Predictive Molding of Precision Glass Optics

Transcription

1 09B-0258 Predctve Moldng of Precson Glass Optcs Shrram Madapus 1, Nam-Ho Km 2 Unversty of Florda Yazd omhe 3 Moore Nanotechnology Systems Copyrght 2008 SAE Internatonal ABSRAC Precson glass moldng process s an attractve approach to manufacture small precson optcal lenses n large volume over tradtonal manufacturng technques because of ts advantages such as low cost, fast tme to market and beng envronment frendly. In ths paper, we present a physcs-based computatonal tool that predcts the fnal geometry of the glass element after moldng process usng the fnte element method. Deformatons of both glass and molds are consdered at three dfferent stages: heatng, moldng, and coolng. A 2D axsymmetrc fnte element model s developed to model the glass moldng process. he proposed modelng technque s more effcent than the all-n-one modelng technque. he molds are assumed to be rgd, except for thermal expanson, at all tme and glass treated as a flexble body durng the compresson. Detals on dentfyng materal parameters, modelng assumptons, and smplfcatons are dscussed. he tool can be used to predct the fnal shape of the molded optc. hs tool eventually can be used to desgn proper mold geometry that yelds the correct shape of the fnal optcal element, thereby elmnatng the teratve procedure for desgnng the molds. INRODUCION In the recent years, asphercal glass optcs s wdely chosen because of ther superor optcal propertes, such as lesser aberraton and lower brefrngence, over plastcs and sphercal optcs. he tradtonal lens manufacturng process s a mult-step process whch requres a seres of materal removal processes [1, 2]. Although, there have been recent advances n optcal fabrcaton technques such as magnetorehologcal fnshng and on beam polshng [3], the complexty of these processes s such that the overall costs are hgh for medum to hgh volume producton of aspherc optcs. Furthermore, the process ncurs envronmental ssues because of the use of grndng fluds and polshng slurres. A potental low-cost and fast method to produce precson glass optcs s a compresson moldng process [4]. In a lens moldng process a glass gob s heated to a temperature above the glass transton temperature and s pressed between two mold halves havng the requred aspherc profle. he formed lens s then ether cooled naturally or by forced conventon to a room temperature resultng n ts fnal geometry. If ths entre process s desgned correctly, t can be easly adopted for hgh volume producton of precson asphercal glass lenses. he flowchart n Fgure 1 shows the use of a numercal smulaton tool to produce the fnal product wthout the conventonal teratve mold desgn. Fg 1: Producton flow of asphercal lenses usng a numercal smulaton tool 1 Research Assstant, Mechancal and Aerospace Engneerng Dept., Unversty of Florda, Emal: shrmad@ufl.edu 2 Assocate Professor, Mechancal and Aerospace Engneerng Dept., Unversty of Florda, Emal: nkm@ufl.edu 3 Moore Nanotechnology Systems, LLC, Emal: tomhe@nanotechsys.com

