Compressive modulus of adhesive bonded rubber block

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1 Songklnkin J. Sci. Tecnol. 0 (, -5, M. - Ap. 008 p:// Oiginl Aicle Compessive modls of desive bonded bbe block Coeny Decwykl nd Wiiy Tongng * Depmen of Mecnicl Engineeing, Fcly of Engineeing, Pince of Songkl Univesiy, H Yi, Songkl, 90 Tilnd. Received Novembe 007; Acceped 0 Apil 008 Absc Te pesen sdy emined e effec of in desive lye on e modls of n elsic bbe block bonded beween wo ples. Te ples wee ssmed o be igid, bo in eension nd flee, nd sbjeced o veicl compession loding. Te en s ppoc ws sed o obin e nlyic defomions of e bbe nd desive. Te nlyic defomions wee en vlided wi e finie elemen model. Tee ws good geemen beween bo meods. Te modls of e bonded bbe block, defined s effecive modls, ws en sdied. Te effecive modls ws incesed by e fco (+ (/ (6 / + -, wic is composed of e spe fco of e bbe block (/, io of e bonded nd nbonded es, nd e se siffness fco ( /, io of modls nd ickness of bbe nd desive. Te effecive modls does no depend on eie fcos, wen e se siffness of e join is ig o / >0. Keywods: bonded bbe block, ples, mecnicl popeies, nlyicl modeling, finie elemen nlysis. Inodcion An elsic bbe block is widely sed s n engineeing componen fo lod being (en 99. enelly, e bbe block is bonded o seel ples nd is ssmed o ve pefec bonding nd o be igid in eension nd flee o povide veicl siffness. Wen e elsic bbe block is compessed de o lod cying, e bbe epnds lelly. Wen lel epnsion is esiced, e veicl siffness of e bonded bbe block is incesed nd e bbe is ssmed o be incompessible (Ko nd Lim, 00. Te sfce inecion vies depending on e se siffness of e desive lye, wic les e siffness of e elsic bbe block. An ndesnding of e effec of se modls nd e ickness of e desive lye bonding beween e wo sfces will fcilie e design of moe sible desive. Te en s ppoc (en nd Meinecke, 970 is sed o deemine e compessive siffness of bbe block bonded o igid ples (Tsi nd Lee, 998. Tis ppoc ssmes e bbe is in e oionl plne, emins pln nd e veicl lines become pbolic, b e noml sess componens in ll diecions e eql o e pesse, nd c bbe wi Poisson s io ecly eql o 0.5 is ssmed o be incompessible. Tis meod is genelly cceped o deemine e veicl siffness of bonded bbe block. Te siffness deived sing is meod s been veified nd ee is good geemen wen comped wi finie elemen nlysis. In is ppe, we popose o se e en s ppoc o sdy e effec of in desive lye on e defomions, pesse, sess, nd veicl siffness of n elsic bonded bbe block. Te en s ppoc ws sed o deemine e nlyic solions of e displcemens. Te defomions of e bbe nd desive wee lso vlided by finie elemen nlysis. To pesen e effec of e in desive lye in e finie elemen nlysis, e sping elemens wee modeled by e in desive lye nlysis (TALA meod (Decwykl e l., 00. *Coesponding o. Emil ddess: wiiy@me.ps.c.

