【机械类毕业论文中英文对照文献翻译】冲压半径和角度对管向外卷曲过程的影响
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【机械类毕业论文中英文对照文献翻译】冲压半径和角度对管向外卷曲过程的影响,机械类毕业论文中英文对照文献翻译,机械类,毕业论文,中英文,对照,文献,翻译,冲压,半径,角度,向外,卷曲,过程,影响
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Int J Adv Manuf Technol (2002) 19:587596Ownership and Copyright Springer-Verlag London Ltd 2002Influence of Punch Radius and Angle on the Outward CurlingProcess of TubesYou-Min Huang and Yuung-Ming HuangDepartment of Mechanical Engineering, National Taiwan University of Science and Technology, Taipei, TaiwanUsing the theory of updated Lagrangian formulation, this studyadopted the elasto-plastic finite-element method and extendedthe increment determination method, the rminmethod, to includethe elements yielding, nodal contact with or separation fromthe tool, maximum strain and limit of rotation increment. Thecomputer code for a finite-element method is established usingthe modified Coulombs friction law. Conical punches withdifferent radii and angles are used in the forming simulationof hard copper and brass tube ends. The effects of variouselements including the half-apex angle of punch (?) and itsradius (R), the ratio of the thickness of the tube wall to themean diameter of tube, mechanical properties, and lubricationon the tubes outward curling, are investigated. Simulationfindings indicate that when the bending radius at the punchinlet (?) satisfies the condition of?c, curling is presentat the tube end. On the other hand, if?c, the tube endexperiences flaring. The variable?cis called the critical bend-ing radius. The value of?cincreases as the value of?increases. Furthermore, the findings also show that?cis neithercorrelated with tube material nor lubrication.Keywords: Elasto-plastic; Finite elements; Half-apex angle;Outward-curling process1.IntroductionThe process of forming convex edges of a metal tube is oftenemployed for connecting two tube parts, linking and lockinga tube part and its components, and connecting fluid pipelinesor combining complementary pipelines with reinforcement atthe tubes end. It is a common industrial technology relatedto tube ends. The tube end processes are generally dividedinto tapper flaring, flange flaring, step flaring, and curl forming.The tube-curling deformation discussed in this paper curls theopening of the tubes end outward in a circle. The simulationCorrespondence and offprint requests to: Dr You-Min Huang, Depart-ment of Mechanical Engineering, National Taiwan University ofScience and Technology, Taipei, Taiwan. E-mail: ymhuang?.twof the tube-curling deformation is a complex and difficulttask because the deformation process is highly nonlinear. Thenonlinear deformation characteristic is due to:1. The large displacement, rotation, and deformation duringmetal deformation.2. The nonlinear material deformation behaviour when metalmaterial experiences large deformation.3. The nonlinear boundary condition generated by the frictionbetween the metal and tool interface, and their contact con-ditions.These characteristics made the finite-element method the mostwidely used of the metal process analyses. To improve theprocessandincreaseindustrialproductivity,thisstudydeveloped an elasto-plastic finite-element computer code usingthe selective reduced integration (SRI) simulation method,which employs four integral point elements within the fournodes of a rectangle. The objective is to simulate the tube-curling deformation process.Nadai 1 studied tube-nosing in 1943. The theoretical induc-tion was an extension of the curved shell theory. Nadai assumedthe friction coefficient to be constant, and ignored the presenceof effective stress variation in the shell. Cruden and Tompson2 conducted a series of tube-nosing experiments to establishthe various limitations of tube-nosing and evaluate the effectsof various parameters. Manabe and Nishimura 37 also con-ducted a series of experiments on both conical tube-flaring andtube-nosing processes to investigate the effects of variousparameters on the forming load and stressstrain distribution.The parameters studied include the conical punch with differentangles, lubrication, material and tube wall thickness. Tang andKobayashi 8 proposed a rigid-plastic finite-element theoryand developed a computer code