CHAPTER 2 REVIEW OF LITERATURE

Transcription

1 CHAPTER REVIEW OF LITERATURE.1 GENERAL Nowadays th application of composit matrials in nginring, particularly in structural applications, is incrasing in scop. As a rsult, nw aspcts of matrials, such as th long-trm bhaviour undr mchanical, thrmal and chmical/nvironmntal loadings ar bing addrssd. Composit matrials provid solutions to varity of tchnical problms facd ths days by many industris. Composit matrials hav bcom mor popular, bcaus of incrasd comptition in th global markt for lightwight componnts of gratr strngth and stiffnss. Among many matrials, composit matrials hav th potntial to rplac widly usd stl and aluminum, and nhanc prformanc. As an xampl, by rplacing stl componnts with composit componnts, on can achiv 60% to 80% saving in componnt wight, whil 0% to 50% wight saving occurs if aluminum parts ar rplacd by composit parts (Mazumdar, 00). Th salint applications of composit matrials ar outlind in Tabl.1. Tabl.1: Applications of Composits (Ptrs, 1998) Major Aras Automobils Applications Combustion chambrs (SiC-SiC), Engin cylindr linrs (Al- SiC), CNG storag cylindrs, Disl Engin pistons (SiCw/Alalloy), Brak rotors, Laf springs (E-glass/poxy), Driv shafts (Al-C), Flywhls, Racing car braks (Al-SiC), Motorcycl driv 1

2 sprockt, Pullys, Torqu convrtr ractor, Shock absorbrs (SiCp/Al-alloy), Radiator nd caps. Sub-Marin Propulsion shaft (Carbon and glass fibrs), Cylindrical prssur hull (Graphit/Epoxy), Sonar doms (Glass/Epoxy), Composit piping systm, Scuba diving cylindrs (Al-SiC), Floats, Boat hulls. Commrcial and Industrial Prssur vssls, Ful tanks, Cutting tool insrts, Laptop cass, Wind turbin blads, Elctric motors, Firfighting air bottls, Artificial ligamnts, MRI scannr cryognic tubs, Whlchairs, Hip joint implants, Eyglass frams, camra tripods, Musical instrumnts, Drilling tubs, Drilling motor shaft, Drill casing, Cran componnts, High prssur hydraulic pip, X-ray tabls, Hart valvs, Hlmts, Crucibls, Bams. Arospac quipmnt and structurs Rockt nozzl (TiAl-SiC fibrs), Hat xchangr panls, Engin parts (B-Al), Wind tunnl blads, Spaccraft truss structur, Rflctors, Solar panls, Camra housing, Hubll spac tlscop mtring truss assmbly, Turbin rotor, Turbin whls (oprating abov 40,000 rpm), Nos caps and lading dg of missils and Spac shuttl. Aircraft Wings, Rotary launchrs, Engin casing, Rings (Al O 3 /Alalloy), Driv shaft, Propllr blads, Landing gar doors, Thrust rvrsr(carbon/bismalimid), Hlicoptr componnts viz. Rotor driv shaft, Mast mount, Main rotor blads (Carbon/Epoxy). 13

3 Sports Tnnis rackts, Golf shafts, Racing bicycl fram (SiCw/6061), Fishing rod, Pool cus.. YIELD CRITERIA Th succssful dsign of any structural componnts rquirs fficint and saf us of matrials. Thrfor, th ability to prdict th yild strngth of a matrial i.. th onst of plastic yilding, undr loading conditions, is of grat importanc for dsign purposs. In th fild of plasticity, it is important to stablish mathmatical rlations for dtrmining th conditions at which plastic yilding bgins whn a matrial is subjctd to any possibl combination of strsss. Furthr, undr uniaxial loading, th plastic flow bgins at yild strss as in cas of simpl tnsil tst. Th yilding undr a stat of combind strsss can b rlatd to som particular combination of principal strsss. Svral rlations hav bn proposd as yild critria that giv us information about th strngth of a matrial undr multiaxial loading. Th yild critria may b xprssd in trms of spcific quantitis, such as th strss stat, th strain stat, a strain nrgy quantity, or othrs. A yild critrion is usually writtn in mathmatical form by mans of a yild function f (, Y ), whr ij dfins th stat of strss and Y is th yild strngth undr uniaxial tnsion (or comprssion). Th yild function is dfind such that th yild critrion is satisfid whn f (, Y ) 0. Whn f (, Y ) 0, th strss stat is ij lastic. Th condition f (, Y ) 0 is undfind. To dvlop a yild function, th ij componnts of th multiaxial strss stat ar combind into a singl quantity known as th ffctiv strss ( ). Th ffctiv strss is thn compard with th ij ij 14

4 yild strss Y, in som appropriat form, to chck inst of yilding. Th trm f ( ij ) is th yild function, a function of six strss componnts of th strss tnsor that dfins th yild surfac in th six dimnsional strss spacs sparating th lastic and plastic domains...1 Yild Critria Usd in th Prsnt Study Th prsnt work dals with th analysis of composit cylindr, which is assumd to b ithr isotropic or anisotropic. In addition, th ffct of rsidual strss is also takn into account in a sgmnt of th study. This sction brifly dscribs th yild critria applicabl for ths thr cass. For isotropic matrials, th yilding would dpnd only on th magnituds of principal strsss, and 1 3. For ths matrials th wll known von Miss (1913) yild critrion, also known as distortion-nrgy thory, might b xprssd as, ( ) ( ) ( 1/ ) (.1) whr is quivalnt strss. Th von-miss critrion implis that th yilding undr both uniaxial tnsion and comprssion starts at th sam lvl of tnsil and comprssiv strsss. Howvr, Arsnault (1987) pointd out that vn in an isotropic mtal matrix composit yilding dos not bgins at th sam lvl of tnsil and comprssiv strsss undr uniaxial loading. Badini (1990) also noticd that th comprssiv yild strngth of 15 vol% SiCw/6061Al composit is highr than its yild strngth in tnsion. Th procssing of mtal matrix composits oftn involvs cooling from highr tmpratur, which rsults in rsidual tnsil thrmal strss in th matrix du to rstraint causd by cramic rinforcmnts. To account 15

