Composite Structures

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1 Compoite Structure 5 (8) Cotet lit vilble t ScieceDirect Compoite Structure jourl homepge: teril tilorig for reducig tre cocetrtio fctor t circulr hole i fuctiolly grded mteril (FG) pel T G.J. Nie b Z. Zhog R.C. Btr b School of Aeropce Egieerig d Applied echic Togji Uiverity Shghi 9 Chi b Deprtmet of Biomedicl Egieerig d echic Virgii Polytechic Ititute d Stte Uiverity Blcburg VA 46 USA ARTICLE INFO Keyword: Fuctiolly grded mteril Circulr hole Stre cocetrtio fctor teril tilorig ABSTRACT By umig tht Youg modulu d Poio rtio of lierly eltic d iotropic mteril vry log the rdil directio i pel with circulr hole d deformed by fr field uixil teile trctio we firt lyticlly fid the tre cocetrtio fctor K t the hole. The problem i olved by uperpoig olutio of two problem oe of uiform bixil teio d the other of pure her. The olutio of the firt d the ecod problem re repectively i term of hypergeometric fuctio d Frobeiu erie. Subequetly we lyticlly tudy the mteril tilorig problem for uiform bixil teio d give explicit vritio of Youg modulu to chieve prepecified K. For the pel loded by fr field uixil teile trctio we how tht the K c be reduced by fctor of bout 8 by ppropritely grdig Youg modulu d Poio rtio i the rdil directio. By plottig K veru the two ihomogeeity prmeter we olve the mteril tilorig problem for pel loded with fr field uixil trctio. The lyticl reult hould erve bechmr for verifyig the ccurcy of pproximte/umericl olutio for ihomogeeou pel.. Itroductio Eve though the mechicl behvior of ihomogeeou mteril h bee tudied ice 95 there h bee tremedou ctivity i thi field durig the lt three decde [ 6]. A heterogeeou mteril with cotiuou ptil vritio of mteril prmeter i ofte clled fuctiolly grded mteril (FG). With the vilbility of 3- D pritig for mufcturig mteril with complex microtructure it i ow feible to fbricte tructure to hve the optimum tre d tri ditributio for ehcig their mechicl propertie uder precribed lod [78]. Oe uch problem i cotrollig the tre cocetrtio fctor K roud circulr hole i pel. It i well ow tht K t circulr hole i ifiite pel compoed of homogeeou iotropic d lierly eltic mteril deformed i uixil teio equl 3 [9]. For orthotropic mteril Lehitii et l. [] hve deduced K for ifiite plte cotiig circulr hole d deformed by remote uixil teile trctio. Bed o Lehitii olutio of the ple eltottic problem uig complex vrible Britt [] Techev et l. [] d Xu et l. [34] repectively foud K for iotropic rectgulr pel with cetrlly locted circulr d ellipticl cutout lmited compoite with circulr hole d compoite lmite with either ellipticl hole or multiple hole. Author of Ref. [5 8] employed the fiite elemet method (FE) to evlute tree i compoite lmite with circulr hole. Kubir d Bhu-Chdr [9] d Eb [] uig the FE foud tht K i igifictly iflueced by the ptil vritio of the mteril ihomogeeity. Oe eed very fie meh er the hole d coduct covergece tudie to deduce reobly ccurte vlue of K tht c be rduou t. Hug d Hft [] d Cho d Rowld [3] optimized fiber oriettio er hole to miimize K d icree the lodcrryig cpcity of compoite lmite. Lope et l. [4] d Gome et l. [5] foud fiber oriettio gle d their volume frctio either to miimize the pe tre roud cutout or to mximize the buclig d the firt-ply filure lod of compoite pel. They poited out tht the optimum vrible-tiffe deig with cetrl hole c hve erly the me iitil buclig lod pel with the me volume frctio of fiber but o hole. By dividig ihomogeeou mteril pel ito erie of piecewie homogeeou lyer d uig the method of complex vrible Yg et l. [67] Yg d Go [8] d Kuhwh d Sii [9] hve how tht whe Youg modulu decree with the ditce from the hole boudry K > 3. ohmmdi et l. [3] lyticlly foud K roud circulr hole i ifiite FG plte ubjected to uiform bixil teio d pure her by umig tht both Youg Correpodig uthor t: School of Aeropce Egieerig d Applied echic Togji Uiverity Shghi 9 Chi. E-mil ddre: gj@togji.edu.c (G.J. Nie). Received 8 Jue 8; Received i revied form Augut 8; Accepted 7 Augut 8 Avilble olie 3 September / 8 Elevier Ltd. All right reerved.

