ANALYSIS AND PREDICTION OF SIZE EFFECT ON LASER FORMING OF SHEET METAL

Size: px

Start display at page:

Download "ANALYSIS AND PREDICTION OF SIZE EFFECT ON LASER FORMING OF SHEET METAL"

Susanna Oliver
5 years ago
Views:

1 ANALYSIS AND PREDICTION OF SIZE EFFECT ON LASER FORMING OF SHEET METAL Peg Cheg, Y.Lawrece Yao, Chao Liu, Duca Pratt, Yaju Fa Departmet of Mechaical Egieerig Columbia Uiversity, New York, NY 7 KEYWORDS Size Effect, Laser Formig, Sheet Metal ABSTRACT Geometric effects play a importat role i laser formig process; however, few ivestigatios have bee reported o the geometric size effect other tha sheet thickess. I this paper, the ifluece of size effect, icludig sheet width ad sheet legth, o laser iduced deformatio is experimetally, umerically ad aalytically ivestigated. A size matrix is desiged to cover a wide rage of sheet width ad legth for experimets ad umerical simulatio. Distictive treds i bedig agle are observed for varyig sheet width, legth or both. The results are iterpreted i terms of heat sik effect ad bedig o-uiformity. A aalytic mod is devoped to facilitate size effect predictio. The mod is based o the solutio to a movig strip heat source over a fiite size sheet ad o the accout for pre-bedig effect amog cosecutive segmets o the scaig path. It is compared with a existig aalytical mod ad umerical simulatio. INTRODUCTION Laser formig is a flexible formig process that forms sheet metal by meas of stresses iduced by exteral heat istead of exteral force. Uderstadig various aspects of laser formig has bee a challegig problem of cosiderable theoretical ad practical iterest. The ratioships of bedig distortio ad process parameters, material properties ad workpiece thickess have bee devoped i aalytical mods. Additioal iformatio, such as ifluece of strai hardeig, strai rate effects ad edge effect have also bee reported i experimetal ad umerical ivestigatios. The geometric effects i laser formig, however, have ot bee fully studied. Most studies focused o the effect of workpiece thickess without cosiderig other geometric attributes. Scully (987) proposed that plate thickess s is oe of the primary factors i laser formig ad he adopted a quatity P /( s V ) that was used i arc wdig to study the effect of plate thickess o bedig agle, but he did ot give a exact ratioship betwee them. Koloma ad Karol (99) proposed a aalytical mod i which the bedig agle is a fuctio of the reciprocal of the square of thickess. However, the bedig agles calculated from this formula are some orders of magitude too large compared with experimets. Vollertse (99a, 99b) proposed a simple two-layer mod based o the Trasactios of NAMRI/SME 39 Volume 3,