to simulate the cold-nosing of105 mm AISI 1018 steel. Huang et al. 9 simulated the cold-nosing process through elasto-plastic and rigid-plastic finite-element methods and compared the simulation results withexperimental data to verify the accuracy of the rigid-plasticfinite element. Kitazawa et al. 10,11 conducted experimentswith conical punches, and copper and brass materials with theassumption that the tube materials were rigid-perfectly plastic.They had explored previously the effect of the arc radius of588You-Min Huang and Yuung-Ming Huangthe punch, angle, and tube wall thickness on the tube-curlingdeformation process. Kitazawa 12 used carbon steel, copper,and brass as the tube materials in his experiment of tube-curling deformation. When the flaring energy increment at thefront edge of the tube end (?Wf) is greater than the energyincrement at the curl edge (?Wc), the tube end materialundergoes outward curling forming because of the absence ofunbending. On the other hand, if ?Wc? ?Wf, the tube endmaterial is attached to the conical punch surface because ofunbending and fails to curl or form flaring. The energy rulewas used to induce the deformation energy at the time of tubeforming for the purpose of comparison and verification, thusestablishing the criterion of outward curling.In this study, the same material constants and dimensionsused in other studies 12 are adopted in the simulation of theoutward curling process. The findings are compared with theresults reported in 12 to verify the accuracy of the elasto-plastic finite-element computer code developed.2.Description of the Basic Theory2.1Stiffness EquationAdopting the updated Lagrangian formulation (ULF) in theframework of the application of incremental deformation forthe metal forming process (bulk forming and sheet forming)is the most practical approach for describing the incrementalcharacteristics of the plastic flow rule. The current configurationin ULF at each deformation stage is used as the referencestate for evaluating the deformation for a small time interval?t such that first-order theory is consistent with the accuracyrequirement.The virtual work rate equation of the updated Lagrangianequation is written asFig. 1. Boundary conditions for the deformation tubes geometry ofoutward curling at an initial and a particular stage. Units, mm;R, punch radius; ?, half-apex punch angle; ?, bending radius atpunch inlet.?V(?*ij 2?ik?kj) ?ij-V +?V?jkLik?Lij-V =?Sft ?i?vi-S(1)where ?*ijis the Jaumann rate of Kirchhoff stress, ?ijis theCauchy stress, ?ijis the rate of deformation which is theCartesian coordinate, viis the velocity, t ?iis the rate of thenominal traction, Lij(= ?vi/?Xj) is the velocity gradient, Xjisthe spatial fixed Cartesian coordinate, V and Sfare the materialvolume and the surface on which the traction is prescribed.The J2flow rule?*=E1 + v?ik?jl+v1 2v?ij?kl3?(E/(1+v) ?ij?kl2? 2(0H?+ E/(1 + v) ?kl(2)is employed to model the elasto-plastic behaviour of sheetmetal, where H? is the strain hardening rate, ? is the effectivestress, E is the Youngs modulus, v is the Poissons ratio, ?ijis the deviatoric part of ?ij. ? takes 1 for the plastic state and0 for the elastic state or the unloading.It is assumed that the distribution of the velocity v in adiscretised element isv = N d(3)where N is the shape function matrix and d denotes thenodal velocity. The rate of deformation and the velocitygradient are written as? = B d(4)L = E d(5)where B and E represent the strain ratevelocity matrix andthe velocity gradientvelocity matrix, respectively. SubstitutingEqs (4) and (5) into Eq. (1), the elemental stiffness matrixis obtained.As the principle of virtual work rate Eq. and the constitutiverelationship are linear Eq. of rates, they can be replaced byincrements defined with respect to any monotonously increasingmeasure, such as the tool-displacement increment.Following the standard procedure of finite elements to formthe whole global stiffness matrix,K ?u = ?F(6)in whichK =?e?V?e?BT(Dep Q) B-V +?e?V?e?ETGE-V?F =?e?S?e?NTt ?