CONTACT BETWEEN FLEXIBLE BODIES IN NONLINEAR ANALYSIS, USING LAGRANGE MULTIPLIERS

Transcription

1 COAC BEWEE FLEXIBLE BODIES I OLIEAR AALYSIS, USIG LAGRAGE MULIPLIERS Dr. Phillipe Jeeur Philippe.jeeur@samcef.com Samech, Parc Scieifiue du Sar-ilma Rue des Chasseurs Ardeais, 8 B-403 Agleur-Liège, Belgium Absrac his paper deals wih coac i oliear aalysis. Oly implici schemes are looked a. I order o solve he coac problem, a augmeed lagragie procedure is used. he resulig sysem of euaios is solved simulaeously for he displacemes ad Lagrage mulipliers. Coac bewee wo flexible bodies ad coac bewee a flexible body ad a rigid sur are reaed. A special care is ake whe he flexible body is modelised by secod order eleme i 3d aalysis. Iroducio I oliear aalysis, here are differe ways o solve a coac problem. Pealy mehod or Lagragie mehod ca be used. he coac problem ca be solved a he same ime as he oher olieariies or i ca be solved i a ucoupled way. I his case, eiher for each srucural ieraio, he coac problem is solved or he srucural ieraios are performed wih froze coac codiios. I his paper, we solve he coac a he same ime as he oher olieariies ad we use a augmeed Lagragie mehod [,]. he resulig sysem of euaios is solved simulaeously for he displacemes ad Lagrage mulipliers. Coac bewee flexible ad rigid bodies ad coac bewee wo flexible bodies are looked a. I he firs case, we aalyze he coac bewee oe slave ode ad a maser rigid sur, i he secod case, we aalyze he coac bewee oe ode ad a maser. Whe secod order elemes are used i 3d aalysis, we add a ode liked o he. ha meas we have brick eleme wih odes ad erahedral elemes wih odes. Coac euaios variaio of ageial displacemes ( u u bewee he curre ieraio ad he begiig of he ime sep. We ca compue he variaio ad icrease of hese values i fucio of he odal ukows. he odal ukows are he displacemes of he slave ode, ad eiher he displacemes of he ode o wich he rigid sur is liked or he displacemes of he odes of he. We wrie : δd δ u = Bδ δ u = Bδ d d = = = = Bδ Bd Bd Bd where is he ormal ad he ages o he sur. he secod sep is o defie he coac codiio. For his purpose, 3 Lagrage mulipliers are iroduced, oe for he coac ad wo for he fricio. We iroduce he 3 uaiies: = kλ pd = kλ p u = kλ p u where k is scalig facor ad p a regularizaio parameer. I order o decide if here is coac or o, we look a. he coac crieria is a mixed bewee forces ad displacemes. If we look a Figure, he soluio a he covergece is he bol lie, eiher he ormal disace or he Lagrage muliplier should be eual o 0. Durig he ieraios, we are o o he soluio, he doed lie divides he space i wo zoes, oe where he ode is said i coac ad he oher oe where he ode is said o i coac. he slope of his doed lie depeds o he regularizaio parameer p. A he covergece, he soluio will o deped o p bu he covergece propery depeds o p. Whe he value of p chages, he crieria o say if a poi is i coac or o will chage. We do o discuss o he value o be give o p. I he umerical applicaio, we ake i as a fucio of he siffess of he srucural elemes. he firs sep is a geomerical sep. We search he projecio of he ode o he maser sur, eiher a rigid sur or a flexible. From his projecio, we ca compue he ormal disace d bewee he ode ad he sur ad he