2 temperature gradiet mechaism (TGM) assumptio. Due to the assumptio that bedig deformatio is uiform alog the sheet legth, the oly geometric factor appeared i those mods is plate thickess. Experimetal ivestigatios ad umerical simulatio, however, have show that sheet size has a effect o laser formig. Vollertse (99a) poited out that the width of the sheet iflueces the coolig coditios ad therefore the bedig agle, ad he also observed the ifluece of the sheet legth o stress state. But he did ot cosider the size iflueces i his aalytical solutios. Hsiao, et al. (997) have foud that for Ico 65 the agular distortio icreases as sheet legth icreases ad he proposed the reaso is the loger plate provides more cold metal to produce thermal stresses. He also proposed that there is a upper limit i the agular distortio with respect to the sheet legth. However, these ivestigatios o size effect are empirical ad did ot lead to aalytical predictio. I this paper, the size effect exhibited i laser formig process is ivestigated. First the experimetal ad umerical ivestigatios aimed at advacig the uderstadig of the causes of the size effect i the straight-lie laser formig are preseted for a wide rage of sheet sizes. Various deformatio patters due to the size effect are explaied. To better quatify the role of size effect o laser formig, a predictive mod is devoped, i which fiite sheet width ad legth are cosidered. The proposed mod is experimetally validated. EXPERIMENTS AND SIMULATION Experimets of straight-lie laser formig (Figure ) were carried out with the sheet legth ad width varyig from 8 mm to mm, i which the size effect was ivestigated i three cases: varyig sheet width, varyig sheet legth, ad varyig square size. Additioal set of samples (L = ~6 mm, W = 8 mm) were also used to further ivestigate the sheet legth effect because it was reported (Vollertse, 99a) that the effect of sheet legth is more proouced whe legth-to-thickess ratio ( L / s ) is small. The sheet thickess is fixed at.89 mm sice effect of thickess has bee wl studied as discussed i the previous sectio. The sheet material is AISI ste. The laser system used i the experimets is a PRC-5 CO laser, with a maximum output power of.5 Kw ad power desity distributio was TEM. The diameter of the laser beam used is mm. I umerical simulatio the laser formig process is moded as a sequetially coupled thermal-mechaical process. More details about the thermal mod ca be foud i Cheg ad Yao (3). I the mechaical aalysis, direct itegratio is used for the oliear trasiet dyamic respose. The body force at a poit ca be expressed i the virtual work equatio as f vdv F vdv u vdv () V V where f is the body force, v the virtual vocity fid, V the volume, F the exterally prescribed force, the desity of the material, u the acceratio fid. With the odal iterpolatio, the iertia force is N M M ( N N dv ) u ; that is, the cosistet mass matrix times the acceratios of the odal variables ( u ), M where N M N or N (N, M=,, up to the total umber of variables i the problem) is a set of N vector iterpolatio fuctios, ad V the referece desity ad referece volume, respectivy.. Sheet thickess s z x y Laser beam d FIGURE SHEMATIC OF STRAIGHT-LINE LASER BENDING ABAQUS was used to complemet the umerical simulatio. A quadratic -ode brick emet was used i the mechaical aalysis because this kid of emet has o shear lockig ad hourglass stiffess ad is also suitable for bedig-deformatio-domiated processes such as laser formig. A user subroutie of dflux was devoped to mod the heat source iput from the Gaussia laser beam. V Sheet legth L V Scaig vocity V ½W Clamp Sheet width W Trasactios of NAMRI/SME Volume 3,

3 EXPERIMENTAL AND NUMERICAL RESULTS AND DISCUSSIONS. Effect of Sheet Width Figure shows the variatio of bedig agle with the sheet width varyig while the sheet legth beig costat. Good agreemet has bee obtaied betwee the simulatio ad the experimet. It is see from that, for carbo ste, whe the sheet width icreases, bedig agle decreases. This differs from the observatio o LF of Ico 65 (Hsiao, et al., 997), where it was reported that bedig agle icreases with both sheet width ad sheet legth ad it was attributed to larger thermal stress caused by larger costrait from the cold plate. The discrepacy is idicative that the size effect o bedig agle may ot be as simple as it was proposed. Whe the sheet width varies, both thermal coductio coditio ad sheet weight vary. I umerical simulatio, the weight effect was removed by detig the gravity term i the mod. Figure 3 shows the bedig agle differece betwee mods with or without cosiderig the gravity. It is foud that, for the sheet width rage ivestigated, the bedig agle differece caused by gravity is almost the same for 8x8 ad 8x samples, which clearly idicates that the gravity effect is ot the major reaso for the bedig agles chage caused by sheet width chage. Further evidece was obtaied by examiig the Z-directio stress at a typical poit ad it is foud that gravity does ot affect the bedig stress sigificatly eve the sheet width icreases from 8 to mm. So the reaso of bedig agle differece caused by differet width may lie i the thermal coductio coditio. Figure shows the simulated time history of temperature. It is see that the peak temperature at the top surface of workpiece drops whe the sheet width icreases, while the temperature at the bottom surface is almost the same. The reduced temperature gradiet i the thickess directio is clearly resposible for the reduced bedig agle whe sheet width icreases. This is obviously due to the fact that a larger width provides a larger heat sik. FIGURE 3 INFLUCENCE OF GRAVITY ON BEND- ING ANGLE WHEN THE SHEET WIDTH VARIES FIGURE EXPERIMENTAL AND NUMERICAL RE- SULTS FOR DIFFERENT SHEET WIDTH FIGURE SIMULATED TIME HISTORY OF TEM- PERATURE FOR DIFFERENT SHEET WIDTH (. SECONDS TIME DELAY BETWEEN EACH CASE FOR VIEWING CLARITY) Effect of Sheet Legth Figure 5 shows that the bedig agle icreases with sheet legth. Figure 6 shows temperature o the top ad bottom surfaces of workpiece. It is see that although the sheet Trasactios of NAMRI/SME Volume 3,