5 for th ffct of diffrnt yild strsss in comprssion and tnsion, som rsarchrs (Schllkns and D Borst, 1990a, b; Bicanic t al, 1994; Moin, 1996) hav mployd th isotropic form of Hoffman yild critrion givn blow, ( ) ( ) Q ( ) 1 0 (.) whr Q is a matrial paramtr. A numbr of othr invstigators (Bicanic t al, 1994; Moin, 1996) hav prfrrd following form of Hoffman s yild critrion that mploys uniaxial comprssion and tnsil yild strsss xprssd by f c and f t rspctivly, ( ) ( ) ( f c f t )( 1 ) f c f t 0 (.3) In both th yild critria dscribd abov (i.. von-miss s and Hoffman s yild critria) it is assumd that th matrial is isotropic. This assumption may hold good at th start of plastic dformation. But this would not b a valid assumption aftr th matrial undrgos apprciabl longation. Furthr, th mchanical strngth of whiskr rinforcd composits ar dpndnt on th orintation of whiskrs in th matrix (Ldrich and Shastry, 198; Crov t al, 1985; McDnls, 1985). Badini (1990) obsrvd that th comprssiv yild strngth in th longitudinal dirction of 15 vol% SiCw/6061Al composit was highr than its yild strngth in th transvrs dirction. Morovr, fibrrinforcd composit or componnts that ar fabricatd by procsss such as forging, rolling or xtrusion xhibits anisotropic proprtis. Thrfor, th us of von-miss yild critrion givn by Eqn..1 would not b an appropriat choic for such cass. Hill (1948) modifid th von-miss yild critrion applicabl for isotropic matrial to incorporat th ffct of anisotropy. Hill assumd th matrial to b 16

6 orthotropic i.. thr xists only thr mutually orthogonal plans of symmtry at vry point insid matrial. Th intrsction of ths plans is rfrrd as th principal axs of anisotropy. Th Hill s yild critrion dos not considr Bauschingr ffct and also assums that th hydrostatic strsss do not affct yilding. Hill s critrion taks th following form whn th principal axs of anisotropy ar takn along x, y and z dirctions, F( y ) z G( z ) x H ( x ) y Lτ yz M zx N xy 1 (.4) whr th paramtrs F, G, H, L, M and N ar matrial constants dfining th dgr of anisotropy. For principal axs of orthotropic symmtry, th Hill s critrion givn by Eqn. (.4) bcoms, F ( ) G( ) H ( ) 1 (.5) whr, and 1 3 ar th principal strsss. In limiting cas, whn anisotropic constants ar qual (i.. F G H ), th Hill s critrion givn by Eqn. (.5) rducs to von-miss critrion for isotropic matrials, Eqn. (.1)..3 CREEP AND CREEP MECHANISMS Crp is th progrssiv tim-dpndnt inlastic dformation undr constant load and tmpratur. Du to crp, a structural componnt undrgos tim-dpndnt changs in stat of strss and strain such as progrssiv dformations, rlaxation and rdistribution of strsss, local rduction of matrial strngth, chang of matrial bhaviour from isotropic to anisotropic tc. Kraus 17

7 (1980) pointd that th crp bhaviour is a function of strss, tmpratur, tim, strss history, tmpratur history and matrial. Crp bhaviour also includs th phnomnon of rlaxation, which is th rduction of strss in a structur with tim whil th total strain rmains constant. Furthr, it also includs rcovry, which is charactrizd by th rduction of inlastic strain with tim aftr th strss has bn rmovd. Th phnomnon of crp is obsrvd in most of th matrials. Th oprating tmpratur in various industrial and structural applications is sufficintly high to caus significant crp as obsrvd in chmical industris, nuclar powr plants, missils, arongins, gas turbins tc. Th xprimntal tsting and valuation of crp in composit matrials ar quit complx, costly as wll as tim consuming. Thrfor, th prdiction and analysis of crp proprtis for assssing srvic lif of componnts mad of composit matrials subjctd to svr thrmo-mchanical loadings is of grat practical importanc. Accordingly, a thorough undrstanding and dvlopmnt of mthods for analyzing strss in componnts undrgoing crp is xtrmly important..3.1 Crp undr Uniaxial Stat of Strss Uniaxial crp tsts ar th basic xprimnts to valuat crp bhaviour of matrial. In this tst, a standard cylindrical spcimn is hatd to a constant tmpratur (T) of th ordr of 0.3 to 0.5T m (T m is th mlting tmpratur of th matrial) and subjctd to a constant tnsil forc F. Th normal strss inducd in th spcimn is usually much lss than th yild limit of th matrial. Th axial strain is noticd with rspct to tim till fractur of th matrial. A typical crp curv obtaind by plotting axial strain with tim is shown schmatically in Fig

8 Fig..1: A typical uniaxial crp curv. As soon as th spcimn is loadd, an instantanous dformation, which could b lastic or lasto-plastic dpnding upon th magnitud of th load, is obsrvd. It is rprsntd by portion OA of th curv. Th rmaining curv rprsnts th crp dformation, which incrass slowly with tim. For analytical convninc, it is dividd into th following thr rgions: (i) primary or nonstady stat crp (AB) in which th crp rat is dcrasing, (ii) scondary or stady stat crp (BC) in which th crp rat is approximatly constant and (iii) trtiary crp (CD) in which th crp rat incrass and ultimatly lads to ruptur at th point D. If th load is rmovd at any tim, curv similar to EF is 19

9 obtaind; som amount of strain is instantanously rcovrd. Aftr that thr is a crtain amount of strain rcovry that bcoms asymptotic with tim axis and thn no mor strain is rcovrd. Th crp strain may thrfor includ prmannt as wll as rcovrabl strain. To analyz th nginring structurs and componnts undrgoing crp, mathmatical modls of crp ar ndd. For this purpos, data obtaind from th tnsil tsts srv as th main sourc of information about crp. Th crp dformation is strongly influncd by tmpratur, strss and tim. Th crp rspons of matrial can b writtn as, f (, t, T) (.6) s whr is th crp strain, is th applid strss, t is th tim priod and T is th tmpratur of application. Th Eqn. (.6) can also b writtn in th following form, f ). f ( t). f ( ) (.7) s 1( 3 T As th tmpratur during crp tst is constant, thrfor, Eqn..7 bcoms, f ). f ( ) (.8) s 1( t Many mpirical xprssions, as a function of f ( ) and f ( ), hav bn 1 t dvlopd in th past. Th most commonly usd strss function law is Norton s law givn as f n 1 ( ) B, whr B and n ar th matrial constants. Th approximation of a crp curv to a straight lin is possibl if scondary crp rgion is prdominant and lastic as wll as primary crp is ngligibly small. Undr such conditions, it can b assumd that crp strain rat dpnds upon strss function only. Th tim dpndnc of crp has bn xprssd in trms of 0

10 Baily s mpirical law givn by: m f ( t) A1t, whr A 1 and m ar th matrial constants Uniaxial Crp undr Variabl Strss In th prvious sction, crp has bn dscribd undr constant uniaxial stat of strss. Howvr, in most of th nginring componnts subjctd to crp, th strss is variabl du to two rasons. Firstly, th load applid on th structur may vary and scondly, th strss distribution may continuously vary with tim undr constant load. Thrfor, th crp laws drivd for constant on dimnsional strss condition cannot b applid to variabl strss conditions and hnc must b modifid. Tim hardning and strain hardning ar two popular thoris, which tak into account th variabl strss and variabl tim. Th tim hardning thory stats that for a constant tmpratur and variabl strss condition, crp rat ( c ) is a function of strss and tim. f (, t) (.9) c Howvr, in cas of strain hardning thory it is assumd that th crp rat is a function of strss and accumulatd strain. f (, ) (.10) c Th particular forms of ths laws can b obtaind by assuming that th crp curv can b rprsntd by Baily-Norton law, which is a common rprsntation of crp in th primary and scondary crp rangs undr isothrmal conditions and is givn blow, n m c A t (.11) whr A, m and n ar constants whos valus dpnd upon th matrial typ. 1

