INTERFACE INCLUSION PROBLEMS IN LAMINATED MEDIUM

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1 JOURNAL OF THEORETICAL AND APPLIED MECHANICS 3,39,21 INTERFACE INCLUSION PROBLEMS IN LAMINATED MEDIUM Bogdn Rogowki Mrcin Pwlik Deprtment o Mechnic o Mteril, Technicl Univerity o Łódź e-mil: [email protected] A moduli perturbtion method i ued to contruct the olution or the contct tine in the incluion problem o compoite lminte under tte o torionl deormtion. The incluion i conidered to be embedded t the interce o the lminte. The ollowing olution hvebeenobtined:i)theexctolutionortheincluioninmedium coniting o two lyer nd two hl- pce,ii) irt-order ccurte olution or lyered medium coniting on + m) contituent,iii) n pproximtive olution or phyiclly inhomogeneou medium the limiting ce o the lyered medium. Key word: niotropy, lyered medi, incluion, perturbtion method, torion interce 1. Introduction The tre nlyi o lminted iber-reinorced compoite mteril h beenubjectoincreingimportnceduetotheexpndedueouchmteril in divere modern engineering ppliction. Thi nlyi i not ey becue compoite lminte very oten contin interlminr incluion, crck or delmintion, which h been oberved common nd unvoidble occurrence in mny prcticl itution. It i well known tht tudie o the incluion problem or bonded multiphe medium require pecil phyicl nd nlyticl conidertion tht re not encountered in thoe correponding to their homogeneou counterpr. In the multiphe ytem, the olution rie not only rom geometric dicontinuity but lo rom the mteril dicontinuity. The plne incluion problem or two iotropic plne olved by men o ingulr integrl eqution method w conidered by Grilickiǐ nd Sulim

2 622 B. Rogowki, M. Pwlik 1975). Some incluion problem nd comprehenive lit o reerence up to the yer 1982 re preented in the book by Alekndrov nd Michitrin 1983). Jevtuhenko et l.1995) preented ome plne contct problem or lyered medium with n interce incluion in the rmework o the homogenition theory. The problem o n interce incluion between diimilr orthotropic hl-pce w conidered by Rogowki1993). In thi pper, the problem o n interce incluion in lyered compoite or in phyiclly nonhomogeneou medium i conidered in the rmework o the modulu perturbtion pproch. Thi method w pplied by Go1991), Fn ey l.1992) to the olution to ome rcture nd inhomogeneity problem, repectively. The nlyi dier rom the previou tudie in number o pect. Firt, by uing the modulu perturbtion pproch cloed-orm olution or the torionl contct tine o compoite which conit o two diimilr lyer nd two diimilr hl-pce i obtined. Thi olution h lo n dvntge over the previou nlye by implicity nd nlyticl orm o the reult. Further, we how tht the perturbtion nlyi ctully provide the irt-order ccurcy olution to the contct tine o n elticlly nonhomogeneou medium with rbitrry, piecewie contntlyered) or continuouly vrying eltic moduli in the depthwie direction. 2. Reerencette We ue cylindricl coordinte nd denote them by r, ϑ, z). Conider n N-lyered compoite lminte contining n intercil rigid circulr thin incluionotherdiu loctedbetweenthetwolyerndtwitedby mllngle ϕbymenothetorque Tppliedtothedic.Theincluion my repreent the reinou or cementing mteril, which i ued to trner the nchoring lod to the geologicl medium, or intnce. A the reerence tte we chooe the olution to the two-phe counterprt problem obtined by Rogowki1993). I the circumerentil xiymmetric diplcement o the rigid dic i then the contct tree re νr,)=ϕ r r 2.1) σ zϑ r,)= 4 π ϕ 1) i µ i r 2 r 2 r< 1) i z 2.2)

