let. - vol. 48 (2002) {t. - no. 11 STROJNIŠKI VESTNIK 11 JOURNAL OF MECHANICAL ENGINEERING strani - pages 569 - 642 ISSN 0039-2480 . StrojV . STJVAX cena 800 SIT galna singularnost torzije kompozitne palice -k--o-m- pozitne The Corner Singularity of Composite Bars Torsion Vpliv zrnate strukture na elasto- plasti~ni odziv polikristalnega skupka ------- The Eff of Grain Structure on the Elastic-Plastic Response of a Polycrystalline Aggregate ptimiranje oblike konstrukcij: ristrani~ni projektni element ---Structural Shape Optimization: Trilateral Design Element 4. timiranje pogonskega mehanizma stiskalnice za globoki vlek ------- Optimization of Link-Drive Mechanism for Deep Drawing Mechanical Press olo~anje kriti~ne obremenitve kro`nih kolobarjev v elasto-plasti~nem obmo~ju ------- Determination 0f the Buckling Loads of Circular Annular Plates in the lastic-Plastic Region bolj{ana razpoznava du{enja z uporabo zvezne val~ne transformacije ------- Enhanced identification of damping usin continuous wavelet transform " Ti-CM -------------------OJ 2 ------------w r». *- O) in A 9770039248001 © Strojni{ki vestnik 48(2002)11,569 © Journal of Mechanical Engineering 48(2002)11,569 Mese~nik Published monthly ISSN 0039-2480 ISSN 0039-2480 Vsebina Contents Strojni{ki vestnik - Journal of Mechanical Engineering letnik - volume 48, (2002), {tevilka - number 11 Uvodnik Editorial Alujevič, A.: Petnajst let Kuhljevih dnevov (1987- Alujevič, A.: Fifteen years of “Kuhelj’s Memorial 2002) 570 Days” (1987-2002) Razprave Papers Mejak, G.: Vogalna singularnost torzije kompozitne Mejak, G.: The Corner Singularity of Composite Bars palice 571 in Torsion Kovač, M., Simonovski, I., Cizelj, L.: Vpliv zrnate strukture na elasto-plastilčni odziv Grain Structure on the Elastic-Plastic Response polikristalnega skupka 580 of a Polycrystalline Aggregate Kegl, M.: Structural Shape Optimization: A Trilateral projektni element 591 Design Element pogonskega mehanizma stiskalnice za globoki Drive Mechanism for Deep Drawing vlek 601 Mechanical Press Bremec, B., Kosel, F.: Določanje kritične Bremec, B., Kosel, F.: Determination of the Buckling obremenitve krožnih kolobarjev v elasto- Loads of Circular Annular Plates in the Elastic- plastičnem območju 613 Plastic Region Slavic, J., Boltežar, M.: Izboljšana razpoznava dušenja Slavic, J., Boltežar, M.: Enhanced identification of z uporabo zvezne valčne transformacije 621 damping using continuous wavelet transform Poročila 632 Reports Osebne vesti 640 Personal Events Navodila avtorjem 641 Instructions for Authors © Strojni{ki vestnik 48(2002)11,570 © Journal of Mechanical Engineering 48(2002)11,570 Uvodnik Editorial ISSN 0039-2480 ISSN 0039-2480 Uvodnik Editorial Petnajst let Kuhljevih dnevov (1987-2002) Slovensko društvo za mehaniko je letos v spomin na akademika prof.dr. Antona Kuhlja, ob stoti obletnici njegovega rojstva, pripravilo svoje šestnajsto znanstveno srečanje. Prvič smo se s tem namenom sestali leta 1987 v Preddvoru, ko je imel uvodno predavanje njegov dolgoletni sodelavec prof.dr. Peter Vencelj. Srečanja so bila nato še na Rogli, v Lipici, Portorožu, Šmarjeti, Radencih, Martuljku, Zemonu, Mokricah, Logarski dolini, Mariboru in na Bledu. Vodili so jih predsedniki društva prof.dr. Miran Saje, prof.dr. Maks Oblak, prof.dr. Boris Stok in prof.dr. Leopold Škerget. Med udeleženci naj omenim prof.dr. Antona Kuhlja ml., profdr.Marka Škerlja, kakor tudi pokojnega prof.dr. Milana Muršiča in preostale neimenovane. Ko je bil akademik prof.dr. Anton Kuhelj še živ, smo se takrat stari in mladi vsak drugi torek v mesecu dobivali zvečer na stari tehniki. Tam smo imeli posamezna “preskusna” predavanja, saj je veljalo načelo, da na kongresih mehanike lahko sodelujemo le, če smo se poprej predstavili v lastni hiši. Posamezno ali v tandemu smo predstavljali svoje delo, starejši pa so potem razpravljali in spraševali (prof.dr. Ervin Prelog, prof.dr. Miloš Marinček, profdr.Srdjan Turk in drugi). Po smrti našega mentorja, ki je leta 1980 umrl po prometni nesreči, smo nekaj časa ostali brez naših stalnih srečanj, po sedmih letih pa smo v njegov spomin pričeli vsak september dvodnevna srečanja, z letno skupščino našega društva in sedaj že tradicionalno večerjo z dobro kapljico. Pri tem velja omeniti tudi sponzorja našega letošnjega srečanja v hotelu Ribno, tj. Elan iz Begunj. Srečanja obsegajo obe poglavji naše dejavnosti - tako trdnine kakor tekočine. Vendar so najbolj uspela tista srečanja, ko nismo strogo ločevali obeh dejavnosti in tudi ne teorije, numeričnega in eksperimentalnega dela, ki skupaj predstavljajo celostno podobo delovanja Slovenskega društva za mehaniko. Pred tridesetimi leti smo imeli samo ljubljansko podružnico skupnega Jugoslovanskega društva za mehaniko (ki ga je tudi na Bledu ustanovil profdr.Kuhelj). To društvo je bilo tudi prireditelj kongresa GAMM 1972 v Ljubljani, ki pa ga je preprečil izbruh epidemije črnih koz. Nekaj let kasneje smo v Ljubljani ustanovili Slovensko društvo za mehaniko kot združbo visokošolskih učiteljev treh ljubljanskih (FS, FAGG in FNT) in ene mariborske šole (VTŠ). Tudi dežurstvo predsedovanja smo “kolobarili”, vendar žal starejši “udje” odhajajo. Tako sem se podpisani ovedel, da sem kar naenkrat postal najstarejši še neupokojeni član našega društva. Nestorju Antonu Kuhlju smo hvaležni za vse, kar nam je dal in zapustil. Prof.dr. Andro Alujevič Odgovorni urednik SV 1 SšnnstsfcflM]! mn stran 570 © Strojni{ki vestnik 48(2002)11,571-579 ISSN 0039-2480 UDK 678.01:620.175:534.013 Izvirni znanstveni ~lanek (1.01) © Journal of Mechanical Engineering 48(2002)11,571-579 ISSN 0039-2480 UDC 678.01:620.175:534.013 Original scientific paper (1.01) Vogalna singularnost torzije kompozitne palice The Corner Singularity of Composite Bars in Torsion George Mejak Materialna matrika kompozitov je na vsaki materialni komponenti nespremenljiva. Ta nezveznost materialne matrike omejuje regularnost rešitve elastomehanične naloge s kompozitnim materialom. Poleg materialne nezveznosti na regularnost rešitve vpliva še geometrijska oblika stične ploskve med sosednjimi materialnimi komponentami. Vsaka medmaterialna geometrična singularnost je vir singularnosti, ki se praviloma manifestira v obliki koncentracije napetosti. Pomembni podatek za izračun koncentracije napetosti je red vogalne singularnosti. V prispevku je za modelni problem torzije kompozitne prizmatične palice predstavljena metodologija določitve reda vogalne singularnosti. V prvem delu prispevka je podan model torzije s popolno in nepopolno vezjo med materialnimi komponentami. Za model popolne vezi je nato z asimptotičnim razvojem v vrsto dokazan obstoj koncentracije napetosti v vogalu. Izrecno je izračunan red vogalne singularnosti v odvisnosti od vogalnega kota in materialnih lastnosti kompozitov. Pomembna ugotovitev je, da je red singularnosti neodvisen od usmeritve materiala v vogalu. V primeru nepopolne vezi je dokazan obstoj koncentracije napetosti v vogalu za dovolj ohlapno vez. Rezultat je dokazan z regularnim asimptotičnim razvojem v okolici popolne medmaterialne nepovezanosti. © 2002 Strojniški vestnik. Vse pravice pridržane. (Ključne besede: kompoziti, palice, problemi moeliranja, analize singularnosti) The material matrix of composite materials is constant for each individual component. This discontinuity sets regularity bounds upon the solution. Besides material discontinuities, the regularity of the solution is also affected by geometrical singularities along the material interfaces. If the interface has corners, we speak about corner singularities of the composite. Mechanical manifestations of these singularities are stress concentrations. One of the most important pieces of information about the corner singularity is the order of the singularity. In this article a method for determining the order of the singularity for the model problem of a composite bar in torsion is presented. In the first part a model of torsion for a composite bar with perfect and imperfect bonds is given. For a perfect interfacial bond the existence of the corner stress concentration is proved by the asymptotical method. The order of the singularity with respect to the angle of the corner and the material constants is explicitly computed. It is found that the order is independent of the material orientation in the corner. For the case of an imperfect bond the existence of the stress concentration is established for a weak bond. The existence is proved by the regular asymptotic perturbation of the no-material adhesion. © 2002 Journal of Mechanical Engineering. All rights reserved. (Keywords: composite, bars, modelling problems, singularity analysis) 0 UVOD V tehnični praksi so konstrukcijski elementi zaradi različnih razlogov pogosto sestavljeni iz dveh ali več različnih materialov. Takim sestavljenim elementom pravimo kompoziti. Pomemben razlog za uporabe kompozitov je doseči želene materialne lastnosti kompozita s primerno izbiro materialov posameznih komponent [1]. Materialna matrika kompozita je nespremenljiva na vsaki materialni komponenti posebej in je tako stopničasta funkcija. Komponente materialne matrike se pojavljajo kot koeficienti elastične energije. 0 INTRODUCTION Construction elements are quite often made of composite materials. The reason for using composites is to obtain a product with the desired properties composed of different materials [1]. The material matrix is constant for each material component, but has jump discontinuities across the material interfaces. As a result, the coefficients of the governing partial differential equations have jumps across the interfaces. It is well known [2] that the regularity of the solution depends upon gfin^OtJJIMISCSD 02-11 stran 571 |^BSSITIMIGC Mejak G.: Vogalna singularnost - The Corner Singularity Znano je, da nezveznost materialne matrike odločilno vpliva na regularnost rešitve [2]. Posebej to pomeni, da moremo pri kompozitih pri prehodu iz ene materialne komponente v drugo pričakovati določeno singularnost. Po drugi strani je znotraj komponent materialna matrika nespremenljiva in je potemtakem rešitev na posamezni komponenti notranje regularna. Poleg nezveznosti materialne matrike pri prehodu iz ene materialne komponente v drugo na regularnost vpliva še geometrijska oblika stične ploskve sosednjih materialnih komponent. Vsaka geometrična singularnost je vir nove singularnosti, ki se praviloma izraža v obliki koncentracije napetosti. Tej singularnosti pravimo vogalna singularnost kompozita. Znano je, da je poznavanje reda singularnosti rešitve pomembno pri numeričnem modeliranju [3], saj vpliv singularnosti na rešitev ni krajeven, temveč praviloma celovit. Uspešnost neposredne metode omejitve vpliva singularnosti, z zgostitvijo diskretizacije v okolici vira singularnosti, je omejena s povečanjem števila prostostnih stopenj in poslabšanjem numerične pogojenosti naloge. Bolj učinkovita je dekompozicija rešitve na singularni in regularni del ter lepljenje singularnega dela z diskretizacijo regularnega dela. Pomanjkljivost te metode je, da moramo singularni del rešitve poznati dovolj natančno. Preprostejša, zato pa še vedno dovolj učinkovita, je metoda dekompozicije prostora aproksimacije na singularni in regularni del. V metodi končnih elementov to pomeni uporabo singularnih elementov v okolici singularnosti rešitve. Za uporabo te metode je dovolj poznati red singularnosti rešitve [3]. V prispevku bomo določili red vogalne singularnosti za modelni problem torzije prizmatične kompozitne palice. Prispevek je razdeljen vštiri razdelke. Po uvodu sledi formulacija problema torzije kompozitne palice v variacijski obliki in v obliki robne naloge. Formulirana je naloga torzije s popolno vezjo med posameznimi materialnimi komponentami palice in nepopolno vezjo, ki dopušča na medmaterialnem stiku dislokacijo v smeri osi palice. V tretjem razdelku je obravnavana vogalna singularnost s pomočjo asimptotičnega razvoja v okolici vogala. Tu se bomo omejili na torzijo palice s popolno vezjo. Izpeljana je karakteristična enačba za lastne vrednosti in dokazan je obstoj koncentracije napetosti z izrecnim izračunom reda vogalne singularnosti. Torzija palice z nepopolnimi vezmi med materialnimi komponentami je obravnavana v četrtem razdelku. Pokazali bomo, da dislokacija ne sprosti napetosti in da ima tudi v tem primeru rešitev singularnost v vogalu. 1 TORZIJA KOMPOZITNE PALICE Napetost pri torziji homogene prizmatične palice m0,l], kjer sta n prerez palice in l dolžina palice, je dana s Prandtlovo napetostno funkcijo x. V kartezičnem koordinatnem sistemu z osjo z v smeri osi palice sta tako edini neničelni komponenti napetosti t13= ^0(8% / dy) in t23= t32=-M9(dx/dx), the regularity of the coefficients. On the other hand, coefficients are constant within the components, and the interior regularity is not affected. Besides material discontinuities, there is another possible source of the singularity: the shape of the interface boundary between the material components. Each geometric singularity of the interfacial boundary is the source of another singularity of the solution. The mechanical manifestation of these singularities is through stress concentrations. If the interfacial boundary has corners, we speak about the corner singularities of the composite. It is well known that accurate numerical modelling depends upon a firm knowledge of the order of the singularities [3], as the numerical solution is globally affected by singularities. The direct approach of the local mesh refinement is hampered by the increase in the number of degrees. Also, the condition number of the problem may be affected by the high ratio of the element sizes. A more effective method is to decompose the solution into the singular and regular parts. However, to do this, the singular part of the solution has to be known in advance. Simpler, and still good enough, is the method of decomposing the discretization space. In the case of the finite-element method this means that singular elements are used around the source of the singularity. To use singular elements one only has to know the order of the singularity [3]. In this paper the discussion is restricted to the model problem of the torsion of a composite bar with perfect and imperfect interfacial bonds. The paper has four parts. After an introduction we have the formulation of the problem. Variational, as well as distributional formulations are given. Attention is given to possible axial dislocations, which arise due to the imperfect bonding. In the next section the corner singularity of bars with a perfect bond is approached by the asymptotic expansion. A characteristic equation is derived and the existence of the stress concentration is established. It is proved that the asymptotic expansion has only one singular term, which gives the stress concentration. In the last section the torsion with the imperfect bond is considered. Stress concentration is proved in the case of the weak bond and thus the axial dislocation does not relax the stress concentrations. 