-S?tIn these equations, K is the global tangent stiffness matrix,Dep is the elemental elasto-plastic constitutive matrix, ?udenotes the nodal displacement increment, and ?F denotesthe prescribed nodal force increment. Q and G are definedas stress-correction matrices due to the current stress at anystage of deformation.Influence of Punch Radius and Angle on Tube Curling589Fig. 2. (a) Deformed geometries, and (b) nodal velocity distributions in the outward curling of tubes at 12 different forming stages. Hard coppertube, n = 0.05; t0= 0.8 mm; ? = 60; R = 3.4 mm.2.2SRI SchemeAs the implementation of the full integration (FI) scheme forthe quadrilateral element leads to an excessively constrainingeffect when the material is in the nearly incompressible elasto-plastic situation in the forming process 13, Hughes proposedthat the strain ratevelocity matrix be decomposed to thedilation matrix Bdiland the deviation matrix Bdev, i.e.B = Bdil+ Bdev(7)in which the matrices Bdiland Bdevare integrated by conven-tional four-point integration. When the material is deformed tothe nearly incompressible elasto-plastic state, the dilation matrixBdilmust be replaced by the modified dilation matrix Bdilthat is integrated by the one point integration, i.e.B = Bdil+ Bdev(8)in which B is the modified strain ratevelocity matrix. Substi-tuting Eq. (8) into Eq. (7), the modified strain ratevelocitymatrix isB = B + (Bdil Bdil)(9)Explicitly, the velocity gradientvelocity matrix E is replacedby the modified velocity gradientvelocity matrix EE = E + (Edil Edil)(10)3.Numerical AnalysisThe analytical model of the tube-curling process is axiallysymmetric. Thus, only the righthand half of the centre axis isconsidered. Division of the parts finite elements is automati-cally processed by the computer. Since there is drastic defor-mation from the bending and curling at the tubes end, a finerelement division is required for this section in order to deriveprecise computation results. The lefthand half shown in Fig. 1denotes the sizes of the part and die at the beginning. Thedetailed dimensions are given in Table 1. In the local coordi-nates, axis 1 denotes the tangential direction of the contactbetween tube material and the tool, while axis n denotes thenormal direction of the same contact. Constant coordinates(X,Y) and local coordinates (l, n) describe the nodal force,displacement and elements stress and strain.590You-Min Huang and Yuung-Ming HuangFig. 3. As Fig. 2, but R = 3.8 mm.Table 1. Punch angle and radius.Punch apexPunch radius R (mm)angle ? (deg.)602.0, 2.4, 2.8, 3.1, 3.4, 3.8, 4.1, 4.4, 4.8, 5.1652.4, 2.8, 3.1, 3.4, 3.8, 4.1, 4.4, 4.8, 5.5, 5.4703.1, 3.4, 3.8, 4.1, 4.4, 4.8, 5.1, 5.4, 5.8, 6.1753.4, 3.8, 4.1, 4.4, 4.8, 5.1, 5.4, 5.8, 6.1, 6.4804.8, 5.4, 5.8, 6.1, 6.4, 6.8, 7.2, 7.6, 8.0, 8.4855.8, 6.1, 6.4, 6.8, 7.4, 9.0, 9.4, 9.8, 10.2, 10.6, 11.0Table 2 gives the material conditions of our simulation. Theexterior radius of the tube remains unchanged at 25.4 mm; butthere are three different tube wall thickness values, namely,0.4, 0.6 and 0.8 mm, used in the experiment and computation.The Poissons ratio and Youngs modulus of the hard coppertube are 0.33 and 110740 MPa, respectively. The Poissonsratio and Youngs modulus of brass are 0.34 and 96500MPa, respectively.3.1Boundary ConditionThe righthand half of the tube shown in Fig. 1 denotes thedeformation shape at a certain stage during the tube-curlingTable 2. Mechanical properties of materials used.MaterialsOuterHeat treatmentnFYield stressdiameterin vacuum(Mpa)?0.2(Mpa) thicknessCopper25.4 0.8As received0.09380220500C 1 h0.5363026As received0.05450280300C 1 h0.09450260400C 1 h0.4661050600C 1 h0.506404270/30 Brass25.4 0.8As received0.18730280? = F?n; ? = stress; ? = strain.deformation. The boundary conditions include the followingthree sections:1. The boundary on the FG and BC sections:?fl? 0, ?fn? 0,?vn= ?v nwhere ?flis the nodal tangential friction forces increment,and ?fnis the normal force increment. As the materialtoolcontact area is assumed to involve friction, ?fland ?fnarenot equal to zero. ?vn, which denotes the nodal displacementInfluence of Punch Radius and Angle on Tube Curling591increment in the normal direction of the profile of the tools,is determined from the prescribed displacement incrementof the punch ?v n.2. The boundary on the CD, DE, EF, and GA sections:?Fx= 0,?Fy= 0The above condition reflects that the nodes on this boundaryare free.3. The boundary on the AB section:?vx= 0,?vy= 0The