外文翻译--使用固体介质膨胀成型的三维有限元模拟 英文版

*Corresponding author.E-mail address: bryan.macdonalddcu.ie (B.J. Mac Donald).Finite Elements in Analysis and Design 37 (2001) 107116Three-dimensional nite element simulation of bulge formingusing a solid bulging mediumB.J. Mac Donald*, M.S.J. HashmiSchool of Mechanical and Manufacturing Engineering, Dublin City University, Dublin 9, IrelandAbstractBulge forming is an innovative manufacturing process which is used to manufacture many industrialcomponents. Common products include T-branches, cross branches and angle branches. The understandingof the process to date has been rather limited. Finite element analysis of the process incorporating the contactphenomenon between the die and the tube reported in literature has been limited, with only very particulararrangements and liquid bulging media being considered. Three-dimensional simulation of the process usinga solid bulging medium has not been reported. This paper presents a three-dimensional simulation of themanufacture of cross-branch components using a solid bulging medium. The e!ect of varying frictionbetween the bulging medium and the tube is examined and the history of development of the bulge and stressconditions in the formed component are presented. The results of the simulation are compared to thosepreviously obtained in simulation of the hydraulic bulging process. The explicit non-linear FE codeLS-DYNA3D is used to perform the simulations. ( 2001 Elsevier Science B.V. All rights reserved.Keywords: Bulge forming； Simulation； Finite element method1. IntroductionBulge forming is a widely used industrial process which is generally used to form complexcomponents from tubular blanks. Components are formed by restraining the blank in a die bearingthe desired shape and applying an internal hydrostatic pressure to the tube via a liquid or solidmedium. Bulge forming by pressure load alone is limited in the type and quality of components itcan produce. If a compressive axial load is applied to the ends of the tube in conjunction with thepressure load, then material can be pushed into the deformation zone during, forming thus0168-874X/01/$-see front matter ( 2001 Elsevier Science B.V. All rights reserved.PII: S 0 1 6 8 - 8 7 4 X ( 0 0 ) 00023-8Fig. 1. Bulge forming of cross branch components.preventing excessive thinning of the walls. Fig. 1 shows cross-sectional views of an arrangement forthe manufacture of cross-branch tubular components.Experimental studies concerning bulge forming using a liquid bulging medium have been widelyreported in literature, and recent numerical works 14 have contributed to a further understand-ing of the process. Bulge forming using a solid medium has been less widely reported. Al-Qureshi etal. 5 described an experimental process for axisymmetrically bulging thin-walled metal tubes. Ina separate presentation, Al-Qureshi 6 compared bulge forming using a polyurethane rod withhydraulic bulge forming. It was concluded, due to the complexity of analysing the polyurethaneforming technique, that the more highly developed hydraulic forming method gave better results.Filho and Al-Qureshi 7 presented an experimental method of forming tee joints from straighttubes using a urethane rod. The deformation was achieved by using repetitive loading and108 B.J. Mac Donald, M.S.J. Hashmi / Finite Elements in Analysis and Design 37 (2001) 107116unloading cycles and by varying the length of the urethane rod between cycles. Thiruvarudchelvanand Travis 8 described experiments to axisymmetrically bulge copper tubes using a urethane rod.Thiruvarudchelvan 9,10 developed an approximate theory for predicting the initial yield pressureand nal forming pressure required for bulging a metal tube using a urethane rod. The theory madeuse of experimentally determined friction characteristics and was compared with experimental results.The above sources have investigated the bulge-forming process using a solid medium in severalways, and some understanding of the process has been accumulated. However, in order to optimisethe design of the manufacturing process and hence the product, a deeper understanding is required,particularly of the internal stress distributions. In order to investigate the bulging and deformationmechanisms, a non-linearnite element analysis has been conducted to simulate the solid mediumbulging process of cross-branch components.2. Finite element modelThe explicit FE code LS-DYNA3D is used in this paper to simulate the bugle forming process.Simulations were carried out to bulge a copper cylindrical tube of 1.03 mm wall thickness,24.12 mm diameter and 107 mm length into a cross joint. The diameter of the branches was equalto that of the main tube. The length of the bulging medium was modelled slightly shorter than thatof the tube. This accurately modelled the actual situation which facilitates insertion of the punchinto the tube.By taking advantage of symmetry it was possible to model one-eighth of the problem.Fig. 2 shows the