1 TORSION OF COMPOSITE BARS For a homogenous prismatic bar Qx[0,l], where n is the cross section and l is the height of the bar, stress components are given by the Prandtl stress function x . In the Cartesian coordinate system with the z axis aligned with the axis of the bar the only non-zero stress components are t13= t31= fi9(dx / dy) 1 BnnBjfokJ][p)l]Olf|i[gO | | ^SsFvWEIK | stran 572 Mejak G.: Vogalna singularnost - The Corner Singularity kjer je m strižni modul palice, J pa je torzijski zasuk palice na dolžino palice. Potencialna energija torzije palice je vsota elastične energije in potenciala površinskih sil. Elastična energija homogene palice je: and t= t=-mJ(dx/dx) where m is the shear modulus of the bar and J is the torsion angle per unit length of the bar. The potential energy of the bar is the sum of the elastic energy and the potential of the surface traction. The elastic energy is: U = le : t dW = lm021l\Vz\2dW, potencial površinskih sil pa je: whereas the potential of the surface traction is: Ut = - r • u ldW - r • udW, JW H JS kjer je u vektor pomika, ki ima na osnovni ploskvi z=0 pomik samo v smeri osi palice, S je plašč palice. Potencial površinskih sil moremo preoblikovati v: where u is the displacement vector, which is at the base z=0, directed along the z axis. The lateral surface of the bar is denoted by S. The potential of the surface traction is rewritten as: U = - 2lmd2 \ zdW + lm&2\ %r¦ ndG -lmd2\ — dG. Potencialna energija palice je tako Up = Ue + Ut. Brezrazsežni zapis potencialne energije je U = U / lm0J2 ) , kjer je m0 referenčni strižni modul. V nadaljevanju bomo uporabljali izključno brezrazsežni zapis. Da bo pisava enostavnejša, brezrazsežnih in razsežnih strižnih modulov mi s pisavo ne bomo ločili. Potencialna energija U kompozitne palice s prerezom W= UiN= 1 W , kjer so Wi disjunktni prerezi posameznih materialnih komponent, je vsota potencialnih energij materialnih komponent palice. Potem je potencialna energija torzije enaka: The potential energy is thus U =Ue+Ut. The corresponding non-dimensional form is U = U / ( lm0J2 ) , where m0 is a reference shear modulus. In the following, only the non-dimensional form will be used and thus, to simplify the notation, we make no notational distinction between the dimensional and non-dimensional moduli. The potential energy U of the composite bar NWi, where W are cross with the cross section W= Ui N= i=1 i1 sections of the individual material components, is the sum of the potential energies of the individual material components. Thus: U 2I M 2b dW i=1 dW Emi I dci ds dG ,(1), kjer sta mi ter Zi strižni modul in napetostna funkcija i-te komponente, ^ pa je pomik i-te komponente v smeri osi palice. Rob dWi materialne komponente je vsota robov do sosednjih materialnih komponent in zunanjega roba. Tu smo vzeli, da je prerez kompozitne palice enostavno povezano območje. V primeru, da ima prerez luknjo, lahko luknjo obravnavamo kot materialno komponento s strižnim modulom, ki limitira proti nič [4]. Zunanji rob prereza je prost, zato imajo napetostne funkcije tistih komponent, ki sestavljajo ovoj na zunanjem robu, nespremenljivo vrednost. Napetostna funkcija je določena do stalnice natančno, zato moremo te stalnice izbrati tako, da imajo napetostne funkcije na zunanjem robu vrednost nič. Na robu med dvema materialnima komponentama velja ravnovesni pogoj recipročne Cauchyjeve relacije. To v zapisu z napetostno funkcijo na skupnem robu i-te in j-te materialne komponente pomeni enakost mi (dZi / ds) = mj (d Ž j / ds). Funkciji Xi in žj se torej na skupnem robu razlikujeta le za stalnico. Pri predpostavki enostavno povezanega prereza Wi in izbire vrednosti napetostnih funkcij na zunanjem robu potem sledi enakost izrazov mi^ in mjZj na where mi and Xi are the shear modulus and the stress function of the cross section W, and i is the dislocation of the i-th component in the direction of the z axis. The boundary 3Wi of the i-th material component is the union of the boundaries between the material components and the part of the outer boundary. We assume here that the cross section of the bar is simply connected. In the opposite case, where the cross section has a hole, the hole can be treated as the limit of the material, with the shear modulus vanishing, [4]. The outer boundary is traction free, and thus the stress functions along the outer boundary are constant. The stress function is determined up to a constant factor, and thus the constants can be arranged such that the stress functions along the outer boundary are all vanishing. Across the interfacial boundary the Couchy reciprocal relation holds. In particular, along the interfacial boundary between i-th and j-th component we have mi (dZi / ds) = mj (dXj / ds). Functions X and ij thus differ along the common boundary for a constant. Due to the arrangement of the constants along the outer boundary it follows then that miZi and mjŽj are equal along the common boundary. Therefore, Mejak G.: Vogalna singularnost - The Corner Singularity skupnem robu. To pomeni, da napetostna funkcija c, katere zožitev na Wi je enaka ci, ni zvezna na W. Pri ravnovesnem pogoju na meji med različnima materialoma je druga vsota v (1) enaka nič. Potem je: N /1 function c, which equals ci on Wi is not continuous on W. It follows from the equilibrium that the second sum in (1) vanishes. Hence: i=1 W ds kjer je Gij skupni rob komponent Wi in Wj, in skok osnega pomika na meji med dvema materialoma V primeru popolne vezi med materialnimi komponentami je ta skok enak nič. Točneje, velja [f\ = 0. Potemtakem je potencialna energija za popolno kompozitno palico enaka: N Če ima pomik f skok na meji med dvema Wi materialoma je vez med materialoma nepopolna. Pri nepopolni vezi se materialne komponente dislocirajo v osni smeri. Najpreprostejši model, glej [5], nepopolne vezi temelji na predpostavki, da je dislokacija v osni smeri sorazmerna napetosti. Z enačbo je f = -am(dc / ds), kjer je a pozitivna stalnica. Pripadajoča potencialna energija je: U = U(c) = Y1 mi\[-^ci - Očitno se (4) za a=0 reducira v potencial popolne vezi (3). Potencial v (4) je definiran na množici: 2 cijdW-Y.mi\f dsi dG (2), where Gij is the ij common boundary of Wi and Wj, and: is the axial dislocation at the interfacial boundary. In the case of perfect bonding there are no dislocations and thus M = 0 . Therefore, the potential energy of the composite bar with perfect bonding is: |Vci| -2ci dW. (3). In the case of the axial dislocations we speak of the imperfect bonding and f has a jump across the interface. The most simple dislocation model is based on the assumption that axial dislocations are proportional to the axial stresses, [5]. Thus \ft\ =-am(dc / ds), where a is a positive proportional factor. The potential energy for the imperfect bonding is thus: 2ci dW + a2>i2J dci ds dG (4). V = \c:c\WieH1(Wi) in ds G eL2(Gij)\, Evidently, for a=0, (4) reduces to (3). The potential (4) is defined on the functional space: dc kjer je H1 (Wi) prostor Soboljeva prvega reda. Tu smo privzeli, da so robovi Gi odsekoma regularni. Napetostna funkcija c ima skok na prehodu iz enega v drugi material. Ta skok moremo s preprosto preslikavo ci=mici regularizirati. Pripadajoči regulariziran potencial je: where H1 (Wi) is the Sobolev space of the first order. In the above it was assumed that the interfacial boundaries Gij are piecewise regular. The stress function c has jumps across the material interfaces. These jumps are eliminated with a simple transformation ci = mici. The corresponding potential has the form: 1 dci ds dG (5), prostor pa: with the corresponding space: dc V = \c:c WieH1(Wi) in ds G eL2 (Gij)\. Prostor V je s skalarnim zmnožkom: The functional space V, equipped with the scalar product: du dv Hilbertov prostor. Omenimo, da moremo pri privzetku regularnosti robov potencial definirati namesto na V na množici: H1(Wf\ [c: ciG e H3/2(Wi)}. Rešitev torzijske naloge kompozitne palice je minimum potenciala torzije na dopustni množici. Rešitev je torej dana z minimizacijsko nalogo: ij ds ds is a Hilbert space. It should be noted that assuming some additional regularity upon Gij the space V can be replaced with H1(W)(\[c: ci Gi \ H3/2(Wi)} A solution of the torsional problem is a solution of the minimization problem for the potential energy. Thus we have: 1 BnnBjfokJ][p)l]Olf|i[gO | | ^SsFvWEIK | stran 574 Mejak G.: Vogalna singularnost - The Corner Singularity min Eksistenca in enoličnost rešitve izhaja iz šibke napol zveznosti in koerkivnosti potenciala na Hilbertovem prostoru V, glej [6]. Minimizacijski nalogi pripada variacijska naloga: poišči ^ V, tako da je: c^V The existence and the uniqueness of the solution follows from the fact that the potential energy is a coercive, weakly lower semicontinuous function defined on the Hilbert space V, see [6]. To the minimization problem is associated a variational problem: find^V, such that: kjer je: in a(c, w) = b(w), za vsak/for all w e V where (6), i=1 m i ds ds and b(w) = 2 w dW. Variacijski nalogi je v porazdelitvenem pomenu enakovredna robna naloga: najdi c e V(D) = vf]{c: DcWi e L2 (Wi)}, tako da velja: 1 The variational problem is equivalent to the following boundary value problem in the distributional sense: find c e V(D) = vf] { c: DcWi e L2 (Wi)} such that: mi 1dc_ m dn Dc = 2 v/in Wi za vsak/for all i = 1,...,N in/and d2c ds2 0 na/on Gij za vsak/for all 1 Meja kristalnega zrna Grain boundary Usmeritev kristalne rešetke Orientation of crystal lattice Mreža končnih elementov Finite element mesh Obremenitev Loading 'V V V V V V YVVVVVVYVN/VYVv p2 Sl. 1. Polikristalni skupek Fig. 1. Polycrystalline aggregate 1 BnnBjfokJ][p)l]Olf|i[gO I | ^SsfiFWEIK | stran 582 Kova~ M., Simonovski I., Cizelj L.: Vpliv zrnate strukture - The Effect of Grain Structure trdnina, pri čemer zanj velja posplošeni Hookov zakon constitutive relations are given by the generalized [13]: Hooke’s law [13]: s C ijkl ' e kl (1), kjer sij pomeni tenzor napetosti 2. reda, Cijkl tenzor togosti 4. reda in ekl tenzor specifičnih deformacij 2. reda. Elastične lastnosti polikristalnega skupka so popolnoma določene z lastnostmi in medsebojnim delovanjem posameznih monokristalov Elastični materialni parametri so pridobljeni iz literature za železo a s telesno osrednjeno kubično mrežo ([14] in [15]). Predpostavljeno je, da majhne količine legirnih elementov ne vplivajo na elastično togost in podajnost monokristalov [15]. Uporabljeni so naslednji elastični snovni parametri: Ciiii = 230 GPa, Ciijj = 135 GPa in Cijij = 117 GPa [15]. 1.3 Kristalna plastičnost Kristalna plastičnost predpostavlja, da je plastična deformacija posledica zgolj zdrsa kristalnih ravnin. Predpostavljeno je, da je Schmidova napetost (strižna napetost v sistemu zdrsa) gonilna sila zdrsa [16]. Sistem zdrsa je določen s kombinacijo ravnine (določena z normalo m(a)) in smerjo zdrsa (s(a)) znotraj kristalne rešetke. Telesno osrednjena kubična mreža ima tri družine ravnin zdrsa: {110}, {112} in {123} ter eno družino smeri zdrsa: <111>, kar pomeni 48 možnih sistemov zdrsa [17]. Osnovni zakon kristalografskega zdrsa se glasi: where a represents the stress tensor of the 2nd rank, Cijkl the ij stiffness tensor of the 4th rank, and ekl the strain tensor of the 2nd rank. The elastic properties of the polycrystalline aggregate are completely defined by the properties of, and interaction between, the crystal grains. The material parameters for the elasticity are obtained from the literature for a-Fe with a body-centered-cubic crystal lattice ([14] and [15]). It is assumed that small amounts of alloying elements do not change the elastic stiffness/ compliance of a crystal grain significantly [15]. The following elastic material parameters were used C = 230 GPa, Ciijj = 135 GPa and Cijij = 117 GPa [15]. iiii 1.3 Crystal Plasticity Crystal plasticity assumes that plastic deformation is the result of crystalline slip only. It is assumed that crystalline slip is driven by Schmid stress (resolved shear stress). The slip system by a defined with combination of a slip plane (defined by its normal m(a)) and a slip direction (s (a)) within the crystal lattice. A body-centered-cubic i crystal lattice has three families of slip planes - {110}, {112} and {123} - and one family of slip directions, <111>, which accounts for 48 possible slip systems. The basic law of crystalline slip is: s& C ijkl ' ( e & kl e & kl ) C ijkl ' e ^1 g & a )( sama+sama ) (2), kjer s & ij pomeni časovni odvod tenzorja napetosti, e & kl ječasovni odvod tenzorja specifične deformacije, e & kpl časovni odvod tenzorja plastične specifične deformacije in g & (a) hitrost zdrsa sistema zdrsa a. Plastičnost neodvisno od hitrosti deformacije lahko obravnavamo kot mejni primer plastičnosti, odvisne od hitrosti deformacije [16]. Hitrost zdrsa g & (a) je določena z ustrezno Schmidovo napetostjo t(a): where &ij is the stress-rate tensor, e& kl is the strain-rate tensor, e & kpl is the plasticity strain-rate tensor and g &(a) is the slipping rate of the a-th slip system. Rate-independent plasticity may be treated as the limit of rate-dependent visco-plasticity [16]. The slipping rate g &(a) is determined by the corresponding Schmid stress t(a) as: g &( a)=a &( a ) av (a) (a) (3), pri čemer so a(a) referenčna stopnja strižne napetosti, n je občutljivostni parameter strižne napetosti in ga) trenutno stanje utrjevanja monokristala. V limiti, ko se n približuje neskončnosti, enačba (3) ustreza enačbi za material, neodvisen od hitrosti deformacije. Trenutno stanje deformacijskega utrjevanja monokristala g(a) je določeno iz hitrosti deformacijskega utrjevanja (a) : where a& (a) is the reference strain rate, n is the strain-rate sensitivity parameter and g(a) is the current strain-hardened state of the crystal. In the limit as n approaches infinity, eq. (3) approaches that of a rate-independent material. The current strain-hardened state of the monocrystal g(a) can be derived from the strain-hardened rate (a) : (b) (4), kjer hab pomeni modul utrjevanja. Modul utrjevanja je obravnavalo več avtorjev ([16] in [18]), vsi pa imajo where hab are the slip-hardening moduli. Other authors have dealt with hardening moduli ([16] and [18]), with | IgfinHŽšlbJlIMlIgiCšD I stran 583 glTMDDC Kova~ M., Simonovski I., Cizelj L.: Vpliv zrnate strukture - The Effect of Grain Structure za osnovo izkustvene modele. V naši raziskavi smo uporabili Peirce et al. [19] ter Asaro ([18] in [20]), zakon o utrjevanju materiala, ki za izračun modula utrjevanja uporablja naslednji enačbi: all of them basing their work on empirical models. The Peirce et. al. [19] and Asaro ([18] and [20]) hardening law is used in our research, which uses the following equations: haa = h(g ) = h0 sech 2 h0 in/and hab=qh(g) (a*b) (5), kjer so: h0 začetni modul utrjevanja, t0 meja tečenja, ki je enaka začetni vrednosti trdnosti materiala g(a)(0), tS je mejna napetost, nad katero se začenjajo velike plastične deformacije, g kumulativni zdrs in q faktor utrjevanja. Plastični snovni parametri so pridobljeni iz literature za kristalno plastičnost, neodvisno od hitrosti deformacije [17], in iz rezultatov nateznega preskusa obravnavanega jekla. Uporabljene so bile naslednje vrednosti: občutljivostni parameter strižne napetosti n = 50, referenčna hitrost strižne napetosti a (a) = 0,001 s1, začetni modul utrjevanja h0 = 70 MPa, mejna napetost velikih plastičnih deformacij tS = 15,5 MPa, meja tečenja t0 = 155 MPa in faktor utrjevanja q = 1. Enačbe (2) do (5) so za uporabo z metodo končnih elementov podane v koračni obliki [16]. Te enačbe so v splošnem zelo toge [13]. Togost sistema narašča z naraščanjem števila sistemov zdrsa. Zaradi togosti potrebuje klasična računska shema zelo majhne korake (in dolge računske čase) za zagotovitev stabilnosti rešitve. where h0 is the initial hardening modulus, t0 is the yield stress, which equals the initial value of the current strength g(a)(0), tS is the the break-through stress, where large plastic flow initiates, g is the cumulative slip and q is the hardening factor. Material parameters for the plasticity are obtained from the literature for rate-independent crystal plasticity (e.g. [17]) and from the results of a standard tensile test of the selected material. The following values were used: the strain-rate-sensitivity parameter n = 50, the reference strain rate a& (a) = 0.001 s–1, the initial hardening modulus h0 = 70 MPa, the break-through stress tS = 15.5 MPa, the yield stress t0 = 155 MPa and the hardening factor q = 1. For the use with the finite-element method, equations (2) to (5) are given in incremental form [16]. These equations are, in general, very stiff [13]. The stiffness of the system increases with the number of slip systems. Due to the stiffness the classical integration scheme needs very small incremental steps (and long computational times) to ensure solution stability. 