MECHANICAL ANALYSIS OF 2D-BRAZED JOINT USING A NEW HYBRID MAX- FEM MODEL

Size: px

Start display at page:

Download "MECHANICAL ANALYSIS OF 2D-BRAZED JOINT USING A NEW HYBRID MAX- FEM MODEL"

Marilynn Grant
5 years ago
Views:

1 MECHANICAL ANALYSIS OF D-BRAZED JOINT USING A NEW HYBRID MAX- MODEL A. IFIS a, F. Blteryst b, M. Noar c a Laboratore d Energétqe et de Mécanqe Théorqe et Applqée, LEMTA CNRS-UMR 756, GIP-InSIC, 7 re d Hellele, 88, Sant Dé des Vosges, France b Laboratore d Energétqe et de Mécanqe Théorqe et Applqée, LEMTA CNRS-UMR 756, GIP-InSIC, 7 re d Hellele, 88, Sant Dé des Vosges, France c Laboratore d Energétqe et de Mécanqe Théorqe et Applqée, LEMTA CNRS-UMR 756, GIP-InSIC, 7 re d Hellele, 88, Sant Dé des Vosges, France a abderrazzaq.fs@gmal.com b francos.blteryst@nsc.fr c mohammed.noar@nsc.fr ABSTRACT Ths wor deals wth the performance of a new approach combnng the nmercal extended Fnte Elements Method X- and the analytcal method of Matched Asymptotc Epansons MAE. The proposed new MAX- model s well adapted for stdyng and modelng the mechancal behavor of medms contanng snglartes sch as thn layers or adhesve jonts wthot any reqred mesh refnement n ther vcnty. The methodology conssts of the constrcton of enrchment parameters wth the MAE technqe and ther ntegraton nto the X formlaton. Correcton matr of stffness s then defned and ntegrated n the comptaton algorthm. To descrbe the mechancal behavor of a proposed strctre wth D brazed jonts, the MAX- hybrd model has been mplemented as an UEL sbrotne nder Abaqs mplct. Compared wth the classcal FE method, the obtaned reslts n terms of stress feld, strans and dsplacements show a good accracy wthot any reqred mesh refnement. Keywords: hybrd technqe, thn layers, Matched Asymptotc Epansons, X-, correcton matr, UEL MAX- sbrotne.. INTRODUCTION Drng the last decade, several methods have been ntrodced for modellng snglar problems sch as thn layers, adhesve jonts, coatng, etc.. Analytcal approaches based on asymptotc assmptons sch as Matched Asymptotc Epansons MAE gve at two dfferent-scales an appromaton of the man solton n strctres contanng snglartes M. Van Dye, 975; P. Schmdt, 8. As shown by the wor of Legllon and Abdelmola, the MAE method has been sed to analyse brazed jonts n order to descrbe the crac propagaton process at the nterface between the jont and the bonded sbstrates. However the dffclty of the nmercal mplementaton of ths method maes ts se very complcated and lmted to some smple cases. In the other sde, the nmercal methods headed by the Fnte Element Method strggle wth snglar problems, where a mesh refnement s reqred n order to tae nto accont snglartes. Besdes, n order to overcome ths lmtaton, partclar nmercal methods have been ntrodced to deal wth ths dffclty. These methods allow to gve a mlt-scale analyss n dscontnos medms as assembly strctres, welded medms, etc.. Two possbltes are gven by these approaches: the frst procedre conssts n mang a local analyss and then project the nformaton abot the snglarty behavor at the large scale. The homogenzaton and the Arleqn methods, descrbed by T.I. Zohd, J.T. Oden and G.J. Rodn, 996 and P.T. Baman and H. Ben Dha, 8 respectvely, are based on ths prncple and are sed n several wors, especally to analyse compostes materals. The second procedre conssts n defnng a correcton of the classcal FE method. The nformaton abot the snglarty s then stored n an added part called enrchment. Thereby, the local analyss s not reqred. X- N. Moes, J. Dolbow and T. Belytscho, 999; H. Bayesteh and S. Mohammad, and G- I. Babsa, U. Banerjee and J. Osborn, 4 are two partton of nty PUM I. Babsa and J.M Melen, 999 methods that are sed n several wors n order to solve varos types of problems especally crac problems.. STATE OF THE ART: METHODS USED FOR SINGULAR PROBLEMS.. X- approach: Problems wth snglartes have been frstly treated sng FE method by pdatng the mesh topology n order to match the geometry of the snglarty. However, the ntrodcton of X- crcmvents ths problem by enrchng a standard appromaton wth specal addtonal fnctons. The form of the enrched appromaton follows the partton of nty. The geometry of the snglarty s then nvolved by pdatng the enrchment scheme; no remeshng of the doman s reqred. The only nteracton between the mesh and the geometry of the snglarty nvolves the constrcton of the enrched bass fnctons. The classcal X- formlaton s gven by: Proceedngs of the Int. Conf. on Integrated Modelng and Analyss n Appled Control and Atomaton, ISBN ; Brzzone, Daphn-Tangy, Jnco and Merryev Eds. 96

