THEOETICAL CONSIDEATIONS AT CYLINDICAL DAWING AND FLANGING OUTSIDE OF EDGE ON THE DEFOMATION STATES Lucian V. Severin 1, Dorin Grădinaru, Traian Lucian Severin 3 1,,3 Stefan cel Mare Univerity of Suceava, Faculty of Mechanical Engineering, Mechatronic and Management, omania, e-mail: everin@fim.uv.ro, gradinaru@fim.uv.ro, everin.traian@fim.uv.ro Abtract. Thi paper preent an analyi of the deformation tate by drawing and flanging outide of edge, in order to determine of the limit between procee depending on the degree of deformity. Analytical relationhip were determined to calculate the main real deformation, relative height increae of drawing piece wall due to radial real deformation by drawing coefficient. elation were etablihed without conidering friction between blankholder, female die and bending radiu of workpiece material on the female die. Analyi howed that the drawing coefficient of between 0.5-0.8 the proceing can be conidered drawing and when i greater than 0.8, relative height increae due by radial deformation i inignificant, proceing can be conidered a outide flanging of edge procee. Keyword: drawing, deformation, drawing coefficient, relative height, outide flanging of edge. 1. Introduction Drawing of a heet metal i a proce that conit in a platic deformation proceing of a plan workpiece in a hollow, or reducing the ize of a cro ection of hollow piece to increae the depth [1]. During proceing by drawing, workpiece flange reduce their dimenion in plan a part cavity i formed under the action of deformation force F (fig1.). If the plan workpiece i related to a cylindrical coordinate axi ρ, θ and z, at any point M of flange acting platic deformation tre of the flange, σρ and σθ, which are the main normal tre and are definite by relation, [1]: c ln ; c 1 ln (1) where: σc i the yield trength of the blank material; - workpiece radiu; ρ point radiu, M. Figure 1. Cylindrical drawing cheme 1- punch; - female die; 3- blankholder; 4- blank 87
Principal normal tre in the z direction, σz, i inignificant in value compared with σρ and σθ, due to the mall value of the thickne blank relative to other dimenion of the blank and the low value of the tre generated by the blank holder force Q. Thu it can be conidered that the tenion tate of the work-piece flange may be conidered a plan tre tate. The total drawing tre ariing in the wall, σρt, i greater than σρ, due to the friction that occur between the workpiece and mould element, and the tre required at the bending and traightening of the drawing of metal heet. The calculation of σρt i [1]: blankholder, female die and bending radiu of workpiece material on the female die. The value of circumferential train εθ in certain M point, on workpiece flange located at the radiu ρ, when it reache on the wall with the average radiu r preed in poition M, i defined by the relation []: r ln (4) If we take into account only the main normal tree that caue platic deformation of the flange connection between the principal tre and real train in platic domain, i defined by the relation [3, 4]: t c ln r D d pl p e d c pl 1 () where: r i radiu meaured average fiber thickne of workpiece; p - preure generated by force Q on the flange workpiece by blankholder; μ - coefficient of friction between the workpiece material and the mould element; pl - rounding radiu of the female die; - thickne of the blank; α - angle correponding arc with pl radiu. The σρt value defined by the relation () can be drawing maximum force F, calculated: F d t (3). Deformation tate analyze of workpiece If a point M of the workpiece flange the tre tate created the by the force F can be conidered a approximately a flat tre tate, the deformation tate i a pace deformation tate due to the pecial deformation condition. In all three direction, ρ, θ and z are produced the real main train ερ, εθ and εz (fig. ). The analye of deformation tate will be made without conidering friction between Figure. Deformation tate at cylindrical drawing z z = ct (5) Becaue it believe that σz 0, the relation (5) become: z z 88
z z (6) According to the law of contancy of volume in platic deformation at any point, in the volume of work-piece material relationhip can be written: ερ+εθ+εz=0 (7) Sytem defined by equation (5), (6) and (7), and taking into account relation (1) follow: r ln ln 1 z (8) ln r ln 1 ln ln (9) The graphical repreentation of the real main train ερ, εθ and εz baed on the poition of the point M in the plane workpiece i hown in figure 3. 3. Calculating thickne and height relative growth of wall piece Analyi of the real main train graph εz along radiu ρ, according to the current point how a thinning of the wall part at proximity to the bottom part, with negative value, then pa through a point that ha the value zero, after which how a thickener to the top urface, it ha poitive value. Figure 3. Graphical preentation of the real main train along radiu ρ of point M poition The real main train ɛz i defined by the equation []: ln z 0 (10) where 0 i the thickne of the plane workpiece at a point M located on the radiu ρ and i the wall thickne at point M located at the radiu r (fig. ). 89