1. 4 Ocena velikosti reprezentativnega 1.4 Estimation of the representative volume element prostorninskega elementa size V literaturi se je za najmanjši vzorec nehomogene snovi, ki je makroskopsko homogena, uveljavil termin reprezentativni prostorninski element (RPE). Velja, da pri vzorcu nehomogene snovi, večjim od RPE, vpliv velikosti na makroskopskem nivoju ni opazen [13]. Velikost RPE polikristalnega skupka je določena s primerjavo makroskopskega tenzorja togosti C* in tenzorja podajnosti D*ijkl. Pri tem so makroskopske veličine povprečene po celotnem polikristalnem skupku. Za polikristalni skupek, večji od RPE, velja ([10] in [21]): The minimum size of a mesoscopically inhomogenous material that is macroscopically homogenous is, in the literature, usually refered to as the representative volume element (RVE). For volumes of mesoscopically inhomogenous material larger than the RVE the size effect at the macroscopic level cannot be observed [13]. The RVE size is defined by a comparison of the macroscopic stiffness C*ijkl and the compliance tensors D*ijkl. The macroscopic quantities are averaged over the entire polycrystalline aggregate. For polycrystalline aggregate larger than RVE stands ([10] and [21]): C * ijkl (D* ijkl )- (6). Enačba (6) v splošnem ne velja za polikristalne skupke, manjše od RPE. Obnašanje takšnih polikristalnih skupkov je odvisno od njihovih velikosti in makroskopskih robnih pogojev [13]: makroskopski tenzor togosti predpostavlja robni pogoj s predpisano napetostjo, makroskopski tenzor podajnosti pa robni pogoj s predpisanim pomikom. Z upoštevanjem razmerij med napetostmi in specifičnimi deformacijami (npr. enačba (1)), je enačbo (6) mogoče poenostaviti z uporabo makroskopskih napetosti ali specifičnih deformacij [22], npr.: Equation (6) is, in general, not valid for polycrystalline aggregates smaller than RVE. The behavior of polycrystalline aggregates is governed by their size and the macroscopic boundary conditions [13]: the macroscopic stiffness tensor therefore assumes a stress-driven boundary condition, while the macroscopic compliance tensor assumes a displacement-driven boundary condition. With general relations between stresses and strains in mind (as for instance described in equation (1)), equation (6) can be simplified by using macroscopic equivalent stresses or strains [22], e.g.: 1 SšnnstsfcflM]! ma stran 584 Kova~ M., Simonovski I., Cizelj L.: Vpliv zrnate strukture - The Effect of Grain Structure Indeksa in d označujeta robna pogoja s predpisano napetostjo oziroma pomikom. Tako poenostavljeno merilo je, navkljub poenostavitvam, dovolj dobra ocena za predstavitev ključne težnje [22]. Nekateri avtorji za določitev velikosti RPE uporabljajo ekstrapolacijo z upoštevanjem velikosti polikristalnega skupka [21]. Razmerje med makroskopskima tenzorjema togosti in podajnosti za polikristalni skupek se lahko izrazi kot [21]: (7). Indexes s and d denote the stress- and displacement-driven boundary conditions, respectively. This simplified condition is sufficient to present the crucial trends [22]. Some authors have used an extrapolation to estimate the RVE size. The extrapolation is based on the size of a polycrystalline aggregate [21]. The relation between stiffness and compliance tensors for that polycrystalline aggregate can be written as [21]: Ci*jkl.D*klmn=Iijmn+O(VlVrve) (8), kjer VRVE pomenijo velikost RPE, V velikost polikristala, manjšega od RPE in O oceno ostanka. Z upoštevanjem enačbe (7) in dejstva, da je število zrn i v polikristalnemu skupku sorazmerno njegovi velikosti, namesto enačbe (8) uporabimo: where VRVE represents the RVE size, V is the size of a polycrystalline aggregate smaller than RVE, and O is the estimate of the residuum. With equation (7) and the proportionality between the number of grains in the polycrystalline aggregate, and its size, in mind, one can use: 1 + O(i/iRVE) (9). Ocenjujemo, da je RPE dosežen, ko je ocena ostanka O manjša od 1% (0,1%). 2 REZULTATI Predstavljeni so nekateri rezultati predlaganega računskega postopka. Predstavljeni so primeri makroskopskega odziva polikristalnih skupkov z različnimi usmeritvami kristalnih rešetk in robnimi pogoji. 2.1 Diagram as Na sliki 2 je predstavljena zveza med makroskopsko primerjalno (Misesovo) napetostjo in We estimate that RVE size is achieved when residuum O is smaller than 1% (0.1%). 2 RESULTS The results of the proposed numerical approach are presented. Examples of the macroscopic response of polycrystalline aggregates with different orientations of the crystal lattice and boundary conditions are shown. 2.1 Stress-strain diagram Figure 2 shows the relationship between the macroscopic equivalent (Mises) stress and the 600 500 400 300 200 100 0 600 500 400 300 200 100 0% 5% 10% 15% Makroskopska primerjalna specifična deformacija [/] Macroscopic equivalent strain [/] 0% 5% 10% 15% Makroskopska primerjalna specifična deformacija [/] Macroscopic equivalent strain [/] Sl. 2. Zveza med makroskopsko primerjalno napetostjo in primerjalno specifično deformacijo pri robnih pogojih s predpisanim pomikom (levo) in napetostjo (desno) Fig. 2. A relationship between macroscopic equivalent stress and macroscopic equivalent strain with displacement (left) and stress driven boundary conditions (right) 0 Kova~ M., Simonovski I., Cizelj L.: Vpliv zrnate strukture - The Effect of Grain Structure makroskopsko primerjalno specifično deformacijo za 30 različnih primerov naključnih usmeritev kristalnih rešetk za polikristalni skupek s 14 zrni (sl. 1). Predstavljena sta primera z robnimi pogoji s predpisanim pomikom in napetostjo. Raztros krivulj zaradi različnih usmeritev kristalnih rešetk je razločno viden. V elastičnem območju krivulje močno sovpadajo, raztros meje tečenja pa je precejšen. Na robu predpisana napetost v povprečju povzroči bolj tog odziv. To je skupaj z že opisanimi težavami vzrok za manjše število analiziranih primerov. 2.2 Ocena velikosti RPE v elastičnem območju Analize v elastičnem območju so bile opravljene na polikristalnih skupkih s 14, 23, 53, 110 in 212 zrni. Vsak polikristalni skupek je bil analiziran za 30 primerov naključne usmeritve kristalne rešetke (z na robu predpisano napetostjo oziroma pomikom). Analize so bile izvedene pri zunanji obremenitvi p1 = 200 MPa in p2 = 100 MPa (sl. 1). Rezultati so primerjani z analitično rešitvijo v izotropni trdnini s snovnimi lastnostmi: E = 210 GPa in v= 0,29 [10]. Povprečne vrednosti so bile izračunane prek vseh 30 primerov usmeritev kristalne rešetke za vsak polikristalni skupek in robne pogoje. Rezultati so prikazani na sliki 3 (levo), pri čemer s in d označujeta robne pogoje s predpisano napetostjo oziroma pomikom, ave povprečno vrednost, naslednja številka pa število zrn v polikristalnem skupku. Prikazana je tudi analitična rešitev (makroskopska primerjalna specifična deformacija = 0,0515 % in makroskopska primerjalna napetost = 96,2 MPa). Opaziti je mogoče trend zmanjševanja raztrosa rezultatov ob večanju števila zrn v 97.0 96.8 96.6 96.4 96.2 96.0 95.8 95.6 95.4 95.2 t d 14 ^ s 14 d 23 s 23 |d 53 s 53 ^d 110 Os 110 • d 212 s 212 -Have d 14 + ave s 14 + ave d 23 + ave s 23 ave d 53 + ave s 53 ave d 110 -Have s 110 ave d 212 ave s 212 analy tical 95.0 0.035% 0.040% 0.045% 0.050% 0.055% 0.060% Makroskopska primerjalna specifična deformacija [/] Macroscopic equivalent strain [/] macroscopic equivalent strain for 30 cases with different orientations of crystal lattice for a 14-grains polycrystalline aggregate (Figure 1). Two cases with displacement and stress boundary conditions are presented. The scatter of the curves due to the different orientations of the crystal lattice is clearly visible. The curves within the elasticity nearly coincide, with a distinctive scatter of yield points. The stress boundary condition causes a stiffer response. This, together with the difficulties mentioned above, is the cause for fewer analyzed cases. 2.2 Estimation of the RVE size within the elasticity Analyses in the elasticity were carried out on polycrystalline aggregates with 14, 23, 53, 110 and 212 grains. Thirty different random orientations of crystal lattices with stress and displacement boundary conditions were analyzed for each polycrystalline aggregate. The analyses were carried out at macroscopic stress p1 = 200 MPa and p2 = 100 MPa (Figure 1). The results were compared with an analytical solution for an isotropic continuum with the material parameters: E = 210 GPa and v= 0.29 [10]. Average values were calculated over 30 different random orientations of the crystal lattice for each polycrystalline aggregate and boundary condition. The results are shown in Figure 3 (left), where s and d refer to the stress and strain boundary conditions respectively ave refers to the average values and the number following the abbreviation denotes the number of grains in the respective polycrystalline aggregates. The analytical solution (macroscopic equivalent strain = 0.0515% and macroscopic equivalent stress = 96.2 MPa) is also shown. A tendency towards a reduced scatter of the results as the number of grains in the polycrystalline 120 _____________ 115 105 100 95 90 85 80 d d ekstrapolirano d extrapolated analitično analytical s ekst rapolirano s ext rapolat ed 0 100 200 300 400 Število zrn v polikristalnem skupku Number of grains in polycrystalline aggregate Sl. 3. Raztros makroskopskih primerjalnih napetosti in deformacij (levo) ter konvergenca makroskopske primerjlane napetosti v elastičnem območju (desno) Fig. 3. Scatter of macroscopic equivalent strain/stress (left) and convergence of macroscopic equivalent stresses in elasticity (right) 1 BnnBjfokJ][p)l]Olf|i[gO | | ^SsFvWEIK | stran 586 s 110 Kova~ M., Simonovski I., Cizelj L.: Vpliv zrnate strukture - The Effect of Grain Structure polikristlanem skupku. Povprečne vrednosti napetosti in specifičnih deformacij se pri povečevanju števila zrn v polikristalnem skupku bližajo analitični rešitvi. Za oceno velikosti RPE v elastičnosti so bile izračunane makroskopske primerjalne napetosti pri makroskopski primerjalni specifični deformaciji = 0,0515 % (v skladu z analitično rešitvijo). Slika 3 (desno) prikazuje makroskopske primerjalne napetosti v odvisnosti od števila zrn v polikristalnem skupku pri robnih pogojih s predpisanim pomikom (označeno z d) in napetostjo (s). Ekstrapolacijske črte so narisane v skladu z enačbo (9). Iz razhajanja povprečnih vrednosti je razvidno, da RPE ni bil dosežen. Nagibanje k analitični rešitvi in zmanjševanju razhajanja povprečnih vrednosti pri povečevanju števila zrn je jasno vidno. Velikost RPE je ocenjena iz enačbe (9). Z omejitvijo ostanka na 1 % znaša ocena velikosti RPE v elastičnem območju 280 zrn (kar ustreza polikristalnemu skupku velikosti približno 0,3 mm). Pri ostanku 0,1 % znaša predvidena velikost RPE 450 zrn (0,5 mm). Ugotovimo lahko, da so ocene v okviru pričakovanih iz literature (npr. [21]). 2.3 Ocena velikosti RPE v plastičnem območju Analize v plastičnem območju so bile opravljene na polikristalnih skupkih s 14, 23, 53, in 110 zrni s po trideset primeri naključne usmeritve kristalnih rešetk (vsak primer z na robu predpisano napetostjo oziroma pomikom). Analize so bile izvedene pri zunanji obremenitvi p1 = 1000 MPa in p2 = 500MPa (sl. 1). V koračnih enačbah, ki popisujejo kristalno plastičnost, je bil uporabljen majhen računski korak (1 % celotne obremenitve), kar je povzročilo veliko časovno zahtevnost izračuna. Navkljub majhnemu računskemu koraku je zaradi togosti enačb prišlo do razhajanja in s tem do predčasnega končanja nekaterih analiz. Te težave niso dovoljevale, da bi bilo analiziranih vseh 30 primerov naključnih usmeritev kristalnih rešetk, toda razpoložljivi rezultati vseeno omogočajo, da predstavimo in pojasnimo bistvene težnje. V prihodnosti bo za izračun potreben še manjši korak ali drugačna računska shema [13]. Povprečne vrednosti so bile izračunane za vsak polikristalni skupek in robne pogoje. Rezultati so prikazani na sliki 4 (levo), pri čemer s in d označujeta robne pogoje s predpisano napetostjo oziroma pomikom, ave povprečno vrednost, naslednja številka pa število zrn v polikristalnem skupku. Videti je mogoče težnjo k zmanjševanju raztrosa rezultatov ob povečevanju števila zrn v polikristalnih skupkih. Povprečne vrednosti napetosti in specifičnih deformacij se pri aggregate increases is visible. The average values of the stresses and the strains show a clear trend towards the analytical solution with an increasing number of grains in the polycrystalline aggregate. To estimate the RVE size in elasticity, macroscopic equivalent stresses were taken at a macroscopic equivalent strain of = 0.0515 % (in accordance with the analytical solution). Figure 3 (right) shows macroscopic equivalent stresses and scatter depending on the number of grains in the polycrystalline aggregate for displacement- (denoted as d) and stress- (denoted as s) driven boundary conditions. The extrapolation lines are drawn in accordance with equation (9). From the scatter of the average values, one can conclude that the RVE has not been readred. The trend towards the analytical solution and the decrease of scatter with an increasing number of grains is again clearly visible. The RVE size is estimated from equation (9). By limiting the residuum to 1 %, the estimation of the RVE size within the plasticity is 280 grains, which corresponds to a polycrystalline aggregate of about 0.3 mm in size. With a residuum of 0.1 %, the RVE size is estimated to be 450 grains (0.5 mm). One can conclude that the results are within those expected from the literature (e.g. [21]). 