2 Decwykl & Tongng / Songklnkin J. Sci. Tecnol. 0 (, -5, 008. Anlyic Meods. Rbbe block bonded wi in desive lye We conside n infinie sip of e bbe block in ecngl Cesin coodine - s sown in Fige. Te bbe lye s wid of nd ickness of. Te boom nd op of e bbe lye e bonded o igid nd siff ples by in desive s wid of nd ickness of. Te meils e ssmed o be isoopic nd line elsic. In is sdy, i is ssmed e desive lye is in comped o e ickness of bbe. Te io of / is bo 0 o 0.. in desive lye ppe ple bbe block lowe ple / / F (0,0 Fige. Tin bbe block bonded wi in desive lyes Te lowe ple is consined in ll diecions. Te ppe ple is consined in e -diecion nd sbjeced o nifom veicl compessive lod F; s en i is fee o nsle only in e - diecion wi e smll displcemen of. In is cse, e ssmpions fom en s ppoc e sed, nd i is lso ssmed ee is no noml sin in e veicl diecion fo e desive lye, becse e desive is vey in nd siff ogo e ickness. Ts, e displcemen fncions of e bbe in e -diecion cn be denoed s w ( wen / < < /. Also, e displcemen fncions of e desive in e -diecion cn be denoed s w (, wen / < < /+ nd -(/+ < < -/. Te displcemen condiions in e veicl diecion e; / / Consideing only > 0, becse of e symmeicl geomey of e displcemen in e -diecion, nd o sisfy ose condiions, e displcemen fncions e: 0 ( = ( + ( = 0 ( ( 4( + 0 ( (4 Becse of e incompessibiliy of e bbe lye, e sm of e noml sins in e nd diecions is eo: w ( + = 0 (5 Sbsie (4 ino (5, en inege nd pply e condiions ( nd (; nd, 0 ( 4 0 ( w ( = ( (6 0 ( 0 ( = ( + (7 Becse w ( is independen of, e ems of 0 (/ nd 0 (/ e consn in (6 nd (7. To deemine ese ems, i is necessy o conside e eqilibim of foces in e -diecion fo e desive nd bbe lyes. Te se of sess dominn in e bbe lye is pesse (p nd se sess (. Now i is ssmed ee is only se sess ( in e in desive lye. Consideing e eqilibim of long sip ny og -/ o /, i is known ; 0 = ( (8 p = = ( w ( + (9 w (/ = w (/ = w (/+ = - ( w (-/ = w (-/ = w (-(/+ = 0 ( Te displcemen fncions in e lel diecion (-diecion e (, fo e bbe nd (, fo e desive. A biy, we cn illse e defomions in e lel diecion s sown in Fige. I is ssmed ee is pefec bonding beween e in desive lye nd e ples, nd e ickness of e desive lyes is in; s, (, is line / < < (/+ nd (/+ < < -/. Te (, is pbolic -/ < < /. Fige. Te defomion of e bbe nd desive lyes in e -diecion

3 Decwykl & Tongng / Songklnkin J. Sci. Tecnol. 0 (, -5, 008 / / w ( ( + d= d (0 By sbsiing (4, (6, nd (8 ino (0, wee nd e e se modls of e desive nd bbe, especively, we fond ; ( 0 = 4 0 ( ( ( ( Sbsie ( ino (7; = ( ( ( = 4( ( ( ( ( Sbsie ( nd ( ino (6; w ( =[( ( ( ( (+ ( ( ( ] (4 To deemine (, nd (,, we ve o deemine 0 ( nd 0 ( by ineging ( nd ( nd pplying e condiions of 0 (0 = 0 (0 = 0. Tese give; =[4( =[( + 4 ( + 4 ( ( + ( ( 4( + 4 ] ] (5 (6 Consideing Eqions (4 o (6, in e cse of pefec bonding, we cn ssme e se modls of e desive ( is ig comped o e se modls of e bbe (. Ten, ose eqions e simplified o w ( =[( ( ] (7 = 0 (8 = [ ( ( 4( ] (9 Consideing Eqions (4 o (6, in e cse of lbicion, we cn ssme e se modls of e desive ( is eo. Tose give; w ( =[ ( ] (0 ± ( / + = 0 ; ± / ( ( = ( = ( ( Fo e cse of lbicion beween e bbe nd ples, we fond (, is independen of o cn be defined s (. Te desive displcemen (, becomes eo = ±(/+ nd eql o ( = ±/.. Vlidion of nlyic solions Te nlyic solions sown in Eqions (4-(6 wee vlided by e Finie Elemen Meod (FEM. Te commecil finie elemen code ABAQUS (ABAQUS 998 ws sed fo e vlidion. Te FEM model ws composed of wo ps s sown in Fige. Te bbe lye wi e spe fco of / = 5 is ceed in lf wid, becse of e symmey. I is mesed ino -D plne sin (CPE8H; 8-node bi-qdic, ybid wi line pesse. In e finie elemen model, e bbe lye is ssmed line-elsic meil nd e Poisson s io is vey close o 0.5. Te in desive lyes bonding beween e ples nd bbe e ceed wi ickness io of / = 0.. To epesen e esicion of e in desive lye, e TALA meod is sed. Tis meod ses sping elemens o simle e modls nd ickness of e in desive lye in ems of siffness. In is sdy, e se modls io of e bbe nd desive / = 0.5 ws sed. A e ppe ple, e bbe block is compessed wi nifom veicl displcemen of -/ = Deivion of e modls of e bonded bbe block Te compession foce of e bonded bbe block is given by e sm of F nd F (Bnks e l., 00. F is e omogeneos compession foce, wic is obined wen e bbe block is compessed beween flly lbiced sfces. F is e foce eqied o keep poins e oiginl posiion in e plnes of e bonding sfce de o se defomion. To deive F, i is ssmed e defomion of e bbe in e long sip is eo o plne sin (Timosenko nd oodie, 987. Eqion ( nd Hook s lw e pplied nd sed wen e Poisson s io (n of e bbe is 0.5 nd e modls of e bbe is E = fo n isoopic nd incompessible meil. ν ε = ( + = ν σ ( E 4 F = σ A= E (4 To deive F, e pesse is obined fom n eqivlen condiion s sown in Eqion (9 nd pplied wi bondy condiion of pesse e edges of e bbe eo o p(±/=0. I is fond : p( = ( [( ( + 6 ( ] (5 Te p( is posiive wen e bbe is nde compession. Using e sme pinciple, we cn deemine e se sess in e bbe lye;