2 modulu d Poio rtio vry expoetilly i the rdil directio. By umig tht Youg modulu h power lw vritio i the rdil directio d Poio rtio i cott Sburlti [3] tudied the eltic repoe of FG ulr rig ierted i hole of homogeeou plte. Kubir [3] ued the method of eprtio of vrible to fid cloed-form expreio for tree d diplcemet i FG plte with d without hole uder ti-ple her lodig d ued o-trditiol defiitio of K. We ote tht there re limited umber of lyticl tudie o the tre cocetrtio roud circulr hole i iotropic FG pel. Furthermore there re o reult o mteril tilorig for reducig K. We lyticl (i) fid K t circulr hole i iotropic FG pel uder fr field uixil teile trctio (ii) lyze the mteril tilorig problem for uiform bixil teio lodig d (iii) ivetigte the effect of mteril ihomogeeity prmeter o K. For fr field uiform teile lodig we provide repoe fuctio to etimte ihomogeeity prmeter for deired vlue of K. The ret of the pper i orgized follow. Sectio d 3 repectively give the formultio d the olutio of the direct problem i which we lyze deformtio of the pel uder precribed fr field urfce trctio. Sectio 3 i divided ito three ubectio tht provide detil of deformtio uder uiform bixil teio pure her d uixil teio repectively. I Sectio 4 we lyticlly olve the mteril tilorig problem for uiform bixil teio lodig. Sectio 5 provide umericl reult tht etblih the ccurcy d the covergece of the erie olutio for the pure her lodig d deliete effect of the vritio of the mteril propertie o K d tre ditributio. Cocluio of the wor re ummrized i Sectio 6.. Formultio of the direct problem We coider iotropic d lierly eltic FG pel with circulr hole of rdiu ubjected to fr-field uixil trctio how i Fig. (). We lyze how the rdil vritio i Youg modulu d Poio rtio ffect the tre cocetrtio t the hole periphery for ple tre deformtio of the pel. We olve the problem by uperpoig olutio of two problem bixil teio d pure her how i Fig. (b) d (c). We ue cylidricl coordite (r θ) with origi t the hole ceter to decribe the pel deformtio. I the bece of body force equilibrium equtio re rr rθ rr θθ r r θ r rθ r r θ r rθ θθ (b) where rr θθ d rθ re tre compoet. Hooe lw reltig tree to ifiiteiml tri ε ε ε rr θθ rθ i vr ε v r ε v r ε E() r [ ( ) ] E() r [ ( ) ] [ ( )] rr rr θθ θθ θθ rr rθ rθ. E() r (-c) We ume tht Youg modulu E ( r) d Poio rtio vr () re give by either (i) geerl power-lw vritio E r E r r () [ ( )] vr () v [ ( )]with < or (ii) expoetil d power-lw vritio (3) E ( r) E exp[ γ ( r ) ] vr ( ) v [ γ( r ) ] (3b) where E lim E( r) d v lim v( r) d γ γ r r ( < { γ γ} < ) re rel umber d i egtive iteger which help fid lyticl olutio of the problem. For homogeeou mteril γ γ. The vritio of E with r/ for 3 5 d.9.9 depicted i Fig. () revel tht E ()/ r E r / 5. Similrly the vritio of E with r/ for γ.9.9 depicted i Fig. (b) implie tht E ()/ r E r / 5. The fr-field boudry coditio for the bixil teio d the pure her problem re: lim rr ( r) r (4) lim ( r θ) θ r θ θ co lim ( ) rr rθ i r r (4b) d boudry coditio t the hole periphery re rr ( θ) rθ ( θ). (4c) Whe olvig the