2.3 Estimation of RVE size within plasticity Analyses in plasticity were carried out on a polycrystalline aggregate with 14, 23, 53, and 110 grains. Thirty different random orientations of crystal lattices and two boundary conditions (stress and displacement boundary conditions) were analysed for each polycrystalline aggregate. The analysis was performed at the macroscopic stress p1 = 1000 MPa and p2 = 500 MPa (Figure 1). A small, incremental step (1 % of the total stress) was utilized in the incremental equations, which are used to describe the crystal plasticity. This caused a high computational demand for the analyses. Despite the small incremental step used, a divergence and therefore a premature end to some analyses appeared due to the equation stiffness. These difficulties did not allow for all 30 different randomly orientated crystal lattices to be analysed. Nevertheless, the available results enable us to show and explain the essential tendencies. An even smaller time step or unconventional time-integration schemes (e.g., [13]) should be used for analyses in the future. The average values of the macroscopic equivalent stresses and strains were calculated for each polycrystalline aggregate and boundary condition. The results are shown in Figure 4 (left), where s and d refer to the stress and strain boundary conditions respectively, ave refers to average values and the number following the abbreviation denotes the number of grains in the respective polycrystalline aggregates. A tendency towards a reduced scatter of the results as the number of grains in the polycrystalline aggregates increases is visible. The average values of the | lgfinHi(š)bJ][M]lfi[j;?n 0211 stran 587 I^BSSIfTMlGC Kova~ M., Simonovski I., Cizelj L.: Vpliv zrnate strukture - The Effect of Grain Structure 450 445 440 435 430 > o > + 0% 1% ^o o 2% ¦ d14 ¦ s14 Ad23 A. s23 ¦ d53 iD s53 ¦ d110 O s110 -have s 14 -Have d 14 ¦a ve d 23 |ave s 23 Jave d 53 ave s 53 Jave d 110 Jave s 110 520 510 500 490 480 470 460 450 440 430 420 410 d eks t rapolirano d extrapolated s ek st rapolirano s extrapolated 3% Makroskopska primerjalna specifična deformacija [/] Macroscopic equivalent strain [/] 0 200 400 600 800 Število zrn v po li kristalnem skupku Number of grains in polycrystalline aggregate 1000 Sl. 4. Raztros makroskopskih primerjalnih napetosti in specifičnih deformacij (levo) ter konvergenca makroskopske primerjalne napetosti v plastičnem območju (desno) Fig. 4. Scatter of macroscopic equivalent strain/stress (left) and convergence of macroscopic equivalent stresses in plasticity (right) povečevanju števila zrn v polikristalnem skupku bližajo skupnemu povprečju. Pri oceni velikosti RPE v plastičnem območju je bil uporabljen enak postopek kakor pri oceni v elastičnem območju. Makroskopske primerjalne napetosti so bile izračunane pri makroskopski primerjalni specifični deformaciji = 1 %. Slika 4 (desno) prikazuje makroskopske primerjalne napetosti v odvisnosti od števila zrn v polikristalnem skupku pri robnih pogojih s predpisanim pomikom (označeno z d) in napetostjo (s). Ekstrapolacijske črte za povprečne vrednosti so narisane v skladu z enačbo (9). Razhajanje povprečnih vrednosti je večje kot 1 %, zato menimo, da RPE ni bil dosežen. Nagibanje k skupnemu povprečju in zmanjševanju razhajanja povprečnih vrednosti pri povečevanju števila zrn je jasno vidno. Z omejitvijo ostanka na 1 % ocena velikosti RPE v plastičnem območju znaša 750 zrn (kar ustreza polikristalnemu skupku velikosti približno 0,6 mm). Pri ostanku 0,1 % znaša načrtovana velikost RPE nad 1000 zrn (0,7 mm). 3 SKLEP V prispevku je bil predstavljen računski postopek za modeliranje elasto-plastičnega odziva materiala, ki združuje najpomembnejše mezoskopske značilnost in združljivost s klasično mehaniko trdnin. Uporabljeno je bilo eksplicitno modeliranje naključne zrnate strukture. Zrna modeliramo kot monokristale z anizotropno elastičnostjo in kristalno plastičnostjo. Postopek je bil uporabljen za oceno velikosti RPE polikristalnega skupka, nad katero makroskopska nehomogenost zrnate strukture danega materiala izgine in zatorej ni pričakovati, da bi povzročala vpliv velikosti. stresses and strains show a clear trend towards a common average value with an increasing number of grains in the polycrystalline aggregate. The same approach to estimate the RVE size as in elasticity was used. The macroscopic equivalent stresses were taken at the macroscopic equivalent strain of = 1%. Figure 4 (right) shows the macroscopic equivalent stresses and scatter depending on the number of grains in a polycrystalline aggregate for displacement- (denoted as d) and stress-denoted as s) driven boundary conditions. The extrapolation lines for the average values are drawn in accordance with equation (9). The scatter of the average values is larger than 1 %, therefore one can conclude that the RVE has not been achieved. A tendency towards a common average value and a smaller scatter of average values, as the number of grains increases, is clearly visible. By limiting the residuum to 1 %, the RVE size within the plasticity is estimated to 750 grains (which corresponds to a polycrystalline aggregate of around 0.6 mm in size). With a residuum of 0.1 %, the RVE size is estimated to be above 1000 grains (0.7 mm). 3 CONCLUSION A numerical approach that models the elastic-plastic material response was presented in this paper. The approach combines the most important mesoscale features and compatibility with the conventional continuum mechanics. Explicit modeling of the random grain structure was used. The grains were regarded as monocrystals (modeled with anisotropic elasticity and crystal plasticity). The approach was used to estimate the RVE size of a polycrystalline aggregate above which macroscopic inhomogeneity disappears and is therefore not expected to cause size effects. 1 SšnnstsfcflM]! ma stran 588 d s Kova~ M., Simonovski I., Cizelj L.: Vpliv zrnate strukture - The Effect of Grain Structure Naključna zrnata struktura je vzrok za vpliv The random grain structure is the cause for velikosti v polikristalnih skupkih, manjših od RPE. size effects in polycrystalline aggregates smaller than Velikost RPE v elastičnem območju je 280 zrn, kar the RVE. The RVE size within elasticity is estimated to ustreza vzorcu velikosti približno 0,3 mm. Velikost RPE be 280 grains, which corresponds to a specimen of v plastičnem območju je 750 zrn, kar ustreza vzorcu about 0.3 mm in size. The RVE size within plasticity is velikosti približno 0,6 mm. Oba podana primera sta estimated to be over 750 grains, which corresponds to izračunana ob predpostavki 1 % ostanka. Dobljeni a specimen of about 0.6 mm in size. Both cases were rezultati so v skladu z rezultati iz literature. calculated with a residuum of 1% in mind. The proposed Predstavljen računski postopek dobro popisuje vplive numerical approach is suitable for describing the zrnate strukture na elasto-plastičen odziv effects of the grain structure on the elastic-plastic polikristalnega materiala. response of a polycrystalline aggregate. V prihodnosti predvidevamo razširitev modela, Broadening of the approach, which will include ki bo vključeval razvoj poškodovanosti materiala in the development of damage to be material and pospešitev izračuna (izdelava drugačne računske speeding up of the calculation (integration of different časovne sheme). time-integration scheme) is foreseen in the future. 4 LITERATURA 4 REFERENCES [I] Needleman, A. (2000) Computational mechanics at the mesoscale. Acta Materialia. 48, 105-124. [2] Watanabe, O., H. M. Zbib, and E. Takenouchi (1998) Crystal plasticity, micro-shear banding in polycrystals using Voronoi tessellation. International Journal of Plasticity. 14(8), 771-778. [3] Kroner, E. (1986) Statistical modelling. in: Gittus, John and Zarka, Joseph, Editors. Modelling small deformations of polycrystals. Elsevier Applied Science; 229-291. [4] Gittus, J. and J. Zarka (1986) Modelling small deformations of polycrystals. London, Elsevier. [5] Frost, H. J. and M.F. Ashby (1982) Deformation-mechanism maps. Pergamon Press; ISBN, 0-08-029338-7. [6] Hafner, J. (2000) Atomic - scale computational materials science. Acta Materialia. 48, 71-92. [7] Parker, A. (1999) Predicting material behavior from the atomic level up. Science and Technology Review. (6), 22-25. [8] Kovač, M. Influence of microstructure on development of large deformations in reactor pressure vessel steel. Ljubljana: University of Ljubljana. [9] Hibbit, Karlsson & Sorensen Inc. ABAQUS/Theory Manual, Version 6.2. Pawtucket, R.I., USA: Hibbit (2001) Karlsson & Sorensen Inc. [10] Cizelj, L. and M. Kovač (2001) Constitutive models for the elastic-plastic polycrystalline aggregate with stohastic arrangement of grains. Rev 0. Ljubljana, Institut Jožef Stefan; IJS-DP-8334. [II] Aurenhammer, F (1991) Voronoi diagrams - a survey of a fundamental geometric data structure. ACM Computing Surveys. 23(3), 345-405. [12] Riesch-Oppermann, H. (1999) VorTess, generation of 2-D random poisson-Voronoi mosaics as framework for the micromechanical modelling of polycristalline materials. Karlsruhe, Germany, Forschungszentrum Karlsruhe; Report FZKA 6325. [13] Nemat-Nasser, S. and M. Hori (1993) Micromechanics: overall properties of heterogeneous materials. Amsterdam: North-Holland. [14] Nye, J. F (1985) Physical properties of crystals. Oxford, Clarendon Press. [15] Grimvall, G. (1999) Thermophysical properties of materials. Amsterdam: North-Holland. [16] Huang, Y.(1991) A user-material subroutine incorporating single crystal plasticity in the ABAQUS finite element program. Cambridge, Massachusetts, Harvard University; MECH-178. [17] Nemat-Nasser, S.; T Okinaka, and L. Ni (1998) A physically-based constitutive model for BCC crystals with application to polycrystalline tantalum. Journal of the Mechanics and Physics of Solids. 46(6), 1009- 1038. [18] Asaro, R. J. (1983) Micromechanics of crystals and polycrystals. Micromechanics in Applied Mechanics. 23, 2-115. [19] Peirce, D.; R.J. Asaro, and A. Needleman (1982) Material rate dependence and localized deformation in crystalline solids. Acta Metallurgica. 31, 1951. [20] Asaro, R J. (1983) Crystal plasticity. Journal of Applied Mechanics. 50, 921. [21] Weyer, S. (2001) Experimentelle Untersuchung und mikromechanische Modellierung des Schadigungsverhaltens von Aluminiumoxid unter Druckbeanspruchung. Karlsruhe, Germany, University of Karlsruhe. | lgfinHi(š)bJ][M]lfi[j;?n 0211 stran 589 I^BSSIfTMlGC Kova~ M., Simonovski I., Cizelj L.: Vpliv zrnate strukture - The Effect of Grain Structure [22] Cizelj, L. ; M. Kovač, Z. Petrič, L. Fabjan, and B. Mavko (2002) Effects of grain structure on elastic-plastic behavior of polycrystalline aggregate. Rev 0. Ljubljana, Institute Jožef Stefan; IJS-DP-8547. Naslov avtorjev: mag. Marko Kovač dr. Igor Simonovski profdr. Leon Cizelj Institut Jožef Stefan Odsek za reaktorsko tehniko Jamova 39 1000 Ljubljana marko.kovac@ijs.si igor.simonovski@ijs.si leon.cizelj@ijs.si Authors’ Address: Mag. Marko Kovač Dr. Igor Simonovski Profdr. Leon Cizelj Jožef Stefan Institute Reactor Engineering Division Jamova 39 1000 Ljubljana, Slovenia marko.kovac@ijs.si igor.simonovski@ijs.si leon.cizelj@ijs.si Prejeto: 24.12.2002 Received: Sprejeto: 31.1.2003 Accepted: 1 SšnnstsfcflM]! ma stran 590 © Strojni{ki vestnik 48(2002)11,591-600 © Journal of Mechanical Engineering 48(2002)11,591-600 ISSN 0039-2480 ISSN 0039-2480 UDK 658.512.2:004.41 UDC 658.512.2:004.41 Izvirni znanstveni ~lanek (1.01) Original scientific paper (1.01) Optimiranje oblike konstrukcij: tristrani~ni projektni element Structural Shape Optimization: A Trilateral Design Element Marko Kegl Prispevek obravnava izpeljavo tristraničnega projektnega elementa za uporabo pri optimalnem projektiranju oblike konstrukcij. Osnova za izpeljavo novega elementa je tristranična Bezierjeva ploskev, ki je običajno parametrizirana z uporabo težiščnih koordinat. V prispevku je uporabljena drugačna parametrizacija, ki je bolj prilagojena postopkom optimizacije oblike. Na podlagi te ploskve je definiran projektni element - Bezierjevo telo, katerega mreža nadzornih točk ima v topoloskem pomenu obliko tristranične prizme. Uporaba izpeljanega elementa je ponazorjena na dveh številčnih zgledih. © 2002 Strojniški vestnik. Vse pravice pridržane. (Ključne besede: projektiranje konstrukcij, optimiranje oblik, elementi projektni) This paper considers the derivation of a trilateral design element for the shape optimization of structures. The element derivation is based on a trilateral Bezier patch, usually being parametrized by barycentric coordinates. Here, another type of parametrization is used, which is more convenient for employment in optimization procedures. Based on this patch the design element is derived - a Bezier body whose control points represent a trilateral prism in the topological sense. The use of the derived element is illustrated by two numerical examples. © 2002 Journal of Mechanical Engineering. All rights reserved. (Keywords: structural design, shape optimization, element design) 0 UVOD Področje optimizacije oblike konstrukcij se je začelo intenzivneje razvijati približno pred dvema desetletjema. Kmalu se je izkazalo, da pri optimizaciji oblike naletimo na nove težave, ki jih pri običajni optimizaciji nismo poznali [1]. Te so bile več ali manj vezane na dejstvo, da se mreža končnih elementov pri optimizaciji oblike spreminja. To pripelje do kvarjenja mreže ter do nenatančnih rezultatov pri analizi odziva in občutljivosti. Stopnja kvarjenja mreže je zelo odvisna od načina njene parametrizacije. Imam [2] je tako v svojem delu predstavil različne zamisli parametrizacije in enega od njih poimenoval tehnika projektnih elementov. Projektni element je v bistvu primerno parametriziran geometrijski objekt, ki končne elemente oskrbuje s potrebnimi geometrijskimi podatki (koordinate vozlišč itn.). Imam je za projektni element predlagal 20-vozliščni izoparametrični element, malo