ࡱ>  q` $bjbjqPqP .(::@@@@@@@T 8 $8 $Th p " rtttttt$h5V9@@@@ d@ @ rr^@@ \ 06 Pvr0~,@hvJ<XTTT TTT TTT@@@@@@ Department of Aerospace Engineering Indian Institute of Technology Kanpur AE 681 Composite Materials Hashins Failure Criteria for Unidirectional Fibre Composites These are interacting failure criteria where more than one stress components have been used to evaluate the different failure modes. These criteria were originally developed for unidirectional polymeric composites, and hence, applications to other type of laminates and non-polymeric composites have significant approximations. Usually Hashin criteria are implemented within two dimensional classical lamination approach for point stress calculations with ply discounting as the material degradation model. Failure indices for Hashin criteria are related to fibre and matrix failures and involve four failure modes. The criteria are extended to three dimensional problems where the maximum stress criteria are used for transverse normal stress component. The failure modes included in Hashins criteria are as follows. Tensile fibre failure for 11 e" 0  EMBED Equation.DSMT4  Compressive fibre failure for 11 < 0  EMBED Equation.DSMT4  Tensile matrix failure for 22 + 33 > 0  EMBED Equation.DSMT4  Compressive matrix failure for 22 + 33 < 0  EMBED Equation.DSMT4  Interlaminar tensile failure for 33 > 0  EMBED Equation.DSMT4  Interlaminar compression failure for 33 < 0  EMBED Equation.DSMT4  where, ij denote the stress components and the tensile and compressive allowable strengths for lamina are denoted by subscripts T and C, respectively. XT, YT, ZT denotes the allowable tensile strengths in three respective material directions. Similarly, XC, YC, ZC denotes the allowable tensile strengths in three respective material directions. Further, S12, S13 and S23 denote allowable shear strengths in the respective principal material directions. Reference: Hashin Z. Failure criteria for unidirectional fibre composites, ASME Journal of Applied Mechanics, Vol. 47 (2), 1980, pp 329-334.     PAGE  #$9IJef+  h k       B D }vvn_jF1M hKzCJUVaJjhDbCU hDbChDbC hDbCH*hDbChGFh*$h*h7uh~+hNhOh_hN 5h_hO5CJ aJ h_h*CJ aJ h_h*5CJ aJ h_hWWCJ aJ hWW5CJ aJ hD5CJ aJ h_5CJ aJ h_h_5CJ aJ "$Jef   J L & ( ` b $a$gds$a$gds $ & Fa$gds$a$gdDbC $ & Fa$gdDbC$a$gdO$a$gd*$a$gd_$D F H J      ( * X Z \ ^ DJTƹ⣖zvrvmv hvH*h^hvjRhhKzEHUjJG1M hKzCJUVaJjhhKzEHUjKG1M hKzCJUVaJ hshsjhDbChKzEHUj5G1M hKzCJUVaJjhsU hsH*hshDbCjhDbCUjhDbChKzEHU)b RT "XYZe$a$gdO$a$gdv$a$gdv $ & Fa$gd>m$a$gds$a$gds $ & Fa$gdTV "$,.026z|(*+./23ö󮪦}u}hhK$BH*hK$Bh(h(H*h(h