Mixed MLPG Staggered Solution Procedure in Gradient Elasticity for Modeling of Heterogeneous Materials

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1 Mxed MLPG Saered Solon Proedre n Graden Elay for Modeln of Heeroeneo Maeral Bor alšć 1 ra Sorć 1 and Tomlav arak 1 Abra A mxed MLPG olloaon mehod appled for modeln of deformaon repone of heeroeneo maeral n raden elay. Heren a heeroeneo maeral doman ompred of wo orop homoeneo par wh dfferen maeral ela propere ondered. The olon for he enre doman obaned by enforn he orrepondn bondary ondon alon he nerfae of he homoeneo doman. For he approxmaon of he nknown feld varable he Movn Lea Sqare (MLS) fnon wh nerpolaory ondon are appled. The ran raden elay baed on he Afan heory wh one mrorral parameer lzed. The ornal forhorder eqlbrm eqaon of raden elay are olved n a aered manner a an nopled eqene of wo e of eond-order dfferenal eqaon. The propoed mxed mehle approah eed and demonraed by a repreenave nmeral example. eyword: Mxed mehle approah olloaon mehod aered olon proedre heeroeneo maeral 1 nrodon owaday a lare nmber of dfferen mehle mehod are lzed for he modeln of maeral deformaon repone. Th de o her benefal haraer n omparon o andard meh-baed mehod. The mehle nmeral approahe are able o overome problem h a elemen doron and me-demandn meh eneraon proe. everhele he allaon of mehle approxmaon fnon de o hh ompaonal o ll a major drawbak. Th defeny an be allevaed o a eran exen by n he mxed Mehle Loal Perov-Galerkn (MLPG) Mehod paradm [Alr L Han (006)]. n he preen onrbon he MLPG formlaon baed on he mxed approah adaped for he modeln of deformaon repone of heeroeneo maeral baed on he ran raden elay heory. A heeroeneo rre on of wo homoeneo maeral whh are drezed by rd pon n whh eqlbrm eqaon are mpoed. n addon he ran raden elay baed on he Afan heory wh only one mrorral parameer ondered. The raden heory ed n order o more araely apre he maeral behavor near he nerfae beween reon wh dfferen maeral propere and o remove jmp n he ran feld ha an be oberved when a laal heory of lnear elay ed. The olon of forh-order dfferenal eqaon arn n non-la heore reqre a hh-order of approxmaon fnon [Ake Afan (011)]. Hene n he Fne Elemen Mehod (FEM) for olvn h ype of problem no a we hoe ne andard formlaon need o poe C 1 onny whh lead o omplaed hape fnon wh lare nmber of nodal deree of freedom even f mxed elemen are lzed [Amanado Arava (00)]. Therefore hee FEM proedre hold no be ed de o her neffeny relaed o hh nmeral o [Ake Afan (011)]. On he oher hand he reqred C 1 onny obanable n a mple and a rahforward manner when he mehle mehod are ondered [Alr (004)]. n he propoed mehod he forh-order eqlbrm eqaon of raden elay are olved a an nopled eqene of wo e of he eond-order dfferenal eqaon [Ake Moraa (008)] for he prpoe of frher derean he onny reqremen of he formlaon. Hene wo dfferen bondary vale problem loal (laal) and non-loal (raden) are ben olved where he olon of he former problem ed a an np n he laer problem. n boh bondary vale problem ndependen varable are approxmaed n mehle fnon n h a way ha eah maeral reaed a a eparae problem [Chen Wan H Ch (009)]. The lobal olon for he enre heeroeneo rre aqred by enforn approprae bondary ondon alon he nerfae of wo homoeneo doman. The applaon of he aered olon heme [Ake Moraa (008)] lzn he mxed mehle approah rel n le omplaed mehle formlaon whh only ha he C 0 reqremen on he approxmaon fnon. A olloaon mehle mehod ed whh may be ondered a a peal ae of he MLPG approah where he Dra dela fnon ed a he e fnon. Sne he olloaon mehod employed he ron form of eqlbrm eqaon employed and me-onmn nmeral neraon proe avoded. The MLS approxmaon fnon [Alr (004)] wh nerpolaory propere (MLS) are appled [Mo Bher (008)]. Th enable mple mpoon of eenal bondary ondon a n FEM. aral bondary ondon on oer ede are enfored va he dre olloaon approah. n he loal 1 Faly of Mehanal Enneern and aval Arhere Unvery of Zareb vana Lčća Zareb Croaa.