kasneje pa sta Braibant in Fleury [3] v svojo formulacijo 2D projektnega elementa vpeljala krivulje zlepkov B. V naslednjih letih smo na področju optimizacije oblike lahko opazili vse pogostejšo uporabo osnovnih zasnov 0 INTRODUCTION The field of structural shape optimization began to develop more rapidly about two decades ago. It soon turned out that shape optimization is accompanied by several new difficulties not known in conventional optimization [1]. These difficulties were more or less caused by the fact that during the process of shape optimization the finite-element mesh changes. This in turn leads to deterioration of the mesh and to inaccuracies in the calculation of the structural response and the sensitivity information. The extent of the mesh deterioration depends significantly on its parametrization. Imam [2] presented in his work several parametrization concepts, one of them he termed the design-element technique. A design element is essentially a parametrized geometrical object acting as the geometrical data (nodal coordinates, etc.) provider for all the finite elements it contains. Imam proposed to employ a 20-noded isoparametric element as the design-element. Somewhat later Braibant and Fleury [3] introduced B-spline curves into the formulation of their 2D design element. In the following years, the fundamental concepts of the design element technique as well as Bezier and B-spline curves and gfin^OtJJIMISCSD 02-11 stran 591 |^BSSITIMIGC Kegl M.: Optimiranje oblike konstrukcij - Structural Shape Optimization tehnike projektnih elementov ter krivulj oziroma ploskev zlepkov B ([4] do [8]). Pred kratkim je bil v [9] predstavljen osnutek parametrizacije oblike z uporabo tehnike projektnih elementov ter uporabe splošnega projektnega elementa. Predlagan osnutek obravnava zvezne in diskretne konstrukcije na enoten način, projektni element je definiran kot Bezierjevo telo. Topološko ima ta element obliko štiristranične prizme, zato ga bomo imenovali kar stiristranični projektni element. Mejna ploskev tega elementa je sestavljena iz šestih štiristraničnih Bezierjevih ploskev Če je treba, lahko katerokoli od teh ploskev degeneriramo v tristranično ploskev s primernim pozicioniranjem nadzornih točk. To je sicer mogoče, vendar pa ni najbolj smotrno, saj pri tem uporabljamo več nadzornih točk, kakor pa jih dejansko potrebujemo. V takem primeru je zato uporaba prave tristranične ploskve mnogo bolj primerna. Glede na to je verjetno primerno definirati (razen štiristraničnega) tudi tristranični projektni element (sl. 1). Topološko je ta element tristranična prizma, njegova mejna ploskev pa je sestavljena iz dveh tristraničnih in treh štiristraničnih Bezierjevih ploskev. (a) surfaces were progressively employed in shape optimization ([4] to [8]). Recently, a concept of shape parametrization was presented [9] using the design-element technique and a general-purpose design element. With the proposed concept, continuum and discrete structures are parametrized in a unified way. The employed design element is a rational Bezier body. Topologically, this convenient element represents a quadrilateral prism, so it will be termed here as the quadrilateral design element. The boundary surface of this element consists of six quadrilateral Bezier patches. If necessary, any of these patches can be degenerated to a trilateral patch by adequate positioning of the control points. This is possible, but not very efficient, since one needs to define and employ more control points than are actually needed. In such a case the use of a genuine trilateral patch represents a much better choice. Therefore, it seems to be advantageous to define (except for the quadrilateral) also a trilateral design element (Figure 1). Topologically, the trilateral element represents a trilateral prism. Its boundary surface consists of two trilateral and three quadrilateral Bezier patches. (b) Sl. 1. Stiristranični (a) in tristranični (b) prizmatični projektni element Fig. 1. A quadrilateral (a) and a trilateral (b) prismatic design element 1 TEHNIKA PROJEKTNIH ELEMENTOV Obravnavajmo konstrukcijo, ki v geometrijskem smislu pomeni telo B v stvarnem 3D prostoru. Pri tehniki projektnih elementov si celo telo B zamislimo kot sestavljeno iz preprostej ših delov D, ki imajo znane geometrijske lastnosti in jih je preprosto parametrizirati (slika 2). Te sestavne dele imenujemo projektne elemente. Naj simbol D označuje poljuben projektni element. Vzemimo tudi, da D v geometrijskem smislu pomeni Bezierjevo telo. Obliko in lego Bezierjevega telesa določajo lege njegovih nadzornih točk, nanj pa lahko gledamo tudi kot na preslikavo f enotske Nadzorne točke Bezierjevega telesa imajo v tej preslikavi vlogo parametrov, ki vplivajo na izračun r oziroma na lego in obliko projektnega elementa D. Torej lahko zapišemo r=f(s,qijk), kjer smo s qijk simbolično označili krajevne vektorje nadzornih točk telesa D. Spreminjanje oblike telesa D dosežemo preprosto tako, da spreminjamo lege njegovih 1 THE DESIGN ELEMENT TECHNIQUE Let us consider a structure, in geometrical sense, being represented by the body B in real 3D space. By the design element technique the whole body B can be thought of as being assembled of simpler parts D, which exhibit known geometrical properties and can be parametrized in a simple way. These individual parts are termed the design elements. Let the symbol D denote a generic design element. Let us also assume that in a geometrical sense D represents a Bezier body. The shape and the position of a Bezier body are determined by the positions of its control points. It can also be thought of as a mapping f from the unit cube U = [0,1]3 into real 3D space (Figure 3). The Bezier body control points act in this mapping as parameters that affect the calculation of r, i.e. the position and shape of the design element D. Thus, we can write r=f(s,qijk), where the symbol qijk was used to denote symbolically the control point position vectors of the body D. The change of the shape of D can simply be achieved by changing the positions of its control 1 BnnBjfokJ][p)l]Olf|i[gO | | ^SSfiflMlGC | stran 592 Kegl M.: Optimiranje oblike konstrukcij - Structural Shape Optimization D1 D2 ,qijk B x Sl. 2. Telo B predstavljata projektna elementa D in D2 Fig. 2. The body B is represented by the design elements D1 and D2 nadzornih točk qijk. Preostane le še, da na primeren način vključimo uporabo ustrezne diskretizacijske metode, na primer metode končnih elementov. Za uporabo te metode potrebujemo mrežo končnih elementov, ki mora pokrivati celotno območje D. Vendar to ni vse, saj moramo imeti v mislih tudi dejstvo, da bomo obliko telesa D spreminjali - oblika mreže končnih elementov mora tem spremembam ustrezno slediti. To najenostavneje dosežemo tako, da mrežo oziroma vozlišča posameznih končnih elementov definiramo v enotski kocki U. Krajevne vektorje vozlišč mreže v stvarnem prostoru pa izračunamo z uporabo preslikave f (sl. 4). Tako bo mreža končnih elementov avtomatično in povsem natančno sledila spremembam oblike telesa D. Glede na zgoraj opisano lahko rečemo, da je projektni element podan s preslikavo f. Z izrazom izpeljava projektnega elementa mislimo torej na izpeljavo splošnega izraza za f ter na izpeljavo vseh potrebnih izrazov, ki so odvisni od f in jih potrebujemo pri izračunavanju vhodnih geometrijskih podatkov za mrežo končnih elementov. V dosedanjem delu [9] smo izpeljali in na raznih primerih uporabili štiristranični projektni element, ta prispevek pa prikazuje izpeljavo tristraničnega elementa. 2 TRISTRANIČNI PROJEKTNI ELEMENT Tristranični element je zasnovan na tristranični Bezierjevi ploskvi. Za razliko od štiristranične variante Sl. 3. Preslikava iz U v D pod vplivom parametrov Fig. 3. A mapping from U into D influenced by the parameters qijk points qijk. What still needs to be done is to introduce, in an appropriate way, the use of an adequate discretization method, for example, the finite-element method. In order to employ this method, we need some finite-element mesh, spanning the entire region of D. In doing so, we have to keep in mind that the shape of D will change - the shape of the finite-element mesh must follow those changes accordingly. The easiest way to achieve this is to define the mesh (i.e. the nodes) in the unit cube U. The nodal position vectors in the real space are then calculated by employing the mapping f (Figure 4). In this way the finite-element mesh will automatically and accurately follow the shape changes of the body D. According to the discussion above one can say that the design element is defined by the mapping f. Thus, by the term design element derivation we actually mean the derivation of some generic expression for f as well as the derivation of all the necessary expressions that depend on f, and need to be employed to compute the geometrical data of the finite-element mesh. In our previous work [9] we derived and successfully employed the quadrilateral design element on several examples. This paper presents the derivation of the trilateral design element. 2 THE TRILATERAL DESIGN ELEMENT The trilateral design element is based on a trilateral Bezier patch. In contrast to the quadrilateral Končni element Finite element z\ y x Sl. 4. Preslikava vozlišč v stvarni prostor Fig. 4. The mapping of the nodes into the real space gfin^OtJJlMISCSD 02-11 stran 593 |^BSSITIMIGC Kegl M.: Optimiranje oblike konstrukcij - Structural Shape Optimization je tristranična Bezierjeva ploskev parametrizirana z uporabo težiščnih koordinat u = [u,v,w]T. Te koordinate med sabo niso neodvisne, saj morajo zadoščati pogoju u+v+w=1. Krajevni vektor p poljubne točke na tristranični ploskvi reda n je podan z [10]: version, a trilateral Bezier patch is conventionally parametrized by employing barycentric coordinates u = [u,v,w]T. These coordinates are not independent, since they must fulfill the requirement that u+v+w=1. The position vector p of a generic point on a trilateral patch of the order n is given by [10] 7 .Bij k qijk (1), kjer sta | i | =i+j+k in ij,k>0. Simbol qijk označuje nadzorno točko, Bnk=Bnk ( u ) pa je bivariantni Bernsteinov polinom reda n, ki je definiran kot: Bn ijk i! j!k ! Vendar taka običajna parametrizacija ni najprimernejša za izpeljavo projektnega elementa. Pri tem je namreč najugodneje, da je parametriziran z neodvisnimi parametri ter da ima primerne karakteristične parametrične smeri v vsaki točki. Da bi zadostili temu pogoju, moramo težiščne koordinate nadomestiti z dvema primernima neodvisnima parametroma. V ta namen najprej preštevilčimo nadzorne točke in pripadajoče polinome: nadzorni poligon tristranične ploskve je definiran z mrežo N = 12 ( n + 1 )( n + 2 ) nadzornih točk qijk. Vpeljimo nov dogovor ter nadzorne točke označimo s q i=1,...,N, kjer je q =q00 , medtem ko so naslednje nadzorne točke tiste, pri katerih indeksi z leve naraščajo najhitreje (sl. 5a). Enak dogovor sprejmimo tudi za polinome, tako da imamo BN, i=1,...,N, kjer je B N 0n Nova neodvisna parametra s1 in svpeljimo tako, da dobimo primerne parametrične smeri, in sicer: u =1-s , v where | i | =i+j+k and i,j,k&0. The symbol qijk denotes a control point and Bnk = Bnk ( u ) is the bivariate Bernstein polynomial of the order n, defined as: (2). This conventional arrangement, however, is not very convenient for the derivation of a design element. A design element is most conveniently defined as a geometrical body, parametrized by three independent parameters and exhibiting three characteristic parametric directions at each point. In order to meet this requirement, the barycentric coordinates of the trilateral patch have to be replaced by two suitable and independent parameters. For this purpose we re-numerate the control points and their corresponding polynomials: the control polygon of a trilateral patch is defined by a triangular scheme of N = 12 ( n + 1 )( n + 2 ) control points qijk. We introduce a new arrangement, denoting the control points with q i=1,...,N so that q1Sq0n, while the subsequent control points are the ones with the leftmost index increasing most rapidly (Figure 5a). The same arrangement is adopted for the polynomials, now denoted by BN, i=1,...,N where BN=Bn 00n . Let us introduce new independent parameters, s1 and s2, so that one gets convenient parametric lines as follows: ss , w=ss (3), kjer sta s1,s2«=[0,1]. S temi zvezami postavimo where s^[0,1]. These relations establish the odvisnost u = u (s1, s2), preprosto pa lahko tudi dependency u = u (s1, s2), and it can easily be verified preverimo, da velja u,v,w^,1] ter u+v+w=1. S tem that u,W [0,1] and u+v+w=1. By adopting the dogovorom lahko poljubno točko na ploskvi zapišemo above arrangement, the position vector of a generic kot: point on the patch can be written as N kjer je SiN =BiN(u(s1,s2)), tako da lahko zapišemo SN =SiN(s1,s2) in posledično p = p(s1,s2). S to neodvisno parametrizacijo dobimo primerne parametrične smeri e1 = dp/ds1 in e2 = dp/ds2 (sl. 5b), ploskev pa s tem postane primerna tudi za izpeljavo projektnega elementa. Za tristranično ploskev potrebujemo shemo N nadzornih točk. Če vzamemo M takšnih shem, dobimo tristranično prizmatično shemo N *M nadzornih točk q , i=1,...,N, j=1,...M. Te nadzorne where SiN=BiN ( u ( s1,s2 )) so that one can write SN =SN ( s1,s2 ) and consequently p =p ( s1,s2. This independent parametrization yields convenient parametric directions e1 = dp/ds1 and e2 = dp/ds2 (Figure 5b), which makes the patch convenient for the derivation of a design element. For a trilateral patch one needs a triangular scheme of N control points. If one takes M such schemes, one gets a trilateral prismatic scheme of NxM control points q , i=1,...,N, j=1,...M. These 1 BnnBjfokJ][p)l]Olf|i[gO | | ^SSfiflMlGC | stran 594 Kegl M.: Optimiranje oblike konstrukcij - Structural Shape Optimization ;;::orq2=q101 q1=q002 (a) q3=q200 s1=const. e1>\ ^\z p xAy (b) Sl. 5. Preštevilčenje nadzornih točk (a) in predlagana parametrizacija ploskve (b) Fig. 5. Renumbering of control points (a) and the proposed parametrization of the patch (b) Sl. 6. Tristranični projektni element z značilnimi smermi v točki r Fig. 6. The trilateral design element with characteristic directions at the point r točke lahko definirajo tristranični projektni element (sl. 6), krajevni vektor njegove poljubne točke pa lahko zapišemo kot: control points can be taken to define a trilateral prismatic design element (Figure 6) by defining the position vector of a generic point on the element by NM (5), i=1 j=1 kjer je UM = UM ( s3 ) j-ti univariantni Bernsteinov gi =&r&si, i=1,2,3 (sl. 5). t Označimo sedaj s s0 =\s10,s0,s30\ točko, ki določa lego poljubnega vozlišča nekega končnega elementa. Nadalje predpostavimo, da sta lahko s0 kakor tudi vse nadzorne točke odvisne od projektnih spremenljivk, zbranih v vektorju b. Torej velja qij= q (b) in s0= s0(b). ' Za analizo konstrukcije potrebujemo v splošnem za vsako vozlišče naslednje geometrijske podatke: krajevni vektor r0 ter morda smerni vektor n0 (lupine, nosilci). Če predpostavimo, da lahko n0 izrazimo v odvisnosti od g0, potrebujemo torej za analizo odziva in občutljivosti izraze za naslednje količine: r0, g0, dr0/db ter dg 0/db. Krajevni vektor izrazimo preprosto kot: where UM = UM ( s3 ) is the jth univariate Bernstein element are given by gi = dr/8si, i=1,2,3 (Figure 5). Let us now denote by s0 = [s0, s0, s31 a point that defines the position of a generic node of some finite element. Further, it is assumed that both s0 as well as all the control points depend on design variables, assembled in the vector b. Thus, we have q= q(b) and s0= s0(b). ij ij For the structural response analysis we need, in general, for each node, the following geometrical data: its position vector r0 and maybe some direction vector n0 (shells, beams). Assuming that n0 can be expressed in terms of g 0, we need for the response and sensitivity analysis expressions for the following quantities: r0, g0, dr0/db and dg 0/db. The position vector is straightforwardly expressed as: NM ZZEiNj M qij i=1 j=1 (6), kje je EijNM=SiNUjM. Smerni vektorji so podani z: where EijNM=SiNUjM. The direction vectors are given by: gfin^OtJJIMISCSD 02-11 stran 595 |^BSSIrlMlGC Kegl M.: Optimiranje oblike konstrukcij - Structural Shape Optimization N M f)EiNM i=1 j=1 ds k =1,2,3 (7), kjer je: where: dEN (8). _____\(dSiNldsk)UM,k = 1,2 Ssk \SiN(eUMj/dsk), k = 3 Odvod krajevnega vektorja vozlišča po The design derivatives of the position vector projektnih spremenljivkah je podan z: are given by: dr0 db ZZ medtem ko so odvodi smernih vektorjev: dEN M ds0k ds db d2EN M ds0 dsds db dg0 N M db i=1 j=1 Izrazi (6) do (10) so vse kar potrebujemo za izračun potrebnih geometrijskih podatkov (ter njihovih odvodov po projektnih spremenljivkah) za poljuben končni element. 