N Nenr N N b Where Nenr s the enrchment terms nmber, the appromated fncton, N the standard fnctons for node, the nnown of the standard part at node, the enrchment fncton and b the enrchment parameter.

2 N Nenr N N b Where Nenr s the enrchment terms nmber, the appromated fncton, N the standard fnctons for node, the nnown of the standard part at node, the enrchment fncton and b the enrchment parameter... MAE approach: MAE approach has been frstly sed for fld mechancs M. Van Dye, 975. It shows ts effcency n treatng pertrbed problems and bondary layers. Ths method based on asymptotc assmptons has been appled for adhesve jont problems as llstrated n Fgre. Fgre. Two-scales partton of Matched Asymptotc Epansons approach. It conssts n solvng problems at two dfferent scales Fgre by ntrodcng two dfferent asymptotc epansons, the oter Eqaton and the nner Eqaton epansons. v,,,..., v, v,... Wth s the stretched varable / and. denotes the dsplacement vales at the both sdes of the nterface. Each epanson s consttted of two man parts: the leadng terms and v whch are the classcal soltons obtaned by fnte element method whle the second part s consdered as a correcton gven by a pertrbed terms and v dependng on ; the characterstc thcness of the brazed jont. In the wors of D. Legllon and R. Abdelmolab, ; D.H. Ngyen et al., 8, the MAE has been sed to compte the jmps epressons of dsplacement feld across the nterface of dscontnty defned by the oter doman. Ths algorthm s based on smltaneos resolton of classcal eqatons of the model eqlbrm eqatons, constttve laws, contnty condtons for each order, and a matchng process Eqaton 4 for both epansons at ther respectve lmts.e. for the oter epanson, and for the nner epanson. v,, lm v,,, 4 lm Usng ths algorthm, the jmp of dsplacement D. Legllona and R. Abdelmolab, ; D.H. Ngyen et al., 8 s compted and epressed by:,, µ µ,, 5,, µ µ µ,,,, µ µ,,,,,, µ µ µ, µ, µ,, As t wll be presented n the net secton, ths vale has been sed n the setp of the MAX- model... Proposed hybrd MAX- Model In ths wor, the hybrd MAX- model has been ntrodced for the thn layers modellng to release mechancal analyss of a D-brazed jont. The MAX model s a Partton of Unty Method PUM where the man solton s formlated sng two man parts: a classcal dscretzaton and the enrchment terms. The proposed PUM formlaton eplots the Matched Asymptotc Epansons MAE n the defnton of the enrchment parameters. Ths procedre lns the two man parts of the PUM formlaton to gve a corrected form of the standard where correcton matr s ntrodced to compte the stffness matr.. MAX- SETUP.. MAX- Prncple By analysng the shapes of the X- and MAE approaches, t s notable that these two methods share the common sbdvson of the nnown solton at two dfferent parts: the classcal solton and the correcton terms Fgre. Thereby the MAX- model feats the smlarty between the two methods to defne a new procedre descrbng the brazed jont behavor. In fact, the jont s assmlated to a dscontnty nterface whch creates a jmp of dsplacement feld. The new confgraton contanng the ntrodced snglarty s Proceedngs of the Int. Conf. on Integrated Modelng and Analyss n Appled Control and Atomaton, ISBN ; Brzzone, Daphn-Tangy, Jnco and Merryev Eds. 97

