INTERFACE DESIGN OF CORD-RUBBER COMPOSITES

Transcription

1 8 TH INTENATIONAL CONFEENCE ON COMPOSITE MATEIALS INTEFACE DESIGN OF COD-UBBE COMPOSITES Z. Xie*, H. Du, Y. Weng, X. Li Ntionl Ke Lbortor of Science nd Technolog on Advnced Composites in Specil Environment, Center for Composite Mterils, Hrbin Institute of Technolog, Hrbin 5000, P.. Chin * Corresponding uthor(iezhm@hit.edu.cn) Kewords: Interfce design, Cord-rubber, Neutrl inclusion, Non-circulr cross-section Introduction Cord-reinforced rubber composites re widel used in tires, hoses, belts nd vrious ttenution constructions. The interfcil properties pl leding role in the performnce nd durtion of the cord-rubber products. Therefore, optiml design of the interfcil properties is of gret importnce for improving nd enhncing the product qulities. More ttention hs been pid to two ws, i.e., the ddition of bonding gents to the rubber compounds nd the dhesive tretment of the cords, to improve the interfce strength[]. However, there hs been little work concerning the mechnicl design of the interfce. Crmn et l. [] proposed n optiml method for the interfcil modulus b minimizing the mimum principl stress nd the strin energ densit in the composite mterils. A zero-thickness interfce nd perfect bonding re usull ssumed in the phenomenologicl pproches for the composite mterils. In recent ers, considerble work hs been focused on the discussion of the imperfect interfce[3,4]. Bsed upon the imperfect interfce nd the neutrl inclusions tht do not disturb the prescribed uniform stress field in the surrounding elstic bod, u [3] presented the interfce design of single neutrl elstic inclusion in mn tpicl cses. In prctice, the neutrl inclusion does not eist if perfectl bonded interfce between inclusion nd elstic bod is ssumed. Bertoldi et l.[4] concluded tht circulr inclusion coted b continuous structurl interfce ws neutrl for fr broder mteril prmeter rnge thn for the liner interfce. In this work the interfcil prmeters of the cord-rubber composites re studied b mens of the concept of neutrl inclusion. Interfce Design for Cord-ubber in Antiplne Sher. A Generl Neutrlit Condition According to the concept of the neutrl inclusion, the embedded inclusion does not ffect the originl stress field. When the given stresses re pplied to the inclusion, the displcement field cn be clculted in the region of the inclusion. Bsed upon the trnsmission conditions long the interfce, one m get the neutrlit condition. u [3] hs studied generl neutrlit condition in this w for the neutrl elstic inclusion. Consider single cord with sher modulus embedded in the rubber with sher modulus in nti-plne sher, where the subscripts nd refer to the rubber nd cord, respectivel. For the ske of simplicit, the rubber nd cord re ssumed to be the liner isotropic mteril. In terms of the u s nlsis[3], the nti-plne displcement w (, ) stisfies the hrmonic equilibrium eqution in rubber mtri (D ) nd reinforcing phse(d ), i.e., wi = 0, in D i( i=,) () The imperfect interfce conditions long the interfce (Γ) re given b h (, )( w w) = = N N () where N is the direction of the outwrd norml to Γ nd h (, ) the interfcil prmeter proportionl to the densit of the dhesive ler. If there eisted neutrlit condition for the cord-rubber composites, h (, ) must be non-negtive everwhere. Integrtion of Equ.() on the interfce ields w (, ) = w (, ) w 0 (3)

2 in which w 0 is n rbitrr vlue corresponding to rigid bod displcement. For convenience, neglected in the following nlsis for it does not contribute to the deformtion. And then, the neutrlit condition is rewritten s, h w= (, )( ) [ cos N(, ) sin N(, )] w 0 (4) This eqution denotes the reltion between the interfce prmeter h (, ) nd the shpe of cord.. Determintion of Interfcil Prmeters The Nlon66 cord with twisting fctor of 400dete/ is commonl used in tire. A scnning electron microscop (SEM) photogrph in Fig. shows non-circulr cross-sectionl construction for single nlon66 cord embedded in the rubber mtri. Thus the cross-section of the cord is depicted b ( ) =,0 ( ), ( ) 0 = (5) nd plotted in Fig. where is the rdius of one strnd. In the cse of the displcement filed w (, ) = B which stisfies the hrmonic equilibrium eqution, the interfcil prmeter h (, ) is given b h (, ) = (6) Since the stiffness of cord is much higher thn tht of rubber, or nmel 0, the interfcil prmeter is reduced to h (, ) = (7) Clerl, the interfcil prmeter is identicl long the interfce boundr nd independent of the prescribed uniform stress field. Fig.4 shows the dimensionless prmeter H(, ) = h(, ) / s is indicted b the height of green shdow with smbol of plus. In the other cse of the displcement field w (, ) = B 3 which lso stisfies the hrmonic equilibrium eqution, the interfcil prmeter is derived, i.e.,,0 h (, ) =, ( ) 0 In the rnge of h(, ) 0 (8) < <, the interfce prmeter h, is out of the non-negtive <, so ( ) restriction. If the interfcil boundr t < < is replced b two lines prllel to the principle is s shown in Fig.3, the interfcil prmeter becomes, < h (, ) =, (9) Furthermore, dimensionless prmeter H(, ) is introduced to illustrte the interfcil prmeter distribution long the interfce Γ s follows, h (, ) H(, ) =, < =, (0) As illustrted in Fig.5, H(, ) is positive on the entire boundr. In ddition, it is found tht the interfcil prmeter is lso independent of the prescribed uniform stress field nd the cord stiffness. 3 Interfce Design for Cord-ubber in Plne Deformtions 3. Governing Eqution In the nlsis of u[3], the imperfect interfce condition long the interfce Γ ws described in the norml nd tngentil directions b