3 ZGLEDA Za ponazoritev teorije bomo obravnavali dva zgleda. Prvi vsebuje preprosto ravninsko konstrukcijo, njegov namen pa je ponazoriti način uporabe tristraničnega projektnega elementa. Drug zgled vsebuje preprosto prostorsko konstrukcijo, njegov namen pa je pokazati uporabo nedegeneriranega tristraničnega elementa. V obeh zgledih sta podatka o materialu enaka: elastični modul znaša E=210 GPa, Poissonov količnik jev=0,3. Oba zgleda sta zapisana v obliki nelinearnega optimizacijskega problema in rešena z uporabo optimizacijskega modula AMOPT ([11] in [12]). Zgled 1. Optimizacija tristranične podporne plošče Obravnavajmo tristranično podporno ploščo, prikazano na sliki 7. Geometrijski in materialni podatki ter podatki o podprtju in obremenitvi so podani na sliki 7a. Naš namen je določiti obliko konstrukcije tako, da bo njena teža najmanjša in bodo hkrati izpolnjeni vsi omejitveni pogoji. Natančneje, želimo spreminjati višino plošče (navpično lego točke B) in obliko obrisa BC tako, da bo teža najmanjša. Postavljeni omejitveni pogoji se nanašajo na navpični pomik točke C in na napetosti vzdolž robov BC in AC. Da bi lahko definirali optimizacijsko nalogo, konstrukcijo razdelimo na dva projektna elementa (degenerirana v ravninska lika). Zaradi zagotavljanja dovolj velike prilagodljivosti, vzamemo dva elementa, vsak z 10x1=10 nadzornimi točkami (NT). Na sliki 7b so prikazane NT prvega projektnega elementa, medtem ko so NT drugega razporejene na podoben način. Vpeljimo sedaj projektne spremenljivke b, s katerimi definiramo neodvisno gibljive nadzorne točke prvega elementa: q=[0,b1]T, q =[ b 2,b ]T in q =[ b4,b ]T. Če naredimo podobno še pri drugem projektnem 1 BnnBjfokJ][p)l]Olf|i[gO | | ^HI^lfWlDGC | stran 596 dqij db (9), whereas the direction vector derivatives are: dEN M dqij k = 1,2,3 (10). dsk db The expressions (6 to 10) are all we need in order to calculate the required geometric data (as well as the design derivatives) for any finite element. 3 EXAMPLES In order to illustrate the theory, two numerical examples will be considered. The first example involves a simple planar structure and its purpose is to illustrate how to employ the trilateral design element. The second example involves a simple space structure, and it illustrates the use of a nondegenerated trilateral element. In both examples the material data is the same: the Young’s modulus is E=210 GPa and the Poisson ratio is v=0.3. Both examples are formulated in the form of a non-linear constrained optimization problem and solved by the optimization routine AMOPT ([11] and [12]). Example 1. Optimization of a trilateral supporting plate Let us consider the trilateral supporting plate shown in Figure 7. The geometrical and material data as well as the support and loading conditions are given in Figure 7a. Our objective is to determine the shape of the structure so that its weight will be minimal and, at the same time, all of the imposed constraints will be fulfilled. More precisely, we can vary the height of the plate (vertical position of point B) and the shape of the contour BC so that the weight will be minimised. The imposed constraints are related to the vertical displacement of the point C and to the stresses along the edges BC and AC. In order to define the design problem, we start by partitioning the structure into trilateral design elements (degenerated to a flat surface). For flexibility reasons we take two elements with 10x1=10 control points (CP) each. Figure 7b illustrates the CP of the first design element, while the second one is defined in a similar way. Then we introduce design variables b by defining the independently movable CP of the first element as: q11=[0,b1]T, q5=[ b2,b3]T and q81= Kegl M.: Optimiranje oblike konstrukcij - Structural Shape Optimization y p = 8 N/mm AD = DC = 75 mm A> n11 D1 O Neodvisna NT / Independent CP O Odvisna NT / Dependent CP ¦ Pritrjena NT / Fixed CP (a) p* (b) Sl. 7. Tristranična podporna plošča (a) in njena predstavitev z 2 projektnima elementoma (b) Fig. 7. The trilateral supporting plate (a) and its representation by 2 design elements (b) elementu, dobimo skupaj 9 projektnih spremenljivk. Omejitvene pogoje postavimo tako: navpični pomik točke C mora biti manjši od 0,2 mm, medtem ko morajo biti Misesove napetosti vzdolž robov BC in AC manjše od 100 MPa. Konstrukcijo modeliramo z 8 vozliščnimi ravninskimi končnimi elementi tipa Serendipity. Obravnavali bomo dva primera: primer I, kjer vrednosti projektnih spremenljivk niso omejene in primer II, pri katerem (na primer zaradi pritrditve) postavimo naslednje dodatne omejitve: b1,b >80 mm in b2,b>20 mm. Z uporabo lastnega optimizacijskega algoritma lahko oba primera rešimo v manj ko 10 iteracijah (6 iteracij za primer I in 8 za primer II). Pri obeh končnih oblikah (sl. 8) imamo hkrati delujoče tako napetostne pogoje kakor tudi pogoj za pomik. Glede na začetno konstrukcijo, se je prostornina optimalne konstrukcije zmanjšala za 48% v primeru I oziroma za 46% v primeru II. V tem zgledu smo uporabili dva projektna elementa z deset NT. Ker imata elementa štiri skupne NT, smo torej skupaj morali definirati šestnajst NT. Za primerjavo rezultatov smo enak problem rešili tudi z uporabo dveh štiristraničnih projektnih elementov, vsak z 4x4x1=16 NT. Rezultati so bili identični, vendar smo v slednjem primeru morali definirati dvaindvajset NT. V primerjavi s šestnajst NT to pomeni povečanje obsega priprave podatkov za 37,5%. To lahko pomeni bistveno razliko pri zahtevnih optimizacijskih nalogah z zapletenimi oblikami in mnogimi projektnimi spremenljivkami. [ b4,b] T. Adopting a similar arrangement for the second element, we finally get a total of nine design variables. The constraints are imposed as follows: the vertical displacement of point C has to be less than 0.2 mm while the Mises stresses along BC and AC have to be less than 100 MPa. The structure is modeled by 8-node plane finite elements of the Serendipity type. Two cases will be considered: Case I where the values of thedesign variables are not limited, and Case II where (e.g. for fixing purposes) we impose the following limits: b1,b3>80 mm and ^20 mm. By employing our own optimization algorithm, both problems could be solved successfully within ten iterations (six iterations for Case I and eight for Case II). At the final design (Figure 8), displacement and stress constraints were active in both cases. Compared to the initial volume, the optimum volumes are reduced by 48% and 46% for cases I and II, respectively. In this example we employed two trilateral design elements with ten CP each. Since four CP are common to both design elements, we had to define a total of sixteen CP. In order to compare the results, we solved the same problem by employing two quadrilateral design elements with 4x4x1=16 CP each. The results were identical, but for this arrangement we had to define a total of twenty-two CP. Compared to sixteen, this is an increase of 37.5%. For complex design problems with sophisticated shapes and many design variables such an increase can represent a substantial difference. Sl. 8. Optimalne oblike z razporeditvijo napetosti Fig. 8. Optimal shapes with stress distribution Kegl M.: Optimiranje oblike konstrukcij - Structural Shape Optimization Zgled 2. Optimizacija tristranične konzole Obravnavajmo tristranično konzolo, ki jo prikazuje slika 9. Konzola je togo podprta pri z=0 (ploskev ABG) in obremenjena na zgornji ploskvi ABCD z eno od dveh enakomerno porazdeljenih sil: (I) p = - 40 N/cm2 in (II) p =30 N/cm2. Obe obremenitvi sta y ločeno upoštevani v okviru enega optimizacijskega problema. Naš namen je določiti obliko robov BG in CH kakor tudi obliko ploskve BCHG tako, da bo teža konzole najmanjša. Pri katerikoli od obeh obremenitev Misesove napetosti ne smejo preseči meje 100 MPa. Za parametrizacijo mreže končnih elementov bomo uporabili en tristranični projektni element s 15x3=45 nadzornimi točkami. Nadzorne točke, ki določajo obliko ploskve BCHG so odvisne od osemnajst projektnih spremenljivk. Konstrukcija je diskretizirana z uporabo 20-vozliščnih končnih elementov tipa Serendipity. Tudi ta problem smo lahko rešili zelo hitro. Postopek reševanja je bil stabilen, optimalna oblika je bila dobljena po petih iteracijah. Kakor je razvidno iz preglednice, je optimizacijski postopek izpolnil postavljene pogoje (na začetku so ti bili prekoračeni) in hkrati zmanjšal prostornino konstrukcije. Optimalna oblika je prikazana na sliki10. Še komentar k izbiri projektnega elementa. Za uporabo tristranične variante smo potrebovali 15x3=45 nadzornih točk. Če bi uporabili štiristranično varianto, bi za enako prilagodljivost potrebovali Example 2. Optimization of a trilateral cantilever Let us consider the trilateral cantilever shown in Figure 9. The cantilever is rigidly supported at z=0 (face ABG) and loaded at the upper face ABCD by either one of two evenly distributed loads: (I) p = - 40 N/cm2 and (II) p =30 N/cm2. Both loads are considered simultaneously, within a single design problem. Our objective is to determine the shape of the edges BG and CH as well as the shape of the face BCHG so that the weight of the cantilever will be minimised. The Mises stresses are allowed to be, at most, 100 MPa when either the first or the second load is applied. In order to parametrize the mesh, one trilateral design element with 15x3=45 control points is employed. The control points that determine the shape of the surface BCHG are design dependent, which results in a total of eighteen design variables. The structure is modeled by 20-node finite elements of the Serendipity type. This example was also solved very quickly. The solution process was stable and a near optimum design was obtained after five iterations. As can be seen in the table, the optimization process fulfilled the constraints (initially being violated), and lowered the volume of the structure. The optimal design is depicted in Figure 10. A comment on the choice of the design element. For the trilateral design element we needed 15x3=45 control points. For the same order of flexibility we would need a quadrilateral design H Pritrjeni robovi / Fixed edges: AB = DC = AG = DH = 40 mm AD = BC = GH = 200 mm Sl. 9. Tristranična konzola Fig. 9. Trilateral cantilever Preglednica 1. Primerjava začetnega in optimalnega projekta Table 1. Comparison of the initial and optimised designs Prostornina v cm3 Volume [cm3] Najv. prekoračitev pogojev v % Max. constraint violation [%] Začetni Initial Optimalni Optimum 160,0 93,5 27,9 0,0 1 BnnBjfokJ][p)l]Olf|i[gO | | ^SSfiflMlGC | stran 598 Kegl M.: Optimiranje oblike konstrukcij - Structural Shape Optimization Sl. 10. Optimalna oblika konzole Fig. 10. Optimal shape of the cantilever element s 5x5x3=75 nadzornimi točkami. To pa je precej več kakor 45, torej je v danem primeru uporaba tristraničnega elementa precej bolj smotrna. 4 SKLEP V prispevku je bil predstavljen tristranični projektni element. Čeprav lahko namesto tristraničnega elementa vedno uporabimo splošnejšega štiristraničnega, tristranična varianta mnogokrat pomeni bolj naravni izbor. Posledica je manjše število potrebnih nadzornih točk, kar pomeni manj priprave vhodnih podatkov in manj geometrijskih pogojev. Tristranične projektne elemente lahko poljubno kombiniramo s štiristraničnimi. element with 5x5x3=75 control points. This is substantially more than 45, making, in this example, the trilateral element a far better choice. 4 CONCLUSION The trilateral prismatic design element was presented in the paper. Although the quadrilateral prismatic element may be used for the same purpose, the trilateral version often represents a more natural choice. The consequence is a smaller number of required control points, meaning that less input data has to be prepared and that fewer geometrical constraints have to be imposed. The trilateral elements can be used arbitrarily in combination with the quadrilateral elements. 