4 legth icreases, the thermal fid is ot visibly affected. This is primarily because the heat sik effect discussed above is over shadowed i the directio of sheet legth by the fast movig laser beam alog the same directio. Figure 7 shows the time history of bedig agle, peak temperature, ad Z-directio stress i a x8 size sample at a typical positio (x = 6.3 mm). It is see that before the laser beam reaches that poit (sigified by the rapid rise of temperature), the bedig agle has reached about /3 of the fial bedig agle. Whe the laser beam reaches that poit, the bedig agle icreases rapidly approximaty util the Z- directio stress stabilizes. The bedig agle cotiues to rise sly after the laser passes to the poit. The existece of the pre-bedig (bedig at a locatio geerated before the heat source arrives at the locatio) ad post-bedig (bedig at a locatio geerated after the heat source passes the locatio) obviously plays a importat role i creatig the fial bedig agle. It is also foud that there is more prebedig effect i the shorter sheet (8x8) tha i the loger sheet (x8), ad the opposite ca be said about the post-bedig effect. I ay case, withi the sheet legth rage ivestigated, the pre-bedig effect is more sigificat tha the post-bedig effect. Varyig Square Sizes Figure 8 shows the average bedig agle treds i case of varyig width, legth ad both. It is see that for the varyig square sizes, the bedig agle icreases before it decreases. This is clearly because of the competig effects of heat sik primarily associated with sheet width ad the pre-/post-bedig effect primarily associated with sheet legth. The pre-/postbedig effect is more domiat iitially ad as a result the bedig agle icreases before heat sik effect takes over ad thus the bedig agle decreases. FIGURE 5 EXPERIMENTAL AND NUMERICAL RE- SULTS FOR DIFFERENT SHEET LENGTH FIGURE 7 SIMULATED TIME HISTORY OF BEND- ING ANGLE, TEMPERATURE AND STRESS FIGURE 6 SIMULATED TIME HISOTRY OF TEM- PERATURE FOR DIFFERENT SHEET LENGTH (. SECOND TIME DELAY BETWEEN EACH CASE FOR VIEWING CLARITY) FIGURE 8 AVERAGE BENDING ANGLE INFLU- ENCED BY SHEET SIZE EFFECT Trasactios of NAMRI/SME Volume 3,