11 On diffrntiating Eqn. (.11) with rspct to tim, th tim hardning law can b obtaind as blow, d c n m1 c Am t (.1) dt From th abov quation it can b sn that th crp rat dcrass with tim sinc 0 < m < 1. Furthr, Eqn. (.1) can b writtn in a form indpndnt of tim t by liminating t btwn Eqs. (.11) and (.1). (1/ m) ( n / m) ma c (.13) (1m ) / m c Equation (.13) indicats that th crp strain rat dcrass with incrasing crp strain ( c ) i.. with th progrssion of crp strain, th matrial hardns. Though, both th laws ar drivd from th sam quation, but it is obsrvd that for varying strss, th tim and strain hardning laws giv diffrnt crp rats. This diffrnc is procdural and not phnomnological. Quit oftn th strain hardning modls giv mor accurat prdictions of xprimntal rsults for stpwis changs of strss. Unfortunatly, strain hardning modls do not always yild accurat prdictions, particularly whn svral stp changs in strss occur during th sam tst (Robotnov, 1969). Pickl t al (1971) also noticd that th strain hardning modl is unabl to accuratly prdict th crp bhaviour rsulting from structural instabilitis. But for structurally stabl matrials, th prdictions by strain hardning modl ar fairly rliabl. Howvr, in th cas of gradually varying strss, both th laws giv approximatly similar prdictions..3. Crp undr Multiaxial Stat of Strss Th uniaxial crp tsts allow us to stablish th basic faturs of crp bhaviour and to stablish rlationship btwn strss, strain rat, tmpratur,

12 3 tim, tc. Howvr, in actual practic, most of th structural componnts oprat undr multiaxial strss conditions. Mtallurgical modls to dscrib crp undr multiaxial strss crp ar limitd and th availabl modls of multiaxial crp ar mainly phnomnological in form (Kraus, 1980; Boyl and Spnc, 1983; Gooch and How, 1986). Thrfor, in ordr to analyz th influnc of th strss stat on th tim dpndnt matrial bhaviour, multiaxial crp analysis is rquird. So it is important to gnraliz our conclusions rgarding crp dformations to includ th multiaxial stat of strss. Finni and Hllar (1959) proposd th gnralizd constitutiv quations for dscribing crp in isotropic matrials undr th influnc of multiaxial strss as givn blow, ) ( 1 z y x x ) ( 1 x z y y ) ( 1 y x z z zx zx 3 xy xy 3 yz yz 3 (.14) whr z y x,, and zx yz xy,, ar rspctivly th normal and shar strss componnts in x, y and z dirctions. Similarly z y x,, and zx yz xy,, ar rspctivly th normal and shar strain rat componnts in x, y and z dirctions

13 rspctivly. Th ffctiv strss ( ) and ffctiv strain rat ( ) givn abov ar xprssd by, ( ) ( ) ( ) 6( ) 1/ 1 y z z x x y yz zx xy (.15) ( ) ( ) ( ) 6( ) 1/ 1 y z z x x y yz zx xy (.16) Th Eqs. (.14) - (.16), ar basd on th assumption that th matrial is isotropic i.. th proprtis of matrial rmain sam in all th dirctions. But in actual practic thr ar som matrials which xhibits diffrnt proprtis in diffrnt dirctions and ar known as anisotropic matrials.g. wood, rolld mtal shts, laminat or sandwich structurs. As a spcial cas, som of th anisotropic matrials hav similar proprtis in a plan known as plan of symmtry. Such matrials ar calld planar anisotropic matrials and th rsulting anisotropy is calld planar anisotropy. On th othr hand, som of th anisotropic matrials show idntical mchanical proprtis about thr orthogonal plans at a point. Ths matrials ar said to b orthotropic matrials and th lins of intrsction of thr orthogonal plans ar known as axs of anisotropy. Th crp dformations occur along prfrrd orintations, which rsult into mchanical proprtis that ar dirction dpndnt (Taira t al, 1965; 1966). As a consqunc, anisotropy is inducd in a matrial undrgoing crp dformations. Taira t al (1965) dmonstratd that th spcimns takn out from thick-walld tub mad of 0.19% carbon stl at 450 o C, 470 o C and 500 o C xhibits diffrnt crp rat in diffrnt dirctions. Th crp rat in axial dirction of th tub is obsrvd to b 10% highr than th tangntial crp rat. Murakami and Yamada (1974) hav also noticd similar rsults. Thrfor, an initially isotropic matrial bcoms anisotropic during crp dformations and th 4

14 5 diffrnc btwn th prdictions basd on isotropic and anisotropic thoris of crp gos on incrasing with th progrssion of crp dformation. In this light, it is xtrmly important for th dsignr of structural componnts to considr anisotropy of th matrial during thir strss analysis for a mor ralistic approach. Bhatnagar and Gupta (1967) proposd a thortical framwork to incorporat th ffct of initial anisotropy on th crp bhaviour of matrial. Th problms of crp in th prsnc of anisotropy hav bn carrid out by som othr invstigators (Pai, 1967; Snddn, 197; Hndrson and Snddn, 197). Constitutiv quations of crp in an anisotropic matrial as proposd by Bhatnagar and Gupta (1967) ar givn blow, z y x x G H H G ) ( x z y y H F F H ) ( y x z z F G G F ) ( ) ( yz yz L ) ( zx zx M ) ( xy xy N (.17) Th ffctiv strss ( ), following Hill s critrion (Hill, 1950), is givn by, 1/ ) ( ) ( ) ( ) ( 1 xy zx yz y x x z z y N M L H G F (.18)

15 6 whr F, G, H, L, M and N ar th anisotropic constants. Th st of constitutiv quations givn by Eqn. (.17) can b transformd in trms of cylindrical polar coordinats r, θ and z as, z r r G H H G ) ( r z H F F H ) ( F G G F r z z ) ( ) ( z z L ) ( zr zr M ) ( r r N (.19) Similarly th ffctiv strss bcoms, 1/ ) ( ) ( ) ( ) ( 1 r zr z r r z z N M L H G F (.0) In cas of vanishing isotropy i.. F=G=H=L=M=N, th Eqs. (.19) - (.0) rduc to thos rportd for isotropic cas as givn in Eqs. (.17) - (.18)..3.3 Crp Law for Al/Al-Alloy Basd MMCs In aluminum basd composits, undrgoing stady stat crp, th ffctiv crp rat ( ) is rlatd to th ffctiv strss ( ) through wll documntd thrshold strss ( o ) basd crp law (Park t al, 1990; Mishra and Pandy, 1990; Mohamd t al, 199; Pandy t al, 199; Gonzalz and Shrby,