problem for tree we employ the followig comptibility equtio: εθθ ε ε ε rr θθ rr εrθ εrθ r r θ r r r r r r θ r θ (5) 3. Solutio of the direct problem We lyticlly olve the problem for the uiform bixil teio i ubectio 3. d ue the Frobeiu erie to lyze the problem for pure her lodig i ubectio 3.. By uperpoig olutio of thee two problem we obti the olutio for the uixil teio problem i ubectio Uiform bixil teio We ote tht the problem geometry the mteril propertie d the Fig.. Schemtic etch of pel with circulr hole ubjected to uiform fr-field teile trctio d it plit ito two problem. 5

3 Fig.. Vritio of Youg modulu with r/ for () two vlue of d two vlue of i Eq. (3) d (b) two vlue of γ i Eq. (3b). fr field lodig re xiymmetric. Accordigly we ume pel deformtio to be xiymmetric for which equilibrium Eq. () reduce to rr r d θθ rr dr Thu owig rr oe c fid θθ from Eq. (6). r We itroduce o-dimeiol prmeter ρ d ubtitute for tri from Eq. () ito the comptibility coditio (5) to get d rr E d ( 3 rr ve E v ) ( ) ρ E ρe ρe ρ rr d where ( ) ( ) Geerl power-lw vritio of E ( r) d vr ()give by Eq. (3) Subtitutig for E ( r) d vr ()from Eq. (3) ito Eq. (7) d olvig the reultig equtio we get rr C C for ρ (6) (7) (8) C rr F ( t t ; t ; ρ) C F ( t t ; t ; ρ) for 3 ρ (8b) C ρ ρ F t t t C ρ ρ rr ( ) t7 ( ; ; ) ( ) t F( t9 t; t; ) for (8c) C ρ ρ F t t t C ρ ρ rr ( ) t ( ; ; ) ( ) t 3 4 F ( t t4; t5; ) for (8d) Here C d C re cott to be determied from boudry coditio (4) d (4c) F( b; c; z) ( ) ( b) /( c) z /! i hypergeometric fuctio d t H H t H H t 3 t 4 H 3 H t 5 H 3 H t 6 t 3 H t H t H 4 t H t H t H t H t H t v H H ( ) H 4 5 H H 5 v v 4 ( ) H 4 ( 5 ). For the olutio c be expreed i term of Beel fuctio J ( z) d Y ( z) repectively of the firt d the ecod id d the modified Beel fuctio I ( z) d K () z. We ote tht the olutio (8) for the homogeeou mteril gree with tht give i Timoheo boo [9]. Cott C i i (9) pperig i differet equtio do ot i geerl hve the me vlue Expoetil d power-lw vritio of E ( r) d vr ()give by Eq. (3b) Subtitutig for E ( r) d vr ()from Eq. (3b) ito Eq. (7) d olvig the reultig equtio we get t F t ρ ρ tρ C U t t exp( 8 )[ ( ; 3; 6 ) ( 3 6 rr ) ρ C 7 7 ] () Here C d C re cott to be determied from boudry coditio (4) d (4c) F( ; b; z) d U( b z) re the cofluet hypergeometric fuctio defied F( ; b; z) ( ) /( b) z /! U( b z) /Γ( ) e zt t ( t) b dt where Γ( ) i the Euler gmm fuctio d 3 (v ) γ vγ γ t6 t6 γ( γ 4 v γ) t7 t8 t () For γ the olutio c be expreed i term of Beel fuctio J ( z) d Y ( z) repectively of the firt d the ecod id. 3.. Pure her deformtio We itroduce Airy tre fuctio φ( r θ) d ote tht φ φ φ φ ( ) r r r θ r r r θ () rr θθ rθ ideticlly tify equilibrium Eq. (). I view of fr-field coditio (4b) we ume tht φ( r θ) φ () r coθ (3) From Eq. () (5) () d (3) we fid tht the comptibility coditio reduce to dφ where z ρ dφ 3 z ρ dφ ( ) z ρ dφ ( ) ( ) z ( ρ) φ 3 4vE 8 ve ( ) 8vE E 4v z ( ρ) ρe ρe ρe ρe 3 ρ ve ve ( ) ve 9E v 9 z ( ρ) z ρ ρe ρe ( ) ρe ρe ρ ρ3 E ( E ) ve E v 9 E E ρe ρe ρ ρ (4) E z3 ( ρ). ρ E (5) We olve the 4th order ordiry differetil Eq. (4) with vrible coefficiet by the method of Frobeiu erie. 5

