【机械类毕业论文中英文对照文献翻译】通过倾斜样品镦粗加工实现成形压力机加载
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通过倾斜样品镦粗加工实现成形压力机加载K. Chodnikiewieza*, S.B. Petersenc, R.Baiendrab, P.A.F. Martinsca波兰华森、85.02-524、那巴特科技大学b斯特拉斯克莱德大学，格拉斯哥，蒙特罗斯大街75号，英国c.葡萄牙Lisboa Codes1096号Rorisco Pais大街、专科高等教育学院摘要：对倾斜样品进行鐓粗是加载成形压力机的一个途径，而通过垂直和水平施力进行鐓粗加工是一种可行的方法。然而，由于样品界面的条件分配不均，需要对该方法的使用范围进行量化。倾斜样品的塑性形变通过使用一种名为PAST2的FE代码进行分析。通过分析，本文提出了试验样品选择及试验范围的指导方针。很明显，润滑条件、样品的减少、及垂直与水平力度比，在变形的整个过程中并非恒定不变。因此，当研究这些统一特性的范围时，有对压力特性进行分必要析。关键词：成形压力,鐓粗,样品,有限元法1.引言鐓粗也许是最常用的金属成形方法。实践中，鐓粗用于单独的成形工序，或用于更加复杂成形工序的初期阶段。在机械测试中，鐓粗用于提取流变曲线、摩擦参数及可加工性。鐓粗试验还常用于确定分析方法和数值方法的可靠性，以描述与金属流等有关的具体现象。为了便于评估压力弹性，对平行表面的金属样品进行鐓粗，从而产生一种与压型压力（通常是垂直的）轴相平行的压力。这种通过对压型压力的弹性偏向进行测量的方法，引申出成形压力的一个垂直的和两角的刚性系数1,2。尽管如此，在许多情形中，成形压力的加载来自垂直和水平两个方向。水平方向的压力FH和垂直方向的压力FV的比例构成了成形压力F，其值可高达0.2；因此，在压力弹性检测的过程中，须对这些加载条件进行模拟实验。可以通过在两个斜垫圈（图1（a）之间的液压千斤顶进行加压，但活塞与气缸的摩擦力会使得该方法有失准确。通过最近提出的另一方法能获得更具代表性的加压条件，这种方法建立在鐓粗倾斜样品的基础上（图1（b）遗憾的是，这种鐓粗方法并不常见：其流程的某些方面Ramaeker和Kals4已进行过报道。两人皆考虑到了材料流动的不稳定性是因工具角度未对准造成的。然而，Ramaekers和Kals并未对此流程进行详述。因此，在使用鐓粗倾斜样品进行压力弹性实验之前，需要加以更全面的分析。2. 求解方法对倾斜样品的鐓粗可使用2-D有限元程序PLAST2进行模拟。该程序建立在变分原理的基础上，由葡萄牙里斯本技术专科高等教育学院开发。该原理要求所有容许速度uj能满足兼容性和不可压缩性条件，并能使如下函数成立：在这个表达式中，是有效应力，而是有效应变率，Tj是表面牵引力，SF是牵引力作用的表面，V是体积。有效应力和有效应变率分别做如下定义：其中和分别表示偏应力和偏应变率。若函数的值不变，其一阶变分消失；即=0.不可压缩性的定义如公式：其中，是体积应变率。为将其考虑在内，使用了补偿函数，这要求对函数进行修改5。修改后的函数的公式如下：其中，K是一个大的正罚常数。若函数的第一个变分消失，得到：那么，平衡方程式和体积固定性约束将会同时得到满足（5）。公式（6）对以上变分方程进行了描述。忽略样品及模具的弹性形变，则有效应变率和偏应力之间的关系可通过Levy-Mises公式进行表达：其中，代表塑性区，与屈服应力相等；的公式如下：其中，C和n表示材料常数，表示有效张力：其中，表示张力。另外，使用了Wanheim和Bay7提出的摩擦力模型。按照该模型，摩擦应力的公式是：当比例小于1.5，工作界面的摩擦力和正应力p成比例；当大于3时，相对摩擦应力=fa接近一个恒定值，该值等于f（图2）。为了消除在中心处摩擦应力突变，得出如下近似值：其中，j是单位矢量，其方向与模具的工作材料速度Us 的方向相反；并且与Us 相比，V0是一个小的正数（6）。8中提供了PLAST2的有关具体信息。使用PLAST2进行的金属流动分析结果与实验结果吻合。3. 假设本分析采用了如下假设： （i）具备平面应变鐓粗条件 （ii）当工作材料相对上模进行运动时，在下模样品界面上主要以黏着摩擦为主。这种假设与实验条件相符，即下模粗糙，而上模及样品较光滑，润滑良好。 （iii）低碳钢的抗屈强度和有效应变速率质检的关系假设为： （iv）样品的初始几何关系可通过Ho/Bo的比例体现，其中Ho是沿着样品中心线测得的原始平均高度，Bo是样品原始宽度，而是模具和样品构成的角度（详见图1）。（v）如下比例用于确定样品高度的减少量，其中H是样品在中心线处的当前高度。 （vi）分析采用了如下基本参数：Ho/Bo = 0.5, f= 0,5,= 10 。对于该分析，在其它参数保持不变的时候，每个参数都会与规定值有所差异。4. 实验结果平形及倾斜样品的变形中，后者的特点可从图3的基本参数及相对压力p/o在样品上表面的分部可以看出。样品与上模之间的间隔受到接触面的摩擦力的影响。倾斜样品的变形描述如下：（i）流变模型并不关于中心线对称。（ii）流向楔形样品较厚一边的金属体积比流向其较薄的一边的金属体积大。（iii）中点N向样品较薄一边位移。在该点上，物质流没有相对于上模的正切分量。（iv）在样品较厚一边处，产生了上模与倾斜样品之间的间隔。倾斜样品的变形是减值e、摩擦力f，角度及Ho/Bo比例的一个复变函数。对于不同减值e及当常数f=0.5, = 10 和 Ho/Bo=0.5时，样品的对应形状如图4（a）所示。N-N线代表了中点在鐓粗过程中所处的位置，而R-R则代表了由于成形力的加载而产生的位移。对于一个较小的减值，中点的相对坐标XN/Bo会保持不变（如图4（b），但是对于较大的减值，XN/Bo比值会明显改变。变形在不同截面的值与上模和样品之间的间隔相关：这种关系在图4（a）中得到明显体现。由于间隔改变样品与上模的接触面积，中点会向样品较薄一边偏移。当e = 20%, = 10 及 Ho/Bo = 0.5时，对于各种摩擦因子的样品形状如图5所示。进行了如下观察：（i）间隔取决于摩擦因子：当f= 0时，在变形初期间隔较大；而当f= 1.0时，则不会产生间隔。图7 比FA/ FV作为过程参数的函数：(a）（FA / FV）（FL）为E和f= 0.5不同的值，f= 0.5；（b）(FH / fvxe）为和常数f = 0.5的值不同，f= 0.5；（C）（FH / fvxe）为F和FL =常数10 f的不同值=0.5；（d）（FH / FV）（E）不同Fm / Fv= 10，F = 0.5（ii）摩擦因子对中点位置的影响比对样品形状的影响大。相对于常数e、f及Ho/Bo比例的实际线性函数XN()/Bo如图6（b）所示。所得的结论与前一种情况类似：即角度影响到中点的位置，而非样品变形后的轴向截面的形状。鐓粗合力F（FH, Fv）倾向中线。因为工作材料从上表面滑向样品的薄边和厚边，其结果是，合力的倾斜角度会比上模的倾斜角度小，即：相对于不同的e，常数f及Ho/Bo比值的函数(FH/Fv) ()如图7（a）所示。其中可见FH/Fv比率并不受e的水平的影响。这种情况中，可使用如下的等式：这种特点符合压型弹性实验，因为它简化了对力比相关数据的使用。所选的及常数f和Ho/Bo的函数(FH/Fv)(e)（如图7（b）所示）证实了如上特点。图7（c）和图7（d）表明，在压型机刚性实验的过程中，应确保降低摩擦力和Ho/Bo比值。如果这点可以保证，FH/Fv比值会实际上不受样品减少的影响。当Ho/Bo = 1.0时（如图7（d）所示），函数(FH/Fv) 偏离于其它曲线趋势是因为使用的样品太大，后者在形变初期阶段产生了弯曲。单位成形压力（与样品长度相关）垂直分量与减值e的关系如图8所示。在图8（b）的基础上，能得出结论：对于相对较小的变形，成形压力的垂直分量实际与模具之间构成的角度无关，包括当= 0时。该特点简化了用于压型机弹性测试的样品选择流程。