【正文】 (103 lb/)Elongation(%) 507018Table 2k and n values of AL202T3k (MN/m2)n689where k and n represent the strength coefficient and strain hardening exponent, respectively. The values of k and n are also taken from the handbook. The maximum load point at the slope of the true stress–strain curve equals to the true stress, from which it can be deduced that for a material obeying the above exponential relationship between εp and n, εp = n at the maximum load point Engineering stress, s, is converted into true stress–strain σ using the below equation:σ = s(1 + ε) = s exp(ε)Fig. 5. True stress–strain curve of AL2024T3.The plastic region of the true stress–strain curve ofAL2024T3 is shown in Fig. 5.. Punch displacementDuring simulation, the punch is programmed to move downwards for a prescribed displacement. As the distance between the punch and die decreases, the work piece experiences an increase in pressive force. Similarly, the punch has a reaction force acting on it. When the work piece is in full contact with both punch and die, the reaction force on the punch rises sharply. If the punch displacement is not properly controlled, the punch will penetrate the work piece and springback putations are affected. Fig. 6 shows the profile of reaction force versus punch displacement for a setup with punch radius of , punch angle of ? and dielip radius of . In order to determine the punch displacement, Fig. 6 is enlarged at the location where the punch reaction force increases sharply (see Fig. 7).Fig. 7 shows the range of punch displacement between and mm. Within this range, the work piece is in full contact with the punch and die. Various punch displacements are tested is this region to obtain their springback values. Figs. 8 and 9 show the dielip and valley springback results for this range of punch displacements respectively. When the punch penetrates the work piece, springback values fluctuate and the work piece gets out of shape. The above figures show that punch displacements after yield springbacks that cause the work piece to go out of shape. Punch displacements before and close to have springbacks that vary insignificantly. Hence, a punch displacement of is used for this setup. For the other setups with different geometry, the punch displacements are determined using the above method.. Springback angleAfter the punch has travelled the prescribed displacement, the simulation removes both punch and die so that the work piece can springback. Springback angles at the dielip and valley regions are calculated using the Abaqus viewer. The viewer obtains the node coordinates after springback.Fig. 10 shows an example of the dielip region after springback. Coordinates of the 4 nodes are taken from the bottom, middle and top layers of the work piece. Thus, there are altogether 12 coordinates.Fig. 6. Plot of punch reaction force vs. punch displacement.For the bottom layer, the angles are calculated for lines that connect nodes 1–2, 1–3 and 1–4. Thus, three angles are obtained for the bottom layer. The average of these three angles is taken. The same method is used to pute the angle for the middle and top layers. Three average angles for the three different layers are obtained and averaged again to obtain an angle for the dielip region. The springback angle φ is defined as:Fig. 7. A closeup plot of the punch reaction force vs. punch displacement.Fig. 8. Dielip springback vs. punch displacement.Fig. 9. Valley springback vs. punch displacement.Fig. 10. Springback calculation.. Convergence testAfter a successful run of the first simulation using coarse element size, element refinement is required to achieve convergence of springback angles. As described earlier that linear elements are more suitable for large deformation in a bending process, Hconvergence will be used to determine the suitable size of the element. The convergence test is done using the tool configurations shown below:1. punch angle (α = 90?)。2. punch radius (R = )。3. dielip radius (rL = rR = )。4. die valley radius (r = R + t)。5. thickness of work piece (t = mm).The appropriate element size is chosen based on the accuracyof springback angles and time taken for the simulation toplete.. Hconvergence testFig. 11 shows the Hconvergence test of the dielip springback. Fig. 12 shows the Hconvergence test of the valley springback angle. The smallest element interval is and coarsest element interval is mm. These two figures show that springback angles for both dielip and valley regions converge as element interval observe from the convergence pattern that large element intervals provide springback angles that are not as accurate when pared to smaller elements. Although small element intervals yield better results, the putation time for the fine meshing is longer. Fig. 13 shows a plot of the simulation time versus element interval. For an element size of , the putation time required is 22443 s on the SGI Origin 2000 coarsest element size is and its putation time is 562 s. Intermediate elements sizes of and are tested. The element has an error of 2% and is times faster than that of the finest mesh , the element size of is used for the below simulations.Fig. 11. Hconvergence test for dielip springback.Fig. 12. Hconvergence test for valley springback.Fig. 13. A plot of simulation time vs. element interval.Fig. 14. Graph of valley springback vs. punch radius.. Effect of punch radius on springback anglesFig. 14 shows the graph of valley springback versus punch radius. Fig. 15 shows the graph of dielip springbackversus punch radius. These two figures provide the below observations:1. valley springback angle decreases as punch radius increases。2. decrement rate of valley springback decreases as punch radius approaches