16 1993; Pandy t al, 1994; Park and Mohamd, 1995; Cadk t al, 1995; Li and Mohamd, 1997; Li and Langdon, 1997a, 1999a; Yoshioka t al, 1998; Tjong and Ma, 000; Ma and Tjong, 001), n ' Q A o xp (.1) E RT whr th symbols A, n, Q, E, R and T dnot rspctivly structur dpndnt paramtr, tru strss xponnt, tru activation nrgy, tmpratur-dpndnt Young s modulus, gas constant and oprating tmpratur. Th thrshold strss basd crp law givn by Eqn. (.1) may altrnativly b xprssd as, M o n (.) whr M 1 ' Q A xp E RT 1/ n Th crp paramtrs M and o apparing in Eqn. (.) ar dpndnt on th typ of matrial and th tmpratur (T) of application. In a composit, th disprsoid siz (P) and disprsoid contnt (V) ar th primary variabls affcting ths paramtrs. Thrfor, ths paramtrs ar functions of disprsoid siz (P), volum contnt of disprsoid (V) and oprating tmpratur (T). But th functional rlations, dscribing dpndnc of M and o on P, V and T, ar not known and nds to b stablishd. Th valus of M and o can b xtractd from th xprimntal crp rsults rportd for Al-SiCp composit undr uniaxial loading..3.4 Valu of Strss Exponnt in Crp Law Th tru strss xponnt (n) apparing in Eqn. (.1) is usually slctd as 3, 5 and 8, which corrsponds to thr wll-documntd crp cass for mtals and alloys: (i) n = 3 for crp controlld by viscous glid procsss of dislocation, 7

17 (ii) n = 5 for crp controlld by high tmpratur dislocation climb (lattic diffusion), and (iii) n = 8 for lattic diffusion-controlld crp with a constant structur (Tjong and Ma, 000). Pandy and coworkrs invstigatd high-tmpratur crp bhaviour of SiCp/Al (Pandy t al, 1990, 199) and TiB p/al (Pandy t al, 1994) composits. Th obsrvd crp data was xplaind by using substructur invariant modl givn by Shrby t al (1977). Pandy t al (199, 1994) argud that during crp in ths composits th subgrain boundaris ar pinnd by rinforcd particulats lik Al O 3 and TiB, thrby yilding a strss xponnt of 8 and an activation nrgy of th ordr of lattic slf diffusion. Gonzalz and Shrby (1993) ranalyzd th crp data of a sris of SiC/Al composits and dmonstratd that th crp in SiC/Al composits may b dscribd by using substructur invariant modl. It is also pointd out by thm that th barrir spacing plays a ky rol during crp of SiC/Al composits. Mishra and Pandy (1990) analyzd th stady stat crp data of SiC/ 6061Al composits, as rportd by Nih (1984), Nih t al (1988) and Morimoto t al (1988), and noticd that th substructur invariant modl (n = 8) could xplain th ntir st of data. It is obsrvd that th basic stady stat crp mchanism rmains unaltrd for diffrnt shaps and volum fractions of SiC rinforcmnt, whos influnc is confind to thrshold strss. Ma t al (1999) analyzd high tmpratur crp bhaviour of 15 vol% TiB /Al composits in th tmpratur rang 53 K and 673 K and obsrvd that a strss xponnt of 8 yilds th bst linar fit btwn 1/ n and for this composit. Howvr, for n = 3 and 5, th data points xhibit a clar curvatur at low applid strss. It is furthr notd that th data points at all th tmpraturs 8

18 may b wll dscribd by th substructur invariant modl. In a similar way, th crp bhaviour of 0 vol% SiCw/Al, 0 vol% (Al O 3 +TiB )/Al and 0 vol% (Al O 3 +TiB )/Al-Cu composits may also b rasonably xplaind by mans of th substructur invariant modl (Tjong t al, 1999; Ma and Tjong, 1998; Ma t al, 1999). Park t al (1990) in thir study on crp bhaviour of 30 vol% SiCp/6061Al composit dmonstratd that a strss xponnt of 5 rathr than 8 xhibits th bst linar fit btwn 1/ n s and, whr s and dnot shar crp rat and applid shar strss, rspctivly. Mohamd t al (199) in thir study on similar composit also rvald that th valu of tru xponnt (n) 5 and a tru activation nrgy for crp (Q) 08 kj/mol. Thrfor, th prvailing crp mchanism in ths composits is climb-controlld dislocation crp corrsponding to strss xponnt of 5. Li and Langdon (1998a) also mad similar obsrvations. Cadk t al (1994a, 1995) ranalyzd th crp data 0 vol% SiCw/6061Al and 0 vol% SiCp/14Al (Nih t al, 1988), 30 vol% SiCp/6061Al (Park t al, 1990), 0 vol% SiCp/Al (Pandy t al, 199), 6 vol% Al O 3 /Al 5%Mg (Dragon and Nix, 199) and 30 vol% SiCp/Al (Cadk t al, 1994b) composits and concludd that th rlationship btwn 1/ n and in ths composits could b linarly dscribd by assuming n = 5. Thy also obsrvd that th linar fitting of crp data obtaind for SiCp/Al by Pandy t al (199) could b achivd satisfactorily for various valus of strss xponnt (n = 3, 5 and 8). It is du to rlativly narrow intrval of xprimntal crp rats, which ar lss than four ordrs of magnitud. 9

19 Pandy t al (1996) invstigatd th crp bhaviour of 10 vol% SiCp/Al- 4% Mg composit and obsrvd that th valus of strss xponnt (n) = 8 and Q 4kJ / mol for this composit at highr strss lvls. Th activation nrgy obsrvd is highr than th activation nrgy for lattic slf-diffusion. Li and Langdon (1997b) during r-appraisal of th sam data noticd that n 3 and Q 15 kj / mol, which is consistnt with th crp controlld by a viscous glid procss and th dragging of magnsium atom atmosphr. It is wll supportd by th fact that th activation nrgy for diffusion of Mg in an aluminum lattic is ~ 130 kj / mol. Th markd diffrnc noticd btwn ths two analyss is attributd to th difficultis associatd with dtrmining th bst valu of n for xprimntal crp data, which span ovr a vry limitd rang of strain rats (Li and Langdon, 1999b). Th rportd fact is in agrmnt with th arlir work carrid out by Cadk and Sustk (1994), which suggsts that th crp data of MMCs should xtnd ovr at last fiv ordrs of magnitud of strain rat to nabl an unambiguous dtrmination of th valu of tru strss xponnt (n). Th crp bhaviour of 6061Al and 7005Al matrix composits rinforcd with 0 vol% of irrgularly shapd Al O 3 indicat that th tru strss xponnt is clos to 3 for Al O 3 p/6061al composit aftr taking thrshold strss into account (Li and Langdon, 1997a, 1998a). In addition, th tru activation nrgy obtaind is clos to that obsrvd for diffusion of magnsium in aluminum matrix (Li and Langdon, 1997a). Th rsults indicat that th crp in 6061 Al matrix composits is controlld by viscous glid procss of dislocations. Howvr, for Al O 3 p/7005al composit, th tru strss xponnt appars to b ~ 4.4 and th crp is controlld by a dislocation climb procss. Thrfor, th crp in mtal matrix composits is controlld by crp of matrix alloys and th crp bhaviour 30

