SHEAR LAG MODELLING OF THERMAL STRESSES IN UNIDIRECTIONAL COMPOSITES

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1 ORA/POSER REFERENCE: ICF00374OR SHEAR AG MODEING OF HERMA SRESSES IN UNIDIRECIONA COMPOSIES Chad M. adis Departet o Mechaical Egieerig ad Materials Sciece MS 3 Rice Uiversity P.O. Box 89 Housto X 7705 ladis@rice.edu ph: ax: ABSRAC A two-diesioal shear lag odel is preseted to aalyze the steady state distributios o stress ad teperature i uidirectioally reiorced coposites. Equatios allowig or variatios i axial displaceet ad teperature alog ay give iber or atrix regio are developed. he derivatio o the goverig equatios is greatly sipliied by the assuptios that displaceets perpedicular to the iber directio ca be igored the axial displaceet ad teperature are uior over the cross sectio o ay iber ad the distributios o teperature ad axial displaceet are biliear withi the atrix regios. hese equatios are the solved or a coiguratio with uiorly spaced atrix cracks ad used to deterie the eective Youg s odulus ad theral coductivity o the cracked aterial. KEYWORDS Coposites theral stresses shear lag. INRODUCION he aalysis o stress ad strai i ulti-iber coposites has bee acilitated by the developet o shear lag odels. hese shear lag odels treat ibers as oe-diesioal load carryig structures that traser loads to oe aother through shear stresses withi the atrix aterial. he origial ulti-iber shear lag odel was orulated by Hedgepeth[] i 96. Sice the a uber o researchers have ehaced this odel by icludig eects like slidig at the iber atrix iterace or the load carryig capability o the atrix see adis ad McMeekig[]. A direct two-diesioal extesio o the Hedgepeth[] odel was proposed by Beyerlei ad adis[3] to iclude the eects o atrix stiess that reduces to the Hedgepeth[] odel i the appropriate liit. he work preseted i this paper will build o the odel o Beyerlei ad adis[3] to iclude the eects o steady state teperature distributios. hese shear lag

2 odels are ost useul i large scale coposite ailure siulatios like those carried out by Ibabdeljalil ad Curti[4] ad adis et al.[5] where ore detailed stress calculatios would be uteable.. GOVERNING EQUAIONS he odel syste cosidered is depicted i Figure with the ibers separated by the atrix i a periodic ashio. Figure : wo diesioal iber/atrix coiguratio. he Youg s odulus area theral coductivity ad theral expasio coeiciet o the ibers are E A k ad. he Youg s odulus shear odulus area logitudial theral coductivity trasverse theral coductivity theral expasio coeiciet ad width o the atrix regios are E G A k k ad W. he thickess o the syste is t hece A = Wt. Followig the procedure outlied by adis ad co-workers [-3] the equatios goverig the axial displaceet ad teperature alog the th iber u ad ad the ceter o the th atrix regio u ad are η d η d η d k ( )= 0 dx dx dx () η d η d η d k ( )= 0 () 3 dx dx dx ρ du ρ du ρ du ρ d ρ d d ( ) = G u u u 3 dx dx dx dx dx dx ρ du ρ du ρ du ρ d ρ d d G u u u ( )= 3 dx dx dx 3 dx dx dx (3) (4) EA ka Gt ρ= η= G = = E A ka E A W k kt (5) kaw

3 he essetial assuptios required to obtai these equatio are that the aterial is costraied to displace i the x directio oly shear deoratio ad trasverse teperature gradiets i the ibers are eglected ad the displaceet or teperature proile alog the trasverse directio i the atrix is biliear. 3. N-INDEPENDEN SOUIONS I geeral Eqs. (-4) represet a large set o coupled ordiary dieretial equatios or the displaceet ad teperature i each iber ad atrix regio. However i the locatios o iperectios i.e. iber or atrix cracks are distributed uiorly over all iber or atrix regios the Eqs. (-4) reduce to our goverig ordiary dieretial equatios. I other words i the distributios o displaceet ad teperature are idepedet o the iber or atrix regio uber the the ollowig equatios gover the syste: η η 4k 0 ( )= 6 (6) η η 4k( )= 0 (7) ρ ρ ρ 4 = ρ u u G u u (8) ρ ρ ρ ρ u u 4G( u u)= (9) where u ad u are the teperature ad displaceet distributios i all ibers ad all atrix regios. he ' ad the " deote irst ad secod derivatives with respect to x. he geeral solutios to Eqs. (6-9) are u = u β x = q x C e C e βx 0 0 η β x η = 0 q0x C e C e η η βx ρ ρ u = u0 0x ε0x q0x ρ ρ x x x C e C e C e C e β β β β u u u ux ρ ρ ρ q x ε x q x 8G ρ ρ ρ x x ρ x ρ C e C e Cu e Cu e ρ ρ u ux β β β β (0) () () (3) βηρ η ρ 4 ρ C C 48G = (4) βρ β ( η ) 4 48 ( ρ )( ρ ) G

