Anisotropic discs loaded by parabolically distributed pressure

Transcription

1 Available lie at Draft ScieceDirect Draft Draft Structural Itegrity Prcedia 00 (016) st Eurpea Cferece Fracture, ECF1, 0-4 Jue 016, Cataia, Italy Aistrpic discs laded by parablically distributed pressure Christs F. Markides, Stavrs K. Kurkulis* Labratry f Testig ad Materials, Departmet f Mechaics, Natial Techical Uiversity f Athes, Zgrafu Campus, Athes, Greece Abstract The cmplex ptetials which describe the elastic equilibrium f a circular disc made f a trasversely istrpic, hmgeeus material are preseted. It is assumed that the disc is laded by a parablic distributi f radial stresses which act alg tw fiite circular arcs, atisymmetric with respect t the disc s ceter. The prblem is here slved withi the frame f liear elasticity assumig plae strai cditis. The cmplex ptetials techique is adpted as it was frmulated by Lekhitskii i his pieerig ctributis. Whe the cmplex ptetials are determied, e ca btai the respective stress field develped all ver the disc fr ay value f the agle betwee the axis f symmetry f pressure distributi ad the plaes f material istrpy. Atteti is fcused at the disc s ceter ad at the laded diameter. Cclusis regardig the applicability f the Brazilia-disc test i case f specimes made f trasversely istrpic materials are draw. 016 The Authrs. Published by Elsevier B.V. Peer-review uder respsibility f the Scietific Cmmittee f ECF1. Keywrds: Aistrpic materials; Trasverse istrpy; Brazilia-disc test; Cmplex ptetials, Stress tesr 1. Itrducti The Brazilia-disc test is the mst cveiet ad widely used substitute f the direct tesi test. It was iitially itrduced fr istrpic materials (Careir 1943; Akazawa 1943; Hdrs 1959). Hwever, mst rck-like materials exhibit sme kid f aistrpy ad therefre the applicability f existig slutis fr stresses ad displacemets becmes questiable. The prblem is usually cfrted umerically (Da & Kietzky 014) r experimetally (Vervrt et al. 014) sice aalytic slutis are very rare (Exadaktyls et al. 001). I this ctext, a attempt t btai such a sluti is described here, usig Lekhitskii s (1968, 1981) cmplex ptetial techique fr rectiliear * Crrespdig authr. Tel.: ; fax: address: stakkur@cetral.tua.gr The Authrs. Published by Elsevier B.V. Peer-review uder respsibility f the Scietific Cmmittee f ECF1.

2 Christs F. Markides, Stavrs K. Kurkulis / Structural Itegrity Prcedia 00 (016) aistrpic materials. The advatage f the sluti itrduced is that the disc is csidered laded by a parablic (rather tha uifrm) distributi f radial stresses, a assumpti much clser t reality Kurkulis et al. (01).. Theretical csideratis Csider a liearly elastic, hmgeeus ad rectiliearly trasversely istrpic bdy. Let E, ν ad G be the Yug s mdulus, Piss s rati ad shear mdulus i the plaes f istrpy, ad E, G the elasticity ad shear mduli i plaes rmal t thse f istrpy. Let als, ν be the Piss s rati defiig the magitude f dilatati i plaes f istrpy fr cmpressi rmally t them. Itrducig a Cartesia referece system {O; x,y,z} with its rigi at the disc s ceter ad its y-axis rmal t the plaes f istrpy, the geeralized Hke s law reads as: 1 1, E E E G x x y z yz yz , E E E G E y x y z xz xz xz , E E E G z x y z xy xy (1) It is see that frm the twelve -zer strai cefficiets α ij ly five (α 11, α 1, α 13, α ad α 66 ) are liearly idepedet. A cylidrical disc f radius R ad thickess w is w cut frm this bdy, with its crss-secti rmal t the plaes f istrpy. This trasversely istrpic disc, deted frm here trastrpic (fr brevity reass), is cmpressed, i cmplete absece f fricti, betwee the jaws f the device suggested by the Iteratial Sciety fr Rck Mechaics (ISRM) fr the implemetati f the Brazilia-disc test by a verall lad P frame actig withi its crsssecti. P frame frms a arbitrary agle ϕ with respect t the material layers. Mrever, that P frame is parablically distributed alg tw symmetric parts f the disc s lateral surface each e f area Rω ο w (Fig. 1), fllwig the law: P frame Half-ball bearig ϕ x y σ r Upper jaw R ω ο O 1.5R Guidig pi ω ο Lwer jaw σ r Fig. 1. The device suggested by the Iteratial Sciety fr Rck Mechaics fr the stadardized implemetati f the Brazilia-disc test.