2 Numercal smulaton of the entre glass moldng process s a complex process. Performng an all-n-one smulaton of entre moldng process would not only be computatonally expensve, but would ncur modelng a lot of unnecessary complex processes. In addton, all-n-one modelng prevents from understandng mportant aspects of each process and fals to provde physcal nsghts of the problem at hand. In ths paper we am to analyze each sub-process that goes nto the moldng process and to create a computatonally smplfed model usng engneerng judgment and assumptons that would gve results accurate enough for manufacturng purposes and smple enough for use for desgn purpose. We analyze and segregate the vtal phenomenon that affect the qualty of the fnal optcal element and treat them quanttatvely. Y et al. [5] showed that t s possble to use the fnte element method to predct the glass moldng process by usng commercal software. However, there s no commercal software avalable customzed for smulatng mult-step moldng processes. A more sophstcated model s requred to accurately predct the manufacturng of aspherc lenses by moldng and a smplfed model s desred for computatonal expense. Hence, n ths paper we am to 1. understand the physcs of a compresson glass moldng process and create a smplfed numercal model 2. develop a computatonal tool usng fnte elements to smulate the glass moldng process for the above model Usng the above tool, one would be able to predct the fnal geometry of the glass after the compresson moldng. From the predcted geometry of the glass, the reverse engneerng of the moldng process can be performed to determne the ntal condtons such as, the mold profle, operatng temperature, and other process parameters that would result n the desred profle of an optcal element. hs would be acheved by executng the tool through an optmzaton routne to produce the desred profle of element by treatng the above quanttes as desgn varables and mnmzng the error between the desred and the resultng geometres. he followng secton explans the glass moldng process. hen the physcs-based model for glass moldng process s proposed, followed by the fnte element model of the smulaton of the process. Optmzaton to predct the mold geometry and results are dscussed towards the end. MOLDING PROCESS OF GLASS OPICS echnology to produce glass optcs va moldng currently exsts commercally, but t s not wdely used. We beleve the major dffcultes encountered are quantfyng and analyzng each process numercally that occur n the moldng process and predctng the fnal shape of the glass element after moldng. Another dffculty s to produce molds whch wll result n geometrcally correct lens after the compresson and coolng processes. he reason for ths s that, durng the heatng and coolng process, the dmensons of both the mold cavty and glass change due to thermal effects. Resdual stresses exst durng the coolng process due to the phenomena of structural relaxaton whch further affects the geometry of the molded glass. In addton, glass s an elastc sold at room temperature, but the prmary deformaton of glass occurs at hgh temperature (550 to 700 C) where glass behaves as a vscoelastc materal and hence the materal propertes change at the moldng temperature. Due to the above challenges, a mold cavty that s geometrcally perfect at room temperature wll not produce a geometrcally correct element. Predctng the geometry of molded optcs from the knowledge of mold geometry s dffcult. As a result, the process of creatng molds s nherently teratve. Fgure 2 outlnes the basc steps n a compresson glass moldng process: Fg 2: Glass Moldng Process a. Intal shape of the glass and the molds at room temperature b. Heatng process: Both glass and molds are deformed due to thermal expanson c. Compresson process: Glass s pressed on thermally deformed mold surfaces d. Coolng process: Molds recover ther orgnal shapes, but fnal geometry of glass s dfferent from that of molds due to thermal contracton and resdual stress DEAILED SEPS IN A GLASS MOLDING PROCESS he typcal glass moldng process can be conducted n the followng steps [9]: 1. Heatng: Heatng s the frst step n the glass compresson process. Durng ths process, the molds and the glass gob are heated above the glass transton temperature. Durng the heatng process the system s contnuously purged wth Ntrogen to prevent oxdaton on the molds. he molds are heated to the commanded temperature by an nducton heatng system around the molds 2. Soakng: Soakng s necessary to acheve the steady state of constant temperature between the molds and the glass. he controller mantans the com-

3 manded temperature for the duraton of soakng cycle tme. On large szed molds and glasses, the soakng tmes can be of the order of few mnutes to ensure unform temperature 3. Pressng: he pressng process commences at the end of soakng cycle. Mantanng the temperature constant, the upper mold s pressed unformly to deform the glass accordng to the mold profle 4. Gradual coolng: he temperature of the glass and the molds are reduced gradually tll the annealng temperature of glass. he gradual coolng s done by njectng heat contnuously to mantan the rate of coolng and usng Ntrogen gas to cool. If needed force can be appled n ths cycle 5. Steep coolng: Once the glass annealng temperature has been attaned, the glass and the molds are cooled to room temperature usng a hgh flow of Ntrogen gas. After ths cycle has been completed, the glass s unloaded from the molds and a new gob loaded Fgures 3 and 4 show the glass moldng apparatus along wth nducton heatng/coolng system. Fg 3: Moldng Setup Fg 4: he Inducton Heatng System he graph n Fgure 5 outlnes the temperature profle for the glass pressng cycle. he unform force for pressng s appled n the pressng cycle whch occurs at tme 3 Fg 5: Glass Pressng Cycle PHYSICS-BASED MODEL FOR HE GLASS PRESSING PROCESS he followng explans the assumptons and the physcal model developed for smulatng the glass pressng process. he WC molds are hghly nert and have hgh structural stffness. Hence they are assumed to be rgd bodes when compared to the stffness of the glass near ts glass transton temperature. However, the molds expand and contract lnearly n accordance wth ther thermal expanson coeffcent durng the heatng/coolng cycle. Glass s assumed to be lnearly elastc untl t attans the glass transton temperature.e. durng the heatng process. Hence glass s also assumed to expand and contract lnearly durng the heatng/coolng process. After the soakng process, durng the pressng cycle, glass behaves as a non-newtonan vscous flud and hence s assumed to behave as a vscoelastc materal. Due to soakng, we assume the system to have acheved steady state, whch means the glass and the molds are at the same temperature durng the pressng process. he gradual and steep coolng s modeled as lnear thermal analyss. Snce the molds and glass are assumed elastc durng that phase, the rate of coolng does not affect the analyss sgnfcantly. he resdual stresses that are present at the end of steep coolng cycle contrbute to the resdual stresses at the end. he vscoelastc stresses would have vanshed due to the gradual coolng. FINIE ELEMEN MODEL he followng descrbed the fnte element model for the smulaton of the glass moldng process. akng advantage of the symmetrcal shapes of aspherc glass lenses and the molds along ther axs of revoluton, the moldng process s modeled as a 2D axsymmetrc analyss. For heatng/coolng process, a lnear thermoelastc model s used. For compresson process, nonl-