3 descrbed sng a specfc MAX- formlaton whle the enrchment parameters are defned sng the jmps compted by the MAE approach Eqaton 5. N Nenr Instead of sng nodal vales of the global feld as sed n formlaton Eqaton and n the wor of Ngyen et al. 8, ths appromaton Eqaton 6 ntegrates the contnos solton. The smlarty between the oter epanson Eqaton and formlaton Eqaton 6 s employed N N b Non-pertrbed term correcton term Fgre. Identfcaton of leadng and correcton terms sng MAX- model... Problem poston The man prpose here s the setp of the MAX model for a D-brazed jont nder mechancal loadng see Fgre. The adhesve jont and sbstrates are consdered as elastc domans. to dentfy the enrchment parameters b a. The b algorthm of dentfcaton s llstrated for lnear strctral element contanng the jont poston Fgre 4: U=,4 mm Steel: E= Gpa, v=, Fgre 5. Lnear strctral enrched element contanng the jont poston L= mm Thness e =,mm Mechancal propretes BN-: E= 74 Gpa, v=, Adhesve thn layers Y, U B= mm X, U Fgre. D brazed jont nder mechancal loadng.. MAX- formlaton The MAE oter epanson Eqaton crshes the thn layer. The latter s trned to an nterface of dscontnty whch completely crosses the wdth of the assembly see Fgre. As a reslt, the problem s assmlated to the strong dscontnty problem wth presence of dsplacement jmp across the nterface. Heren, the MAX- wll consder the formlaton Eqaton and adapts t to the consdered applcaton. In ths contet, the formlaton proposed n eqaton 6 ses the same strategy as the oter epanson of MAE whle eepng the basc shape of X: N N Nenr N H b 6 Where b and are the enrchment parameters whch mst be defned and H s the Heavsde fncton by: defned n the pont H Eqatons 6 and are sed to compte the nodal vales of the global felds Eqaton 7 and Eqaton 8 as follow: Frstly, by sng the formlaton Eqaton 6: f f a a a 4 4 a4 b b b 4 4 b4 7 And secondly, by sng MAE oter epanson Eqaton : Then, the dsplacement dfference between two nodes located on the same rdge and crossng the jont poston s compted wth the two appromatons Eqaton 7 and Eqaton 8 as follow. Frstly, by sng the formlaton Eqaton 7: 4 4 a4 a a a 4 4 b4 b b b Proceedngs of the Int. Conf. on Integrated Modelng and Analyss n Appled Control and Atomaton, ISBN ; Brzzone, Daphn-Tangy, Jnco and Merryev Eds. 9 98