2 problem he laal lnear ela bondary vale problem for eah homoeneo maeral drezed by n he ndependen approxmaon of laal ran and laal dplaemen. n order o derve he fnal loed yem of laal drezed eqaon wh he laal dplaemen a only nknown he approxmaed laal ran are expreed n erm of laal dplaemen n approprae knema relaon. n he mlar manner for he drezaon of he non-loal bondary vale problem ndependen approxmaon of he raden dplaemen dplaemen and he dervave of raden are lzed. Heren o oban he fnal k olvable yem of drezed raden eqaon he approxmaed dervave are wren n erm of raden dplaemen a he olloaon node. The mxed MLPG olloaon mehod for he modeln of deformaon repone of a heeroeneo maeral n raden elay preened and explaned a lare n Seon. The propoed mehod eed and analyzed by ondern a problem of he lamped heeroeneo plae bjeed o nform dplaemen a he rh end n Seon 3. n Seon 4 onldn remark and frher reearh delne are ven. Mxed MLPG Mehod for Graden Elay The wo-dmenonal heeroeneo maeral whh ope he lobal ompaonal doman rronded by he lobal oer bondary ondered. The bondary repreen he nerfae beween wo bdoman and wh dfferen homoeneo maeral propere. eparae he lobal doman n h a manner ha and. j dnh wheher he laal or raden bondary vale problem ben olved. The ame analoy apple o all oher bondare where ome knd of bondary ondon prerbed e.. he nerfae bondary n he laal bondary vale problem denoed a whle n he raden one denoed. Hene he ypal heeroeneo maeral ben analyzed now porrayed n F. 1. The overnn eqaon for he preened example are he ron form D eqlbrm eqaon whh have o be afed whn he lobal ompaonal doman dvded no and. Aordn o he aered olon proedre derbed n [Ake Moraa (008)] wo e of eondorder paral dfferenal eqaon an be lzed o derbe he deformaon of he heeroeneo maeral. Thee eqaon are here wren for eah homoeneo maeral eparaely. Th he fr eqaon e repreenn he laal bondary vale problem eqal o σ b 0 whn (1) j x j σ b 0 whn () j x j Whle he eond eqaon e for he non-loal raden problem lze a mrorral parameer l and expreed a l whn (3) mm l whn. (4) mm A evden frly he laal bondary vale problem olved whoe olon hen ed a an np on he rh hand de of he raden eqaon. n h operaor-pl proedre he laal and raden bondary ondon need o be afed on he oer bondare of he heeroeneo rre dependn on whh problem rrenly ben olved. Hene a n [Alr L Han (006)] he laal bondary ondon nlde he dplaemen and raon eqal o on (5) on (6) n on (7) j j Fre 1: Two-dmenonal heeroeneo maeral Frhermore ne n he aered proedre wo dfferen bondary vale problem are olved one afer he oher he lobal bondary an be denoed a or o n on (8) j j whle he raden bondary ondon an be he dplaemen and eond-order normal dervave of

3 dplaemen R [Polzzoo (003)] where jk denoe he hrd-order enor ompred of eond dervave of dplaemen on (9) on (10) R n j nk jk R on + n R n j nk jk R on. n (11) (1) Frhermore o aqre he olon for he enre rre he ondon on he nerfae bondare and need o be enfored for boh he laal and he raden problem. Aordn o [Ake Moraa (008)] f he laal elay problem olved hee bondary ondon are he onny of dplaemen and reproy of raon 0 on (13) + n n 0 on. (14) + + j j j j On he oher hand f he raden problem ondered he nerfae