD in his upper bound analysis but he also did not consider the effects of the outer curved corner angle. Alkorta and Sevillano [10] have analyzed the pressure needed for non-friction ECAE of perfectly plastic and strain hardening materials using an upper bound and a FEM solution. They have compared the results achieved from these solutions. Since in their investigations the effect of friction was not considered, a further development is needed on the analysis to take into account the effect of friction. Using the upper bound method, Luis Perez [15] has analyzed a configuration of ECAE dies called equal fillet radii angular extrusion (EFRAE) being slightly different from the general ECAE dies. Because of the design considered by Luis Perez [15], he actually did not consider the effect of outer curved corner angle in his analysis. Altan et al. [11] have analyzed the deformation of the material in a 90° ECAE die using the upper bound theorem. Although their model includes the effect of friction between the sample and the die walls, the radius of inner corner of the die, and the dead metal zone on the deformation pattern during ECAE, they have not taken into account the effects of outer curved corner angle of the die. Moreover, their investigation has been restricted just to 90° ECAE dies.
In the present study, an upper bound solution is presented to consider the effects of die geometry and friction coefficient on the total strain and extrusion pressure. Extrusion pressure is measured in two dies of the same angle but of different outer curved corners and the results are compared with the results of the theoretical model.
Section snippets
Analysis
In this research, in order to develop an upper bound solution, a simple deformation model considering the effects of outer curved corner angle, introduced by Alkorta and Sevillano [10] and then used by Altan et al. [11], is utilized. In this deformation model, the ECAE die is divided into three regions as shown in Fig. 1. In Region I, the material moves rigidly downward with a velocity of V0. Region II, called the “deformation zone”, is where the material undergoes continuous plastic
Experimental procedure
A commercial grade aluminum alloy (AA 6070) was used in the ECAE tests carried out in this work. The composition of the alloy is shown in Table 1. Rods of the above material were received in hot extruded condition. To obtain uniform metallurgical properties in all samples, they were solution treated at 550 °C for 2 h and cooled in the furnace. Then, the Cook and Lark compression testing [18] was conducted to determine the stress–strain curve of the alloy. Specimens of three different heights to
Work hardening constants and friction coefficient determinations
From the force-stroke data recorded during compression testing of the billet material, the true stress and true strain values were calculated and the flow curve was determined. The strain hardening exponent, n, and the strength coefficient, K, were determined by fitting the data to the equation σ = Kɛn. The values of K and n were found to be 179.3 MPa and 0.26, respectively. With respect to the extruded material and the die material and using the procedure proposed by Khan et al. [19], the
Conclusions
An upper bound analysis was carried out in order to investigate the plastic deformation behavior of the material during the ECAE process with outer curved corner. From the results of the analysis the following conclusions are made:
- (1)
As decreasing the die angle has a greater effect on increasing the total strain and extrusion pressure than the outer curved corner angle of the die and because increasing the friction coefficient leads to an increase in the extrusion pressure and has no effect on
Acknowledgements
The authors would like to thank the Iranian National Science Foundation (INSF) and Sharif University of Technology, Tehran, Iran, for the financial support and research facilities used in this work.
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