4 Secondly, by sng MAE oter epanson Eqaton 8: The eqalty between Eqatons 9 and gves the condtons that the enrchment parameters have to satsfy: a a 4 4 a a b b 4 4 b b To compte these parameters vales, t s not necessary to have an eplct epresson of the pertrbed term n each node. In fact, by sng the MAE algorthm ntrodced n wors D. Legllona and R. Abdelmolab, ; D.H. Ngyen et al., 8 the term for nodes and 4 respectvely 4 for nodes and are appromated to and 4 h,, h, respectvely. h s the heght of the enrched element whle respectvely 4,, denotes the dsplacement jmp crossng the nterface n the common abscssa between nodes and 4 respectvely and. Tang nto accont the prevos appromaton, the epresson proposed for the parameters b s gven bellow: 4 h, for, 4 b h, for,.4. Stffness matr The epresson Eqaton 5 shows a lnear dependence between the jmp and the leadng dervatve terms. From ths dependence rses a transfer matr between the jmp and the npertrbed stran vector:,, A,, BS,,,, C A µ µ, µ s the leadng stran vector, B and µ µ S µ. A C BS, By ntegratng ths reslt n the enrchment parameters epresson, the latters are lned to the leadng term nodal vales as shown n Eqaton 4. 4 hc B, for, 4 b hc B, for, 4 [C] s the enrchment matr defned n Eqaton and [B] s the stran matr lnng the stran vector to the nodal vales of the leadng term. The epresson Eqaton 4 s njected n the X- formlaton Eqaton 6 creatng then a transfer matr between the global nodal dsplacement vales : and the leadng term one Ι H C B 8,8 5, I 8,8 s the dentty matr and H s called Heavsde matr and epressed below: H t 4 H H H H 4 H H H H By consderng that the global problem and the leadng oter one are assmed nder the same eteror loadng F F, the global stffness matr K s lned to the one of the leadng problem K as follow: t K B D B I H C B 8,8 Ω, dω 6 Conseqently, the global stffness matr can be defned as a correcton of the standard stffness matr sng a correcton matr CM defned by: Proceedngs of the Int. Conf. on Integrated Modelng and Analyss n Appled Control and Atomaton, ISBN ; Brzzone, Daphn-Tangy, Jnco and Merryev Eds. 99

5 CM I H C B, 8,8 7 Fnally, the lnear system to consder s: K F 8 combnaton of X- and MAE, and wll gve as a fnal reslt the corrected form of standard. Ths correcton s presented by the matr CM Eqaton 7 whch stores the reqred nformaton for the jont behavor. As a reslt, the strctre of the program Fgre 4 wll be consttted by the standard whle the correcton matr wll be compted n an nternal sbrotne dependng on the consdered applcaton. The enrchment parameters b and the leadng term of the fnte elements formlaton Eqaton 6 do not appear eplctly n the fnal system Eqaton 8. Ths s de to the MAE reslts whch reveal a lnear dependence lng the enrchment parameters to the leadng term dervatves. Also, the contnty of the fnte elements formlaton Eqaton 6 allows to defne a correcton matr CM Eqaton 7 and to Standard program Gass ponts Shape fnctons Shape Matr] Stffness, condctvty and capacty matrces Standard stfness, condctvty, capacty matr Standard RHS Correcton matrces Stfness, correcton correcton Stffness Correcton Condctvty correcton Capacty correcton Effort correcton ln the global stffness matr K to the standard one. The comptaton of the leadng term and the enrchment parameters s then not needed, and the global dsplacement feld s drectly compted by nmercal resolton of the system Eqaton 8. It can be noted that only the correcton matr CM s Corrected jacoben AMATRX compted and then njected n a standard program. The obtaned solton covers the oter domans. Ths means that only the nflence of the thn layer on the global strctre s characterzed. In ths wor, the local analyss of the thn layer s not consdered. However, f necessary, the model can tae nto accont the comple behavor n the vcnty of the jont, ncldng crac, delamnaton or other nonlnear aspects for eample. Ths can be possble by resolvng the nner problem Eqaton of the MAE. Followng ths approach, the local analyss s restrcted n the enrched elements whch are epanded to zoom on the jont and the contact nterfaces. The resolton of the nner problem can be performed by sng the estng methods, X-, etc. 4. NUMERICAL IMPLEMENTATION OF MAX MODEL Ths secton provdes a valdaton stdy for the proposed MAX- hybrd model. The obtaned reslts are compared to those gven by the standard by meshng fnely the thn layer. In ths wor, ABAQUS code s sed to mplement the model sng an UEL sbrotne. The same code s also sed to compte the reference solton for D-brazed jont n fgre Corrected effeort RHS Fgre 5. Algorthm strctre of the mplemented UEL MAX- sbrotne The reslts gven n the setp of the model are mplemented n the UEL MAX- sbrotne followng the strctre presented n Fgre 6. The correcton matr CM Eqaton 7 and the corrected stffness matr Eqaton 6 are compted for the D-brazed jont presented n Fgre. The comptaton s lmted to D plane stress state. Thereby, the enrchment matr C Eqaton needed for the comptaton of the correcton matr CM Eqaton 6 s wrtten as: C E 9 E By resolvng the system Eqaton 8, the dsplacement feld s obtaned. Then, other mechancal felds are dedcted. The reslts gven by the UEL MAX- sbrotne are compared to ones obtaned by standard nder the same code. The consdered mesh, dsplacement, stress, stran and error are presented for the two methods. Mesh 4.. Nmercal mplementaton The model developed here has been mplemented n ABAQUS as a ser element sng the UEL sbrotne. The procedre of the development of ths sbrotne s based on a prevos wor ntrodced by Gner et al. 9. The model starts from a Proceedngs of the Int. Conf. on Integrated Modelng and Analyss n Appled Control and Atomaton, ISBN ; Brzzone, Daphn-Tangy, Jnco and Merryev Eds.