成形压力的垂直分量可以按照表面平形的样品鐓粗常用的等式计算而得。合力的第三个特点是作用点i。该点的位置在图4（a）已经以R-R线的形式进行了确定。正式的作用点与中点一致。因此，对于中点所得出的结论，同样适用于合力的作用点。图8.的关系，C：，：成形力的FvL作为过程参数的函数：（一）皮质部分的和P = 10T选定的值Ho / B O = 0.5；（b）的选定值/我和F = 0.5，H = 0.5.5. 结论 1. 对于样品的固定摩擦和固定相对高度Ho/Bo，成形压力水平分量与垂直分量的比例FH/Fv主要取决于样品对上模的倾斜度。 2. 可以选择一个倾斜样品，以便FH/Fv比例不会受到样品的减少的影响。为了达到该目的，应满足如下条件：（i）样品的减少应保持相对较低，如当f= 0.5时，其值不应超过20，而当摩擦较高时，允许较大的减少值；（ii）样品相对高度Ho/Bo应小于0.5；（iii）样品与上模界面之间的摩擦应保持相对较低。参考文献I E. Doege, Static and dynamic stiffness of presses and someeffects on the accuracy of workpieces, A,. CIRP, 29 (1980).2 DIN 55 189, Ermittlu,g ton Kenmrerten fiir Pressen der Blechrer-arbeitung hei stati.sher Belastung, Teil 1,2, 1985.3 K. Chodnikiewicz, Balendra R. and T. Wanheim, A new conceptfor the measurement of press stiffness, J. Mater. Process. Tech-nol., 44 (1994) 293-299.4 J.A.H. Ramaekers and J.A.G. Kals. lnstable material flow inextrusion and upsetting, Amt. CIRP, 31 (1992).51 O.C. Zienkiewicz and K. Morgan. Finite Element and Applica-tip, Wiley, New York. 1983.6 S. Kobayashi, S.I. Oh and T. Altan, Metal piThing mtd theFinite-Element Method, Oxford University Press, New York,Oxford. 1989.7 N. Bay and T. Wanheim, Real area of contact and friction stressat high pressure sliding contact. Wear. 38 11976) 201 - 209.8 P.A.F. Martins, J.M.C. Rodrigues and M.J.M. 8arata Marques,Numerical and experimental simulation of cold forging pro-cesses, XIII SemintJrio National de Forjamento, UFRGS, PortoAlegre, Brazil, 1993.9 M. Arentoft, S.B. Petersen, J.M.C. Rodrigues, P.A.F. Martins,R. Balendra and T. Wa-heim, Review of the res-arch on theinjection forging of tubular material, J. Meter. Process. Tech-nol., 52 (1995) 460-471.9Jeurmd or Materials Processing Technology ELSEVIER Journal of Materials Processing rcchnolog 68 (1997 13 I . . . . . . . Loading of forming presses by the upsetting of oblique specimens K. Chodnikiew;z ,*, S.B. Petersen , R. Balendra b P.A.F. Martins ll2trs, Unirerity o/TcImoh,gv ul.Vmhutta $5. 02-524 lVtr.a,. P, hmd h Utlircrsil) d Stralhclvdc. 75 .llotlrose Street. Glasgow GI IX J, I.:K bl.titulO Superior Tc;cnico.41. Rori.co Pai. 1096 Lishoa C,&,. Porlu,tttl Received 21 September 1995 Abstract The upsetting of oblique specimens as a means of loading forming presses with vertical and horizontal forces is a feasible concept, however, due to the changing distribution of conditions at the specimen intertace, the scope of this approach needs to be quantified. The plastic deformation of cbaque specimens was analysed using an FE code named PLAST2. The analysis suggests guidelines for the sciectiou of test specimeas and the scope of such tests: it is apparent that the lubrication conditions and specimen reduction, and the ratio between the vertical and horizontal forces, do not remain constant over the entire course of the deformation. Hence, the analysis of measurements of the behaviour of the press would be meaningful if the range of uniform behaviour of such tests was obse,wed. , 1997 Elsevier Science S.A. Kevword*: Forming presses: Upsetting: Specimen: Finite element method 1. Introduction Upsetting is. perhaps, the most commonly exploited metal forming process. In practice, upsetting is used either as a separate forming process or as the primary stage of more complex forming operations. In mechani- cal testing, upsetting is used for extracting flow curves and friction-dependent parameters