discretised model. Some points are indicated for further discussion in the resultssection. Both the die and the tube were modelled using eight-node solid elements. A total of 2875elements describe the model. The interfaces between the die and the tube, and between the bulgingmedium and the tube, were modelled using an automatic surface to surface contact algorithm. Thealgorithm uses the material properties of both contacting surfaces to calculate the sti!ness of thecontact elements. An elastic coulomb friction law was assumed and a coe$cient of friction of 0.15was assigned between the die and the tube. This value is representative of values measuredexperimentally 11. The coe$cient of friction between the bulging medium and the tube wasvaried in order to determine its e!ect on the process.The tube material properties were determined from a compression test on annealed copper.A piecewise linear elasto-plastic material model was used with the following parameters: Youngsmodulus124103 MPa, yield strength160 MPa, tangent modulus925 MPa, Poissonsratio0.3, density8.9106 kg/ mm3. The material parameters for the bulging medium werechosen to approximately represent lead. Soft metals such as lead are widely used as a bulgingmedium in industrial applications and are capable of producing far more deformation thanelastomers.3. Loading and solutionAs one-eighth of the problem was modelled by taking advantage of symmetry, the tube andbulging medium nodes were constrained in the appropriate directions. The die was constrained asB.J. Mac Donald, M.S.J. Hashmi / Finite Elements in Analysis and Design 37 (2001) 107116 109Fig. 2. The nite element model.Fig. 3. Loading pattern used.a rigid body. The internal pressure was generated by applying a prescribed displacement to thenodes at the end of the ller material. The axial deformation in the main tube was generated byapplying a prescribed displacement to the nodes at the edge of the tube end. In the actual formingprocess the axial loads on theller and tube are applied by a shouldered punch, where the length ofthe shoulder controls the amount of deformation of theller material prior to axial deformation ofthe tube. In this case the punch was not modelled, instead the nodes of the tube under the punchwere constrained in the radial and circumferential directions, which is equivalent to the punchbeing in place. The loading pattern used in all simulations is shown in Fig. 3. This pattern wasdeveloped in order to maximise branch height.110 B.J. Mac Donald, M.S.J. Hashmi / Finite Elements in Analysis and Design 37 (2001) 107116Fig. 4. Distribution of Von-Mises stress in the tube at 20% of axial displacement.In order to gain a better understanding of the process, a number of simulations were carried outin which the process parameters were varied. In all simulations the axial load was appliedsimultaneously to the ller material and tube. The rst simulation was carried out using a zerocoe$cient of friction between the bulging medium and the tube. In subsequent simulations thecoe$cient of friction was varied between zero and 0.5.4. Results and discussion4.1. Development of the bulgeIn order to study the development of the bulge the loading case in which the value of frictionbetween the ller material and tube was set as 0.3 was chosen. The case was taken as beinga representative of the values measured experimentally 11. Figs. 47 show the development of thebulge in the tube. It can be seen from the gures that the bulge initially develops as a hump justahead of the die bend. As the pressure exerted by the bulging medium increases then the bulgebegins to become more regular. By the time 40% of the axial displacement has been applied, thehump has disappeared and the branch top is almost#at, as shown in Fig. 5. From this stage, due toincreasing internal pressure and axial displacement of the tube, the bulge develops in a regularfashion as illustrated in Figs. 6 and 7.It is interesting to note the development of stress in the tube as the bulge develops. At 10% of theloading the maximum stress is located at point A. As the process continues, the location ofmaximum stress does not move, but, the stress gradient around the area of maximum stressbecomes more concentrated as loading increases. It can be seen from the deformation of theelements in this region that the high stress is due to a combination of compressive axial strain andtensile hoop strain.B.J. Mac Donald, M.S.J. Hashmi / Finite Elements in Analysis and Design 37 (2001) 107116 111Fig. 5. Distribution of Von-Mises stress in the tube at 40% of axial displacement.Fig. 6. Distribution of Von-Mises stress in the tube at 70% of axial displacement.4.2. Ewect of frictionThe e!ect of varying the friction between the bulging medium and the tube was investigated bycarrying out four simulations which used friction factors of 0, 0.15, 0.3 and 0.5, respectively. Themaximum branch height produced in each case was around 11 mm and decreased with increasingfriction, with