3 Interce incluion problem in lminted medium 623 ndtherigidrottion ϕ othedicigivenby ϕ = 3T 16µ 1 +µ 2 ) 3 2.3) Ineqution2.2)nd2.3)µ i ithevergehermoduluotheorthotropic mteril,i.e. µ i = G ri G zi, G ri nd G zi rethemterilhermoduli,nd i=1,2reertobodie1nd2,repectively.thetre σ zϑ r,z)ndthe diplcement ν r,z)inidethebimterilmediumregivenbyormule Rogowki, 1993) ν i) ξ i,η i )= 2 π π ϕ r 2 tn 1 ξ i ξ ) i 1+ξi 2 σ i) zϑ ξ i,η i )= 4 π ϕ µ i η i ξ 2 i +η2 i 1 η 2 i 1+ξ 2 i 2.4) Thetwoetotheobltepheroidlcoordinte ξ i,η i )rereltedtothe cylindricl coordinte r, z) by the eqution r 2 = 2 1+ξ 2 i )1 η2 i ) iz=ξ i η i ξ i 1 η i 1 2.5) where i = G ri /G zi ithemeureotheorthotropynd i =1repreent n iotropic mteril. Solution2.1)-2.4) correpond to the bimteril ininite medium, which i choen the reerence tte or the th order olution. 3. Moduli perturbtion nlyi 3.1. Compoite coniting o two diimilr lyer nd two diimilr hlpce Nowthequetionihowthetorque Tndtherottion ϕwillberelted withechotherintheceolyeredmedium.theexctnwertothi quetion would require the exct olution. A review o the exiting literture revel tht the exct nlyticl olution to thi problem i not vilble. In thi pper we obtin cloed-orm olution to the problem without retriction on the rnge o pplicbility, uing the moduli perturbtion nlyi. Conider ninterceincluioninytemotwolyerbondedtotheincluionnd to two hl-pce. The rnge o convergence o the perturbtion olution will

4 624 B. Rogowki, M. Pwlik be without n retriction i we chooe bimteril ininite medium with two verge moduli correponding to the upper nd the lower ilm the reerence tte,i.e. µ i =µ i) +µ i) )/2,where µ i) nd µ i) rethevergeher moduli o the ith ilm nd o the ith ubtrte, repectively. Then introducing mteril prmeter κ i = µ i) µ i) µ 1) +µ 1) +µ 2) +µ 2) 3.1) wehve κ i 1,1).Theprmeterκ i denotethertioothechngeothe vergehermoduluothe ithlyerilm)tothemoduluothereerence bimterilmedium.alterntively κ i denotethertioothechngeothe verge her modulu o the ith ubtrte to the modulu o the reerence medium. During the trnormtion, the pplied torque T i kept contnt butthetwitngleϕndthetrinenergyrellowedtochnge.theenergy conervtionlwrequirethttheextrworkdonebythetorque Tbeequl totheenergychngeinthewholebody.totheirt-orderccurcyinthe modulivrition κ i or < z <h i nd κ i or z >h i theequtiono the energy conervtion red 1 2 Tδϕ = κ i i=1 A i) σ i) zϑ ν i) da i + A i) σ i) zϑ ν i) da i ) 3.2) where δϕ itherottionchnge, σ i) zϑ nd ν i) retheknowntreend diplcementorthereerencebimterilmedium,nd A i) denotethe,ai) boundry urce o the trnorming region. For the preent o geometry A i) conitothethreehorizontlplne: z= + nd z=h 1,z= h 2 nd A i) conitothetwoplne: z=h+1 + nd z= h + 2.There integrlontheplne z=iequl Tϕ.Theright-hndideoEq.3.2) denote the irt order energy vrition due to the moduli trnormtion. Eq. 3.2)reducetotheollowingexpreionor δϕ /ϕ δϕ = ϕ 2 i=1 [ κ i 1+ 2π σ i) zϑ Tϕ ν i) ) z=±h ± i ] rdr 3.3) SubtitutingEq.2.4) 1 nd2.5)with µ i =µ i) +µ i) )/2ndnoticingtht i ziequl 1) or <z h 1 nd 1) z+h 1 1) 1) )or z h 1 i.e. ξ 1) = 1) h 1/,ndlterntively ξ 2) = 2) h 2/ortheplne z=h 1