AB section is the fixed boundary at the bottom end ofthe tube. The displacement increment of the node at thislocation along the y-axis direction is set at zero, whereas nodeB is completely fixed without any movement.As the tube-curling process proceeds, the boundary will bechanged. It is thus necessary to examine the normal force ?fnof the contact nodes along boundary sections FG and BC ineach deformation stage. If ?fnreaches zero, then the nodeswill become free and the boundary condition is shifted from(1) to (2). Meanwhile, the free nodes along the GA and EFsections of the tube are also checked in the computation. Ifthe node comes into contact with the punch, the free-boundarycondition is changed to the constraint condition (1).3.2Treatment of the Elasto-Plastic and ContactProblemsThe contact condition should remain unchanged within oneincremental deformation process, as clearly implied from theinterpretation in the former boundary condition. In order tosatisfy this requirement, the r-minimum method proposed byYamada et al. 14 is adopted and extended towards treatingthe elasto-plastic and contact problems 15. The increment ofeach loading step is controlled by the smallest value of thefollowing six values.1. Elasto-plastic state. When the stress of an element is greaterthan the yielding stress, r1is computed by 15 so as toascertain the stress just as the yielding surface is reached.2. The maximum strain increment. The r2term is obtained bythe ratio of the defaulted maximum strain increment ? tothe principal strain increment d?, i.e. r2= ?/d?, to limit theincremental step to such a size that the first-order theory isvalid within the step.3. The maximum rotation increment. The r3term is calculatedby the defaulted maximum rotation increment ? to therotation increment d?, i.e. r3= ?/d?, to limit the incremen-tal step to such a size that the first-order theory is validwithin the step.4. Penetration condition. When forming proceeds, the freenodes of the tube may penetrate the tools. The ratio r416is calculated such that the free nodes just come into contactwith tools.5. Separation condition. When forming proceeds, the contactnodes may be separated from the contact surface. The r5term 16 is calculated for each contact node, such that thenormal component of nodal force becomes zero.6. Sliding-sticking friction condition. The modified Coulombsfriction law provides two alternative contact states, i.e.sliding or sticking states. Such states are checked for eachcontacting node by the following conditions:vrel(i)l vrel(Il)l? 0(a)if ?vrel(i)l? ? VCRI, then f1= ?fn, the node is insliding state(b)if ?vrel(i)l? ? VCRI, then f1= ?fn(vrell/VCRI), the nodeis in quasi-sticking state.vrel(i)l vrel(I1)l? 0.The direction of a sliding node is opposite to the directionof the previous incremental step, making the contact node asticking node at the next incremental step. The ratio r6is thenobtained here, r6= Tolf/?vrel(i)l?, which produces the change offriction state from sliding to sticking, where Tolf (= 0.0001)is a small tolerance.The constants of the maximum strain increment ? andmaximum rotation increment ? used here are 0.002 and 0.5,respectively. These constants are proved to be valid in the first-order theory. Furthermore, a small tolerance in the check pro-cedure of the penetration and separation condition is permitted.3.3Unloading ProcessThe phenomenon of spring-back after unloading is significantin the tube-forming process. The unloading procedure isexecuted by assuming that the nodes on the bottom end of thetube are fixed. All of the elements are reset to be elastic. Theforce of the nodes which come into contact with the tools isreversed in becoming the prescribed force boundary conditionon the tube, i.e. F= F.Meanwhile, the verification of the penetration, friction, andseparation condition is excluded in the simulation program.4.Results and DiscussionFigure 2 shows the simulation results of curling forming ofcopper tubes with the punch semi-angle, ? = 60, and the arcradius, R = 3.4 mm. Figure 2(a) denotes the geometric shapeof the deformation, in which the last shape is the final shapeof the part after unloading. Figure 2(b) shows the nodal velocitydistribution during deformation. The last diagram shows thenodal