4 near, large deformaton, vscoelastc, four-node quadrlateral element s mplemented. he nonlnear system of equatons s solved teratvely usng Newton-Raphson method. HERMAL ANALYSIS Lnear thermal analyss s mplemented to quantfy the deformaton of the molds and glass durng the heatng and coolng process. For glass, the thermal expanson of ts radus s calculated analytcally by multplyng the ntal radus wth ts thermal expanson coeffcent. For molds, lnear thermoelastc analyss s performed usng fnte elements. Fgure 6 shows the meshng of the lower axsymmetrc mold. he top profle s the aspherc profle. From the thermal analyses, the surfaces of the molds are extracted and modeled as rgd surfaces for the next cycle;.e., the pressng analyss. Fg 6: Meshng of the Lower Aspherc Mold he profle of the deformed surface s obtaned by addng the nodal dsplacements of the above analyss to the orgnal coordnates of the mold as P d P new (1) thermal old where P new s the new profle of the mold surface, d thermal s the vector of dsplacements from the thermal analyss, and P old s the ntal profle of the mold. he thermally expanded profle s used for the pressng analyss. Fgure 7 shows the orgnal and the deformed geometry of the surface of the mold after thermal analyss. he temperature of the WC mold has been rased by 550 C. he upper mold surface s the thermally expanded mold surface. he expanson has been magnfed by a factor of 50 to contrast the expanson aganst the orgnal shape. he unts are n mllmeters. Fg 7: Expanson of Mold Surface durng Heatng Cycle VISCOELASIC MAERIAL MODELING FOR GLASS Snce the strans ncurred are qute large, small stran vscoelastcty coupled wth large deformaton effects s adopted. Glass near ts transton temperature can be consdered to be an ncompressble vscous non-newtonan flud [14]. Hence we assume the followng propertes of glass durng the compresson phase: 1. Glass s ncompressble near ts transton temperature; whch mples the Posson s rato to be nearly he relaxaton occurs only n the shear modulus and that there s no relaxaton n the bulk modulus 3. As a result of above, there s relaxaton n the devatorc component of stress, and no relaxaton n volumetrc component of stress due to the constant bulk modulus Generalzed Maxwell model [13] s used to formulate the vscoelastc materal behavor. Fgure 8 shows the modelng of a vscoelastc materal usng a generalzed Maxwell model. he Maxwell model s mplemented only n the shear modulus and hence there s relaxaton only n devatorc stress. he Maxwell parameters are determned by expermental compresson tests whch wll be addressed later. Fg 8: Generalzed Maxwell Model

5 In Fgure 8, G s the elastc shear modulus at nfnte tme, N s the number of Maxwell elements to be modeled, G1... G N s shear modulus of each Maxwell element, s the tme constant of each Maxwell element. and 1... N In the Maxwell model, the relaxaton of the shear modulus can be expressed by the followng equaton [13]: N G( t) G G exp t / (2) 1 At nstantaneous tme;.e., t 0, the effect of all dashpots s neglgble and the nstantaneous shear modulus can be obtaned as G(0) G G G (3) 0 N 1 On the other hand, at nfnte tme, t, the effect of all sprngs vanshes and the shear modulus at nfnte tme becomes G( ) (4) G Due to the presence of the dashpots, the response s tme dependent. Snce glass s assumed to as a non- Newtonan flud near the transton temperature, t s ncompressble. Hence the vscoelastc relaxaton s consdered only n the shear modulus. here s no relaxaton n bulk modulus and s constant durng the analyss. Due to the ncluson of the effects of large deformaton, the consttutve equatons are formulated n terms of the rotated stress R R, where R s the rotaton arsng from the polar decomposton of the deformaton gradent F. Let R R p1 where s the devatorc part and p s the pressure part. he consttutve relaton for a vscoelastc response takng the rotaton nto account s gven as [10, 11] t t 2G G exp ( R dr) d (5) 0 n 1 where t s the pseudo or the past tme, G s the elastc shear modulus, n s the number of Maxwell elements, G s the shear modulus of th Maxwell element, s the relaxaton tme of th Maxwell element, and d s devatorc component of R. Assumng the stress at n th load step s known, we obtan the recursve formula for calculatng the current stress by the mdpont ntegraton of the above ntegral from 0 to t n+1 resultng n: t S exp R n 1 S R n t 2exp GR en R 2 1/2 1/2 1/2 where, R = Rn 1Rn, R1/2 Rn 1Rn 1/2, n 1/2 s the stran tensor at n+1/2 and en 1/2 s the devatorc component of n 1/2 stran tensor. (6) he above stress s the vscoelastc stress whch wll relax wth ncreasng tme due to the presence of the decayng exponental term. he elastc shear stress arsng from the nstantaneous modulus s obtaned by ntegraton procedure smlar to above: S 1 4 n R S n R G R 1/2 e n1/2 R 1/2 (7) Snce there s no relaxaton n bulk modulus, the pressure response s gven by: P1 n 1 P1 n 2K v 1 (8) n 1/2 where K s the bulk modulus, v n s the volumetrc 1/2 stran, and 1 s second order dentty tensor. By combnng Eqs. (6) (8), the total Cauchy stress at load step n+1 s gven as: P1 S S (9) n 1 n1 n1 n1 1 N Hence we can see that the vscous stresses would relax wth passng tme, but the stresses due to the pressure and the nstantaneous modulus of elastcty would be present. hey would result n resdual stresses. BOUNDARY CONDIIONS he fnte element model s developed such that the glass and the center of the mold surface have ntal contact on the axs of symmetry. Hence the common node at the orgn (common to the flexble and rgd surface) s fxed to prevent any rgd body dsplacement. he lower rgd nodes are fxed n all degrees of freedom and the upper mold surface s specfed wth the dsplacement necessary for compresson. For the thermal analyss, the center of the aspherc profle along the symmetry axs s fxed to prevent rgd body rotaton. Fxng ths node also preserve the orgn of the model even though the mold surface changes ts profle. IDENIFICAION OF MAERIAL PROPERIES Cylndrcal compresson test s performed to dentfy the Maxwell parameters of the glass. he cylndrcal compresson test s performed by mposng an nstantaneous stran on a cylndrcal glass sample and notng the decay of the stress wth passng tme, whlst mantanng a constant stran. Snce the compresson s comparatvely small, small stran vscoelastc consttutve equaton s used for fttng the parameters. he small stran vscoelastc stress response s gven by [11]: t de 2 G( t ) d Kv1 (10) d 0 where,