h 6hHkh hJH*hJhYhOh*h|QhsjhDbChKzEHUjHG1M hKzCJUVaJ hvH*jhDbChKzEHUjIG1M hKzCJUVaJhvjhvU.WZlmn$$$ $ $ $ $ $$$}$~$$$$$$ɶîh*1h6CJaJ"jh6CJUaJmHnHuh6CJaJh60JmHnHuU h60Jjh60JUh6hojhoUhCh1hDbCh hK$BhhK$BH*($$ $$Z$}$$$$$$$$$a$gd*1 $h]ha$gd6h]hgd6 &`#$gdo PAGE 2 Department of Aerospace Engineering, Indian Institute of Technology Kanpur Course: AE 681 Composite Materials Instructor: PM Mohite $$$a$gdO,1h/ =!"#$% Dd  b  c $A? ?3"`?2\fM,Dj`!\fM, "8xڵV]hY>ޙ&iժD*`Z"]IN54%RRWe_B7mPا}7>G?ϙ; UZorON9ι?3jqG`?l:vXSchik*.A^ׁZW(AP~h,5"0P? >%#a7kulH[Mp}&^q!!dbv̯Ih\<4N~9 X 'Xh9c0<J޲B?V Y8D=Ծ1ǩ(jL+UQ`mRJV%'er+DeP,MTmƿPNh.m|9*]@35Z1UC8AM;ȃsVd=s],X%[VCԗ`aĒ*$?㤅bru:ͳ+%Vӗ@x Zu9nxO؞@v:8-yYk~:~_G˛^D}У|YV>]Z5~|iAԇ'-z[;;CCC\gف~`NJROU؃ =" 2Q-UrBBșǫhpr|E>3esf6z}[֛G-ß-mI45e^C|Xa cA X W?N.ҟ)d;ߧY.C}1'=0-3;x\v8,Y/oQ`3,M@u.vz9 #]((( Q%a0B"!C%g fw 2㨦aku#pfq @g' Eڛ-8Dd b  c $A? ?3"`?26dDeT9^j`!V6dDeT9\8$xڥUOSA}i E(`)ȭT}& -#b1&`7xՄ&&ēZ۾}BN3;ٝe @Á3VT*ZFnsqps7#QVbp.rPtz87{'\Z0.SLzOkFM@\jס,Z4ȤI=]Чd4o|^i_G=QУ L ߛoA`2(6¡BqJ,OLet o3H@ @ܸ!cPVfß_{v ;}R9Y]fY17`1K^N'^m摵0VlwZ5^;hal.[ͮ;[Wk^6u +>K/v-]H?ln!v(mb=R^׾r&ZaM42 ``x(|Xx*HG+U=gWWT2q=CJ*Camd& Mu/F|y_H Tl;P,+#yA)iO*|(@|`)Ñ`a cB QQ4A})`/RuKT*If`K=q?NlPQ =R>/b8(.K\09ĐI"!DzO)RerlM0!$/61}o"ckMU{au0>T`Dd h b  c $A? ?3"`?2(m+=6 j`!~(m+= @"5LxڵW_h[U97Oss7+Y;дTLo5?5-lN/>@_ ꣂ{Ȳ)߹{{o~{)>v M 1{m7FL[?qALwxw}GQA)._[nYx>{Q<R<|έUH ˟Z^o`aA`Q` 8&x^ c1} \d#?Fo{o_ڎU}|lyKMWOk+Jځ75yXyѓY e~2'uU'41{$0rHIJb_c>[uu~{,,/P,뚊B߂ϡ 7o| G=r8%y1G2_̋U($9Ȥwhk@JJvTJ A<2R0(fN*ve]_2}…](ě2oyS݈L82UfuKlRow8>Kb ϰ~kD%w$=#ҍ9Y{)3[r5Dd pb  c $A? ?3"`?2/~7hHkhbK#cy[j`!S/~7hHkhbK#cy2N!xڵXO[U9Að,(/#آ#@7ڵ$&mf!Ɖ\|~1~آ_L,&hN|:K{Ͻeh[O={NAP @-@=1BlFϦGC 4"#%$&'(*)+.,-0/12;3<56789:>?@ABCDEFGHIJRoot Entry# F<6!Data WordDocument".(ObjectPool% 06<6_1295075020F0606Ole CompObjiObjInfo  !"#$%&'()*-0123456789:;<=>?@CFGHIJKLORSTUVWXZ[\]^_abcdeg FMathType 6.0 Equation MathType EFEquation.DSMT49qUtR DSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A   s 1Equation Native q_1295075125 F0606Ole CompObj i1 X T () 2 ++ s 122 ++s 132 S 122 ==e"1failure    <<1no failure{ FMathType 6.0 Equation MathType EFEquation.DSMT49qlR DSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A   s 11 X C () 2 ==e"1failObjInfo Equation Native _1295075147F0606Ole ure    <<1no failure{ FMathType 6.0 Equation MathType EFEquation.DSMT49q"|R DSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_CompObjiObjInfoEquation Native >_1295075146F0606APAPAE%B_AC_A %!AHA_D_E_E_A   s 22 ++s 33 () 2 Y T2 ++ s 232 "-s 22 s 33 S 232 ++ s 122 ++s  13 2 S 122 ==e"1failure    <<1no failure{ FMathType 6.0 Equation