3 Christs F. Markides, Stavrs K. Kurkulis / Structural Itegrity Prcedia 00 (016) r P Pc 1si si, Pc P, P 0 () max The elastic equilibrium f that trastrpic disc is t be determied. I this ctext, the disc s crss-secti is csidered lyig i the z=x+iy=re iθ cmplex plae (Fig. ). Assumig that w is cmparable t R, plae strai cditis are adpted. Takig w it accut Lekhitskii s apprach fr aistrpic cylidrical bdies, with their faces beig plaes f elastic symmetry (Lekhitskii 1981), the equilibrium equatis, Hke s geeralized law, the equatis f cmpatibility ad the respectice budary cditis, fr zer bdy frces, are reduced respectively t: x xy x y x y xy x 0, 0 u v u v,, x y y x x 11 x 1 y y 1 x y xy 66 xy y x xy 11x 1y 1x y 66xy 0 cs, x cs, y X, cs, x cs, y Y (6) x xy xy y (3) (4) (5) where u, v are the Cartesia cmpets f displacemet, X, Y are the cmpets f σ r L ad β ij =α ij α i3 α j3 /α 33 are the s-called reduced elastic cstats, which frm Eqs.(1) equal: , 1, 1, 66 E E E E G (7) Accrdig t the thery f elasticity, Eqs.(3) imply the existece f the Airy s fucti F(x,y) s that: P frame y ω ο σ r = P(θ) x Plae f elastic symmetry Plaes f istrpy w z=re iθ r θ O ϕ ο R L y Ε ν G x ω ο z Ε, ν, G Ε, ν, G z P frame Fig.. The islated trastrpic disc uder parablic pressure: Cfigurati f the prblem ad defiiti f symbls.

4 4 Christs F. Markides, Stavrs K. Kurkulis / Structural Itegrity Prcedia 00 (016) F F F xy x, y y, x xy (8) Eq.(5) prvides the geeralized biharmic equati i F: F F F x x y y (9) with a characteristic equati: (10) Eq.(10) has fur purely imagiary rts, the s-called cmplex parameters, as fllws: 1 i 1, i, 3 1 i 1, 4 i ; (11) , 1 ; (1) ver-bar detes the cmplex cjugate. The, fllwig Lekhitskii (1981) it ca be writte: F x,y F z F z (13) with z 1 =x+μ 1 y ad z =x+μ y the s-called cmplicated cmplex variables, ad: z F z, z F z (14) the Lekhitskii s cmplex ptetials (where prime detes the first derivative). Φ 1 ad Φ are btaied frm the give values f stresses the disc s budary. Namely, stresses (Eqs.(8)) ad their values L (Eqs.(6)), expressed i terms f Φ 1 ad Φ as (Lekhitskii 1981): x 1 1 z1 z, y 1, xy 11 (15) S S (16) z z Y ds, z z X ds with S the arc legth L. Accrdig t Lekhitskii (1968), Φ 1 ad Φ are sught i series frm as: z A A z A P z, z B B z B P z ; 0 1 1, 1, 1, 1, 1, 1, 1, 1, 1, P z R 1 i z z R 1 z z R 1 (17) (18)