and for defining workability. Frequently, upsetting experiments are per- formed to determine the reliability of analytical and numerical methods, to illustrate specific phenomena which relate to metal flow etc. In order to evaluate press elasticity, metal specimens with parallel faces are upset, this producing a force which is parallel to the main (normally vertical) axis of the press. This method of loading together with mea- surements of the elastic deflections of the press enables the development of definition of a vertical and two angular stiffness coefficients of the press 1,2. However, in many practical cases, the press is loaded not only vertically but also horizontally. The ratio between the horizontal F H and vertical Fv components of the forming force F may be as high in value as 0.2; * Corresponding author. Fax: + 351 I 8474045. 0924-0136/97/$17.110 1997 Elsevier Science S.A. All rights reserved. Pll S0924-0136(96)02517-4 hence, these loading conditions would have to be simu- lated during press elasticity measurements. It is possible to load the press with a hydraulic jack located between two oblique washers (Fig. I(a), but friction between the piston and the cylinder renders the mcthod inaccu- rate. A recently proposed 3 alternative method which is based on the upsetting of oblique specimens (Fig. l(b) provides a more representative loading condition. Unfortunately, this form of upsetting is not common: some aspects of this process have been reported by Ramaekers and Kais 4, who considered the unstable L I I I Ram i Ram I L Upper Die spo i ,;-.-fl i p I I I I I (a) (b) Fig. 1. Methods of loading a press with vertical and horizontal forces: (a) using a hydraulic jack; Ib using the upsetting of an oblique specimen. 14 K. Chodnikiewicz et aL / Journal oJ Materials Processhlg Technology 68 (1997) 13-18 the functional/7 5; the modified functional, /Tin, hav- t ing thivfrm: J fs 1.0 ( 1 . I, x/k ,0 o9 f Ht= 0dV+K 2-(e) .-dV- Tju/dS (5) 08 - 0.8 i. - 0. where K is a large positive penalty constant. Now, if the 06 -o6 first variation H in the functional / vanishes, then: o, ; f o, o., 6/7= g6dV+K vt,dV- T;ujdS=O (6) - 0.3 02 and both the equilibrium equations and the volume o.z constancy constraint will be, approximately, satisfied 0.1 p/Go simultaneously 5. The above variational equation is 0 ! 2 3 , 5 Fig. 2. Relative frictional stress as a function of normal stress and friction factor 71. flow of material which results from the angular mis- alignment of the tools. However, Ramaekers and Kals did not provide a detailed insight to the process, thus prior to the use of the upsetting cf oblique specimens for press elasticity experiments, a more comprehensive process analysis is required. 2. Method of solution The upsetting of oblique specimens was simulated using the 2-D finite-element program PLAST2 which was developed at lnstituto Superior Tcnico, Lisbon. This program is based on the variational principle which requires that all admissible velocities uj, which satisfy the conditions of compactibility and incompress- ibility, make the functional: /7= fvatdV- s TjujdS (I) F stationary. In this expression 0 is the effective stress, is the effective strain-rate, Tj is the surface traction, SF is the surface on which tractions are prescribed and V is the volume. The effective stress and effective strain-rate are defined