2 coac Wihou coac λ Crieria soluio d For he fricio behavior, here is hree saus. he firs oe is he simples oe, here is o fricio coefficie. I his case, we jus impose ha he Lagrage mulipliers liked o he fricio are eual o 0. We ake as eleme forces: ( δλ kλ δλ kλ δ F= If here is fricio, i remais wo saus. Eiher he ode is sick o he sur or he ode is slidig. I order o see wha is he saus, we firs compue he followig uaiy: Sick =. o coac Figure Coac codiio If is larger ha 0, here is o coac ad we impose ha he Lagrage mulipliers liked o coac ad fricio is eual o 0. he odal forces a he eleme level are compued from:. Coac ( δλ kλ δλ kλ δλ kλ δ F= If is smaller ha 0, here is coac. he odal forces a he eleme level are compued from: F= δd ( pd kλ = B δλ kd ( pd kλ δλkd A he euilibrium, he ormal disace is eual o 0 (euaio liked o he Lagrage muliplier ad i ca be see ha he regularizaio parameer has o ifluece o he soluio. I ca also be see ha he Lagrage muliplier is he coac force divided by he scalig parameer k. From his euaio, we ca compue a ieraio marix which is symmerical. If he derivaive of B is egleced, his ieraio marix is compued from: pb B δλ k B kb d 0 dλ We see ha k is iroduced i order o have he same order of magiude for he euaio liked o he Lagrage muliplier ad he euaios liked o he displacemes. I is ake eual o he siffess of he srucural elemes..3 Fricio We compare o he ormal force. he saus is sick if: µ 0 I his case, we impose ha he variaio of ageial displacemes is eual o 0. he eleme forces are compued from: δ F= ( p u kλ ( p u kλ δλ k u δλ k u As for he coac, he regularizaio parameer has o ifluece o he soluio. We ca build a ieraio marix which is symmerical. Slip he saus is slip if: µ 0 We firs defie he direcio of slidig : v = / v = / he eleme forces are compued from: F= δλδλ µ F v µ F v ( µ F v kλ k p ( µ F v kλ k p We ca check ha he fricio Lagrage mulipliers are parallel o he variaio of slidig displacemes. A he euilibrium, he wo las euaios are proporioal o p. Bu as hey are local euaios, p has o ifluece o he resuls. If we iroduce:

3 δv δv v v v pδ u kδλ pδ u kδλ = = v v v pδ u kδλ pδ u he ieraio marix is eual o: µ v µ v df= δ uδλδλ µ v k p µ v k p ( pdd kdλ kδλ Some are posiive, some are egaive. For a cosa pressure, we should have he same disribuio o he coac zoe. Wih a coac sraegy ode o, i will o be he case. he forces will go oly hrough he mid-side odes ad ohig o he corer odes. So, wih sadard 0 odes eleme, i is o possible o have he exac soluio for his small problem. he soluio is o add a ode a he middle of he. Is shape fucio is eual o: ( ξ ( η ( ζ = µ p δ uδλδλ µ k I his case, i is o symmerical. µ k k µ I p dλ dλ if he coac is he upper oe. We have also o chage he shape fucio of he odes belogig o he. For a corer ode or a mid-side ode, we have simple relaioship bewee he ew shape fucio ad he classical oe: ew, corer ew, midside = = classical, corer classical, midside 4 3 Projecio of he slave odes Local ewo ieraios are performed i order o fid he projecio of he ode o he sur, excep if he is a riagle a degree. We will o discuss his poi here. Jus recall ha whe we compue he B marix i he rigid sur case, here is a maser ode o which is liked he sur ad wih whom he sur is movig. For flexible o flexible coac, he s see by a ode chage durig he compuaio. We updae hese s a each ieraio. 4 Secod order eleme i 3d aalysis We cosider wo brick elemes wih 0 odes (Figure. he firs oe is fixed alog z a he boom. he rigid body modes i he xy plae of he wo elemes are fixed. Coac is iroduced bewee he wo elemes. A cosa pressure is applied a he op of he srucure. Oce he shape fucios are defied, i is a classical isoparameric eleme wih odes. he odal forces cojugaed o a cosa pressure are ow eual o: F = hey have he same sig. For he small problem, we ow have he exac soluio. For he riagular s, we adop he same sraegy ad ge erahedral eleme wih odes. 5 umerical applicaio he mehod has bee iroduced i he Samcef Mecao sofware [3,4]. 5. Bucklig of a cylider z y x Figure wo bricks elemes If he area of he upper ad he pressure are eual o, he odal forces cojugaed o he pressure are eual o: F = Figure 3 Bucklig of a cylider