bondary ondon nlde he onny of dplaemen and reproy of fr-order normal dervave of dplaemen 0 on (15) + n n + 0 on +. (16) The wo-dmenonal heeroeneo onnm drezed by wo e of node 1... and M 1... P where and P ndae he oal nmber of node whn and repevely. Heren he ame e and poon of he node are ed for he drezaon of boh he laal and he raden bondary vale problem. ow for eah ondered drezaon node he MLPG onep [Alr (004)] appled wheren he Dra dela e fnon hoen a he weh fnon n loal weak form and he loal approxmaon doman are defned arond eah node n order o ompe he onnevy beween he node. For he node pooned on he nerfae bondare he approxmaon doman are rnaed n h a manner ha he drezaon node from one homoeneo maeral nflene only he node belonn o ha maeral. For he drezaon of boh bondary vale problem he mxed olloaon proedre [Alr L Han (006)] lzed. All nknown feld varable are approxmaed eparaely whn bdoman and where he ame approxmaon fnon are employed for all feld omponen. For he hape fnon onron he well-known MLS approxmaon heme [Alr (004)] employed. The nerpolaory propere of he MLS approxmaon fnon are aheved by lzn he weh fnon aordn o [Mo Bher (008)]. Sne he drezaon of he laal bondary vale problem n he mxed MLPG approah well domened n he enf lerare he derpon of he obaned eqaon for he laal problem here kpped and he reader referred o [alšć Sorć arak (017)] where h approah derbed n deph. n h onrbon he man fo hfed o he drezaon of he raden bondary vale problem and he orrepondn bondary ondon. Here he dplaemen and dervave of dplaemen are nknown feld varable. Th for he node whn he maeral and node pooned on he bondare and hee approxmaon are wren a ( h) n 1 ( x) ( x )( ) wh (17) ( h) G 1 ( ) ( x) ( x )( ) whn (18) where repreen he nodal vale of wo-dmenonal hape fnon for node and for he nmber of node whn he approxmaon doman whle and G denoe he nodal vale of he dplaemen and dervave of dplaemen omponen. ow frly he overnn eqaon of he raden problem (3) and (4) are rewren n her marx form a he drezaon node n he doman and [ ( )] l (19) +T [ ( )] l (0) T M M M where T ( ) denoe he Laplaan operaor wren n marx form. Hene he operaor +T and are eqal o ( ) ( ) ( x) 0 ( x) 0 x1 x ( ) ( ) 0 ( x) 0 ( x) x1 x (1)

4 T ( ) ( ) ( xm) 0 ( xm) 0 x1 x. ( ) ( ) 0 ( xm) 0 ( xm) x1 x () The overnn eqaon (19) and (0) are now mlaneoly drezed by he approxmaon (17) and (18) reln n +T l [ ( G)] 1 1 (3) T l [ ( G)] M 1 1. (4) n he above eqaon G and G denoe he veor of nknown dervave of dplaemen defned by (5) x x x x [ T 1 1 G ] [ ] 1 1 (6) x x x x T 1 1 [ G ] [ ]. 1 1 A obvo he eqaon (3) and (4) repreen an nolvable yem ne he lobal nmber of nodal nknown larer han he nmber of eqaon. Th he yem of eqaon here loed mply by enforn he ompably a eah node beween he approxmaed nodal dervave of dplaemen ( h) G G ( h) G G ( x ) and ( x ) and he nodal dplaemen and repevely. Hene he ompably eqaon wren n knema dfferenal operaor D and D are D (7) G D. (8) G Eqaon (7) and (8) are now aan wren a every drezaon node and drezed by (17) whh yeld G ( 1 1 (9) D x ) G G 1 1 (30) D ( x ) G where G G x and G G x ndae he mare onn of he fr-order dervave of hape fnon wren analooly o operaor n (1) and (). nern he drezed ompably relaon (9) and (30) no he drezed overnn eqaon (3) and (4) a olvable yem of lnear alebra eqaon wh only he nodal dplaemen a nknown aaned F wh n (31) F wh n (3) M M where he raden nodal oeffen mare are eqal o T + l and S [ G G ] (33) T M M l M S [ G G ]. (34) + Heren he mare S and S M are he daonal mare omprn of nodal hape fnon vale + ( x) 0 S 0 ( x ) S M ( xm) 0. 