5 Interce incluion problem in lminted medium 625 nd z= h 2,repectively,ndthenintegrtingwerrivettheollowing expreionor δϕ /ϕ δϕ ϕ = κ 1 [ 2I 1) h 1 ) [ 1 ] κ 2) 2 2I h 2 wheretheintegrl Iξ i),ξi) = i) h i/,ioundtobe 1+ 2π σ i) zϑ Tϕ ν i) ) z=±h ± i =1 6 π ξi) ξ i) Iξ i) )=2 π tn 1 ξ i) ξ i) = i) h i rdr= π 2 tn 1 ξ i ξ i 1+ξ 2 i + ξi) π ) 1 ξ 2 i ξi) )2 ) ξi 4 [ 3+2ξ i) i) 1+ξ )2) ln )2 ξ i) )2 ) ] 1 3.4) dξ i =2Iξ i) ) 1 ) ] 2 3.5) Theirtorderpproximtiono ϕ,ϕ 1 =ϕ +δϕ i κ i ),1torderolution ϕ 1 =ϕ [ 1 κ 1 2Iξ 1) ) 1 ) κ 2 2Iξ 2) ) 1 )] 3.6) The correponding higher order olution re obtined recurively rom the previouolution,otht ϕ 2 =ϕ +δϕ 1,i κ 2 i ),2ndorderolution { ϕ 2 = ϕ 1 κ 1 2Iξ 1) ) ) 1 κ 2 2Iξ 2) )+ ) 1 + [κ 1 2Iξ 1) ) 1 ) +κ 2 2Iξ 2) ) 1 )] 2 } Ingenerl,oneh κ n i ), nthorderolution, n>1 ] n ϕ n =ϕ [1+ 1) k[ κ 1 2Iξ 1) ) ) 1 +κ 2 2Iξ 2) )] k ) 1 k=1 3.7) 3.8) Itieenthttheumconvergetotheexctolution n inthernge o convergence, i.e. [ ϕ= lim ϕ n=ϕ 1+κ 1 2Iξ 1) ) n ) 1 +κ 2 2Iξ 2) )] 1= ) 1 3.9) = 3T 16 3 [ µ 1) 1 Iξ 1) ) ) +µ 1) Iξ1) )+µ2) 1 Iξ 2) ) ) +µ 2) ] 1 Iξ2) )

6 626 B. Rogowki, M. Pwlik i κ1 2Iξ 1) ) 1 ) +κ 2 2Iξ 2) ) 1 ) <1 3.1) Theweightingunction Iξ i) ),ξi) = i) h i/,vnih h i ndpprochunity h i,ndmonotonicllyincreewith ξ i).theinequlity ineq.3.1)itiiedince κ 1 +κ 2 <1nd 2Iξ i) ) 1 1,1)orny vlue o the mteril nd geometric prmeter. The perturbtion olution or ϕ in Eq.3.9) perectly mtche the exct olutioninbothlimitingbimterilce,i.e. i) h i/ nd i) h i/, correponding to the ce o the bimteril ininite medium, with the interce incluion, mde entirely o the hl-pce mteril or, lterntively, o two lyerbytrnormingtothetwohlpceregion z >h i,repectively. ItcnbeeenromEq3.5)nd3.9)thttheweightingunction Iξ i) ) nd 1 Iξ i) )givethertioothetrinenergytoredinthetrnorming region o the ilm nd o the ubtrt, repectively, to the totl trin energy tored in the reerence bimteril medium. The eective contct tine hould hve the orm µ e =µ 1) 1 Iξ 1) ) ) +µ 1) Iξ1) )+µ2) 1 Iξ 2) ) ) +µ 2) Iξ2) )3.11) Hence,itcnbeidtht µ e ithevergehermoduluweightedbythe trin energy denity ditribution Nonhomogeneou medium with piecewie contnt moduli The modulu perturbtion pproch provide nturl chnnel or extenion to more complicted problem. Uing the previouly preented nlyi, itcnbehownthtthetwitngleotheinterceincluioninthen+m)th lyer o the compoite lminte cn be obtined with the eective tine [ µ e =µ 1) 1 I + n i=1 zn 1) )] n µ 1) [ zi 1) ) i i I I ϕ= [ +µ 2) 1 I z i 1 1) i 1 T 16 3 µ e 3.12) )] + zm 2) )] m + n i=1 3.13) µ 2) [ zi 2) ) i z i 1 2) )] i 1 i I I