4 3... Geerl power-lw vritio of E ( r) d vr ()give by Eq. (3) We ume tht φ ( ρ) ρ b ( )( ρ) with b ( ) (6) where b re give by the recurrece formule the vrible i to be determied d equl the totl umber of term i the erie. Subtitutig from Eq. (3) d (6) ito Eq. (4) d equtig to zero coefficiet of differet power of ρ we get recurrece formule ((7) d (7b)) for b d the idicil Eq. (7c) for : B() b() b() ( 4)( )( )( ) (7) b () B() b() B3() b () ( 4)( )( )( ) (7b) ( 4)( )( ) (7c) The legthy expreio for Bi ()( i 3) re lited i Appedix A. Root 4 of the idicil Eq. (7c) re idepedet of the ihomogeeity prmeter d. Accordig to the Frobeiu method [33] olutio of the tre fuctio for differet vlue of re φ ( ρ) C ρ b ( ) ( ρ) C b ( ) ( ρ ) C3 b () ( ρ) C ρ b() ( ρ) for ρ 4 < d db φ ρ C ρ ρ db () () ( ) { { l (l ρ) b ( )} ( ρ) } d d db [ ( )] C { { l ρb [ ( )]} ( ρ ) } d C3 b () ( ρ) C ρ b() ( ρ) for ρ 4 d 4 4 d b φ ( ρ) C { ρ [( ) ( )] { l ρ[( ) b ( )]} ( ρ ) } d C C b () ( ρ) 3 b() ( ρ ) ρ (8) (8b) Cρ b() ( ρ) for 4 (8c) where b() ( ) b () d Ci i 34 re cott to be determied from boudry coditio (4b) d (4c) Expoetil d power-lw vritio of E ( r) d vr ()give by Eq. (3b) For E ( r) d vr ()give by Eq. (3b) we lo ue the method of Frobeiu erie to olve Eq. (4). Lettig i Eq. (6) we hve φ ( ρ) ρ b ( )( ρ ) with b ( ) (9) Subtitutig from Eq. (3b) d (9) ito Eq. (4) d equtig to zero coefficiet of differet power of ρ we get followig recurrece formule (() (b) (c)) for b ( ) d the idicil equtio (d) for : D () b () b() ( 5)( 3)( )( ) () D() b() D3() b() b () ( 6)( 4)( ) (b) b 3 D4() b() D5() b () D6() b () () ( )( 3 )( 5 )( 7 ) 3 (c) ( 4)( )( ) (d) The legthy expreio for Di ()( i 6) re lited i Appedix B. The idicil Eq. (d) i the me the idicil Eq. (7c) d i idepedet of the ihomogeeity prmeter γ d γ. Thu db φ ρ C ρ ρ db () () ( ) { { l (l ρ) b ( )} ( ρ ) } d d db [ ( )] C { { l ρb [ ( )]} ( ρ ) } d C3 b () ( ρ ) C ρ b() ( ρ ) ρ () where b() ( ) b () d Ci i 34 re cott to be determied from boudry coditio (4b) d (4c). From the expreio (Eq. (8) or Eq. ()) of the tre fuctio φ ( ρ) wefid tree by uig Eq. () d (3) Stre cocetrtio fctor K for the uixil teio problem We ote tht for the uiform bixil teio d the pure her π problem θθ h the mximum vlue t θ. Deotig by K K d K repectively the tre cocetrtio fctor correpodig to the bixil teio the pure her d the uixil teio problem (ee Fig. ) we hve [34] K θθ ( ρ) ρ θθ ( ρ θ) ρ θ π K K K K ( b c) 4. Alyticl olutio of mteril tilorig for uiform bixil teio We c lyticlly olve the mteril tilorig problem for uiform bixil teio lodig but ot for the imple her problem ice it olutio i i term of Frobeiu erie. Aume tht the tre tte i ρ m ρ m ( m ) rr θθ [ ( ) m] with < (3) For m θθ i cott i the pel d equl the fr field xil tre. Alo the boudry coditio t the hole periphery i exctly tified. For the tre tte how i Eq. (3) Eq. () give K m > (4) We ow fid E ( ρ) d vρ ( ) to chieve K m. Subtitutig for the tre rr from Eq. (3) ito Eq. (7) we obti [ E ρ ρ ( m ) ] ( ) E m ( ρ) ( ρm )[ vρ ( ) v ( ρ)] mm ( ) ρm E( ρ) E( ρ) (5) We hve oe ordiry differetil equtio for two uow 5