By analyzng these reslts, t can be noted that the model reprodces the oter dsplacements and stress felds gven by the method.

Overvew of the classcal dscretzaton and the MAX- meshng User elements are not represented by ABAQUS post-processor. However, n order to reach ths am, a post-processng sbrotne shold be reqred.

and MAX- mesh data MAX- Rato /MAX Nodes 74 65,7 nmber Elements 75 55,76 nmber Mnmm, mm mm element sze Fgre 8 shows that the crve gven by the model fts the one obtaned by standard from the frst

6 By analyzng these reslts, t can be noted that the model reprodces the oter dsplacements and stress felds gven by the method. In order to gve an accrate comparson, a plot of the dsplacements n the secton = mm are presented n Fgre : U_ ABAQUS U_ MAX- U_ ABAQUS U_ MAX- U_ ABAQUS U_ MAX- MAX- Fgre 6. Overvew of the classcal dscretzaton and the MAX- meshng User elements are not represented by ABAQUS post-processor. However, n order to reach ths am, a post-processng sbrotne shold be reqred. Ths wll be performed n forthcomng developments. The comparson between the two mesh strateges s smmarzed n Table : Table. and MAX- mesh data MAX- Rato /MAX Nodes 74 65,7 nmber Elements 75 55,76 nmber Mnmm, mm mm element sze Fgre 8 shows that the crve gven by the model fts the one obtaned by standard from the frst enrched node. Conseqently, a sngle MAX- element reprodces the same solton gven by several elements. The lne lnng the two enrched nodes s a lnear nterpolaton and does not present the solton n the vcnty of the jont. The latter can be obtaned by the resolton of the nner problem of MAE. To analyze the accracy of the model, three error epressons are compted tang the solton as reference: MAX Relatve error: errrelatve Mamm error: errma ma MAX Norm L: err MAX These epressons are sed to estmate the error nto the secton = mm by sng the prevos reslts of Fgre 8 % From Table, t can be noted that the model presents an mportant mesh optmzaton. Ths shold be more notable n forthcomng transent analyses where the tme ncrement for sch comptatons depends on the sze of elements. Stress Fgre 9. and MAX- dsplacements n the secton = mm MAX- Fgre 7. and MAX- stress dstrbtons Dsplacement Fgre. Relatve error n the secton = mm Table. error appromaton U U MAX- Fgre 8. and MAX- dsplacements maerrrelatve errma err,6 %,6 %,9,8,9-7,95-7 By analyzng these error appromatons, t becomes trval that the hybrd model presents accrate reslts wthot meshng the thn layer. However, the Proceedngs of the Int. Conf. on Integrated Modelng and Analyss n Appled Control and Atomaton, ISBN ; Brzzone, Daphn-Tangy, Jnco and Merryev Eds.