20 of composits could b dividd into diffrnt two classs: (i) class M (mtal-typ) with n 3 and (ii) class A (alloy-typ) with n 5, as obsrvd in solid solutions (Li and Langdon, 1998b). Similar obsrvations ar also rportd by Li and Langdon (1999a) during ranalysis of th availabl crp data of 0 vol% SiCp/04Al composit (Gonzalz and Shrby, 1993). It is quit vidnt from th litratur consultd so far that though, som of th rsarchrs (Mishra and Pandy, 1990; Pandy t al, 199; Gonzalz and Shrby, 1993; Pandy t al, 1994) usd a tru strss xponnt of 8 to dscrib stady stat crp in Al-SiCp, w (subscript p for particl and w for whiskr) composits but a numbr of othr rsarchrs (Park t al, 1990; Mohamd t al, 199; Park and Mohamd, 1995; Cadk t al, 1995; Yoshioka t al, 1998; Li and Mohamd, 1997; Li and Langdon, 1997a, 1999a) obsrvd that a strss xponnt of ithr ~3 or ~5 rathr than 8 provids a bttr dscription of stady stat crp data obsrvd for discontinuously rinforcd Al-SiC composits..4 FUNCTIONALLY GRADED MATERIALS Functionally gradd matrials (FGMs) ar a nw gnration of nginrd matrials that ar gaining intrst in rcnt yars. FGMs wr initially dsignd as thrmal barrir matrials for arospac structural applications and fusion ractors (Hirai and Chn, 1999; Chan, 001; Umura, 003). FGMs also find applications in structural componnts oprating undr xtrmly high-tmpratur nvironmnts (Noda t al, 1998; Librscu and Song 005). FGMs ar th composit matrials in which th contnt of rinforcmnt is gradually varid in som dirction to achiv gradint in proprtis. Du to gradd variation in th contnt of constitunt matrials, th proprtis of FGMs undrgo smooth and continuous chang from on surfac to anothr, thus liminating intrfac 31

21 problms and diminishing thrmal strss concntrations. As an xampl FGMs basd on cramic rinforcmnt in mtal matrix, ar abl to withstand hightmpratur nvironmnts du to bttr thrmal rsistanc of cramic constitunts, whil th mtal constitunts nhanc thir mchanical prformanc and rduc th possibility of catastrophic fractur. Th application of concpt of FGMs to Mtals Matrix Composits (MMCs) has ld to th dvlopmnt of componnts dsignd with th purpos of mploying slctiv rinforcmnt in crtain rgions whr nhancd proprtis lik incrasd modulus, strngth and /or war rsistanc ar rquird (Jolly, 1990; Hirai, 1996; Takzono t al, 1996; Koizumi, 1995, 1997; Akira and Wataban, 1997; Pattnayak t al, 001; Zhu t al, 001; Kiback t al, 003)..4.1 Applications of Functionally Gradd Matrials Functionally gradd matrials (FGMs) hav a high potntial for divrsifid ara of applications. Som of th salint applications ar rportd in Tabl. blow: Tabl.: Potntial Applications of FGMs Application Aras Functionally gradd polystr-calcium phosphat Rfrnc Schillr t al (000) matrials for bon rplacmnt with a controllabl in vitro polystr dgradation rat. Smart structurs, whr pizolctric ffct could b usd to supprss th dynamic or static rspons, whil grading may rsult in rducd strsss and dformations Ootao and Tangiwara (000); Lim and H (001); Wang and Noda (001); 3

22 of pizolctric FGM structurs Ding t al (003) Functionally gradd layr btwn th Cr MO shank and Cho and Park (00) cramic tip of a cutting tool improving th thrmal strngth. Functionally gradd pizolctric actuators. Li t al (003) FGM mtal/cramic armor. Liu t al (003) Functionally gradd hatd floor systms. Takuch t al (003) Functionally gradd prosthsis joint incrasing adhsiv Quin and Datta (004) strngth and rducing pain. Functionally gradd Thrmal Barrir Coating (TBCs) for combustion chambrs, whr it is ncssary to protct Lushak t al (004); Ivosvic t al (006) mtallic or composit componnts in aircraft ngins and powr gnrators from xcssiv tmpratur Functionally gradd thrmal protction systms for Cooly and Palazotto (005) spaccraft, hyprsonic and suprsonic plans. Rotating blads in hlicoptrs and turbomachinary applications Oh t al (005); Librscu and Song (005).4. Fabrication of FGMs Th various tchniqus hav bn proposd for manufacturing of FGMs such as lctrophortic dposition (Put t al, 003; Vanmnsl t al, 005), chmical vapor dposition (Kim t al, 005), spark plasma sintring (Shn and 33

23 Nygrn, 00; Tokita, 003) and cntrifugal casting (Bishuvl and Vrwij, 000; Vlhinto, 003). Ths mthods ar usd to manufactur FGMs with th proprtis varying in th thicknss dirction. To achiv in-plan variation of proprtis in FGMs ultraviolt irradiation mthod is usd (Lambros at al, 1999). Th rduction of gas consumption and wight of a car ar th motivating forcs for growth and innovation in th automotiv industry. A significant rduction in wight can b achivd by producing cylindr linrs in Al matrix composit by mploying cntrifugal casting tchniqu. Th war rsistanc of this composit undr propr working conditions is suprior to cast iron, which is commonly usd for th production of linrs (Bonollo t al, 004). Discontinuously rinforcd composits viz. cramic particls rinforcd in aluminum alloy matrix producd by cntrifugal casting can b considrd as FGMs as thy posss varying distribution of rinforcmnt in th radial dirction owing to cntrifugal forc ffct. Th diffrnt dnsitis of cramic particls and aluminum alloy matrix lad to a cntrifugal sparation whr th highr dnsity constitunt movs towards th outr zons and vic vrsa. Th concntration profil of cramic particls in th radial dirction can b controlld and optimizd by adjusting th paramtrs such as mould rotation spd, mould tmpratur, contnt and siz of cramic particls and tmpratur of moltn aluminum. Th hardnss profil of such FGM is dirctly proportional to th amount of hard cramic particls (Kang and Rohatgi, 1996; Liu t al, 1996; Gao and Wang, 000). Bonollo t al (004) usd cntrifugal casting tchniqu to manufactur cylindr linrs in Al matrix to produc functionally Gradd (FG) composit (Al- 34