4 4 Decwykl & Tongng / Songklnkin J. Sci. Tecnol. 0 (, -5, 008 Ts, = ( ( ( (6 ( + 6 F / = p( d= 6 ( + / (7 Te compessive foce is given by e sm of Eqions (4 nd (7 F = F + F E 6 = 4 [+ ( ( + ] (8 Eqion (8 cn be wien ino e effecive modls (E c of e bonded bbe block: E E c 4 6 = [+ ( ( + ]. Resls nd Discssion. Vlidion of e nlyic solions (9 Te displcemen fncions s sown in Eqions 4 o 6 wee vlided sing finie elemen nlysis. Fige pesens e veicl displcemen og e ickness of e bbe lye =0. Te displcemen is nomlied by, sown in e y-is, nd e veicl posiion is nomlied by, sown in e -is. Te mimm veicl displcemen is e ppe ple ( / = 0.5, wee e bbe block is compessed, nd e displcemen becomes eo e lowe ple, wee e block is consined. A e mid-plne ( / = 0, e veicl displcemen is lf e mimm of e veicl displcemen. Te plo in Fige sows ee is good geemen beween FEM nd e nlyic solions fo veicl displcemen. w ( veicl displcemen og e ickness of bbe lye FEM--=0 Anlyicl--= Fige. FEM nd nlyicl vlidion of veicl displcemen w (/ og e ickness of e bbe lye = (,,0 0 lel displcemen e midplne of bbe lye FEM--=0 Anlyicl--= / lel displcemen long e in desive lye FEM--=/ Anlyicl--=/ Fige 4. ( FEM nd nlyicl vlidion of e lel displcemen (,0/ e mid-plne of e bbe lye = 0 nd (b FEM nd nlyicl vlidion of e lel displcemen (,// long e in desive lye = / Fige 4 nd b pesen e lel displcemens in e bbe lye e mid-plne nd in e desive lye bonding beween e ppe ple nd bbe, especively. Te lel displcemen is nomlied by sown in e y-is, nd e veicl posiion is nomlied by sown in e -is. Te lel displcemen inceses linely fom / = 0 o e nbonded sfce of e bbe block (/ = 5. Te mimm lel displcemen occs e mid-plne of e bbe block. Te plos eveled ee is geemen fo e lel displcemen beween e FEM nd nlyic solions 0 < / < 4.5. Te discepncies ppenly occ ne e fee o nbonded sfce (4.5 < / < 5, becse is poin is close o e fee sfces of e bbe nd bonding e, wic is e singliy poin.. Effec of in desive lye on e bonded bbe block Te plos of Eqion (9 in Fige 5 sow e effec of e desive on e effecive modls of e bonded