velocity distribution after unloading. These figures showthat the tube end material enters smoothly along the punchsurface at the initial stage. After that, the tube end materialgradually bends outward and curls up, resulting in the so-called outward curling forming. Figure 3 shows the simulationresult in the case of R = 3.8 mm. The tube end material stillshows the so-called flaring forming after passing the arc induc-tion section and conical face forming.Figure 4 shows the result of curling simulation of coppertubes in the case of ? = 75 and R = 6.1 mm. In contrast,Fig. 5 shows the simulation result of flaring forming in thecase of R = 6.4 mm.Figure 6 shows the strain distributions corresponding tothose of Figs 2 and 3 at a punch travel of 12 mm. Figure 7592You-Min Huang and Yuung-Ming HuangFig. 4. As Fig. 2, but ? = 75; R = 6.1 mm.shows the strain distributions corresponding to those of Figs4 and 5 at a punch progress of 14 mm. The deformation ismostly the result of pulling in the circumference direction,bending in the meridian direction and shearing in the thicknessdirection. The maximum diameter of the tube end is the flaringforming. Thus, its strain value in the circumference directionis far greater than that of outward curling. The reduction intube wall thickness is also more significant in flaring forming.Thus, under the same progress, fracture of the tube end occursmore easily in flaring forming than in other situations.Figures 8 and 9 show the simulation results for hard copperand brass, respectively. The same tube wall thickness, tuberadius, lubrication condition, and half-apex of the punch areused in both simulations. However, different values of thepunch arc radius are used, which generate different types offorming, namely, flaring and curling. That is, if the arcradius, ?, at the punch entrance is greater than the criticalbending radius, ?C, then the tube end material forms flaringalong the punch arc induction section and conical face. Onthe contrary, if ?C? ?, the tube end material leaves thepunch surface and forms curling. Both figures show that thecritical bending radius increases as the half-apex of thepunch increases. A comparison shows that the results of ournumerical simulation are identical to the experimental datareported in 12. The so-called critical bending radius, ?C,can be expressed as follows:?C= RCt02=(RCmax+ Rfmin)2t02(11)whereRC= punch radius corresponding to ?CRfmin= minimum value of punch radius required to flareRCmax= maximum value of punch radius required to curlt0= wall thickness of tubesFigures 10 and 11 show the correlation between the non-dimensional critical bending radius (? C) and the half apexangle of thepunch withtwo differentwall-thickness-to-diameter ratios in the cases of hard copper tube and brasstube, respectively. The value of ? Ccan be defined as follows:? C=?C?(t0d0)(12)If ? Cis unrelated to the wall-thickness-to-diameter ratio, thenthe geometrical similarity law of the critical bending radiusInfluence of Punch Radius and Angle on Tube Curling593Fig. 5. As Fig. 2, but ? = 75; R = 6.4 mm.Fig. 6. Comparison of calculated strain distributions under curling andflaring processes. Hard copper tube, n = 0.05; t0= 0.8 mm; ? = 60.exists. Judging from the fact that the two curves shown inFigs 10 and 11 are almost identical, we can be certain of theexistence of the geometrical similarity law of the criticalbending radius of the tube materials.Fig. 7. As Fig. 6, but ? = 75.Figures 12 and 13 show the relationship between themaximum punch radius and half-apex angle of the punch whencurling occurs in the numerical simulation of wall thicknessvalues of 0.4, 0.6, and 0.8 mm in the cases of hard coppertube and brass tube, respectively. When values of both the594You-Min Huang and Yuung-Ming HuangFig. 