24 SiC; Al-Al O 3 ). Thy also mad an attmpt to analyz th rol of procss paramtrs in cntrifugal casting to optimiz th rinforcmnt distribution at th innr surfac of th linr. Th idal rinforcmnt distribution was achivd for som combination of main procss paramtrs, including casting tmpratur, mould tmpratur and contnt of rinforcmnt. Kiran t al (009) usd Al/17wt%Si alloy to fabricat and charactriz a FG Al/Si in situ matrial. A spcially dsignd cntrifugal casting procss was mployd for th purpos of fabricating th FGM. Th, microstructural charactristic and hardnss profil of th fabricatd FGMs was valuatd..4.3 Rsidual Strss in FGMs Whn a FGM part is coold down from its procssing or hat tratmnt tmpratur to room tmpratur, rsidual strsss ar gnratd in th composit du to th mismatch in cofficint of thrmal xpansion of its constitunts, which significantly dtriorats th prformanc of structural componnts mad of FGMs (Dlfoss t al, 1997). Th rsidual strss thus inducd is rsponsibl for diffrnt yild strngth of th composit matrial in tnsion and comprssion. Povirk t al (1991) invstigatd th ffct of thrmal rsidual strss on mchanical proprtis and ductility of Al-SiC composits. Th bhaviour of composit undr uniaxial loading was obtaind with and without th prsnc of rsidual strss. Th study rvals that th rsidual strss has a small ffct on th prdictd ductility of composit. Th rsidual strsss rdistribut as intrfacial failur is approachd. A clos nd-to-nd fibr spacing givs gratr flow strngth in comprssion than in tnsion and th rsidual strss inducd during thrmomchanical procssing tnds to nhanc this ffct. 35

25 Williamson t al (1995) prformd FEM studis to invstigat th ffcts of intrlayr and crp on th rsidual strss in nickl-alumina mtal cramic joints. Th study rvald that crp strain significantly influncs rsidual strss and joint bhaviour abov 700 K. Ravichandran (1995) prsntd on-dimnsional computation schm for stimating thrmal rsidual strsss, which inducd during fabrication of a FGM systm. Th FGM systm invstigatd consists of cramic (Al O 3 ) and mtal (Ni) phass having variation in only on dirction. Svral functional forms of gradation of constitunts hav bn xamind and it is obsrvd that th linar variation in composition from fully cramic to fully mtal rgion show th last rsidual strss. Th rsidual strsss ar found to incras whn fully cramic and/or fully mtal rgions ar includd in th structur, adjoining th gradd zon. Liw t al (003) analyzd thrmal strsss in FGM hollow cylindrical shlls by subdividing th shll into a numbr of homognous sublayrs. Th strsss and displacmnts within ach sublayr wr dtrmind from th homognous solution for th layr subjctd to th continuity and boundary conditions at th layr intrfacs and at th surfac of th shll. Frnandz t al (004) conductd an xprimntal and thortical study of th strngth diffrntial ffct (SDE) obsrvd during uniaxial tnsil/comprssiv tsting of discontinuously rinforcd mtal matrix composits. Th prdictions wr compard with th xprimntal data obtaind for 6061Al 15 vol% SiCw composits. Th analysis of rsidual strss stat was carrid out by using an xtndd Eshlby quivalnt inclusion mthodology. Th study indicats that th 36

26 rsidual strss and th mchanism of load transfr ar th origin of SDE in discontinuously rinforcd mtal matrix composits. Ann t al (006) stimatd th strngth and rsidual strsss of functionally gradd Al O 3 /ZrO discs by conducting biaxial strngth tsts. Th study indicats that th strngth of such discs, is almost doubl than th disc mad of pur Al O 3. Th incras in strngth is attributd to th comprssiv rsidual thrmal strsss in th Al O 3 surfac layr, originating du to composition gradint. Zhang t al (006) dvlopd analytical modls to prdict th distribution of thrmal rsidual strsss causd by diffrntial thrmal contraction within th multilayr coatings with gradd proprtis and compositions. Th study rvals that th rsidual strss distribution within th functionally gradd coating could b adjustd by controlling th compositional gradint or slcting a propr gradint xponnt of th coating. Bouchafa t al (010) dvlopd an analytical modl for stimation of thrmal rsidual strsss, originatd du to fabrication of xponntial functionally gradd matrial (E-FGM) matrials. Thy assumd that th thrmo-mchanical proprtis of FG layrs vary xponntially through th thicknss. Th study rvald that th magnitud of rsidual strsss incrass whn fully cramic and/or fully mtal rgions ar includd in th structur, adjoining th gradd zon. Howvr, th ffcts of tmpratur dpndnt lastic modulii and thrmal xpansion charactristics of constitunts on rsidual strss ar small. 37

27 .4.4 Crp in FGMs In th past fw yars, th lastic strsss in FGM subjctd to thrmomchanical loading hav bn analyzd by many rsarchrs (Arai t al, 1990; Erdogan and Wu, 1993; Fukui and Yamanaka, 199; Hirano and Traki, 1993; Obata and Noda, 1994; Tanigawa, 1995; Yang, 1998; You t al, 007). Howvr, th studis prtaining to crp in FGMs ar rathr scant. Zhu and Millr (1999) xamind crp bhaviour of FGM, providd with a thrmal barrir coating of Zirconium. In ordr to xamin crp bhaviour of this FGM, th thrmal gradint was producd by hating th cramic surfac with lasr. Th cramic layr was obsrvd to hav primary crp. Th tim, tmpratur and strss dpndnt dformation rsults in coating shrinkag in th loading dirction and lads to strss rlaxation. Sadananda t al (1999) invstigat comprssion crp proprtis of FG fiv layrd MoSi -Si 3 N 4 composit at 100 o C. Th fiv layrs consist of 0, 0, 40, 60 and 80 wt% of Si 3 N 4 with th corrsponding dcras in MoSi contnt. Each layr is mm thick and has uniform distribution of Si 3 N 4. Th study indicats that crp rats in a singl layr dcras with incrasing contnt of Si 3 N 4. Yang (000) proposd an analytical solution to obtain strsss in a FG layrd cylindr by taking into account th lastic and crp bhaviour of th matrial. Zhai t al (005) invstigatd th crp phnomnon of FGMs subjctd to high tmpratur by using computational micromchanical mthod (CMM). Basd on th ral microstructur of FG intrlayr with diffrnt volum fractions, 38

28 th mulation xprimnt was implmntd for th crp tst numrically and th crp paramtrs wr stimatd. Th numrical rsults indicat that th crp phnomnon is obvious not only for th mtal-rich intrlayrs but also for th cramic-rich intrlayrs. Th crp strain rat of th cramic/mtal intrlayr is largr than th corrsponding on of pur mtal undr th sam load whn th modulus of cramic phas is lowr than that of th mtal phas..5 CYLINDER A cylindr is a commonly usd componnt in various structural and nginring applications. In most of th applications, such as prssur vssl for industrial gass, transportation of high-prssurizd fluids and piping of nuclar ractors, th cylindr has to oprat undr svr mchanical and thrmal loads, rsulting in significant crp and thrby dcrasing its srvic lif (Gupta and Pathak, 001; Tachibana and Iyoku, 004; Hagihara and Miyazaki, 008). As an xampl, in th high tmpratur nginring tst ractor th tmpratur rachs of th ordr of 900 o C (Tachibana and Iyoku, 004). Th piping of ractor cooling systm ar subjctd to high tmpratur and prssur and may b damagd du to high hat gnratd in th ractor cor (Hagihara and Miyazaki, 008). Thrfor, th prdiction of long-trm crp bhaviour is xtrmly important for ths applications..5.1 Crp in Cylindr mad of Monolithic Matrial Crp analysis of thick-walld cylindr mad of isotropic monolithic matrial subjctd to intrnal prssur has bn prsntd by Wir (1957) and King and Macki (1967). Davis and Connlly (1959) studid plastic dformation of a thick-walld hollow cylindr and suggstd that for a prfctly plastic matrial, th solution obtaind corrsponds to stady stat crp condition. 39