5 Decwykl & Tongng / Songklnkin J. Sci. Tecnol. 0 (, -5, Fige 5. Effec of e desive lye io ( on e effecive modls (E c /E fo diffeen S (spe fco bbe block. S is e spe fco fo e long sip, wic is / (bonded e/fee e, nd e se siffness fco is = / (io of modls nd ickness of bbe nd desive. In is sdy, / is vied fom 0, wic is pefec bonding condiion, o 000. Te spe fco (S is vied fom o Discssion nd Conclsions Te se modls nd ickness of e desive lye e ken ino ccon o nlye e defomions of e bonded bbe block. Te en s ppoc is sed o deemine e nlyic displcemen solions. Eqions (4 o (6 e e defomions of e bbe nd desive, wic wee vlided by e finie elemen nlysis. Tee is n geemen fo e displcemen beween e finie elemen nlysis nd nlyic solions. Tee wee smll discepncies in e lel defomions of e bbe nd desive e cone of e ming sfces beween e bbe nd bonding e, de o singliy e cones. Te nlyic solions indice e se siffness fco /, io of e modls, nd ickness of e bbe nd desive on e sme e ffec e veicl nd lel defomions, pesse, nd se in e bbe nde compessive lod cying. Te siffness fco llows fo e bonding sfce o esis lel nd veicl defomions nd o le pesse nd se sess in e bbe. In e cse of lbiced sfce, i is ssmed ee is no desive, =0 o /. Wen e bbe is com- pessed, e bbe long e inefce e is no consined nd flly moves in e lel diecion. To minin consn volme de o e incompessibiliy of e bbe, e pesse is en consn, nd ee is no se sess in e bbe. In e cse of pefec bonding, i is ssmed ee is vey siff desive, o / =0. Wen e bbe is compessed, e bbe ne e inefce e is flly consined in e lel diecion, wees e bbe in oe es is no, wic en cees se sess nd pesse gdien in e bbe. Te modls of e bonded bbe block defined s effecive modls (E c is incesed by e fco of (+(/ (6 / + -, wic is composed of e spe fco (/ nd e se siffness fco ( /. Wen ee is pefec bonding o / =0 e inefce, e effecive modls is iges, becse e fll consin e inefce eds e bbe epnsion. Te effecive modls of e bbe gdlly deceses nd conves o vle of 4/E, s e inefce is moe complin in e lbiced condiion o /. Te spe fco lso inceses e effecive modls. Te effecive modls depends only on e spe fco o e desive lye wen e se siffness of e join is ig o / >0. Refeences ABAQUS/Sndd Vesion 5.8 Use s Mnl 998. Hibbi, Klsson & Soensen, Inc., Rode Islnd, U.S.A. Bnks, H.T., Pine,.A. nd Yeo, O.H. 00. Anlysis of bonded elsic blocks. Memicl nd Compe Modelling. 6, Decwykl, C., Rbin, C.A. nd Hn,.T. 00. Anlysis of e effecs of in seln lyes in icf scl joins. AIAA J. 4, 6-8. en, A.N. 99. Engineeing wi bbe: ow o design bbe componens, Ofod Univesiy Pess, New Yok, U.S.A. en, A.N., Meinecke, E.A Compession, bending nd se of bonded bbe blocks. Polym. Eng. Sci. 0, Ko, C.. nd Lim, H.L. 00. Anlyicl solion fo compession siffness of bonded ecngl lyes. Inenionl Jonl of Solids nd Sces. 8, Timosenko, S. nd oodie, J.N Teoy of elsiciy d Ediion, Mcw-Hill, New Yok, U.S.A. Tsi, H.C. nd Lee, C.C Compessive siffness of elsic lyes bonded beween igid ples. Inenionl Jonl of Solids nd Sces. 5,

Some algorthim for solving system of linear volterra integral equation of second kind by using MATLAB 7 ALAN JALAL ABD ALKADER

Some algorthim for solving system of linear volterra integral equation of second kind by using MATLAB 7 ALAN JALAL ABD ALKADER . Soe lgoi o solving syse o line vole inegl eqion o second ind by sing MATLAB 7 ALAN JALAL ABD ALKADER College o Edcion / Al- Msnsiiy Univesiy Depen o Meics تقديم البحث :-//7 قبول النشر:- //. Absc ( /