4 As firs example, we sudy he bucklig of a cylider. he radius is eual o 00, he heigh o 00 ad he hickess. he Youg modulus is eual o 0000, he Poisso coefficie o 0.3 ad he iiial yield sress 500. A axial force is iroduced. Firs a liear aalysis is performed followed by a bucklig aalysis. We iroduce as imperfecio for he oliear aalysis he 5 firs bucklig modes. We impose he axial displaceme of he upper. I his case, we have flexible o flexible coac ad we also iroduce a flexible o rigid coac wih rigid plae a he op ad boom of he cylider. We oly look a he axisymmerical behavior. A saic aalysis is performed. Figure 3 show he deformed shape for differe load level. he axial force i fucio of he axial displaceme is show o Figure 4. maximum of 6 ieraios per sep is eeded. A he maximum load, odes are i coac. O Figure 6, we ca see he disribuio of he coac pressure i he coac area. Figure 6 coac pressure 5.3 Key problem he ex example is a pi iserio, i was proposed by S. Adeff [5]. Boh par are flexible. Figure 4 bucklig of a cylider load displaceme curve 5. Herz coac problem he secod example is a classical Herz coac problem. We sudy he coac bewee a flexible cylider (radius m, Youg modulus Pa, Poisso coefficie 0.4 ad a rigid plae. We use plae srai hypohesis ad secod degree elemes. By symmery, we sudy a half model wih 500 eleme. I he coac area, we use a 8 by 8 regular mesh, elsewhere we use a free mesh. he applied load is eual o 4. Figure 5 shows he disribuio of vomises sresses i he cylider Figure 7 key problem Figure 7 shows he differe posiios of he pi. I should be looked sarig from he boom a he righ ad urig i a rigoomeric way. he key is pushed (righ side from boom o op ad afer ha is pulled back (lef side, from op o boom. Figure 5 - vomises sress disribuio he soluio is close o he heoreical soluio. he load is applied i 5 seps, good covergece is observed. A I his case, he s see by he slave odes chage durig he aalysis. We have o updae he opology a each ime sep.

5 5.4 Pre-sressig of a cylider he purpose of his example is o iroduce a cylider i a hole he radius of which is smaller ha he radius of he cylider. he hole belog o a srucure which has a cyclic symmery, so oly /6 h of he srucure is modelised. 3d secod degree elemes are used, cyclic codiios are iroduced a he boudary. here is some compressio i he lier, bucklig ca occur. composie lier Reacio Figure 9 SEOR ak displaceme Figure 0 bucklig mode he purpose of he sudy is o see for which size of imperfecio bucklig will occur. For a imperfecio eual o he hickess of he lier, we have a isabiliy. he displaceme a he ed of he aalysis are show o figure 0. 6 Coclusio Figure 8 presressig of a cylider Figure 8 shows differe posiios of he ier cylider ad he reacio force correspodig o differe displacemes. 5.5 SEOR ak, sabiliy wih coac he SEOR (Saellie de élécommuicaio e Eudes de ouvelles echologies e Orbie program has as objecive o pu i orbi a saellie wih ew echology i order o es i. I his program, Aerospaiale Mara develops a ew high pressure ak. he ak is made of a cylidrical par ad wo hemispherical pars (Figure 9. he exeral srucure is made of composie maerial ad he ieral par is a meallic lier. he lier is o glued o he composie, here is oly coac wih fricio bewee boh maerials. A pressure is iroduced i he ak. he composie maerial has a liear behavior, plasiciy appears i he lier. Whe he pressure is removed, residual sresses are prese i he srucure. I his way, uder curre load, he behavior of he ak will be liear. A geeral mehod has bee preseed i order o hadle coac i oliear aalysis. A special care has bee give o secod degree eleme i 3d aalysis. Oly saic examples are performed i a implici way. 7 Referece [] A. Heege, P. Alar : A fricioal coac eleme for srogly curved coac problems. I. Jl um Meh. Eg., Vol. 39, 65-86, 996 [] P. Alar, A. Curier : A mixed formulaio for fricioal coac problems proe o ewo like soluio mehods. Comp. Meh. App.Mec. ad Eg., Vol. 9, , 99 [3] Ph. Jeeur, A. Heege : Coac wih fricio, flexible o rigid srucure. Samech repor 69, 998 [4] Samcef : user maual, versio 8.. Samech repor, 000 [5] Adeff, S.E. oliear Aalysis of a Pi isersio, 998 Abaus User s Coferece, ewpor, Rhode Islad, May 998