29 Attia t al (1954) invstigatd rsidual strsss producd in cast iron hollow cylindrs by th crp rlaxation of thrmal strsss and th work is rlatd with failurs of combustion chambr parts in larg intrnal combustion ngins. Th study considrd thick hollow cylindrs subjctd to a radial flow of hat du to hating of th bor whil th outr diamtr was watr-cooling for a chosn priod of tim, during which th thrmal strsss got rlaxd by crp. Th study indicats that th crp strains ar confind to th innr layrs of th wall and rsults in a stp rsidual strss gradint within thos layrs. Th maximum rsidual strss at th bor is 4.5 tims that at th outsid diamtr. Rabotnov (1969) studid non-stady crp in rotating solid and hollow cylindrs. Th analyss assumd that th matrial of th cylindrs is isotropic and rmains so during crp. Rimrott (1959) usd gnrally accptd assumptions of constant dnsity, zro axial strain and distortion nrgy thory and drivd quations for crp rat, crp strains and crp strsss in a thick-walld, closd nd, circular hollow cylindr subjctd to intrnal prssur. Th cylindr was assumd to b mad of a homognous isotropic matrial and th strains wr considrd larg. It has bn shown that by considring finit strains, th crp rat in a thick-walld cylindr subjctd to intrnal prssur incrass, although, th crp rat in th sam matrial subjctd to constant tru strss in simpl tnsion is constant. Th assumption of infinitsimal strains lads to a prdiction of wall thicknss that is on unsaf sid. Dv (1961) obtaind th solution for a rotating hollow cylindr considring infinitsimal strains, isotropic yild critrion, and crp-strain law whil assuming incomprssibl matrial. 40

30 Finni (1960) dvlopd quations for prdiction of stady stat crp strsss and strain rats in a thick-walld tub undr combind intrnal prssur and axial load. Th numrical solutions of ths quations wr obtaind for a numbr of tubs, crp laws and loading conditions. Simpl approximat solutions, which apply for th cass in which th additional axial load is small or larg rlativ to th axial load du to intrnal prssur, ar also providd in th study. Johnson t al (1961) prsntd th analysis of strsss and strains in a thick-walld cylindr subjctd to both intrnal and xtrnal prssurs. Th chang in distribution of strss and strain was analyzd with rspct to tim. Th thortical rlations btwn strss, strain rat and tim wr drivd and applid to a numbr of mtallic alloys at tmpraturs within thir practical working rang. Pai (1967) solvd th problm for orthotropic cylindrs by using picwis linar modl. In this study it is assumd that th strains ar infinitsimal and th dformation is rfrrd with rspct to original dimnsions of th cylindr. Sim and Pnny (1971) analyzd a numbr of thick-walld tubs oprating undr crp conditions for diffrnt loadings such as intrnal prssur, xtrnal surfac loading and inrtia loading. Simonian (1979) obtaind th solution for a long thick-walld cylindr oprating in a tmpratur fild and subjctd to a cntrifugal forc, axial forc and intrnal-xtrnal prssur. It is obsrvd that th matrial of th cylindr dforms according to th non-linar thory of hrdity. Ishikawa and Hata (1980) thortically invstigatd th strsss in an infinit thick-walld tub subjctd to rapid hating at th innr-surfac. Th 41

31 matrial of th tub was assumd to hav tmpratur dpndnt proprtis and charactrizd by th Rambrg-Osgood s strss strain rlation and Norton s law for scondary crp. Gupta and Dharmani (1981) usd Sth s transition thory to driv crp strss quations in rotating hollow and solid circular cylindrs mad of transvrsly isotropic matrial. Th rsults obtaind by thm matchs with thos obtaind by Rimrott and Luk (1961) and Dv (1961), stimatd by using von- Miss yild critrion and Norton s crp law. Mishra and Samanta (1981) invstigatd finit crp in orthotropic thickwalld cylindrical shlls oprating at high prssur and tmpratur. It is obsrvd that th tmpratur variation has a significant ffct on th strain as wll as th strain-rat, particularly whn th molcular anisotropy of th matrial is takn into account. Cdrbaum and Hllr (1989) usd thory of thick orthotropic shlls to analyz cylindr subjctd to dynamic loads. Th formulation includs shar dformation and rotating inrtia ffcts similarly to th first-ordr, shar dformation, laminatd plat thory. Shukla (1997) obtaind th xprssions for lastic-plastic transitional strsss and axial strain in a comprssibl cylindr subjctd to intrnal prssur by using Sth s transition thory. Th study indicats that th prsnc of comprssibility having lssr valu at th intrnal surfac of th cylindr rducs th axial contraction and th strsss, incrass th prssur rquird for th initial yilding and dcrass its valu for fully plastic stat. Gupta and Pathak (1999) stimatd lastic-plastic strsss in a rotating non-homognous cylindr using Sth's transition thory. Th ffct of angular spd on non-homognity was discussd and dpictd graphically. Th study 4

32 indicats that a rotating cylindr mad of non-homognous matrial with nonhomognity incrasing radially rquirs a high prcntag incras in angular spd to bcom fully plastic than to its initial yilding in comparison to a homognous cylindr. Th non-homognous cylindr is much safr compard to a homognous cylindr. Gupta t al (000) analyzd crp strsss and strain rats in a rotating non-homognous thick-walld cylindr by using Sth's transition thory. Th study indicats that for a rotating non-homognous cylindr, with comprssibility incrasing radially, th circumfrntial strss is maximum at th xtrnal surfac at lssr angular spd but at som highr angular spd it bcoms maximum at th intrnal surfac. Th comprssiv valu of axial strss, obsrvd at th xtrnal surfac, incrass with an incras in angular spd. Gupta and Pathak (001) also usd Sth s transition thory to stimat crp strsss and strain rats in a thick-walld circular cylindr mad of comprssibl/incomprssibl matrials and subjctd to intrnal prssur. It is obsrvd that th maximum circumfrntial strss at xtrnal surfac of th cylindr mad of incomprssibl matrial ar rlativly highr than thos obsrvd in a cylindr mad of comprssibl matrial. Th introduction of thrmal ffcts rducs th strsss at th xtrnal surfac of cylindr whn compard to ffct of prssur alon. Strain rats ar obsrvd to b maximum at th intrnal surfac of a cylindr mad of comprssibl matrial, which tnd to dcras with radius. In th prsnc of thrmal ffcts, th crp rats ar found to incras mor at th intrnal surfac than at th xtrnal surfac, whn th rsults ar compard with cylindr subjctd to prssur alon. 43