5 fuctio E ( ρ) d vρ ( ). Oe wy to olve the problem i to ume polyomil expreio for E ( ρ) d vρ ( ) d fid uow i them by miimizig the vlue of the qure of the left hd ide of Eq. (5). Here we ume tht vρ ( ) v cott d olve Eq. (5) for E ( ρ). Eq. (5) h the olutio m ρ v ρ E ( ρ) E ( ( ) m ( m ) m ) m v m where E E( ρ) ρ. We ote tht E E ρ E v lim ( ) ( ) ρ m m m v (6) (7) Fig. 3. The required vritio of Youg modulu with the rdil coordite to chieve give tre cocetrtio fctor K (v.5). Thu for m Eq. (4) d (6) give E ( ρ) E d K which equl tht for homogeeou mteril. For K 3 d 4 the required vritio of E ( ρ)/ E tig v.5 diplyed i Fig. 3 repectively hve E / E Thu de ( ρ ) > (< ) for K < (> ). ρ Tble Covergece of the tre cocetrtio fctor for differet term i the erie olutio (v.5). K Tble For 5 comprio of the tre cocetrtio fctor obtied from the erie olutio d the exct olutio (v.5). K Exct Serie Numericl reult We firt etblih i ubectio 5. tht the Frobeiu erie olutio coverge to lyticl olutio of the problem d the i ubectio 5. combie it with the lyticl olutio of the bixil teio problem to get olutio of the fr field imple teile lodig. We ue it i ubectio 5.. to certi the effect of mteril ihomogeeity prmeter o the tre cocetrtio fctor K. Tble 3 For γ comprio of the tre cocetrtio fctor obtied from the erie olutio d the exct olutio (v.3). K γ.8 γ.3 γ.3 γ.8 Exct Serie Tble 4 Vritio of the tre cocetrtio fctor K with d for fr field uixil teio (v.5) Accurcy d covergece of the olutio for the pure her problem We firt fid the umber of term i the Frobeiu erie olutio tht give coverged olutio for the pure her problem. We ote tht whe oly v vrie i.e. i Eq. (3) or γ i Eq. (3b) the lyticl olutio (4) for the geerl power-lw vritio of E ( r) d vr ()give by Eq. (3) i φ ρ C v ρ v ρ 4 ( ) ( ) F3( d; d; ) C v ρ v ρ ( ) F3( d3; d4; ) C v ρ v ρ v ρ 3( ) F3( d5; d6; ) C4 F3( d7; d8; ) (8) The olutio for the expoetil d the power-lw vritio of E ( r) d vr ()give by Eq. (3b) i φ ρ C v γ v γ v γ ( ) ( ) F 3 ( d 9 ; d ; ) CG 4 ( ρ ρ ρ 3 CG v γ d d 3 4 ( ρ 4 ) 4 v γ d d CG 4 4 ( ρ 4 ) d d 4 ) (8b) where cott Ci i 34 re determied from boudry 53

6 coditio (4b) d (4c) Gpq m p ( x ) i the eijer G fuctio b bq d d { N N} d { } d { N N} d { } 4 d 5 { N N } d { } 6 d 7 { N N} 4 7 i 3 d { } d 7 i 3 d i( 3 i) d d 8 i( 3 i) 9 with N. For homogeeou mteril i.e. i Eq. (3b) the olutio (4) i γ γ C φ ( ρ) Cρ 4 Cρ C3 ρ 4 i Eq. (3) or (9) Fig. 4. Vritio of the tre cocetrtio fctor K with for v.5. Oe get the tre cocetrtio fctor K 4 for the pure her problem. Vlue of K lited i Tble for differet vlue of ugget tht 5 i the erie olutio give reobly ccurte vlue of K d the erie olutio rpidly coverge for the coidered vlue of d. For geerl power-lw vritio of E ( r) d vr ()give by Eq. (3) vlue of K lited i Tble for differet vlue of computed uig i the erie olutio d thoe from the lyticl olutio (Eq. (8)) eure ccurcy of the