5 Iertia effect To ivestigate size effect o bedig agle, iertia effect eeds to be examied sice it varies with sheet size. It was foud that the static mod, i which the iertia effect is igored, uderestimates the bedig agle. This is because the iertia force cosidered i the dyamic mod makes the bedig force larger ad thus the bedig agle larger tha that of static mod. It was also foud that whe sheet width icrease, the bedig agle differece betwee the dyamic ad static mod decreases. This is because the iertia force is maily depedet o the differece betwee laser-iduced thermal load ad gravity effect. Whe the sheet width icreases, the iertia force decreases due to the larger gravity ad the smaller thermal load caused by larger heat sik effect. Whe sheet legth icreases, the iertia effect icreases due to the more domiat thermal load due to the loger heat iput path. PREDICTIVE MODEL DEVELOPMENT As metioed before, laser formig is a fully coupled, o-liear, thermo-mechaical process, ad is ofte simulated by a full threedimesioal fiite emet method. However, aalytical mods are more ituitive ad depict the cause-effect ratios directly. But cosiderable simplificatios are required to succeed i devopig such aalytic mods. The objective of this sectio is to devop a aalytic mod which ca predict the size effect o bedig agle by properly balacig betwee essetial simplificatios ad mod fidity. Existig Aalytical Mods ad Proposed Approach The most widy used mod is perhaps the oe devoped by Vollertse (99), which assumes the temperature gradiet mechaism (TGM) 3lh tht B () s th where B is bedig agle, l h is the width of a heated zoe, is the thermal expasio coefficiet, T is the temperature rise i the heated zoe, ad s is the sheet thickess. This mod was arrived at by usig a two-layer assumptio. The bedig agle is assumed to occur at oce ad therefore is uiform alog the scaig path. The thermal fid is obtaied by AP T (3) Vc plhs where A is absorptio coefficiet, P is laser power, V is laser scaig vocity, c is heat capacity, ad is the desity. Heat coductio is ot cosidered i Equatio. To take ito accout of heat coductio, Scully (987) ad Masubuchi (99) adopted the solutio to the movig lie heat source problem (Rosethal, 97) to determie T q V T( r) e K( Vr) () kg where q is power of the lie source, g is the height of the lie source, k is thermal coductivity, /( ) ad is diffusivity, V is the movig speed alog the x-axis, x Vt is the distace to the heat source alog the movig path, r is the distace to the source ceter i scaig plae, K is modified Bess fuctio of type zero. Eq., however, is obtaied based o ifiite boudary coditio ad is ot suitable for ivestigatig the size effect. As experimetally ad umerically show early, the sheet width effect o bedig agle is primarily associated with the heat sik effect, while the sheet legth effect is primarily associated with pre-/post-bedig effect. The proposed approach i this paper cosists of () derivatio of a T expressio for fiite sheet width W, takig ito accout of the fiite laser beam size; ad () adaptatio of Eq. for a small legth of the scaig path (istead of the etire scaig path) ad derivatio of a B expressio for the scaig legth L. Thermal Effect The fiite width coditio is itroduced i the solutio of the movig lie heat source (Eq. ) by the method of images (Rosethal, 97), q V T e K( Vr ) (5) kg Takig ito accout of the fiite boudary coditio T / y as y ad y a = W/, a solutio ca be obtaied by trasformig Equa- p Trasactios of NAMRI/SME 3 Volume 3,

6 tio (5) ito a Fourier series. The obtaied solutio, however, is based o a lie source without ay lie width ad the temperature value at the source ceter is ifiite. To avoid this, a assumptio that the lie source has a fiite width is made ad this assumptio is realistic for the laser beam source. It is assumed the movig lie source has a width which equals to the laser beam diameter o the sheet. By a series of derivatio, the temperature withi the lie width d / d / is q d e T [ ( ) c agvd V e ( ) p e ( ) ( ) V ( d / ) V ( ) V ( d / ) V V ( d / ) V y cos a y cos ] a (6) where g is the height of the lie source (equal to half of the sheet thickess uder the two-layer assumptio), ad ( ). Please ote Va that T is ow a fuctio of sheet width a. To obtai a represetative T for the two-layer mod (Eq. ), the temperature icrease i the square domai of l h is averaged. From the calculated temperature at the heat source ceter by the aalytic mod, the decreasig tred with icreasig sheet width is observed ad is cosistet with FEM simulatio results. There is some discrepacy i slope liky due to the several assumptios uderlyig the proposed aalytic mod, such as temperature idepedet material properties ad the twolayer assumptio. Mechaical Effect As show before, the sheet legth effect o bedig agle ca be accouted for primarily through the pre-/post-bedig effect alog the scaig path, ad the pre-bedig is more domiat tha the post-bedig i creatig the fial o-uiform bedig agle. Therefore, oly pre-bedig is cosidered i the folig derivatio. To cosider the o-uiform bedig alog the scaig path, the scaig legth L is divided ito segmets. Each segmet is L/ log ad assumed to have a uiform bedig alog the segmet (Fig. 9). For the segmets that heat source has passed, permaet plastic deformatio occurs. For those segmets that heat source has ot arrived, o plastic strai occurs ad oly the segmet folig the plasticdeformed segmets is affected by the previous segmet. s FIGURE 9 SCHEMATIC OF MECHANICAL MODEL Startig the aalysis from the first segmet, sice o pre-bedig effect, the bedig agle i the first segmet ca be obtaied through a two-layer assumptio (Vollertse, 99a), that is lh ( up) (7) s where up ad are strais of the upper ad er layer, respectivy. Assumig that the thermal expasio is fully coverted ito the plastic compressio, we have F M s up pl ( ) ( tht ) (8) EA EI F M s (9) EA EI where F ad M are the force ad momet caused by differece i compressio betwee the upper ad er layer of the segmet, E is the astic modulus, A is the cross-sectioal area of each layer of the segmet ad L A s, ad I is the iertia mometum ad L 3 I s. Substitutig Eqs 8 ad 9 ito Eq. 7 ad otig s the equilibrium equatio F obtaied that s Laser beam ½ W symmetry plae l h l h V Segmet i Segmet i- Segmet Segmet M p up p F (- F ) EI M, it is l h y L z x Trasactios of NAMRI/SME Volume 3,