33 Naybi and Abdi (00) dvlopd a numrical mthod to invstigat th stady stat crp bhaviour of thick-walld cylindrical prssur vssls subjctd to cyclic prssur and/or tmpratur. Shi t al (00) usd finit lmnt analysis to prsnt a mthod for assssmnt of crp lif of stam turbin parts. Th mthod taks into considration both stady and unstady crp strsss of rotors, pips, casings and valv housings. Thy usd modls with stpwis chang in thicknss to obtain stady crp strsss in rotors and discs whil pips and casings wr modld as thick-wall cylindrs. Th study prsntd quantitativ calculation along with practical xampls. Filho t al (004) drivd a lowr bound to th crp ruptur tim of intrnally prssurizd cylindrs. Thy dscribd matrial bhaviour by a phnomnological crp ruptur thory, which accounts for crp of all th phass and full coupling btwn th dformation and damag procss. Altnbach t al (008) analyzd stady stat crp in a prssurizd thick cylindr mad of alloy stl and oprating at 600 o C for linar and powr law rangs. Thy prsntd an approximat solution to illustrat strss rdistributions in cylindr as a rsult of crp procss undr plain strain condition. Th analysis shows that for a crtain rang of intrnal prssur both linar and powr law crp must b takn into account. Th rsults stimatd corrsponding to xtndd constitutiv modl diffrs from th classical ons. Hagihara and Miyazaki (008) dvlopd a mthod to prdict high tmpratur short-trm crp dformation of piping of th ractor cooling systm. Thy applid crp constitutiv quation including trtiary crp basd on isotropic damag mchanics to nuclar-grad cold-drawn stainlss stl (SUS 44

34 316) pip. Finit lmnt analysis was mployd for stimation of local crp dformation of th coolant piping undr isothrmal condition and circumfrntial tmpratur gradint. Th study indicats that th pip failur can b prdictd by intgrating th damag variabl into th crp constitutiv quation, and failur occurrd from th outsid of th pip wall. Sharma (009) stimatd crp strsss in a non-homognous thickwalld cylindr undr th combind ffct of angular spd and tmpratur. Th study concluds that a cylindr mad of lss comprssibl matrial at th intrnal surfac and highly comprssibl matrial at th outr surfac is on th safr sid of th dsign for diffrnt angular spds and tmpratur, as compard to cylindr having highly comprssibl matrial at th intrnal surfac and lss comprssibl matrial at th outr surfac. Sharma t al (009) usd Sth s transition thory to obtain th lasticplastic strsss in a transvrsly isotropic thick-walld rotating cylindr subjctd to intrnal prssur. Th study rvald that th yilding occurs at th intrnal surfac of isotropic cylindr at a highr prssur than that obsrvd for transvrsly isotropic cylindr. Thrfor, circular cylindr mad of transvrsly isotropic matrial is on th safr sid of dsign as compard to cylindr mad of isotropic matrial Crp in th Prsnc of Anisotropy Som matrials possss dirctionally dpndnt proprtis and ar known as anisotropic matrials.g. wood, rolld shts of mtal tc. Furthr, on th basis of xprimntal work conductd by various rsarchrs, it is concludd that initially isotropic matrial bcoms anisotropic during crp dformations. Th diffrnc btwn th prdictions basd on isotropic and anisotropic thoris of 45

35 crp gos on scalating as th crp dformation progrsss. In this dirction, various studis concrning crp dformations of cylindrs considrd anisotropy of th matrial. Davis (1959) analyzd crp in an isotropic thick-walld cylindr subjctd to intrnal prssur. Th rsults obtaind wr compard with th xprimntal rsults. Th discrpancy obsrvd btwn th analytical and xprimntal rsults is attributd to th dvlopmnt of anisotropy during crp. Bhatnagar and Gupta (1969) obtaind th solution for thick-walld cylindr mad of an orthotropic matrial and subjctd to intrnal prssur using constitutiv quations of anisotropy crp and Norton s crp law. Th strsss and crp rats wr stimatd undr th assumptions of plan strain, gnralizd plan strain and plan strss conditions. Th rsults stimatd for plan strain cas wr compard with thos obtaind by Pai (1967). Th study indicats that anisotropy has significant ffct on th crp bhaviour of cylindr. Bhatnagar and Arya (1974) usd finit strain thory to analyz crp bhaviour of prssurizd (intrnal, xtrnal and both intrnal as wll as xtrnal) thick-walld cylindr mad of anisotropic matrial undr larg strains. It is obsrvd that anisotropy xhibits significant ffct on th axial strss, strain and strain rat. Arya and Bhatnagar (1976) prsntd crp analysis of thick-walld anisotropic cylindr subjctd to combind intrnal and xtrnal prssurs by considring lastic strains. Th tim hardning law was usd to obtain th fundamntal quations of crp in an orthotropic cylindr. Th quations wr solvd for a particular cas of planar anisotropy. A numrical xampl prtaining to intrnally prssurizd cylindr was workd out and th rsults obtaind for th 46

36 anisotropic cylindr wr compard with thos stimatd for isotropic cylindr. It is rvald that th radial and tangntial strain rats in an anisotropic cylindr ar lss than thos noticd in an isotropic matrial at all radial locations and at all tims. Th strain rats ar obsrvd to dcras with incras in matrial anisotropy. Th study also rvals that th diffrnc in valus of strain rats at th innr radius is mor pronouncd, which diminishs with incrasing radius. Bhatnagar t al (1980) analyzd th strss and strain rat distributions in a hollow thick circular cylindr rotating about its own axis with a constant angular spd. Th cylindr was assumd to b mad of a homognous and orthotropic matrial and th stady stat crp was dscribd by Norton s law. Th ffct of anisotropy and strss xponnt (n) was invstigatd on th crp bhaviour of cylindr. Th study concluds that th strss and strain-rat distributions in th cylindr ar significantly affctd by anisotropy of th matrial and th valu of xponnt. Th ffctiv strain rat obsrvd in anisotropic cylindr is much lowr than thos obtaind for isotropic cylindr. Th study suggsts that anisotropic matrial may b bnficial for th manufactur of cylindrs bcaus of longr lif and ability to sustain largr forcs without a risk of failur undr crp. Arya t al (1983) invstigatd non-stady stat crp in a thin circular cylindrical shll mad of homognous, incomprssibl and orthotropic matrial. Th rsults indicat that th prsnc of anisotropy could lad to a bttr shll dsign with lowr strains and longr srvic lif. Bhatnagar t al (1986a) analyzd crp in a rotating thick-walld orthotropic cylindr and also confirmd that th matrial anisotropy may hav bnficial ffct on th distributions of strss and strain rat in th cylindr. Th rsults obtaind using small strain thory ar found to b on unsaf sid as 47