0 ( x ) The raden nodal fore veor M F and + M M (35) (36) F n (31) and (3) are ompoed of he known vale of laal dplaemen. A obvo by lzn he aered proedre and he preened mxed mehle raey he oeffen mare and + are aembled n only he fr-order dervave of hape fnon. All approxmaon fnon n h onrbon poe he nerpolaon propery a he node. Coneqenly he eenal bondary ondon are enfored rahforwardly analooly o he proedre n FEM. Therefore by drezn he dplaemen bondary ondon (9) and (10) wh he approxmaon (17) we oban on (37) 1 on (38) The naral bondary ondon (11) and (1) on he bondare and are mpoed n he dre + olloaon approah. Here n order o derve he drezed eqaon of he naral bondary ondon dependen only on he nodal vale of nknown dplaemen he ompably beween eond-order and fr-order dervave of dplaemen a he olloaon node mpoed. Hene for he heeroeneo rre h M ompably an be wren n dfferenal operaor and D eqal o SG G D D (39) D (40) SG G

5 SG + where û SG and denoe he veor of nknown nodal eond-order dervave of dplaemen. ow by employn he eqaon (39) and (40) and he ompably beween he fr-order dervave and he dplaemen defned by (7) and (8) alon wh he dplaemen approxmaon (17) we oban he follown drezed expreon for raden naral bondary ondon + SG+ + on 1 1 R H G (41) SG M M M n 1 1 R H G o. (4) n he above eqaon he mare H and H M onne he eond- and fr-order dervave of dplaemen va he fr-dervave of hape fnon H H ( x) 0 ( x) x1 x ( x) 0 ( x) x x 0 ( ) ( x ) 0 x T 1 F x x T F (43) ( xm) 0 ( xm) x1 x ( xm) 0 ( xm) x1 x 0 ( xm ) x ( xm ) 0 x (44) whle he mare G and G are analoo o he one defned by (9) and (30). Thee eqaon are now nered no he lobal oeffen marx n he row orrepondn o he rren node pooned on and repevely. For he node on he bondary he nerfae ondon (15) and (16) are drezed by n approxmaon (17) and (18) whle alo lzn he drezed ompably ondon (9) and (30) n he reproy of naral bondary ondon. Hene he fnal form of he drezed nerfae ondon of h proedre ae on (45) G+ + G M M 1 1 G G on (46) G+ G where and M denoe he mare ompoed of he n normal veor aoaed o he fr-order dervave of dplaemen. 3 meral Example 3.1 Plae nder nform dplaemen A heeroeneo plae lzed n order o e he ably of he propoed mehod o remove donne from he ran feld. The maeral propere of he lef par of he plae are aken a E 1000 and 0.5 whle he maeral daa of he rh de are E and 0.3. The eomery of eah homoeneo bdoman defned by he lenh L 3 and he heh H 3. The lef de of he plae fxed whle he n dplaemen mpoed on he rh de. The eomery and he bondary ondon are defned and deped n F. and F. 3. Fre : Plae wh laal bondary ondon Fre 3: Plae wh raden bondary ondon For he verfaon of he preened mxed olloaon approah he drbon of he ran omponen and xy alon he lne y 0.9 are porrayed n F. 4 and F. 5 for wo dfferen vale of he mrorral parameer l. The plae drezed by he nform nodal drbon n boh x and y dreon n 4 node where h defne he horzonal and veral dane beween node. The eond-order MLS fnon are appled for he olon of he problem wh he ze of he approxmaon doman eqal o r.4h. A evden from he drbon of he ran x