as follows: -2 , t 12 0 = t aoaj (2) = (23 eoe) ;2 (3) where a and v. are the deviatoric stress and the deviatoric strain-rate, respectively. When the functional H is given a stationary value, its first order variation H vanishes; i.e. / = 0. The definition of incom- pressibility has the form: , = kk = 0 (4) where b is the volumetric strain-rate. In order to take this condition into account, the penalty function method was applied, which requires the modification of presented in detail in 6. The elastic deformation of the specimen and dies is neglected and the relationship between the effective strain-rate and the deviatoric stresses is provided by the Levy-Mises flow rule: 3, in which 0 for the plastic region is equal to the yield stress ao which, in turn, is defined by the equation: ao = C(g) (8) where C and n denote the material constants, and g is the effective strain: g = ( ee,/) 12 (9 in which e e is the strain. Further, the Wanheim and Bay 7 friction model was applied. According to this model, the frictional stress r is described by: r =j- k (10) in which f is the friction factor, is the ratio between the real and apparent contact areas, and k is the shear yield stress of the work-material, defined by: k = (l l) The frictional, , and normal, p, stresses at the die- workpiece interface are proportional for p/ao 3 the relative frictional stress z/k = f approaches a constant value which is equal to f (Fig. 2). In order to eliminate the sudden changes of the frictional stress at the neutral point, the following ap- proximation was defined: : . tan - t (12) /r in which j is the unit vector in the direction opposite to the velocity us of the work-material relative to the die and vo is a small positive number in comparison with us 61. Detailed information about PLAST2 is provided in 8. Metal-flow analysis achieved using PLAST2 com- pares well with the results of experiment 9. K. Chmhtikwwic- tt al. .lourmd ftl Materials Proccsing Techmdogy 6g (19971 I3-18 15 ,1 ! 1+/ i _-._. _,_= . .=-:- :. _ -_-:- -. Oo (a) p/Co Z / 3 ,;- t ! ,. ,/ -.-. o I . , - . - _ - ._=,. B (b) Fig. 3. Distorted mesh for the upseuing of: la) a parallel pecimen: and (b) a edge-shaped specimen: together :ith the distribution of normal p and frictional r stresses. .-.- . / . : 1:) . ,j o (a) x N i) o :; - QI - ttl 0 30 4, 50 0 , -q -01- -02 - - -0.3 - I -0.4 :- (b) , +o.-.- Fig. 4. Presenting: (a) specimen shapes and positions of the neutral point N; and (b) the function Xn(e)Bo for.f= 0.5, fl = 10 and H,:B, = 0.5. 3. Assumptions The following assumptions apply to the analysis: (i) Plane-strain upsetting conditions prevail. (it) Sticking friction prevails on the lower die-speci- men interface, whilst the work-material slides relative to the upper tJie. This assumption corresponds to experi- mental condi0ons where the lower die is rough whilst the upper die and the specimen are smooth and well lubricated. (iii) The relationship between the yield strength and the effective strain rate for mild steel has been assumed to be: ao= 740() TM (MPa) (13) (iv) The initial geometry of the specimen is character- ised by the ratio Ho/Bo in which Ho is the original average height measured along the centre line of the specimen, B0 is the original width of the specimen and fl is the angle between the die and the specimen (refer to Fig. 1). (v) The ratio: Ho-H e - - - 100% (14) Ho is used to define the reduction in height of the speci- men, where H is its current height at the center line. (vi) The following basic parameters were used for the analysis: Ho/Bo = 0.5, f= 0,5, fl = I0 . For the analysis. each parameter was varied about these prescribed val. ues, whilst other parameters were retained constant. 