7 ad 3l h tht s () tht () For the secod segmet that heat source has ot reached, astic strai also occurs due to the compatibility. The astic strai at the boudary betwee the first segmet ad the secod segmet equals to the astic strai of the first segmet, ad the astic stai at the boudary betwee the secod segmet ad the third segmet is zero. So the average astic strai i the secod segmet (the affected segmet) is up tht 8 tht () 8 The the equivalet astic forces o the upper ad the er layer of the secod segmet are p EA p (3) up up Ad it ca be assumed that the forces are loaded at the ceter poit of each applied plae. The the pre-bedig agle of the secod segmet ca be obtaied: pup s 3 lh l T () h th EI 8 s The fial bedig agle of the secod segmet equals to the summatio of the bedig agle caused by heat flux ad the pre-bedig agle: (5) I the similar way, whe the heat source moves out of the secod segmet ad has ot etered the third segmet, the pre-bedig agle of the third segmet is, where is the bedig agle of the secod segmet. Similarly, the bedig agle of the i-th segmet is (6) where i i i is the bedig agle of the (i-)th segmet. The the average bedig agle of a plate with the legth of L ad cosistig of segmets is [ ( )] ( ) (7) Substitutig the expressio of, it is obtaied that lh ( ) tht (8) 3 s where the temperature icremet T ca be obtaied by the thermal mod itroduced i former sectio. Eq. 8 predicts the bedig agle of a fiite sized sheet. The effect of sheet width is reflected i T ad the effect of sheet legth is reflected i. To apply Eq. 8 to a specific laser formig process ivolvig a specific material, ad l h eed to be determied, both are rated to the extet of the plastically deformed zoe. Usig temperature distributios predicted by the proposed thermal mod uder various coditios, the rage of the plastic-deformed zoe i which the temperature is higher tha critical temperature (~7K) is approximaty determied ad rated to the lie eergy P/V as l h.5( P / V).5 (9) L / 3( P/ V) 7.5 () I above equatios, the uit of legth, power ad vocity is mm, watt ad mm/s, respectivy. Substitutig Eqs. 9 ad ito Equatio (8), oe ca predict the bedig agle of a fiite sized sheet. Validatio of the Predictive Mod Figs. ad show the validatio of the proposed mod. It is see that the bedig agle values obtaied from the proposed mod show a similar tred as the experimets ad simulatio, while the existig aalytic mod is uable to predict the size effect. Although the proposed mod over estimates the bedig agle, its agreemet with experimetal ad simulatio values is sigificatly better compared with the other aalytical mod. The mai reaso for this over estimate comes from the omissio of yid stregth cosideratio i the proposed aalytical mod. The fractio of the heat eergy absorbed by the zoes, where the heated temperature is bow the critical temperature to effect appreciable plastic deformatio, is almost 5% of the total iput eergy (Vollertse, 99a), while the mod assumes all the iput eergy is used for deformatio. I the case of varyig sheet legth, the mod predicted tred of bedig agle is wl agreeable with the experimetal ad simulatio values. The tred is cosistet with Vollertse s (99a) experimetal observatios, that is, whe the ratio of legth ad thickess is small, the icreasig tred of bedig agle with sheet legth is Trasactios of NAMRI/SME 5 Volume 3,