4 [ ] βηρ η ρ η η ( ρ ) ( ρ 4) C C 48G = (5) βρ β ( η ) 4 48 ( ρ )( ρ ) G β 48k η = kw η 4 48G ρ βu = EW ρ 4 (6) where 0 q 0 u 0 ε 0 C C C u ad C u ust be deteried ro boudary coditios. he geeral solutio listed i Eqs. (0-6) ca be used to deterie the eective Youg s odulus ad eective logitudial theral coductivity o a coposite with a uior atrix crack spacig. Cosider the crack geoetry illustrated i Figure. Figure : Uior atrix crack spacig coiguratio. he atrix cracks are assued to be tractio-ree ad perectly isulatig. o aalyze the eective theral coductivity o the cracked coposite oly the teperature solutio eeds to be cosidered ad to deterie the eective odulus oly the displaceet solutio eeds to be cosidered. Sice the distributio o teperature or displaceet is biliear across a atrix regio oly the average heat lux or stress at a atrix crack ca be speciied to be zero. So or the theral coductivity calculatio the average teperature gradiet is set to zero at x=0 ad at x=. he solutio or the teperature distributio i a give seget o aterial betwee two atrix cracks is η coshβxcoshβ x = 0 q0 x β sihβ (7) = 0 q0 x β coshβ( x ) coshβx sihβ (8) is the average teperature across the atrix at a give x locatio. he eective where = theral coductivity o the coposite is the just the average heat lux through the coposite divided by the average teperature gradiet across the sae regio. Note that care ust be take to iclude teperature jups across the atrix cracks i these averages. he eective theral coductivity o the cracked coposite is

5 k c ka ka η coshβ = A A β sihβ (9) Eq. (9) is idetical to the result obtaied by u ad Hutchiso [6]. Fro a solutio siilar to Eqs. (7-8) or the displaceets the eective Youg s odulus is E c EA E A ρ coshβu = A A βu sihβu (0) Notice that whe the crack spacig goes to iiity the the coposite coductivity or odulus is the area ractio weighted average o the ibers ad atrix. Whe goes to zero the the coductivity ad odulus o the coposite are kc = ka ( A A) ad Ec = EA ( A A). Aother siple solutio or the uior atrix crack coiguratio is the distributio o stress i the ibers ad atrix whe the teperature o the coposite is uior. he iber stress σ ad the average atrix stress σ are σ = E σ = E ρ sihβu( x ) sihβux ρ sihβu sihβu( x ) sihβux ρ sihβu () () Notice sice there is o applied load oly teperature that σ A σ A =0. 4. DISCUSSION he siple solutios preseted i Eqs. (0-6) do ot exploit the true value o the shear lag odel. Geerally solutios to the equatios o theroelasticity could be obtaied with series solutios or the siple geoetry cosidered i Figure leadig to ore accurate estiates o the coposite coductivity ad odulus. However it is diicult to obtai exact solutios or the distributios o stress ad teperature aroud eve a sigle iber break. Cotrarily siple superpositio techiques exists that ca solve Eqs. (-5) or ay iite uber o arbitrarily located iber or atrix cracks. Hece the sipliied shear lag odel is ost useul i large ailure siulatios like those carried out i [4] ad [5] where rapid solutios or the iteractios o cracks with oe aother ad the loadig are iportat. REFERENCES. Hedgepeth J.M. (96) NASA N D-8.. adis C.M. ad McMeekig R.M. (998) It. J. Solids Structures Beyerlei I.J. ad adis C.M. (999) Mechaics o Materials Ibabdeljalil M. ad Curti W.A. (997) It. J. Solids Structures adis C.M. Beyerlei I.J. ad McMeekig R.M. (000) J. Mech. Phys. Solids u.j. ad Hutchiso J.W. (995) Philos. ras. R. Soc. odo Ser. A