5 Christs F. Markides, Stavrs K. Kurkulis / Structural Itegrity Prcedia 00 (016) Whe z=re iθ is L (r=r), the z 1, =R[(1 iμ 1, )+(1+ iμ 1, )s 1 ]/ with s=e iθ. Substitutig i Eqs.(16) frm Eqs. (17), (18), expadig the right-had sides f Eqs.(16) i Furier series frm ad cmparig cefficiets f the same rder f s=e iθ, the fllwig systems f equatis are btaied fr determiig the real () ad imagiary ( ) parts f cefficiets f Φ 1 ad Φ f Eqs.(17): 1 t 1 t a b 11 t1 1 t1 1 t 1 t a b 11 t1 1 t1 B 1t 1t 1t 1t A 1 1 1, 3,5,... B 1t 1t 1t 1t A (19) with t 1, =(1+ iμ 1, ) /(1 iμ 1, ) ad: a 1 P R 1 si cs si 4 si si e b i c i 3 si e si 4 cs si 4 si cs 4 i i e, 3 si 3 a 1PR si 1 1 si 1 b i 1 1 c si 1cs si 1 si cs e i 1 i 1 1 PRsi 1 c 1 si 1 i 1 si 1 e 1 cs si 1 si cs e 5,7,... i 1 i 1 e (0) (1) the Furier series cefficiets f the parablic pressure iduced the disc (a =b =0, =0,,4, ). Fr =1, e btais the fllwig relatis (Lekhitskii 1968): A B A B a a R A B A B a a Ri b b R A B A B b b Ri () with the respective Furier series cefficiets f parablic pressure give as: i a 1 1PcR si i e si cs si e si b1 i si si (3)

6 6 Christs F. Markides, Stavrs K. Kurkulis / Structural Itegrity Prcedia 00 (016) Clearly, the three Eqs.() ly prvide the fur terms A,B,A adbapart 1 frm a arbitrary real cstat; fr =0, A 0 ad B 0 remai cmpletely arbitrary. Thus, the cmplex ptetials Φ 1, Φ characterizig the equilibrium f the elastic trastrpic disc have bee determied (with the ly excepti f a real ad tw cmplex cstats) ad therefre the prblem shuld be csidered slved, at least ccerig the cmpets f the stress field. Regardig parameters P c ad ω, appearig i the abve frmulae, they ca be arbitrarily predefied assumig that P frame remais cstat. Hwever, a alterative apprach is prpsed here, i rder t achieve a mre accurate apprximati f the actual values f these quatities. I this directi, the frmulae itrduced by Markides ad Kurkulis (01) fr the respective disc-jaw ctact prblem fr istrpic materials (Kurkulis et al. 01), are here further develped, i rder t accut als fr a trastrpic disc. It is the ccluded that: 3P 6K P 1 J 1 P, Arcsi, K 3K Rw Rw 4G 4G frame frame c J (4) with κ(ϕ ), κ J ad G, G J the Muskhelishvili s (1968) cstats ad shear mduli f the disc s ad jaw s crsssectis, respectively, as if bth f them were made f istrpic materials. Mrever, fr the plae strai cditis csidered here ad assumig that κ(ϕ )=3 4ν(ϕ ), it ca be see that: E E cs si cs si E E (5) 3. Results ad Discussi Itrducig Φ 1 ad Φ btaied befre i the geeral frmulae f Eqs.(15), the stress-field cmpets ca be calculated at ay pit f the trastrpic disc. As a example, a disc f radius R=0.05 m ad thickess w=0.01 m is csidered here. The disc is made f a trastrpic serpetius schist. Its mechaical prperties were prvided by Barla ad Iaurat (1973) ad read as: E=58 GPa, E =7 GPa, ν=0.34 ad ν =0.1. The disc is cmpressed by a verall frce equal t P frame =0 kn. Ft the umerical calculatis it was csidered that =59 fr the additial terms i all previus frmulae. Fially, P c ad ω are calculated frm Eqs.(4) ad (5) whereas G is determied usig the fllwig frmulae (Lekhitskii 1981): EE G E1 E (6) As a first step, atteti is fcused at the disc s ceter, give that the stress tesr at this pit is crucial fr the sud determiati f the tesile stregth accrdig t the iitial ccept f Careir (1943) ad Akazawa (1943). The rati, ξ, f the rmal stresses, i.e. the rati ξ= σ r /σ θ f the radial- ver the trasverse-stress at the specific pit is pltted i Fig. 3. It is clear frm this figure that ξ strgly differs frm the respective value fr the istrpic disc, which is equal t 3 (Hdrs 1959), almst idepedetly f the actual stress distributi alg the laded rim (Fairhurst 1964; Hbbs 1965; Huds et al. 197; Kurkulis et al. 013; Markides & Kurkulis 01). Fr the specific cmbiati f mechaical prperties, the value f ξ varies frm a maximum value equal t abut 5.0 fr ϕ =0 t a value equal t abut. fr ϕ =90. Equally imprtat is the fact that the shear stress cmpet τ rθ at the disc s ceter is -zer, i spite f the gemetry ad ladig symmetry. Althugh the magitude f these stresses is a rather small prti f the respective trasverse stress σ θ, it is by meas igrable fr the whle rage f fr ϕ agles. The variati f the τ rθ /σ θ rati versus the agle ϕ is pltted i Fig.4. It is bserved that this variati is t mtus. A clear extremum appears fr ϕ =30. This -mtus behavir shuld be expected, sice fr ϕ =0 ad ϕ =90 the shear stress must be zered due t the additial symmetry f the rietati f the material layers with respect t the ladig axis. Frm a quatitative pit f view, the shear stress attais values equal t almst e furth (τ rθ /σ θ =0.38) f the respective trasverse stress at ϕ =30.