erie olutio i Eq. (8). Similrly for expoetil d power-lw vritio of E ( r) d vr ()give by Eq. (3b) the cloee of vlue of K lited i Tble 3 for differet vlue of γ computed uig i the erie olutio d thoe from the lyticl olutio (Eq. (8b)) verify the ccurcy of the erie olutio i Eq. (). Thu vlue of to get coverged olutio deped upo vlue of γ d γ. 5.. Fr field uixil teile lodig 5... Stre cocetrtio fctor For the uixil lodig vlue of K for differet vlue of d i Eq. (3) lited below i Tble 4 revel tht Fig. 5. Vritio of the tre cocetrtio fctor K o the -ple for v.5. Tble 5 Vritio of the tre cocetrtio fctor K with γ d γ (v.3). γ γ (i) for > ( < ) the K for the FG pel i lrger (mller) th tht for the homogeeou mteril pel (ii) whe the mteril prmeter mootoiclly icree ( < ) with the ditce from the hole ceter the preetly computed reductio i K for the FG pel gree with tht reported i [9] d (iii) the mximum (miimum) vlue of K equl 4.83 (.39) tht i.6 (.3) time the vlue of K for homogeeou mteril. Thu the tre cocetrtio fctor c be reduced by fctor of bout 8 ( 3/.39) by properly grdig Youg modulu d Poio rtio i the rdil directio. For differet vlue of d the vritio of K with i how i Fig. 4. Here we hve tcitly umed tht K cotiuouly deped upo ice the lyticl olutio h bee foud oly for iteger vlue of. It i ee from Fig. 4 tht K icree (decree) with icree of the bolute vlue of for > ( < ). From vlue of K lited i Tble 4 d plotted i Fig. 4 we coclude tht the vlue of ( ) tht ffect the rdil vritio of Poio rtio (Youg modulu) h very little (igifict) ifluece o K. Tble 6 Vritio of the tre cocetrtio fctor K with de ( ρ ) ρ (E ( ρ) ρ.9)..9 γ ρ K

7 Tble 6b Vritio of the tre cocetrtio fctor K with de ( ρ ) ρ (E ( ρ) ρ.4). γ ρ K I order to better how the ifluece of the vritio of E o K we et d fid let-qure fit to obti the followig reltiohip mog K d. K while (3) The vritio of K o the -ple i diplyed i Fig. 5. Oe c fid vlue of thee two vrible for deired vlue of K d hece deig the FG pel. For E d v vryig ccordig to Eq. (3b) vlue of K for differet vlue of γ d γ re lited i Tble 5. We ote tht for γ > (γ < ) K for FG pel i lrger (mller) th tht for homogeeou mteril pel. The mximum (miimum) vlue of K equl 4.88 (.75) tht i.63 (.58) time the vlue of K for homogeeou mteril pel. Vlue of γ i the rge [.9.9] hve egligible effect o K. Bed o the dt i Tble 5 we lo fid let-qure fit to obti the reltiohip betwee K d γ K 3..74γ.385γ.34γ while γ (3) For give vlue of K the olier lgebric Eq. (3) c be itertively olved forγ Depedece of K upo E ( ρ) ρ d de ( ρ ) ρ We ow ivetigte the depedece of K upo E ( ρ) ρ d ( ). We et v.3 d lit vlue of K i Tble 6 d 6b d 7 de ρ ρ d 7b for two rbitrrily choe vlue of E ( ρ) ρ d de ( ρ ). ρ We coclude from vlue of K lited i Tble 6 d 6b tht K lowly decree with icree i de ( ρ ) for both vlue of E ( ρ) ρ 3 ρ d it i le th tht of homogeeou mteril whe >. ρ Tht i the ig of de ( ρ ) re oppoite for imulteouly icreig ρ vlue of K dk. Vlue of K lited i Tble 7 d 7b ugget tht K icree grdully with icree i the vlue of E ( ρ) ρ for the two vlue