8. The effect of tube thickness on the relationship between criticalbending radius and punch angle. Hard copper tube, n = 0.05.Fig. 9. As Fig. 8, but brass tube, n = 0.05.Fig. 10. Non-dimensional critical bending radius half-apex angle ofpunch relationships in outward curling of tubes. Hard copper tube,n = 0.05.Fig. 11. As Fig. 10, but brass tube, n = 0.18.Fig. 12. Maximum radius of punch to curl for various half-apex anglesof punch. Hard copper tube, n = 0.05.Fig. 13. As Fig. 12, but brass tube, n = 0.18.Influence of Punch Radius and Angle on Tube Curling595Fig. 14. Dependence of the critical bending radius on the work harden-ing exponent. ? = 60; t0= 0.8 mm.radius and the corresponding half-apex angle of the punch fallon the curve, the conditions will lead to flaring. On the otherhand, when both values fall below the curve, the conditionswill lead to curling. These figures indicate that the maximumpunch radius increases as the half-apex angle of the punchenlarges.Figure 14 shows the correlation between the materials workhardening exponent (n) and the critical bending radius. Regard-less of the variation in n value between 0.03 and 0.53 owingto different tube materials, results of the numerical simulationindicate that the value of the critical bending radius remainsconstant, thus forming a straight line. These findings are almostidentical to the experimental data shown in 12, and it impliesthat the critical bending radius at the time of curling isunrelated to the work hardening exponent.Figure 15 shows the effects of lubrication on the relation-ships between the critical bending radius and half-apex angleof the punch. The critical bending radius increases with theincreasing half-apex angle of the punch in the case of ? =0.05 and ? = 0.20. Furthermore, the result revealed that thereis a negligible difference in the magnitude of the criticalbending radius between the two lubricant conditions.Fig. 15. Dependence of the critical bending radius on lubricating con-ditions. Brass tube, t0= 0.8 mm.5.ConclusionAn elasto-plastic finite-element computer code was developedfrom the updated Lagrangian formulation to simulate tube-curling with a conical punch process. The high nonlinearity ofthe process was taken into account in an incremental mannerand an rmintechnique was adopted to limit the size of eachincrement step for a linear relation.The modified Coulombs friction law developed is a continu-ous function. As discussed earlier, this function can handle thesliding and viscosity phenomena of the tool and metal interfacethat are difficult to describe with the ordinary discontinuousfrictionmodel.TheSRI finite-elementisthefour-node-quadrilateral element in four integration points. Then the SRIcoupled with the finite-element large deformation analysis isfurther applied successfully to the analysis of the conical tube-curling formation process. According to the above analysesand discussion, the following conclusions can be drawn:1. A critical bending radius, ?C, exists. If the arch radius, ?,at the punch entrance is larger than the critical bendingradius, ?C, then the tube end material travels along thepunch arch induction part and the conical face to formflaring. On the other hand, if ?C? ?, the tube end materialleaves from the punch surface and results in curling. Thevalue of the critical bending radius increases as the half-apex angle of the punch increases.2 The critical bending radius, ?C, follows the geometricalsimilarity law. The value of ?Cincreases as the half-apexangle of the punch increases. We also learned that the valueof ?Cis not correlated with the tube material, lubricationconditions or the wall-thickness-to-diameter radius ratio.3. The circumference strain of flaring is greater than that ofcurling. Flaring also has greater reduction in thickness thancurling. Thus, tube end fracture occurs more easily inflaring. In contrast, products with curling can guaranteehandling safety and reinforce the end portion of tubematerials.4. The die shape is expressed as a numerical function. Thus,the finite-element model developed in this study can beused in the continuous shape of any tools and employed tosimulate all types of flaring or curling processes.References1. A. Nadai, “Plastic state of stress in curved shells: the forcesrequired for forging of the nose of high-explosive shell, formingof steel shells”, Presented at the annual meeting of the ASME,New York, USA, November 23December 1, 1943.2. A. K. Cruden and J. F. Thompson, “The end closer of backwardextruded cans”, NEL Report NO.511, National Engineering Lab-oratories, Glasgow, Sweden, 1972.3. K. Manabe and H. Nishimura, “Forming loads and forming limitsin conical nosing of tube
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