a di!erence of only $1 mm between each simulation. The maximum stress at pointA in each case varied between 300 and 328 MPa with the level decreasing with increasing friction.The maximum stress in the branch top varied between 185 and 200 MPa and again decreased withincreasing friction. An examination of the thinning behaviour of the branch top showed that for112 B.J. Mac Donald, M.S.J. Hashmi / Finite Elements in Analysis and Design 37 (2001) 107116Fig. 7. Distribution of Von-Mises stress in the tube at 100% of axial displacement.Fig. 8. Development of stress in the top central node of the cross branch during simulation 2 (Friction0.15).a friction factor of 0 the branch top had thinned to 90%, of the original thickness. The simulationsfor increased friction exhibited almost no thinning of the branch top. The thickening behaviour ofthe main tube, ahead of the punch, varied from 120% to 102% of the original thickness. Lowerfriction values produced less thinning of the branch top and more thickening of the main tube.B.J. Mac Donald, M.S.J. Hashmi / Finite Elements in Analysis and Design 37 (2001) 107116 113Fig. 9. Development of stress in the top central node of the cross branch during simulation 3 (Friction0.3).The development of stress was also in#uenced by the level of friction between the bulgingmedium and the tube. Fig. 8 shows the development of stress in the top central node of the crossbranch during simulation 2 (friction0.15). The stress development is quite erratic indicatingsome stickslip behaviour between the bulging medium and the tube. When compared with Fig. 9,which shows the development of stress by simulation 3 (friction0.3) it can be seen that increasingfriction results in a much smoother development of stress in the bulge.4.3. Comparison with hydraulic bulgingThe authors in a previous work 4 have simulated hydraulic bulge forming of cross-branchcomponents using the same blank geometry and material properties as used in this analysis. Inorder to compare the solid medium bulging process with the hydraulic bulging process, resultsfrom 4 were compared with results here.Clearly, the use of a solid medium allows for greater branch heights and less thinning of thebranch top. In order to compare stress conditions a further simulation was run to bulge a compon-ent to the same height as that experienced in the simulation of the hydraulic process (11.5 mm).Figs. 10 and 11 show the distribution of Von-mises stress in the formed components usinghydraulic and solid bulging processes, respectively. When the stress distributions were compared itwas noticed that the solid medium bulging process produced less stress in the formed component.The distribution of stress in the formed component was quite similar in the area around the point of114 B.J. Mac Donald, M.S.J. Hashmi / Finite Elements in Analysis and Design 37 (2001) 107116Fig. 10. Distribution of Von-Mises stress in hydraulically bulged cross branch 4. (height11.5 mm).Fig. 11. Distribution of Von-Mises stress in solid medium bulged cross branch. (height11.5 mm).maximum stress for both simulations. The main tube has stressed a great deal in the simulation forsolid medium bulging. This is due to friction between the bulging medium and the tube causinggreater axial stress to be imparted on the tube than was present in the hydraulic simulation. Animportant advantage of the solid medium process noticed during the comparison was that it canproduce the same branch height as the hydraulic process but with far less reduction in length of theunbulged tube. This is again due to friction between the bulging medium and the tube causingmaterial to be pulled into the bulging zone 57.B.J. Mac Donald, M.S.J. Hashmi / Finite Elements in Analysis and Design 37 (2001) 107116 1155. ConclusionThis paper presents the results from thenite element simulation of solid medium bulge formingof cross-joint components. The simulation results indicate that(1) The use of a solid bulging medium allows for greater branch height, less thinning of thebranch top and less stress in the formed component when compared to the hydraulic bulgingprocess.(2) Increasing friction between the bulging medium and the tube results in less stress in theformed component, more thinning of the branch top, less thickening of the main tube anda smoother stress development during the forming process.(3) Increasing friction between the bulging medium and the tube has little e!ect on the resultingbranch height in the formed component.(4) When compared to a cross branch of similar height obtained by the hydraulic bulgingprocess, it is clear that the solid medium bulging process results in far less reduction in thelength of the tube.References1 H. Bauer, FE simulation of the production process of builder camshaft, O.C. Zienkievicz, J.L. Chenot, R.D. Wood(Eds.), in: Numerical Methods in Industrial Forming Processes, Balkema, Rotterdam, 1992, pp. 585600.2 M. Ahmed, M.S.J. Hashmi, Three-dimensionalnite element simulation of