7 Interce incluion problem in lminted medium 627 Here I )ideinedbyeq.3.5), z i ithe zcoordinteotheinterce betweenthe ithnd i+1)thlyerilm),othtthelyerthickneigiven by h i =z i z i 1,i=1,2,...n,with z =.Arigorounlyiotheerror bound or the perturbtion ormul in Eq.3.13) i not yet vilble. Hence, it i necery to exmine the ollowing inequlity 1 { n µ 1) +µ 2) i=1 n + µ 2) [ i I i=1 µ1) µ 1) +µ 2) zi 2) i I µ 1) [ i I zi 1) ) i I ) z i 1 2) i 1 I zn 1) ) n µ 2) µ 1) +µ 2) z i 1 1) i 1 )] + )]} 3.14) I zm 1) ) m < Nonhomogeneou medium with continuouly vrying moduli When the eltic modulu i unction o ptil coordinte then the problem become more complicted compred to the homogeneou ce. It i noted tht the perturbtion olution cn be obtined in tht ce under ome retriction. The olution to the problem o rigid interce dic in the bimteril nonhomogeneou medium with continuouly vrying moduli G i) zz),i=1,2,ndcontntprmeter i), G i) r z)= 2 i Gi) zz),cnbe contructedbytkingthelimit z i z i 1 nd z i z i 1 nd n in Eq.3.13). The eective contct tine h then the integrl orm µ e = 1) di 1) z/) G 1) z z)dz+ 2) dz di 2) z /) G 2) z z)dz 3.15) d z where di 1) z/) dz = 3i) [1+2 i) ) 2 z 2 ) 2 + i) ) 2 z 2 ) ] π 2 ln i) ) 2 z ) When the eltic medium conit o two nonhomogeneou lyer o the thickne h 1 nd h 2 ndthehermoduli G 1) z z)nd G2) z z )with G i) r z)=i) )2 G i) z z),i) = cont bonded to two homogeneou hl-pce withthemoduli G 1) z, G i) r = i) ) 2 G i) z thentheolutionorthecontct tine cn be obtined

8 628 B. Rogowki, M. Pwlik [ 1) µ e =µ 1) 1 I h 1)] + 1) h 1 di 1) z/) dz [ 2) +µ 2) 1 I G 1) z z)dz+2) h 2 h 1 di 2) )] + z /) d z G 2) z z )dz 3.17) When rigid incluion i embedded in nonhomogeneou ininite medium withthehermoduli G z z)nd G r z)= 2 G z z)thenthecontcttine igivenbytheeqution µ e = 32 π [1+2 2 z 2 ) 2 ln 1+ 2 ) ] 2 z 2 2G z z)dz 3.18) The diiculty with the perturbtion method i tht in pecil ce the expreionorthecontcttinegivenbyeq3.15)or3.18)myreducetodivergent integrl. We expect tht thee ormule hve imilr rnge o vlidity Eq.3.13) Two-contituentcompoite Rogowki1995) perormed n integrl eqution nlyi to tudy n xiymmetric torion interce incluion, which pper between boundry lyer nd diimilr hl-pcefig. 2). In thi nlyi the olution o the irt-order ccurcy h the orm o Eq.3.12), wherein our nottion) µ e =µ 1 +µ 2 ) [1 η3) 3πξ 3 η2m+1)= µ 1 µ 1 +µ 2 + η5) 5πξ 5 ] +... µ1 µ ) 2 n 1 1 µ n=1 1 +µ 2 n 2m+1 Inprticulr,i µ 1 =µ 2 theneq.3.19)yield [ µ e =2µ πh πh ] +... ξ = h ) 3.2) Solution3.18)nd3.19)revlidi h 1 />1.Themoduliperturbtion pproch, ee Eq.3.11), give µ e =µ 2 +µ 1 I 1 h) 3.21)

9 Interce incluion problem in lminted medium 629 Fig. 1. Incluion geometry, coordinte nd nottion Fig. 2. Two-phe counterprt o the interce incluion problem

10 63 B. Rogowki, M. Pwlik Fig. 3. Torionl tine o the rigid dic on two-phe orthotropic hl-pce; Eq.3.11)or µ 2) ==µ 2), µ1) =µ, µ 1) =µ, ξ 1) =ξ = h/ oror µ 1 =µ 2 [ 1 h)] µ e=µ 1 1+I Eqution3.21) nd3.22) cn be rewritten ollow { µ e =µ 1 +µ 2 ) 1 µ [ 1 1 h)]} 1 I µ 1 +µ ) 3.23) ndor µ 1 =µ 2 { µ e=2µ [ 1 h)]} 1 I 3.24) 2 Forthickboundrylyer h 1 /)>1Eq3.23)nd3.19)or3.24)nd 3.2) give the reult which dier le thn one percent. Perturbtion olution 3.21),3.22)revlidloormllvlueo h 1 /.Forexmple,Eq.3.21) yield µ µ 1 or 1 h/=.1 µ e = µ µ 1 or 1 h/= ) µ µ 1 or 1 h/=1. Itheincluioniembeddedttheiniteditnce hromthebimteril interceotheininitemediumwiththehermoduli µ 1 nd µ 2 inthe mteril 1 ) then Eq.3.11) yield 1 h [ 1 h)] µ e =µ 1 [1+I )]+µ 2 1 I 3.26)