MathType EFEquation.DSMT49q/tR DSMT6WinAllBasicCodePagesOle +CompObj,iObjInfo.Equation Native /KTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A   Y C 2S 23 () 2 "-1[] s 22 ++s 33 Y C ()++ s 22 ++s 33 () 2 4S 232 ++ s 232 "-s 22 s 33 S 232 ++ s 122 ++s  13 2 S 122 ==e"1failure    <<1no failure{ FMathType 6.0 Equation MathType EFEquation.DSMT49qlR DSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E__1295075145F0606Ole ACompObjBiObjInfoDEquation Native E_1295075144F0606Ole MCompObj NiA   s 33 Z T () 2 ==e"1failure    <<1no failure{ FMathType 6.0 Equation MathType EFEquation.DSMT49qObjInfo!PEquation Native Q1Table=SummaryInformation($Y|R DSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A   s 33 Z C () 2 ==e"1failure    <<1no failure{Oh+'0\   $ 0<DLT P.M. MohiteNormal P.M. Mohite35Microsoft Office Word@P@b1i@_o6,f4& 9j_E )lS{-:<(0l`ehӂ8/!fy}=wYՌVz񊳁ZwOrɑ$ Nu7,|uz?!ǰYF" YAphi?)LZ:7vs FӤxqzNT|~C/ghI ? V+n HA"ORU9c/:< ]I)ntrtWK„L/'%Lj$ SJ)y@Cd%`uGRKòXRPGa^HGHnVJ]C'2T*pTJGC}C4PUB37+wUX2X"9|I0kHQ8Wǡ{$ qỊ0JM\!(T<+PT2.ra]I%"Ix9SӾ8D[%W%'r晌.6A`mDs $  Ņ'ŧӺC! y n9t*onǬh{)LCyD?313)'ZkvUϐ\sJ٫dco֡mF0 YGXc=v ţ6_&Z]ZΝ5՞%SRt.bٷe(_,Su}*7*3o# ƒx| LG# xNʚ{tLOHH3@O¾b80Zh?-6P.*y܃ShמTPRbs1PxH߀ŰH (h菠<|%2Yx%U̍îz.$5p% gR? Je lqe^XF4}a6lHH44f'2RUEJ2e+ wCoW'NCbXbw)=@|E17TuzAS;Dd b  c $A? ?3"`?2F^I Ic?aj`!YF^I Ic?\8'xڥU_HQ?oN?W4)S(Ї9^J\#b#e=/AO=z |1֔9پHvο9{/L(P2xT2vXr]\[D8<8/w,*Ur@dPvLHsAh~W\ ԮªhFJQ3K>]L%"IxNM6]8.ȵ+g2v w 1T DYfNrn+(-h$H߆(䁸QBЦc[rL43 } 4lPA'3^%e#~o60+8sz}֑0Yjݪݯvة czv"L-ΪJբNX^adBŽqc Mye0uw{ϖbIQh7`P}C_%xo$L]Cz{xEyͽ`j"Ӥ20OG}þ\8Zo9(ra< hOUPRB}}n"nǂniT?rh³H1%ULz:(50w%s'R? vJez lqeFo]@N4}a*THH4f'2R7tWDJ2e+sCoW'NC°nf"\}#żRWoUZDocumentSummaryInformation8`|CompObjfq՜.+,D՜.+,4 hp   IIT Kanpur  TitleH 6> MTWinEqns   FMicrosoft Office Word Document MSWordDocWord.Document.89q@@@ NormalCJ_HaJmH sH tH DA@D Default Paragraph FontRi@R  Table Normal4 l4a (k(No List4@4 *1Header  !4 @4 *1Footer  !.)@. 6 Page Number ( $JefAB^_;hiMNOZOr000000000 0000 0000 0000 00000000 000 000000000I00I00I00I00I00I00I00I00A00 5@0I0 0 @0A0 0A0 0@0@0@0@0@0I00D ;hiMK05I0I0I0K0 D T$ b $$ $ BZ\79i::::::"$!!@  @ 0(  B S  ?(  HB  C DHB  C D'!u!uԭj\;ufvf>CC"HHII $$NN9*urn:schemas-microsoft-com:office:smarttagsplace8*urn:schemas-microsoft-com:office:smarttagsCity9*urn:schemas-microsoft-com:office:smarttagsState qBJZ333A_;g5/Th ^`hH.h ^`hH.h pLp^p`LhH.h @ @ ^@ `hH.h ^`hH.h L^`LhH.h ^`hH.h ^`hH.h PLP^P`LhH.5/         ,+l|N *$~+~z36< =5>K$BCDbCGF OYhO|QGXjcHkgs7u_*O !okvWW>mN*1*sDKz1[( +^J@Lq4@ @H@UnknownGz Times New Roman5Symbol3& z Arial"1h%&&#8,,4 2QHX?O2 P.M. Mohite P.M. Mohite