6 omponen he e of he mrorral parameer larer han zero ae he hane n he ran feld a and arond he nerfae of he homoeneo doman. Fre 4: Drbon of ran x for y 0.9 Fre 5: Drbon of ran xy for y 0.9 For l 0 no donny n he ran feld oberved a he nerfae bondary. Aordnly an be onlded ha he mehod able for moohn he ran feld. 4 Conlon The mxed olloaon mehod baed on he Mehle Loal Perov-Galerkn (MLPG) onep ha been propoed and appled for he modeln of deformaon repone of heeroeneo maeral baed on raden elay. The problem olved n a aered manner n he Afan ran raden heory wh only one nknown mrorral parameer whereby frly he bondary vale problem of laal elay olved whoe olon hen ed a he np for he orrepondn raden bondary vale problem. Boh problem are derbed by he eond-order eqaon nead of he ornal forh-order dfferenal eqaon. By employn he mxed MLPG onep he neeary dervave order of approxmaon fnon frher reded n he eqaon. Gven ha a olloaon mehod ed here no need for nmeral neraon. Th he applaon of he aered olon heme and he mxed mehle approah rel n an arae and able nmeral formlaon where only he fr-order dervave of hape fnon need o be allaed. The raden heory ed here n order o more araely apre he maeral behavor near he nerfae beween reon wh dfferen maeral propere and o remove jmp n he ran feld ha an be oberved when a laal heory of lnear elay ed. Th enable more phyal derpon of he ranon of he ran drbon beween varo homoeneo maeral reon nde heeroeneo rre. n frher reearh he derbed mehle ompaonal raey wll be exended o he modeln of damae naon n he zone where he ran loalzaon preen and ondered for he e n mehle mlale ompaon alorhm. Aknowledemen: Th work ha been flly ppored by Croaan Sene Fondaon nder he proje 516. Referene: Amanado E.; Arava. (00): Mxed Fne Elemen Formlaon of Sran-raden Elay Problem. Comper Mehod n Appled Mehan and Enneern vol. 191 pp Ake H.; Afan E. C. (011): Graden Elay n Sa and Dynam: An overvew of formlaon lenh ale denfaon proedre fne elemen mplemenaon and new rel. nernaonal ornal of Sold and Srre vol. 48 pp Ake H.; Moraa.; Afan E. C. (008): Fne Elemen Analy wh Saered Graden Elay. Comper & Srre vol. 86 pp Alr S.. (004): The Mehle Mehod (MLPG) for Doman & BE Drezaon. Teh. Sene Pre Foryh USA. Alr S..; L H. T.; Han Z. D. (006): Mehle Loal Perov-Galerkn (MLPG) Mxed Colloaon Mehod for Elay Problem. CMES: Comper Modeln n Enneern & Sene vol. 14 no. 3 pp Chen.-S.; Wan L.; H H.-Y.; Ch S.-W. (009): Sbdoman radal ba olloaon mehod for heeroeneo meda. nernaonal ornal for meral Mehod n Enneern vol. 80 pp alšć B.; Sorć.; arak T. (017): Mxed Mehle Loal Perov-Galerkn Colloaon Mehod for Modeln of Maeral Donny. Compaonal Mehan vol. 59 pp Mo T.; Bher C. (008): ew Conep for Movn Lea Sqare: An nerpolan onnlar Wehn Fnon and Wehed odal Lea Sqare. Enneern Analy wh Bondary Elemen vol. 3 no. 6 pp Polzzoo C. (003): Graden Elay and onandard Bondary Condon. nernaonal ornal of Sold and Srre vol. 40 no. 6 pp