7 Christs F. Markides, Stavrs K. Kurkulis / Structural Itegrity Prcedia 00 (016) σ r /σ θ (at the disc's cetre) 4 3 τ rθ /σ θ (at the disc's cetre) ϕ [deg] ϕ [deg] Fig. 3. The radial- ver the trasverse-stress at the disc s ceter agaist agle ϕ. Fig. 4. The shear- ver the trasverse-stress at the disc s ceter agaist agle ϕ. Bth pits discussed i the previus tw paragraphs raise serius questis ccerig the applicability f the Brazilia-disc test fr the determiati f the tesile stregth f trastrpic materials. Ideed, it has bee defiitely clarified by may researchers (see fr example Jaeger (1967) ad Fairhurst (1964)), that i rder fr the quatity determied by the Brazilia-disc test t be a reasable represetati f the material s tesile stregth, fracture must start frm the disc s ceter ad fr this t be guarateed ξ shuld lie withi specific limits. As a ext step the variati f the stress cmpets all alg the laded diameter (i.e. alg the symmetry axis f the parablic distributi f the radial stresses actig alg the laded rims) is csidered. Adptig the same as abve umerical values fr the prblem parameters, the stress cmpets are calculated ad pltted frm r=0 t r=r i Fig.5. Three cases are csidered fr agle ϕ equal t 15, 45 ad 75. I the embedded figure a magified view f the distributi f the stress cmpets arud the disc s ceter is shw. It is see that frm a quatitative pit f view (ad besides the presece f shear stresses) the distributis f the rmal stresses clsely resemble τrθ τrθ σr σθ σr ϕ =15 σθ Fig.5. The distributi f the rmal ad shear stresses alg the laded diameter fr three characteristic values f the agle ϕ. The embedded figure is a magified view f the stress distributi relatively clse t the disc s ceter.