5 LITERATURA 5 REFERENCES [I] Haftka, R.T., R.V.Grandhi (1986) Structural shape optimization - a survey. Computer Methods in Applied Mechanics and Engineering, 57, 91 106. [2] Imam, M.H. (1982) Three dimensional shape optimization. International Journal for Numerical Methods in Engineering; 18, 661 673. [3] Braibant, V., C. Fleury (1984) Shape optimal design using B splines. Computer Methods in Applied Mechanics and Engineering; 44, 247 267. [4] Shyy, Y.K., C. Fleury, K. Izadpanah (1988) Shape optimal design using high order elements. Computer Methods in Applied Mechanics and Engineering; 71, 99 116. [5] Maute, K., E. Ramm (1995) Adaptive topology optimization. Structural optimization; 10, 100 112. [6] Ohsaki, M, T. Nakamura, Y. Isshiki (1998) Shape size optimization of plane trusses with designer’s preference. Journal of structural engineering; 124, 1323 1330. [7] Kegl, M., H. Antes (1998) Shape optimal design of elastic space frames with non-linear response. International Journal for Numerical Methods in Engineering; 43, 93 110. [8] Wang, X., J. Zhou, Y. Hu (1999) A physics based parametrization method for shape optimization. Computer Methods in Applied Mechanics and Engineering; 175, 41 51. [9] Kegl, M. (2000) Shape optimal design of structures: an efficient shape representation concept. International Journal for Numerical Methods in Engineering; 49, 1571 1588. [10] Farin, G. (1993) Curves and surfaces for computer aided geometric design (2nd edn). Academic Press, New York. [II] Kegl, M., M.M. Oblak (1997) Optimization of mechanical systems: on non linear first order approximation with an additive convex term. Communications in Numerical Methods in Engineering; 13, 13 20. [12] Kegl, M., B.J. Butinar, B. Kegl (2002) An efficient gradient based optimization algorithm for mechanical systems. Communications in Numerical Methods in Engineering; 18, 363 371. gfin^OtJJlMISCSD 02-11 stran 599 |^BSSITIMIGC Kegl M.: Optimiranje oblike konstrukcij - Structural Shape Optimization Avtorjev naslov: doc. dr. Marko Kegl Fakulteta za strojništvo Univerza v Mariboru Smetanova 17 2000 Maribor marko.kegl@uni-mb.si Authors’ Address: Doc. Dr. Marko Kegl Faculty of mechanical eng. University of Maribor Smetanova 17 2000 Maribor, Slovenia marko.kegl@uni-mb.si Prejeto: Received: 23.12.2002 Sprejeto: Accepted: 31.1.2003 1 SsfinKitsfcflM]! mn stran 600 © Strojni{ki vestnik 48(2002)11,601-612 © Journal of Mechanical Engineering 48(2002)11,601-612 ISSN 0039-2480 ISSN 0039-2480 UDK 621.983.3:621.979:62-86 UDC 621.983.3:621.979:62-86 Izvirni znanstveni ~lanek (1.01) Original scientific paper (1.01) Optimiranje pogonskega mehanizma stiskalnice za globoki vlek Optimization of Link-Drive Mechanism for Deep Drawing Mechanical Press Bojan Vohar - Karl Gotlih - Jo`e Fla{ker V prispevku se ukvarjamo z optimiranjem večzgibnega pogona paha stiskalnice za globoki vlek pločevine. Sedanja konstrukcija ne izpolnjuje vseh postavljenih zahtev, zato jo želimo čimbolj prilagoditi idealnim zahtevam tehnološkega postopka. Osnovni namen je prilagoditi sedanjo hitrostno karakteristiko paha zahtevam delovanja v določenem območju. Zato je bilo treba izdelati analizo pogona in njegov matematični model ter izvesti optimizacijo. Uporabljena metoda za nelinearno optimizacijo je sekvenčno kvadratno programiranje. Ker je postopek časovno odvisen, optimizacijskega modela ni moč uporabiti neposredno, ampak je treba primer prevesti v časovno neodvisno obliko, ki je primerna za reševanje s standardnim optimizacijskim postopkom. Cilj optimiranja je določiti takšne izmere pogonskega mehanizma, ki bi čimbolj izpolnile zahteve. V sklepu je prikazana primerjava doseženih rezultatov optimirane konstrukcije večzgibnega pogona z začetnim stanjem pred optimizacijo. © 2002 Strojniški vestnik. Vse pravice pridržane. (Ključne besede: globoki vlek, stiskalnice za globoki vlek, optimiranje pogoniv, modeliranje) This paper deals with an example of a link-drive for a deep-drawing mechanical press. The existing design has proved unsatisfactory and does not meet all the demands and constraints which are required for this metal-forming process. Optimization of the drive is therefore necessary. The intention of this optimization is to achieve the required velocity characteristics in a defined area of movement. Firstly, the drive is analysed and a mathematical model is made. The whole process is time-dependent, so it cannot be used directly in the optimization algorithm. This mathematical model has first to be transformed into a form suitable for the standard non-linear optimization procedure and then the optimization is carried out. We use the method of sequential quadratic programming. The final objective of the optimization process is to find the dimensions of the link-drive members such that the given requirements are satisfied in the best possible manner. In conclusion, the results are described and compared with the initial design. © 2002 Journal of Mechanical Engineering. All rights reserved. (Keywords: deep drawing, deep drawing press, link-drive mechanizm, optimization, modelling) 0 UVOD Globoko vlečenje je zahteven preoblikovalni postopek. Največji vpliv na potek vlečenja ima oblika končnega izdelka in posledično oblika orodja ter vrsta materiala, ki ga obdelujemo. Poleg teh in drugih tehnoloških dejavnikov na kakovost in pravilen potek vlečenja zelo vpliva hitrost vlečenja pločevine. Odvisna je od stiskalnice, na kateri se vlečenje opravlja. Vsak material ima neko optimalno vlečno hitrost. Največja hitrost vlečenja pločevine je tako ena izmed pomembnejših omejitev pri izbiri stiskalnice. Želje po večji produktivnosti preoblikovalnih strojev narekujejo iskanje novih konstrukcijskih rešitev in izboljšav. Najlažji način povečanja produktivnosti stiskalnic za globoki vlek je povečanje 0 INTRODUCTION Deep drawing is a complex metal-forming process. The quality of the products made with this process is mainly influenced by their required shape and the material used. Many technological parameters influence the quality and the course of the whole process, for example, friction contact and lubrication of the tool. However, the slide velocity is one of the most important factors; this velocity depends only on the press design parameters. Each material has its optimum drawing velocity; therefore, maximum drawing velocity is one of the deciding factors when selecting an appropriate press. The demands for increased productivity encourage the search for better solutions and improvements in the production of metal-forming machines. gfin^OtJJIMISCSD 02-11 stran 601 |^BSSITIMIGC Vohar B., Gotlih K., Fla{ker J.: Optimiranje pogonskega mehanizma - Optimization of a Link-Drive njihove obratovalne hitrosti, torej vrtilne frekvence pogonskega motorja. Vendar hitrosti ni moč poljubno povečevati, ker prevelike vlečne hitrosti povzročajo trganje materiala in druge težave [3], saj material nima na voljo dovolj časa za zadostno tečenje. Hidravlične stiskalnice te probleme zaradi učinkovitega krmiljenja zlahka odpravijo, njihova pomanjkljivost pa je višja cena v primerjavi z mehanskimi stiskalnicami ter drago in zahtevno vzdrževanje. Pri mehanskih stiskalnicah je to težje, ker je pot paha omejena z vnaprej določenim gibom, ki ga določa kinematika in izmere pogonskega mehanizma. Ena takšnih izboljšav v mehanskih stiskalnicah je večzgibni pogon paha [3]. Takšen pogon je zaradi svojih karakteristik primernejši od običajnega ročičnega pogona, saj je njegovo delovanje mogoče veliko bolje prilagajati zahtevam posameznih preoblikovalnih postopkov. Imenujemo ga tudi mehanizem s pospešenim povratnim gibom. Njegova glavna lastnost je, da je hitrost paha med delovnim gibom tudi za polovico manjša od običajnih ročičnih stiskalnic pri enaki obratovalni hitrosti ter zelo pospešena pri vračanju v izhodno lego. Takšna hitrostna karakteristika je zelo podobna hidravličnim stiskalnicam, le da celoten krog poteka pri mehanskih stiskalnicah občutno hitreje. Ker je največja hitrost vlečenja pločevine hkrati omejitev obratovalne hitrosti stroja, lahko stiskalnice s takšnim pogonom obratujejo veliko hitreje kakor običajne, saj te hitrosti ne bodo prekoračile. Tak primer prikazuje slika 1 [3], kjer je prikazana razlika v obratovalni hitrosti med običajno ročično stiskalnico ter stiskalnico z večzgibnim pogonom. Razvidno je, da lahko v stvarnem primeru na sliki stiskalnica z večzgibnim pogonom obratuje s 37 gibi na minuto, medtem ko običajna ročična stiskalnica obratuje z največ 20 gibi na minuto (da ne pride do prekoračitve vlečne hitrosti); torej dosežemo z večzgibnim pogonom za 85 % večjo produktivnost (na minuto 17 kosov več). S slike 1 [3] je razvidna še ena prednost večzgibnega pogona, namreč skoraj nespremenljiva vlečna hitrost v delovnem področju, kar omogoča občutno boljše razmere za tečenje materiala, izboljša kakovost izdelka ter podaljša dobo trajanja orodja. Takšen večzgibni pogon obravnavamo v prispevku, prikazan je na sliki 2. Pogon je izveden prek pogonskega zobnika z izsrednostjo, na katerega je vezana ojnica, ki je na eni strani prek veznega droga povezana z okrovom stiskalnice, na drugi strani pa se gibanje prenaša na drsnik. Na tega je nato prek drogov pritrjen pah. Mehanizem je 6-zgibni ročični s končnim drsnim členom in je v bistvu sprememba običajnega 4-zgibnega ročičnega mehanizma z drsnikom. Želeno hitrostno karakteristiko dobimo zaradi podaljšane ojnice in vezave le-te na vezni drog in okrov, kar spremeni sinusoidni potek hitrosti v že prej omenjeni in prikazani krog, značilen za večzgibne pogone. ^BSfirTMlliC | stran 602 The simplest way to increase the productivity of mechanical presses is to increase their operating speed (the rotational velocity of their driving engine). However, such an increase has its limits, since too high drawing velocities cause tearing of the material and some other problems that occur when there is not enough time available for the material to flow [3]. The issue is not relevant in the case of hydraulic presses due to efficient control and slide control, but their high price and maintenance costs are the main disadvantages. This sort of slide control is hard to implement with mechanical presses, because the motion of the slide is constrained and defined by the dimensions and kinematics of the driving mechanism. However, during the past few years press manufacturers have begun to incorporate special link-drives into their presses [3]. Because of their characteristics these drives (also called quick-return drives) are much more appropriate than conventional crankshaft and eccentric gear drives, they are more flexible and easier to adapt for each individual metal-forming process. Their main advantage is a much lower slide velocity during the working part of the cycle. This kind of velocity characteristic is very similar to hydraulic presses, except that the whole process runs much faster in mechanical presses. Since the maximum drawing velocity is also the limit of the press’s operational speed, the presses with the link-drive can operate at higher speeds than conventional versions without exceeding the maximum drawing-velocity limit. Figure 1 demonstrates the difference in the velocity characteristics between a classic crankshaft press and one with a link-drive [3]. Whereas a link-drive press operates at 37 strokes/min, the crankshaft version can run at 20 strokes/min – at most – before the maximum drawing-speed limit is exceeded. That means 85% more production with the link-drive press (17 pieces/min more than with the conventional press). Another important feature of link-drive presses, which is also evident in Figure 1, is the almost constant slide velocity in the working part of the cycle. This considerably improves the conditions for material flow, the quality of the products, and prolongs the lifetime of the tool. Figure 2 presents the link-drive that is considered in this paper. The drive of the press is accomplished through an eccentric driving gear that is linked with a coupler link. On one side the coupler is connected to the press frame via an additional link, and the other end is connected to an output slider-link combination. The mechanism is a 6-bar slider-crank mechanism, which is a modification of a standard 4-bar slider-crank version. A modified velocity characteristic (lower velocity in the working part of the cycle) is achieved through an extended coupler link and its connection to the press frame. Vohar B., Gotlih K., Fla{ker J.: Optimiranje pogonskega mehanizma - Optimization of a Link-Drive hitrost paha slide velocity 43 [m/min] 24 pričetek vlečenja KS pričetek vlečenja SVP draw start CCP draw start LDP \/ ^NV|\ SML / ^\ \ltp X ZML 1 1WV /Si 1 223° \~/ _ KS 20 gibov/min CCP 20 strokes/min KS 37 gibov/min CCP 37 strokes/min SVP 37 gibov/min LDP 37 strokes/min 0° 360° 90° 180° 270° kot zavrtitve pogonskega zobnika driving gear rotation angle Sl. 1. Primerjava enega kroga klasične stiskalnice (KS) in stiskalnice z večzgibnim pogonom (SVP) Fig. 1. Comparison of one cycle of the classic crankshaft press (CCP) and the link-drive press (LDP) +Y \ Y R2(r2) + I.Af2 f_l R5 Sl. 2. Shema večzgibnega pogona Fig. 2. Scheme of deep drawing mechanical press link-drive Želimo optimirati dani pogon glede na določene kriterije, torej poiskati optimalne izmere ročic in lege vrtišč, da bo ustrezal naslednjim zahtevam: - Hitrost v delovnem območju naj bo čimbolj nespremenljiva, predvsem v predpisanem območju od okoli 75 do 150 mm pred spodnjo mrtvo lego. - Mehanizem naj doseže spodnjo mrtvo lego v območju zasuka izsrednika od 190 do 230 stopinj (običajni ročični mehanizem pri 180 stopinjah) - s Sl. 3. Kinematična shema mehanizma Fig. 3. Kinematic scheme of the link-drive mechanism The final objective of the optimization is to find the dimensions of the link-drive members such that the following requirements are satisfied in the best possible manner: - Slide velocity in the working part of the cycle should be as constant as possible, especially in the range from 75 to 150 mm before the lower toggle point (bottom dead centre), - The mechanism should reach the lower toggle point in the range of the eccentric gear rotation angle from 190° to 230°, so as to achieve a longer gfin^OtJJIMISCSD 02-11 stran 603 |^BSSIrTMlGC Vohar B., Gotlih K., Fla{ker J.: Optimiranje pogonskega mehanizma - Optimization of a Link-Drive tem je zagotovljen daljši čas dejavnega dela delovnega giba (kakovostno tečenje materiala). - Hitrost v delovnem krogu naj bo čim manjša (manjša stična hitrost in manjša preoblikovalna hitrost). - Če je le mogoče, naj bodo spremembe izmer čim manjše; izmere ročic naj variirajo v območju okoli ±20%, prav tako naj se čim manj spreminja gib paha. Za sam postopek optimiranja je treba izdelati matematični model mehanizma ter izbrati namensko funkcijo, ki jo bomo optimirali. Zato moramo najprej opraviti kinematično analizo mehanizma, kjer bodo prikazane funkcijske odvisnosti med vsemi spremenljivkami, ki bodo udeležene v optimizacijskem postopku. 