3 INTEFACE DESIGN OF COD-UBBE COMPOSITES σ σnn = 0, σnt = 0 () = p(, ) u, σ = q (, ) u () nn n nt t where * = (*) (*) denotes the jump cross Γ, p(, ) nd q (, ) re non-negtive norml nd tngentil interfce prmeters, respectivel. The stresses of σ nn nd σ nt s well s the displcements of u nd u re in the norml nd tngentil n t directions, respectivel. Consider the following uniform stress field on the rubber mtri, i.e., σ = A, σ = B, τ = C (3) () () () in which A, B nd C re the constnts. The displcement field cn be esil obtined in the frmework of theor of elsticit, A νb u = M D E B νa C v = ( M) F E (4) where ν is Poisson s rtio, nd M, D nd F re relted to the rigid bod motion. For the neutrl inclusion, the inclusion hs the sme stress field with the mtri, σ = A, σ = B, τ = C (5) () () () The superscripts () nd () refer to the rubber nd cord, respectivel. Thus, the displcement field in the cord hs the form, A ν B u = M D E B ν A C v = ( M) F E (6) Substitution of Equs.(4) nd (6) into Equ.() leds to, p (, )[ lu ( u) mv ( v)] = l A m B lmc q (, )[ lv ( v) mu ( u)] = lm( B A) ( l m ) C (7) In the mnipultion, the rigid bod motion is not tken into ccount, i.e., M, D nd F re zero. B ppliction of Equ.(7), the interfce propert nd the cord shpe under n ssumption of the neutrl inclusion will be discussed in the net sections. 3. Equl-biil Tension For cord with given shpe s shown in Fig. under the equl-biil tension, σ = σ = A= B, τ = C = 0 (8) () () () And the interfce prmeters re derived from Equ.(7), ν ν = ( )( ) p (, ) E E q (9) In view of the pproimte incompressibilit of the rubber nd the significnt difference in stiffness between the cord nd the rubber, the interfcil prmeters re reduced to, p (, ) E q = ( ) (0) After non-dimensionliztion s done in section., one m get P (, ) = = p(, ) Q E () Fig.6 shows the dimensionless norml prmeter P (, ) s indicted b the height of green shdow long interfce boundr Γ. Obviousl, the interfce prmeters re non-negtive on the entire interfce. It should be noted tht the interfce propert is lso independent of the cord stiffness. 3.3 Uniil Tension 3

4 In the stte of the uniil tension prllel to the - is, σ = A, σ = 0, τ = 0 () () () () The interfce prmeters re lso derived from Equ.(7) nd given b p (, ) [( ) ] = ( ) E q (, ) 3 = ( ) E (3) Furthermore, the dimensionless quntities of P (, ) nd Q (, ) re epressed b, ( ) P (, ) = ( ) ( ) Q (, ) = 3 (4) As indicted in Fig.7, the red region with smbol of minus shows the negtive interfcil prmeters, which is phsicll unvilble. Hence, the neutrl cord does not eist under the uniil tension, no mtter wht the interfce prmeter is. To obtin neutrl cord under the uniil tension, the interfce boundr, long which the interfcil prmeters re negtive, is replced b two lines prllel to the stress direction. As shown in Fig.8, the norml nd tngentil interfce prmeters re non-negtive in the replced region. Consequentl, for the designed cross-section of the cord, there eists neutrl cord in the nti-plne sher nd plne deformtions. In fct, this designed interfce contour m be chieved b processing s shown in Fig.9. 4 Conclusions Bsed upon the imperfect interfce nd the concept of neutrl inclusion, neutrlit condition for cord-rubber composites is obtined, nd then the interfcil prmeter is discussed in the nti-plne sher nd plne deformtions. It is found tht the interfce design for two-strnd cord is vilble under n ssumption of the neutrl inclusion. In ddition, the interfcil prmeter is independent of the cord stiffness. Fig.. A cross-sectionl construction for single Nlon66 cord in rubber mtri. θ Fig.. A cross-section of cord θ Fig.3. Designed cross-sectionl construction. H(,) Fig.4. The dimensionless interfcil prmeter under nti-plne sher, w (, ) = B. N N

5 INTEFACE DESIGN OF COD-UBBE COMPOSITES H(,) P(,) Q(,) Fig.5. The dimensionless interfcil prmeter under nti-plne sher, w (, ) = B. P(,) 3 Fig.8. The dimensionless interfcil prmeters under uniil tension. Fig.6. The dimensionless norml interfcil prmeter under equl-biil tension. P(,) Q(,) Fig.7. The dimensionless interfcil prmeters under uniil tension. Fig.9. A SEM photogrph of single Nlon66 cord in rubber mtri Acknowledgement The uthors grtefull cknowledge finncil support b the Ntionl Science Foundtion of Chin under Projects No eferences [] L. Job nd. Joseph Studies on rubber-to-nlon tire cord bonding. Journl of Applied Polmer Science, Vol. 7, No. 7, pp 97-0, 999. [] G.P. Crmn,.C. Averill, K.L. eifsnider nd J.N. edd Optimiztion of fiber cotings to minimize stress concentrtions in composite mterils. Journl of Composite Mterils, Vol. 7, No. 6, pp 589-6, 993. [3] C.Q. u Interfce design of neutrl elstic inclusions. Interntionl Journl of Solids nd Structures, Vol. 35, No. 7-8, pp , 998. [4] K. Bertoldi, D. Bigoni, W.J. Drugn Structurl interfces in liner elsticit. Prt II: Effective properties nd neutrlit. Journl of the Mechnics nd Phsics of Solids, Vol. 55, No., pp 35-63,