16 K. Chodnikiewicz et al./Journal oJ Materials Processing Technology 68 (1997) 13-18 .,S 7- f=oo ,- _ _11 11 /I , f=o.ol,./ - . L-oo ,.I, I . 1 O: (b) (a) X _ 3 _ . B0. ! 0.2 z 0. - 0.25 0.5 0.75 , i 17 . . -o.t L -0.2 -0.3 -0.4 r I .o -f Fig. 5. Presenting: (a) specimen shapes and the position of the neutral point N; and (b) the function X(J)/Bo for e = 20%, # = 10 and Ho/Bo = 0.5. , / t.;/= X N . 0 0 0 i i i .5 10 lfi (a) (b) Fig. 6. Presenting: (a) specimen shapes and the position of the neutral point N; and (b) the function XN(fl)/B o for e = 20/,. fl = 0 and Ho/Bo = 0.5. 4. Results The deformation of parallel and oblique specimens, the latter being charat:terised by basic parameters, are shown in Fig. 3, together with distributions of relative normal pressures P/go on the upper face of the speci- men. Gaps which can be observed between specimens and the upper dies are due to the difference in friction at the contact surfaces. The deformation of the oblique specimen may be described as follows: (i) The flow pattern is not symmetrical about the center line. (ii) A larger volume of metal flows toward the thicker side of the wedge-shaped specimen than towards its thin side. (iii) The neutral point, N, at which material flow has no tangential component relative to the upper die, is displaced towards the thin side of the specimen. (iv) A gap between the upper die and the oblique specimen is created at the thick side of the specimen. The deformation of the oblique specimen is a com- plex function of the reduction e, the friction factor f, the angle # and the ratio Ho/Bo. Specimen shapes are shown in Fig. 4(a) for different reductions and constant f=0.5, fl= 10 and Ho/Bo=0.5. The line N-N repre- sents the position of the neutral point over the course of the upsetting whilst the line R-R refers to the displacements location of the resultant forming force. For a small value of reduction, the relative coordinate XN/Bo of the neutral point remains practically constant (Fig. 4(b), but for greater reductions, the ratio Xr/Bo changes significantly. The trans-sectional value of de- K. Chodnikiewic: et al./Journal of Materials Processing Iechnolog), 68 (1997) 13-18 17 formation refers to the development of the gap bet,veen the upper die and the specimen: this is shown magnified in Fig. 4(a). Since the gap changes the contact area between the specimen and the upper die, the neutral point moves towards the thin side of the specimen. Specimen forms tar e = 20%, fl = 10 and Ho/Bo = 0.5 are shown in Fig. 5 for various friction factors. The following observations are made: (i) The creation of the gap depends on the friction factor: for f= 0 the gap prevails at an early stage of the deformation whilst for f= 1.0 the formation of the gap is not initiated. F.I. FdF. FH,rL, c 2s 0 2 0 05- X - s (a) 0 deg 1 C 25 02 0:5 01 0 05 0 o- . -. . . . . . -o- -o,- S e=lO% 20% 30% 40% 50% J - 15 - 50 -X- 0 - 10 x - t0 2c (b) o ,o C %1 so 025 - FJFv 0 2- ._._.-.- -o- t=-o o - 050 ooS 1.0 -o- 0.25 -x- 0.75 0 + I I SO ,o 20 (c) = ,o e%l 025 . 