1 KINEMATIČNA ANALIZA POGONA Na podlagi kinematične sheme (sl. 3) izpeljemo kinematično analizo mehanizma. Z R 2 do R 6 smo označili vse ročice (R3 in R5 sta skupaj ena ročica -ojnica R3-5), R1 pomeni podlago, 's pa vektor lege paha (dolžina 's se spreminja s časom). V oklepajih so označene dolžine posameznih vektorjev. Mehanizem ima samo eno prostostno stopnjo - torej je njegovo gibanje moč opisati s funkcijo ene same spremenljivke; v našem primeru bo to kot pogonske ročice R4-j 4. Zanima nas potek gibanja paha ter njegove hitrosti in pospeškov, zato moramo poiskati zvezo med vhodno in izhodno veličino, torej funkcijsko odvisnost s-s(4) . The kinematic analysis is made with complex-numbers notation. The mechanism is analyzed in two steps: the first step is the analysis of the 4-bar mecha-nism (R1-R2-R3-R4); and the second is the analy-sis of the 4-bar slider-crank mechanism (R4 -R5 -R6 -s), where the output results of the first analysis are the input data for the second analysis. For a closed loop in the first step we can write the following relation (figure 3): R3 -R4 =0 r1eij1 +r2eij2 or in complex-numbers notation: reij3 -r eij4 =0 Po ločitvi na realni in imaginarni del dobimo sistem dveh enačb z dvema neznankama (kota j2 in j3), znane so vse dolžine ročic r2 do r4, j4 je kot pogonske ročice r4 in je odvisen od časa t in vrtilne frekvence co (j4=«t). Rešitev tega sistema je: (1) (2). After separation of the real and imaginary parts we get a system of two equations with two unknowns (rot. angles j2 and j3). The member lengths r2, r3, r4 are known, the rotational angle of the driving eccentric gear j4 is the input variable, which is dependent on the time t and the rotational velocity w (j4=wt). The solution of this system is: 1 SšnnstsfcflM]! ma stran 604 Vohar B., Gotlih K., Fla{ker J.: Optimiranje pogonskega mehanizma - Optimization of a Link-Drive -B + ^B2 -4 AC 2A 4 B = 2 (K4 sin cp1 - sin 4 D = K5+K4 cos #>1 + K3 cos (4 E = 2(K4sin^ + sin#>4) C = K5-K4 cos #>1 + K3 cos (4 Za vsak kot (j2 in j3) sta mogoči dve rešitvi (obe realni in enaki, kompleksno konjugirani ali realni in različni), ustrezno določimo s kinematično shemo. Prva stopnja je v celoti določena, za poljubni čas t lahko izračunamo vse potrebne parametre in sledi analiza druge stopnje. Iz kinematične sheme 2. stopnje (sl. 4) je razvidno, da je kot j5 vezan na kot j3, saj ročici R3 in R5 sestavljata eno ročico - ojnico R3-5. Zato ga zapišemo kot: ,j3 2arctan e±4e2-4df 2D (3), where the following notations are used: r2 + r2 -r2 +r2 K1 K2 K3 K4 K 2r2r4 (4). -r +r 2r3r4 For every angle j2 and j3 two solutions are possible (both real and equal, complex conjugated or real and different). The appropriate solution is chosen according to the kinematic scheme. The first stage of the mechanism is completed, for an arbitrary time t all the parameters can be computed. The second step of the kinematic analysis follows according to the kinematic scheme of utvhe secounv d stage of the mechanism (Fig. 4). The membeuvrs R 3 and R 5 together form one member – the coupler link R3-5, therefore the angle j5 is dependent on the angle j3, and can be written as: j3 + j35 - p (5), kjer je j35 kot trikotne ojnice, torej kot med ročicama R3 in R5. Izhodna veličina, katere funkcijsko where j35 is the angle ofuv the triuvangular coupler (the angle between members R 3 and R 5 ). The output quan- Sl. 4. Kinematična shema 2. stopnje mehanizma Fig. 4. Kinematic scheme of the second stage I IgfinHŽšlbJlIMlIgiCšD I stran 605 glTMDDC Vohar B., Gotlih K., Fla{ker J.: Optimiranje pogonskega mehanizma - Optimization of a Link-Drive odvisnost potrebujemo, je gib oz. pozicija paha tity, whose functional relation we are looking for, is (drsnika) - dolžina vektorja s . Zapišemo lahko naslednje zveze: s = y6 +y7, y6=r4sinj4+r5sinj the slide position – the length of the vector s . The following relations can be written: y7 x6 d = arcsin y7 tan d 6 x6 r4 cosj4 + r5 cosj5 x6 = r4 cosj4 + r5 cosj5 tan d (6), tan 6 J J oziroma: hence: r4 sinj4 + r5 sinj5 + r4 cosj4 + r5 cosj5 tan (7). Vse odvisne veličine na desni strani enačbe (7) so funkcije kota j4 in smo tako dobili želeno zvezo med vhodno ter izhodno veličino: s = f(j4). Hitrosti in pospeške paha je sedaj moč preprosto dobiti z odvajanjem (7). Iz rezultatov analize je razvidno, da je zveza (7) nelinearna, kar pogojuje način optimiranja Rezultate kinematične analize prikazujejo krivulje: gib, hitrost in pospešek pehala v odvisnosti od vhodnega kota j4 (in s tem posredno časa t) na sliki 5. Iz grafov je vidna tipična karakteristika večzgibnega pogona: SML se pojavi kasneje kakor pri običajnem ročičnem pogonu, v območju delovnega giba je hitrost manjša in ima enakomernejši potek, čemur sledi skokovito povečanje hitrosti pri vračanju paha navzgor. Za obravnavani pogon smo izdelali računalniški program v programskem jeziku v The desired relation s= f(j4 ) is derived, and all the quantities on the right-hand side of the equation are dependent only on the angle j4. The slide velocity and the acceleration can now be easily obtained with a derivation of (7). It is clear that the relation (7) is highly nonlinear and therefore a nonlinear optimization procedure has to be chosen. The results of the kinematic analysis are shown in Figure 5. The typical characteristics of the link-drive are clear from the graphs: the lower toggle point occurs much later than with the classic crankshaft drive; the slide velocity in the working part of the cycle is lower and more constant; there is a rapid velocity increase in the returning part of the cycle. A FORTRAN algorithm was made which, for given input gib paha, stroke s mm 800 600-400-200- a) lega paha, slide position hitrost paha, slide velocity s& mm/s 400 200 b) hitrost paha, slide velocity kot pogonske ročice driving gear rot angle j4 rad -200 -400 3p 2 2p kot pogonske ročice driving gear rot. angle j4 rad 2p območje enakomernejše hitrosti paha, area with more uniform slide velocity pospešek paha, slide a mm/s2 tionf 200- c) pospešek paha, slide acceleration -200 -400 -600 2p kot pogonske ročice, driving gear rot. angle j4 rad Sl. 5. Rezultati kinematične analize večzgibnega pogona (a - lega, b - hitrost, c -pospešek) Fig. 5. Results of the kinematic analysis (a - position, b - velocity, c - acceleration) 1 BnnBjfokJ][p)l]Olf|i[gO I I ^SSfuTMlGC I stran 606 p Vohar B., Gotlih K., Fla{ker J.: Optimiranje pogonskega mehanizma - Optimization of a Link-Drive FORTRAN, ki za dane vhodne podatke izračunava kinematične veličine (lego, hitrost, pospešek in gib paha, lego SML, največji absolutni pospešek in njegovo lego). Izračunani podatki za obravnavani mehanizem so naslednji: - SML pri j4 = 208°, - gib mehanizma = 903,48 mm, - predpisano območje (okoli 75 do 150 mm pred SML): j4 = 155°-176°, - največji absolutni pospešek na predpisanem območju = 198,8 mm/s2 (j 4 = 176°). Kinematična analiza pogona je končana, znani so vsi potrebni parametri, ki jih potrebujemo za optimizacijo. 2 OPTMIRANJE POGONA A) Splošni optimizacijski model Osnovni cilj vsakega optimizacijskega postopka, ki temelji na metodah matematičnega programiranja, je poiskati odgovor na vprašanje “kaj je najboljše?” pri problemih, pri katerih lahko kakovost odgovora izrazimo kot numerično vrednost. Ali drugače: poiskati takšno kombinacijo parametrov (projektnih spremenljivk), ki bodo minimizirali izbrano veličino (namensko funkcijo). Optimizacijski problem je najprej treba spremeniti v matematično obliko. Splošni model optimizacijskega problema zapišemo v obliki: poišči tak vektor projektnih spremenljivk x e Rn , ki bo minimiziral namensko funkcijo f(x,z) e Rn, ob upoštevanju pogojev: data (the drive geometry), calculates the required kinematic quantities (slide position, velocity and acceleration, stroke, position of the lower toggle point, maximum absolute acceleration and its position) in the required resolution step. The calculated values are: - lower toggle point at j4 = 208° - stroke = 903.48 mm - desired working part of the cycle approximately 75 to 150 mm before the lower toggle point): j4 = 155° to 176° - maximum absolute slide acceleration in the de-sired range = 198.8 mm/s2 (j4 = 176°) The kinematic analysis of the drive is finished, and all the parameters required for the optimization procedure are known. 2 OPTIMIZATION OF THE LINK-DRIVE A) General optimization model The basic objective of every optimization procedure based on methods of mathematic programming is to find an answer to the question “What is the best?” concerning problems where the quality of the solution can be expressed and evaluated as a numerical value. In other words: to find such a combination of parameters (design variables) that will minimize the chosen quantity (cost function). But first, the optimization problem has to be transformed into a mathematical formulation. The general optimization model can be expressed in the following form: find such a vector of design variables x e Rn which minimizes the cost function f(x, z) e Rn , subjected to following constraints: 1,2,...,k (omejitve projektnih spremenljivk/constraints of design variables, lower/upper bounds) g (x,z)<0 j=1,2,...m (omejitvene funkcije/inequality constraints) (8), hl(x,z) = 0 l = 1,2,...m' (odzivne enačbe sistema/equality constraints, system-response equations) kjer z pomeni vektor odzivnih oz. sistemskih spremenljivk (hitrosti, pospeški, reakcije itn.), x pa vektor projektnih spremenljivk (parametri, katerih optimalne vrednosti iščemo, da zadostimo zahtevanim kriterijem). Namenska funkcija je tista, ki jo izberemo kot merilo za uspešnost optimiranja in katere minimum iščemo. Če je katerakoli od definiranih funkcij (namenska, omejitvene itn.) nelinearna, imamo opravka z nelinearnim optimiranjem. Optimizacijski problem rešujemo iterativno, za njegov zagon pa potrebujemo začetno točko x0 (ocena ali sedanja varianta projekta, ki ga želimo izboljšati). Optimizacijski problem v obliki (8) ni neposredno uporaben, kadar imamo opravka z dinamičnimi sistemi, kakor je npr. večzgibni pogon, ki ga obravnavamo. Pri takšnih sistemih se pojavlja nova neodvisna spremenljivka - čas, zaradi česar postane where z is the state variable vector of the system-response variables (velocity, acceleration, reaction forces, etc.) and x is the vector of the design variables (parameters for which optimum values are to be found to satisfy given criteria and constraints). The cost function is the one which is chosen accordingly as a measure of the efficiency of the optimization procedure. If any of the defined functions (cost, constraint, etc.) are nonlinear, the nonlinear optimization algorithm has to be used. The optimization problem is solved iteratively, for its start the starting point x0 is required (current project status or estimation). However, the optimization model in the form of (8) cannot be used directly in the cases of dynamical systems, for example, in a link-drive. With these systems a new independent variable arises, time t, which makes the vector of the system variables, z, Vohar B., Gotlih K., Fla{ker J.: Optimiranje pogonskega mehanizma - Optimization of a Link-Drive vektor odzivnih spremenljivk z odvisen od časa t. Obstaja več metod, kako časovno odvisen problem predelati v splošno obliko, ki bo primerna za reševanje. Ena od možnosti ([1] in [2]) je npr., da tak problem prevedemo v zaporedje časovno neodvisnih problemov. V našem primeru smo izbrali metodo z uvedbo nove spremenljivke [1]. Ker nas pri analizi dinamičnih sistemov v nekem časovnem razponu običajno zanima največja vrednost odziva sistema, ki jo želimo zmanjšati oz. optimirati (največji pospeški, hitrosti itn.), lahko namensko funkcijo za tak problem zapišemo kot: time dependent. There are various methods for transforming time-dependent problems into a suitable form for the general optimization model. Some methods ([1] and [2]) translate the time-dependent problem into a sequence of time-independent problems. In our case the method of introducing a new artificial variable [1] was chosen. In the analyses of dynamical systems the maximum values of a measure of the system response in a certain time interval are usually the required quantities (maximum velocity, accelerations, etc.) and these also tend to be the quantities to be optimized. Therefore, the cost function for such a dynamic system can be expressed as: y0 = max f0(x,z(t),t) (9). Da bi takšno namensko funkcijo lahko uporabili v splošnem optimizacijskem modelu (8), se moramo znebiti funkcije max iz namenske funkcije ter časovne odvisnosti iz omejitev. To dosežemo z uvedbo nove umetne spremenljivke xk 1, ki pomeni zgornjo mejo f0, za katero velja: f0(x,z(t),t)- S tem postane vektor konstrukcijskih spremenljivk oblike: To use the above form of the cost function in the general optimization model (8), the maximum value function from the cost function and the time dependency of the constraint functions have to be removed. This can be achieved with the introduction of a new, artificial variable xk+1, which represents the upper limit f0, subjected to: <0 00 0 ;f(t)<0 The transformed problem is now stated in the following form: find such a vector x e Rn, which mini-mizes xk+1 e R, subjected to following constraints: (13). y = }(f0(x,z,t)-xk+1dt = 0 0 t yj=\{gj (x,z,t))dt = 0 0 hl(x,z(t),t) = 0 i =1,2,...,k,k +1 j = 1,2,...m l =1, 2,...m' (14). Tako smo dobili obliko, ki jo predpisuje splošni optimizacijski model (8) in jo lahko uporabimo v postopku optimiranja. The optimization problem is now formulated according to the general optimization model (8), and can be used with the standard optimization procedure. 1 BnnBjfokJ][p)l]Olf|i[gO | | ^SSfiflMlGC | stran 608 Vohar B., Gotlih K., Fla{ker J.: Optimiranje pogonskega mehanizma - Optimization of a Link-Drive B) Optimizacijski model večzgibnega pogona Najprej je treba izbrati namensko funkcijo. Osnovni cilj našega optimiranja je določiti parametre mehanizma tako, da bo hitrost paha v predpisanem območju čim bolj nespremenljiva. Zato bi morali biti pospeški na tem območju enaki nič ali pa se temu čim bolj približati. Kot namensko funkcijo zato določimo največji absolutni pospešek na predpisanem območju, naš cilj pa je ta pospešek zmanj sati v največji mogoči meri. Namenska funkcija za obravnavani problem se torej glasi: B) Link-drive optimization model First, the cost function has to be chosen. The primary objective of this link-drive optimization is to determine the link-drive parameters in such a way that the slide velocity in the defined range of movement is as constant as possible. This means that the slide acceleration in this range has to be zero or close to zero. Therefore, a suitable cost function for this problem is defined as the maximum absolute slide acceleration, and the purpose is to lower this acceleration as much as possible. The cost function is: f = max &s& (x,j4 ) j4 <(D < ppl. Ker je izbočitveni koeficient za elastično izbočitev kel znan, lahko izračunamo mejno vitkost in s tem najmanjšo debelino preskušanca: 2 Elasto-plastic buckling occurs when the buckling load pcel is greater than the load at the beginning of plastification p l or p el>ppl. Because the elastic buckling coefficient is known, the slenderness limit and the minimum thickness of the specimen can be calculated: spl (1 - n2)V 3(1 -d2) 2k,, , h > 2b min h m Največjo debelino preskušanca izračunamo iz pogoja, da kritična obremenitev ob izbočitvi ne preseže največje obremenitve preskuševališča P . Ker je kritična obremenitev v elasto-plastičnem območju manjša od tiste za elastično območje, najbolj neugoden primer izberemo tako, da predpostavimo elastično izbočitev kolobarja. Ocena za največjo debelino preskušanca oziroma najmanjšo vitkost, ki izhaja iz pogoja pcel h