8 more proouced ad there is a upper limit of bedig agle whe the sheet legth keeps icreasig. I the compariso of varyig square size, the predictive mod also captures the variatio tred. Withi the sheet size rage as wl as process coditio used i this ivestigatio, the size effect o laser formig of carbo ste sheet ca be summarized as fols. Whe sheet legth is costat ad sheet width icreases, the bedig agle slightly decreases. This is primarily due to the icreased heat sik effect with sheet width. Whe sheet width is costat ad sheet legth icreases, the bedig agle slightly icreases. This is primarily due to the fact that the bedig agle alog the scaig path is o-uiform ad iflueced by the pre-/postbedig effect. Whe both the sheet legth ad sheet width icrease as square sheets, the bedig agle icreases before decreases due to the competig effects. Based o the experimetal ad umerical results ad physical iterpretatios above, a aalytical mod is devoped to predict the size effect. Sheet width effect is accouted for by modig a movig strip heat source over a fiite width sheet. Sheet legth effect is moded by dividig a fiite scaig path ito small segmets ad takig ito accout the pre-bedig effect betwee cosecutive segmets. The aalytic mod compares favorably with the existig aalytical mod i terms of agreemet with experimetal measuremet ad umerical simulatio. FIGURE ANALYTIC PREDICTION OF BENDING ANGLE FOR VARING SHEET WIDTH + REFERENCES Cheg, P, ad Yao, Y. L, (3), The Ifluece of Sheet Metal Aisotropy o Laser Formig Process, Proceedigs of ICALEO 3, Sectio E, pp.-. Koloma, K., ad Kraol, P., (99), Forme durch locale Erwarmug, Metal Formig-Theory ad Practice: Proceedigs of the 5 th Iteratioal Coferece o Metal Formig, Tisza, M., ad Kardos, K. (Eds), pp Hsiao, Y. C., Maher, W., et al., (997), Fiite Elemet Modig of Laser Formig, Proc. ICALEO 97, Sectio A, pp. 3-. FIGURE ANALYTIC PREDICTION OF BENDING ANGLE FOR VARING SHEET LENGTH CONCLUSIONS Masubuchi, K., (99), Studies at MIT rated to applicatios of laser techologies to metal fabricatio, Proc. of LAMP 9, pp Rosethal, D., (97), The Theory of Movig Sources of Heat ad Its Applicatios to Metal Treatmets, Trasactios of the ASME, Joural of heat trasfer, Vol. 68, pp Scully, K., (987), Laser Lie Heatig, Joural of Ship Productio, Vol. 3, No., pp Vollertse, F., (99a), A Aalytical Mod for Laser Bedig, Lasers i Egieerig, Vol., pp Vollertse, F., (99b), Mechaisms ad Mods for Laser Formig, Laser Assisted Net Shape Egieerig, Proc. of the LANE 9, Vol., Meisebach Bamberg, pp Trasactios of NAMRI/SME 6 Volume 3,

DYNAMIC ANALYSIS OF BEAM-LIKE STRUCTURES SUBJECT TO MOVING LOADS

DYNAMIC ANALYSIS OF BEAM-LIKE STRUCTURES SUBJECT TO MOVING LOADS Ivaa Štimac 1, Ivica Kožar 1 M.Sc,Assistat, Ph.D. Professor 1, Faculty of Civil Egieerig, Uiverity of Rieka, Croatia INTRODUCTION The vehicle-iduced