Taking into account by the relation (8) and equation (10) by calculu the relationhip of the wall thickne drawing piece i determined at a current point M, on the plane work-piece with 0 thickne and ρ radiu located, with the following relation: r ln ln 1 ln 0 e (11) Equation (11) how that the thickne of the workpiece doe not change if: ln 1 0, or ln 1 from the reulting relationhip known: ρ 0.60653 (1) It i eaily een from equation (1) that the poition of point with radiu value of ρ lower than thoe determined by the relation (1) ha been a thinning of the wall, and above thi value i a thickening. Thickening i maximum at the poition point ρ =. For thi poition of the point M, by equalizing the relationhip (8) with (10) reult: ln 0 r 1 ln (13) If noted by m the report r/, report entitled drawing coefficient, can be obtained thickne of the top wall defined by the relation: 0 (14) m If we conider the limit value of drawing coefficient a 0.55 to 0.8, for the lower limit and upper limit, uing equation (14) give a 9 % thickening wall piece repectively 11.8%. For flange flat blank area to get to the cylindrical wall of piece, it i neceary compreion with real main circumferential train εθ. Simultaneouly, the material of flange flow to radial direction becaue i tretched with real main radial train, ερ. Therefore height of the piece h will be greater than (D-d)/. Increaing the height Δh becaue the radial extent can be determined from the relationhip: d or Δdρ= ερ dρ (15) d where ερ i defined by relation (9). By integration of relation (15) between r- reulting increae height Δh of the ize in the radial direction defined by the relation: h d (16) r To olve the integral (16) i developed in erie of function ερ around 0 value, uing Mac Laurin formula [5], retaining the firt two term of development and it get: r 3r (17) 3 By ubtituting ερ defined by equation (17) into (16) and performing calculation follow: 1 h r ln r ln (18) r 3 3r Equation (18) can be expreed in relative value written in the form: h 1 1 mln mln (19) m 3 3m 1 or: h 1 1 ln 1ln (0) r m 3m 3m 1 Graphic repreentation of the relative height according to the coefficient of drawing i hown the increae Δh/ and Δh/r according to decreae of drawing coefficient and i hown in figure 4. It can be eaily een that along decreae of drawing coefficient take place the growth of the relative height. 90
Figure 4. Graph of relative height growth of piece along drawing coefficient. Thi i explained by the growth of real main radial train ερ with decreaing drawing coefficient becaue the material volume in the flange zone reache in wall drawing piece i greater. 4. Deformation tate at flanging outide of edge Proceing the flanging outer edge of piece with a convex hape i characterized by a tre and train tate imilar to the hallow drawing, without holding []. In thi proce the volume of material diplaced from the flange obtained by the reflection in the wall of the blank i le than the cae of piece formed by drawing with high degree of deformation, and due to thi caue the real main train ερ, εθ and εz are maller. Literature doe not clearly pecify the reaon not to that degree of deformation of the work-piece can be conidered the outide flanging of edge and where it can be een by drawing proceing. Analyzing of graph how that relative height growth piece (Δh/ and Δh/ r) i produced at drawing coefficient with mall value i determined. Thu correponding relation (18) and (19) for m = 0.8 reult Δh/ = 0.01 and Δh/r = 0.015. Thi how that the elongation of the material in the radial direction i very mall compared to the ize of the work-piece dimenion or piece. It can be aid that if the plan workpiece dimenion proceing by flanging outide of edge i determined by calculating the average fibber length, piece height deviation are quite mall for great value of drawing coefficient. Therefore it can be conidered for drawing coefficient greater than 0.8, becaue lengthening in radial direction i between the dimenional allowance provided for tolerance of the free dimenion obtained at flanging outide edge proceing 91
5. Concluion Analyi of deformation tate at drawing heet proce allowed analytical relationhip for calculating the real main train cylindrical coordinate. The tranlation relation have been etablihed wa determined formula for the calculation of wall thickne and increaing the height and relative height of the piece due to the radial deformation in the drawing proceing. Thu the coefficient drawing value in the range 0.5-0.8, the drawing technological calculation will be made with drawing pecific relation For large coefficient drawing, m 0.8, drawing proceing can be conidered external reflection proceing margin becaue of high growth rate i within the tolerance obtained free dimenion proceing on cold platic deformation. eference [1] Teodorecu, M., Al., Tehnologia preării la rece. Editura didactică şi pedagogică, Bucureşti, 1980; [] omanovkii, V., P., Ştanţarea şi matriţarea la rece. Editura tehnică, Bucureşti, 1970; [3] Zgură, Gh., Ciocârdia, C., Bude, G., Prelucrarea metalelor prin deformare platică la rece. Editura tehnică, Bucureşti, 1977; [4] Ponomariov, S., D., ş.a., Calculul de rezitenţă în contrucţia de maşini. Editura tehnică, Bucureşti, 1960. [5] Bachman, K., H., Berthold, G., ş. a., Mică enciclopedie matematică. Editura tehnică, Bucureşti, 1980. 9