0.2 . # . . / HO/BO=0 25 005 - -O- 1.0 o ,o e%1 so 0 2o (d) 0 Fig. 7. The ratio F,/Fv as a function of process parameters: ta) (Fa/Fv)(fl) for different values of e and = 0.5, Ho/Bo = 0.5; (b) (FHFvXe) for different values of # and constant f= 0.5, Ho/Bo = 0.5; (c) (FH/FvXe) for different values of f and constant fl = 10 Ho/Bo = 0.5; (d) (FH/Fv)(e) for different Ho/8o and constant # = 10 , f= 0.5. (ii) The friction factor has a greater influence on the location of the neutral point than the specimen fonn. The practically linear function (,(/S)/B, foi con- stant e, f and Ho/Bu is shown in Fig. 6(b). A con- clusion drawn is similar to that in the previous case: the angle / influences the position of the neutral point rather then the form of the axial section of the defonned specimen. The resultant upsetting force F(FH, Fv) is inclined relative to the central line. Since the work-material slides along the upper surface towards both the thick and the thin sides of the specimen, it follows that the angle of inclination of the resultant force 7 would be smaller in value then the angle fl of the inclination of the upper die, i.e.: F. ;=tan-J-f-vl (151 The function (FH/Fv)(fll for different values of e, constant f and Ho,Bo is shown in Fig. 7(a), from hich it can be observed that the ratio FH:Fv does not depend on the level of e. In this specific case the following equation applies: F _ _ H = 0.0132 fl deg (16 Fv This feature suits press elasticity experiments, as i simplifies the use of data pertaining to the force ratio The function (FH/Fv)(e) for selected fl and constan f and Ho/Bo (Fig. 7(b) confirms the above feature Fig. 7(c) and 7(d) show that, during the press stiff hess experiments, both low friction and a low valu of Ho/Bo should be ensured. If it is the case, the ratk FH/Fv remains practically independent of the speci men reduction. The departure of the trend (FwFv)(e for Ho/Bo = 1.0 (Fig. 7(d) from the other graphs re suits from using too large a specimen, the latter un dergoing bending at the early stage of deformation Relationships between the vertical component o the unit forming force (related to the specime: length) and the reduction e are shown schematicall in Fig. 8. On the basis of Fig. 8(b), it can be con eluded that, for a relatively small deformation, th vertical component of the forming force is practicail independent of the angle between the dies, includin the case /= 0. This feature simplifies the procedm for the selection of specimens for press elasticity me surements. The vertical component of the formin force may be calculated from the commonly-know equation for the upsetting of a parallel-faced spec men. The third feature of the resultant force is the point ofi application. The position of this point has been defined Fig. 4(a) in a form of line R-R. The point of the for. application follows the neutral point. Hence, conclusio which have been drawn for the neutral point app K. Chodnikiewk: et aL / Journal of Material, Processing Technology 68 (1997) 13- 18 18 Fv/L kN/mm Fv/L + f=1.0 60 . 60 T -M 0 75 50 40 30 20- 10- 0 0 0.5 /11 50 -o- 0 25 / .-o- o.o /, 4o 2O 10 I I l i t 0 10 20 30 40 50 (a) kN/mm Q- = 15 -e- 100 ? _ 5 0 / I I , I I i 1 o 20 30 40 50 (b) Fig. 8. The relati,c :,:rtical component of the forming force Fv/
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