6 N t G( t ) G G exp (11) 1 and e s the devatorc stran tensor, s the past tme or pseudo tme, K s the bulk modulus, and v s the volumetrc stran. he frst stress component s the shear stress, whch has the relaxaton functon and the second component s the volumetrc stress wth no relaxaton. In the compresson then, the nstantaneous stran s appled lnearly and then the stran s held constant whle notng the decay of stress. Hence the above ntegral s ntegrated analytcally to obtan analytcal expressons for the stress. hen the analytcal stress expressons are ftted to the expermental data to determne the Maxwell parameters whch mnmze the error [12]. CONAC ANALYSIS Contact between glass and molds are modeled usng slave-master concept [15]. In ths model, a node on the surface of the glass model cannot penetrate nto lnear segments of model surface. Snce the elastc modul between glass and molds are sgnfcantly dfferent, the mold surface s assumed to be rgd body. Penalty method s used to mpose the mpenetrablty constrant. RESULS AND FUURE WORK Fgure 9 shows the fnal geometry of a compressed sphercal N-BK7 glass wth an aspherc lower mold and flat upper mold. he ntal glass gob s sphercal n shape wth dameter of 2.0mm and s compressed by 1.2mm at a unform rate. he moldng temperature s 550 C and the ambent temperature s 25 C. he results are verfed usng commercal software ANSYS. In the future, the smulaton results wll be valdated usng experments. Fg 9: Fnal Geometry of Glass after Moldng Process Fgure 10 shows the profles of the fnal geometry of the glass and the ntal shape of the mold profle, whch s the desred glass profle at the end of the moldng process. We can clearly see the sgnfcant dfference n the profle due to the varous processes that go n the glass moldng process. he dfference of the fnal glass profle s scaled by 20% to brng out the vsual contrast between the profles. he longer profle s that of the fnal deformed glass, and the other profle s the ntal mold geometry. It s desred that both the profles are dentcal, but the same s not acheved. Fg 10: Dfference between the fnal Glass and Intal Mold Profle he developed smulaton tool s computatonally cheap and can execute the smulaton much qucker than a complex commercal software analyss. Hence the tool can be optmzed to predct the ntal mold geometry and the process parameters requred n order to obtan the desred glass profle. CONCLUSIONS AND FUURE PLANS In ths paper, nstead of a complcate all-n-one glass moldng analyss, a smplfed mult-step analyss process based on engneerng analyss s presented. A physcal model and a computatonal tool are successfully developed to predct each sub-process n a glass moldng process quanttatvely. he analyses of the sub processes are ntegrated to result n a tool whch can predct the fnal geometry of glass the end of the compresson moldng process. Wth the help of ths tool, the fnal geometry of the glass can be predcted after the moldng process. Usng the fnal geometry of the lens that s obtaned above, the mold geometry that would result n a gven desred optcal geometry can be obtaned mnmzng the error between the generated shape of the lens and the desred shape. hs s acheved by settng up an optmzaton routne that would teratvely fnd the mold geometry and the process parameters requred to produce an optcal element of desred profle. ACKNOWLEDGEMEN hs research was funded by NSF and Moore Nanotechnology Systems, LLC. We are thankful for ther support.

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