8 8 Christs F. Markides, Stavrs K. Kurkulis / Structural Itegrity Prcedia 00 (016) the respective es f the istrpic disc. The mai differece is the fact that fr r=r, i.e. the disc s periphery, the rmal stresses (radial ad trasverse) are t equal t each ther, i.e. σ θ σ r, while i the case f a istrpic disc, fr r=r, it hlds that σ θ =σ r (Markides ad Kurkulis 01). Equally imprtat is the fact that as e appraches the laded rim (i.e. fr r R), the shear stress cmpet starts icreasig abruptly befre it becmes zer fr r=r. 4. Cclusis The cmplex ptetials characterizig the equilibrium f a elastic circular disc made f a trasversely istrpic material were determied aalytically. The disc was csidered uder the acti f a parablic distributi f radial stresses alg tw atisymmetric arcs f its periphery, which clsely resembles the pressure iduced the disc i case it is squeezed betwee tw curved metallic jaws. Mrever, the legth f the laded arcs was assumed t be a fucti f the lad iduced ad the mechaical prpreties f the materials f bth the disc ad the jaws. Takig advatage f the cmplex ptetials, it was pssible t explre the stress field at the disc s ceter ad als all alg the laded diameter. It was ccluded that the applicability f the Brazilia-disc test becmes questiable fr tw reass: The rati f the trasverse-t-rmal stress at the disc s ceter is t cstat ad als shear stresses appear, which fr specific values f the agle betwee the laded diameter ad the material layers may eve reach e furth f the respective trasverse stress. As a result, it cat be a-priri guarateed that facture starts frm the disc s ceter (a requiremet sie-qua- fr the test t prvide reasable results). It is thus strgly suggested t avid usig the Brazilia-disc test fr the determiati f the tesile stregth f trastrpic materials uless a prper fracture criteri is first applied. Ackwledgemets The fiacial supprt f the Natial Techical Uiversity f Athes thrugh the research prject is mst kidly ackwledged. Refereces Akazawa, T., New test methd fr evaluatig iteral stress due t cmpressi f ccrete (the splittig tesi test) (part1). J Japa Sc Civil Egieers 9, Barla G., Iaurat N., Idirect tesile testig f aistrpic rcks. Rck Mechaics 5, Careir, F.L.L.B., A ew methd t determie the tesile stregth f ccrete. I: Prceedigs f the 5 th Meetig f the Brazilia Assciati fr Techical Rules, 3d. Secti, (i Prtuguese). Da, D. Q., Kietzky, H., 014. Numerical simulatis ad iterpretatis f Brazilia tesile tests trasversely istrpic rcks. Iteratial Jural f Rck Mechaics ad Miig Scieces 71, Exadaktyls, G.E., Kaklis, K.N., 001. Applicatis f a explicit sluti f the trasversely istrpic circular disc cmpressed diametrically. Iteratial Jural f Rck Mechaics ad Miig Scieces 38(), Fairhurst, C., O the Validity f the Brazilia Test fr Brittle Materials. Iteratial Jural f Rck Mechaics ad Miig Scieces 1, Hbbs, D. W., A assessmet f a techique fr determiig the tesile stregth f rck. British Jural f Applied Physics 6, Hdrs, G., The evaluati f Piss s rati ad the mdulus f materials f a lw tesile resistace by the Brazilia (idirect tesile) test with particular referece t ccrete. Australia Jural f Applied Scieces 10, Huds, J. A., Brw, E. T., Rumel, F., 197. The ctrlled failure f rck discs ad rigs laded i diametral cmpressi. Iteratial Jural f Rck Mechaics ad Miig Scieces 9, Jaeger, J. C., Failure f rcks uder tesile cditis. Iteratial Jural f Rck Mechaics ad Miig Scieces 4, Kurkulis, S. K., Markides, Ch. F., Chatzistergs, P. E., 01. The stadardized Brazilia disc test as a ctact prblem. Iteratial Jural f Rck Mechaics ad Miig Scieces 57, Kurkulis, S. K., Markides, Ch. F., Hemsley, J. A., 013. Frictial stresses at the disc-jaw iterface durig the stadardized executi f the Brazilia disc test. Acta Mechaica 4(), Lekhitskii, S. G., Aistrpic Plates (Eglish traslati by Tsai S. W.), Grd ad Breach, New Yrk. Lekhitskii, S. G., Thery f Elasticity f a Aistrpic Bdy, Mir, Mscw. Markides, Ch. F, Kurkulis, S. K., 01. The stress field i a stadardized Brazilia disc: The ifluece f the ladig type actig the actual ctact legth. Rck Mechaics ad Rck Egieerig 45(), Muskhelishvili, N. I., Sme Basic Prblems f the Mathematical Thery f Elasticity. Grige, Nrdhff. Vervrt, A., Mi, K. B., Kietzky, H., Ch, J. W., Debecker, B., Dih, Q. D., Frühwirt, T., Tavallali, A., 014. Failure f trasversely istrpic rck uder Brazilia test cditis. Iteratial Jural f Rck Mechaics ad Miig Scieces 70,