of de ( ρ ). Bed o thee obervtio we recommed tht ρ ρ > for FG pel Stre ditributio For the FG mteril defied by Eq. (3) we hve plotted i Fig. 6() (c) the vritio of the o-dimeiol tree log rdil directio for pecific vlue of the gulr coordite θ d the hoop tre v. θ for ρ. It i cler tht o the lie θ π/ for poitive (egtive) vlue of d the hoop tre i mximum (miimum) o the hole urfce. However whe d hve oppoite ig the the ig of i the expreio for Youg modulu determie the vlue of the hoop tre. For.9the hoop tre i erly uiform o the hole urfce. The decy of the hoop tre with ρ ertheholeithefgpeliteeperththtithehomogeeou mteril pel. Stree re eetilly cott for ρ > 5. Eve though the rdil vritio of Poio rtio h very little effect o the tre ditributio d hece the tre cocetrtio fctor demotrted i [3536] it oticebly ffect the diplcemet field. Fig. 6. For 5 v.5 vritio of the () rdil tre o the lie θ (b) the hoop tre o the lie θ π/ d (c) the hoop tre o the hole circumferece ρ. 6. Cocluio We hve lyticlly foud the tre cocetrtio fctor d the tre ditributio i pel with circulr hole d mde of 55

8 Tble 7 Vritio of the tre cocetrtio fctor K with E ( ρ) ρ (.9 ρ )..9 γ E ( ρ) ρ K Tble 7b Vritio of the tre cocetrtio fctor K with E ( ρ) ρ (.36 ρ ). γ E ( ρ) ρ K iotropic fuctiolly grded mteril. Youg modulu E d Poio rtio re umed to vry oly i the rdil directio either geerl power-lw fuctio or the former expoetilly d the ltter power lw. Vritio of E gover tree er the hole periphery d their rte of decy log rdil lie. The tre cocetrtio fctor K c be decreed by fctor of 8 by uitbly grdig the mteril propertie. Vlue of K lowly decree with icree of the vlue of ρ ρ d it i le th tht for homogeeou mteril whe >. Furthermore K icree grdully with icree i the vlue of E ( ρ) ρ. Thee reult ugget tht oe c tilor the rdil vritio of E to chieve pre-pecified vlue of K d eetilly uiform hoop tre i the etire pel. For the mteril problem explicit expreio for E ( ρ) i give for remote bixil lodig of pel. For remote uixil teile lodig we hve ytheized the computed reult to expre K i term of the mteril ihomogeeity prmeter. Oe c ue thee reltio to fid vlue of the ihomogeeity prmeter for deired vlue of K. Acowledgemet GJN' d ZZ' wor w upported by the Ntiol Nturl Sciece Foudtio of Chi (No d 777). RCB wor w prtilly upported by the US Office of Nvl Reerch Grt N with Dr. Y. D. S. Rjpe the Progrm ger. GJN i viitig RCB' group durig 8. Appedix A Expreio for Bi ()( i 3) pperig i Eq. (7) d (7b) re give below. B( ) ( v 4 v ) ( v v v ) (4v 4v v v v) B () ( ) ( ) ( v 4v v ) (v v v v v v ) (4v 4 v ) ( v 3 v v 3 v v v v ) B ( ) ( ) 3 ( ) ( ) ( ) ( ) v ( ) v ( ) v Appedix B Expreio for Di ()( i 6) pperig i Eq. () (c) re give below. D( ) ( 3 8v ) γ (5 3 v ) γ ( v 6) γ ( 3 8) vγ D ( ) ( v 4 v ) γ ( 5 6) vγ γ D3( ) ( 3 v 9) γ ( v ) γ (5v 3) γ ( 5 ) vγ D () ( 4) v γ γ 4 D5( ) ( 5v ) γ ( v 3) γ ( v 3) γ (7 ) v γγ ( 7 ) vγγ D ( ) (3 3) γ 8( v ) γ (8 v ) γ (8 v 6 ) γ (8 v ) γ (7v 43) γ (6 7v 43) γ ( 7 8) v γ 6 (7 ) v γ 56

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