11 Interce incluion problem in lminted medium 631 Thi eqution yield 1.88µ µ 2 or 1 h/=.1 µ e = µ µ 2 or 1 h/= µ µ 2 or 1 h/= ) 4. Concludingremrk We hve preented modulu perturbtion cheme or determining eltic contct tine ditributed by inhomogeneitie. Although inite element method, ee Luren nd Simo1992) or intnce, or boundry element method, ee e.g. Telle nd Brebbi1981), cn hndle thee type o inhomogeneity problem, the preent perturbtion procedure till how it dvntge by implicity nd nlyticl orm o the reult. The preented cloed-orm perturbtion olution3.11) my be pplicble without ny retriction, i.e. or ny combintion o the eltic contnt o mteril contituent, while the irt-order ccurte olution3.13),3.15), nd3.17) my be pplied in moderte rnge o mteril combintion o prcticl igniicnce. The eective eltic contnt o the lminted medium re given by Achenbch1975) nd Chritenen1979). I we conider lminted compoite coniting o lternting plne prllel lyer o two homogeneou iotropic mteril, then Achenbch nd Chritenen reult give G z = G 1 G 2 G 2 δ 1 +G 1 1 δ 1 ) G r =G 1 δ 1 +G 2 1 δ 1 ) where δ 1 =h 1 /h 1 +h 2 )ndwhere h 1,h 2 rethethickneend G 1, G 2 thehermoduliothetwoelticlyer.prlleltothebovetudieo eltic contct problem homogenized model with microlocl prmeter h lo been developed to evlute the eective tine o lyered body. The relted work re given by Kczyńki nd Mtyik1988, 1992), Mtyik nd Woźnik1987). From the olution preented in thi pper we conclude, tht eltic propertie o the boundry lyer inluence trongly the eective tine o lyered bodyndtheolutiondependloonthertioothelyerthicknetothe rdiu o contct region. The ormule mentioned in the literture do not decribe thee eect. By replcing the her modulu with the eective her modulu in eqution2.1),2.2) nd2.3) we obtin the olution.

12 632 B. Rogowki, M. Pwlik The mechnicl torion ield o lminted orthotropic medium under nd bove the interce incluion re given by: rottion 3T ϕ= 16µ 1e +µ 2e ) 3 4.1) diplcement [ ν i) ξ i,η i ) = ϕr1 2 tn 1 ξ i + ξ )] i π 1+ξi 2 = = ϕ 2 in π ϕr 1 r r 1 2 r 2 ) 4.2) z= ± r z= ± r tree σ i) zϑ ξ i,η i ) = 4 π ϕµ ie = η i ξ 2 i +η2 i 1 η 2 i 1+ξ 2 i 4 r π 1)i ϕµ ie z= ± r< 2 r 2 z= ± r> = 4.3) σ i) r 2 2 rϑ ξ i,η i ) = 4 π ϕgi) r 1+ξi 2)2 +ξi 2+η2 i )2= 4.4) z= ± r< = 4 3 π ϕgi) r r 2 z= ± r> r 2 2 treconcentrtionctor i=1,2) ξ i L I) zϑ = 3T 4π 5 1)i µ ie µ 1e +µ 2e z= ± r L I) rϑ = 3T 4π 5 i) Gr z= ± r + µ 1e +µ 2e 4.5) Intheboveolution µ ie,i=1,2,itheeectivecontcttineo the lower nd upper nonhomogeneou hl-pce, repectively. The oblte

13 Interce incluion problem in lminted medium 633 pheroidlcoordinteocitedwiththemterilprmeter i byeq. 2.5)recontinuouttheinterceincetheplne z=z i,r igivenby ξ i ξ i = iz i η i = ξ i ξ i 4.6) Thediplcement νndthetre σ zϑ recontinuouinthelminted medium,butthetre σ rϑ hjumpintheinterce,whichregivenby [σ i) rϑ ]= 4 π ϕ[gi+1) r G i) ξ r ]r2 i ξi 2)2 +ξi 4+ξ 2 i ) ξ i ξ i = iz i 4.7) We oberve tht both tre component hve qure root ingulritie r or r +,repectively.theeingulritierepreentedbythe tre concentrtion ctor, ee Eq.4.5). Whenthetorionlorcereditributedlongthecircle r=r,z=z ) intheinteriorothe ithlyer z i 1 <z z i )thentherottion ϕothe rigid incluion will be ϕ= 3T [1 2 16µ 1e +µ 2e ) 3 tn 1 ξ i + ξ )] i π 1+ξi 2 4.8) where ξ 1,η i,i=1,2,...,nrereltedto r=r nd z=z nd i, how Eq2.5), nd T i the reultnt torque o the pplied torionl orce. The imilrity between ormule4.2) nd4.8) reult rom Betti reciprocl theoremcompre the olution in the pper by Rogowki1998)). I the upper lyered hl-pce i lo loded in the me mnnerymmetriclly) then the rottion ϕotherigidincluionwillbe ϕ= 3T [2 2 16µ 1e +µ 2e ) 3 tn 1 ξ i + ξ i π 1+ξi 2 +tn 1 ξ j + ξ )] j 1+ξj 2 4.9) where ξ j iocitedwith i,i=1,2,...,n,nd ξ j iocitedwith j, j=1,2,...,m. Reerence 1. Alekndrov V.M., Michitrin S.M., 1983, Kontktnye zdchi dl tiel tonkimi pokrytymi i prolonkmi, Nuk, Mokw