7 accracy of the model depends on several parameters and the model can have some lmtatons and crtcal ponts. Among these lmtatons, there are bondary problems whch affect the reslts accracy near to boards. These problems are especally de to the MAE method. The rles sed to ln the oter and nner soltons stll avalable n an nfnte nner doman. However, n the boards, the solton has to satsfy the bondary condtons. The enrchment strategy sed n ths wor cannot then reprodce the solton n the vcnty of boards. In addton to board problems, other parameters may affect the model accracy. The three ponts below are the most mportant:. The rato he / lnng the element heght and the jont thcness. Actally, more ths rato s smaller; more the reslts hghlght a good accracy.. The nmber of enrched elements: n ths wor, a sngle element s enrched n the jont thcness drecton. However, t s trval that ncreasng that nmber wll ncrease the model accracy. Besdes, a new enrchment strategy has to be developed. 5. CONCLUSION The hybrd model ntrodced n ths wor combnes the MAE technqe whch s sed to defne the enrchment parameters and the X- formlaton. Ths new approach called MAX- model has been establshed and sed to solve the mechancal snglar problem of two-dmensonal brazed jonts. The MAX- approach leads to a corrected form of standard where a correcton matr s sed to compte the man solton n the whole strctre. Once set p, the model has been mplemented nder ABAQUS code sng the UEL sbrotne. The reslts gven by the model are compared to those obtaned sng standard for the D- brazed jont. From ths applcaton, t s notable that MAX- model provdes accrate reslts n terms of dsplacement and temperatre felds wthot any reqred mesh refnement n the vcnty of the thn layer. However, the model has ts lmtatons near to boards where the proposed enrchment s not avalable. New enrchment strateges have to be developed n ftre wor to deal wth ths lmtaton. Also, even f t s not developed here, the model can tae nto accont other behavors of the jont sch as damage, cracs and delamnaton. These behavors have to be consdered n the comptaton of jmps sng MAE approach. Besdes, f t s necessary, the local behavor of the jont can be descrbed by the resolton of the nner problem. Thereby the analyss can be developed n two scales nstead of the global analyss presented n ths wor. Fnally, t can be sad that the model needs to be enhanced and generalzed to be sed for more complcated applcatons. REFERENCES I. Babsa, J.M Melen, The partton of nty fnte element method, Int. J.Nmer.Meth.Engrg I. Babsa, U. Banerjee, J. Osborn, Generalzed fnte element methods: Man deas, reslts, and perspectve, Int. J. Compt. Methods 4-7 P.T. Baman, H. Ben Dha, N. Elhodja, J.T. Oden, S. Prdhomme, On the applcaton of the Arleqn method to the coplng of partcle and contnm models, Compt. Mech., H. Bayesteh, S. Mohammad, X fractre analyss of shells: The effect of crac tp enrchments, Comptatonal Materals Scence, Gner E, Smar N, Tarancon JE, Fenmayor FJ. An Abaqs mplementaton of the etended fnte element method. Eng Fract Mech., D. Legllona, R. Abdelmolab, Mode III near and far felds for a crac lyng n or along a jont, Internatonal Jornal of Solds and Strctres, N. Moes, J. Dolbow, T. Belytscho, A fnte element method for crac growth wthot remeshng, Int. J. Nmer. Meth. Engrg D. H. Ngyen, F. Blteryst, M. Lazard, P. Lamesle, G. Dor, Coplng of the extended Fnte Element Method and the matched asymptotc development n the modellng of brazed assembly, Internatonal Jornal of Materal Formng,, 8, pp. 9- P. Schmdt, Modelng of adhesvely bonded jonts by an asymptotc method, Internatonal Jornal of Engneerng Scence, M. Van Dye, pertrbaton methods n fld Mechancs 975 T.I. Zohd, J.T. Oden, G.J. Rodn, Herarchcal modelng of heterogeneos bodes. Compter Methods n Appled Mechancs and Engneerng, , Proceedngs of the Int. Conf. on Integrated Modelng and Analyss n Appled Control and Atomaton, ISBN ; Brzzone, Daphn-Tangy, Jnco and Merryev Eds.

Numerical Solution of Deformation Equations. in Homotopy Analysis Method

Numerical Solution of Deformation Equations. in Homotopy Analysis Method Appled Mathematcal Scences, Vol. 6, 2012, no. 8, 357 367 Nmercal Solton of Deformaton Eqatons n Homotopy Analyss Method J. Izadan and M. MohammadzadeAttar Department of Mathematcs, Faclty of Scences, Mashhad