14 634 B. Rogowki, M. Pwlik 2. Achenbch J.D., 1975, A Theory o Elticity with Microtructure or Directionlly Reinorced Compoite, Springer, Berlin 3. Chritenen R.M., 1979, Mechnic o Compoite Mteril, Whiley, New York 4. Fn H., Keer L.M., Mur T., 1992, Inhomogeneity Problem Reviited vi the Modulu Perturbtion Approch, Int. J. Solid Structure, 29, 2, Go H., 1991, Frcture Anlyi in Nonhomogeneou Mteril vi Moduli- Perturbtion Approch, Int. J. Solid Structure, 27, 2, Grilickiǐ D.V., Sulim G.T., 1975, Pieriodicheky zdch dl uprugoǐ plokoti tonkotennymi vkluchenymi, Prikl. Mtem. i Mekh., 39, 3, Jevtuhenko A.A., Kczyńki A., Mtyik S.J., 1995, Npryzhennoe otoynie loitovo uprugovo kompozit tonkim lineǐnym vklyucheniem, Prikl. Mtem. i Mekh., 59, 4, Kczyńki A., Mtyik S.J., 1988, Plne Contct Problem or Periodic Two-Lyered Eltic Compoite, Ingenieur Archiv, 58, Kczyńki A., Mtyik S.J., 1992, Modelling o Mechnicl Behviour o SomeLyeredSoil,Bull.otheInt.Aoc.oEng.Geology 1. Luren T.A., Simo J.C., 1992, A Study o Mechnic o Microindenttion Uing Finite Element, J. Mter. Re., 7, Mtyik S.J., Woźnik C., 1987, Micromorphic Eect in Modelling o Periodic Multilyered Eltic Compoite, Int. J. Eng. Sci., 5, Rogowki B., 1993, Internl Point Torque in Two-Phe Mteril. Interce Crck nd Incluion Problem, J. o Theoret. nd Applied Mechnic, 31, 1, Rogowki B., 1995, Interce Crck or Incluion in Two-Phe Hl-Spce Under Concentrted Torque, Zezyty Nukowe PŁ, Ser. Budownictwo, 46, Rogowki B., 1998, A Stticl Problem o Reiner-Sgoci Type o n Internlly Loded Orthotropic Hl-Spce, The Archive o Mechnicl Engineering ABM), XLV, 3, Telle J.C.F., Brebbi C.A., 1981, Boundry Element Solution or Hl- Plne Problem, Int. J. Solid Structure, 17, 12,

15 Interce incluion problem in lminted medium 635 Zgdnieni międzypowierzchniowej inkluzji w ośrodkch wrtwowych Strezczenie Z pomocą metody perturbcji modułów znleziono rozwiązni określjące kontktową ztywność w zgdnienich inkluzji umiezczonej n powierzchni klejeni wrtwowego kompozytu bedącego w tnie krętnej deormcji. Otrzymno ntępujące rozwiązni:i) dokłdne rozwiąznie dl inkluzji w ośrodku kłdjącym ię z dwóch wrtw i dwóch półprzetrzeni,ii) rozwiąznie o dokłdności pierwzego rzędu dl ośrodk n + m)-wrtwowego,iii) przybliżone rozwiąznie dl ośrodk izycznie niejednorodnego z modułmi ścinni zmienijącymi ię w poób ciągły, otrzymne jko przejście grniczne w rozwiązniu dl ośrodk wrtwowego. Mnucript received December 5, 2; ccepted or print Jnury 24, 21