A Finite Element Analysis of a Crack Penetrating or Deflecting into an Interface in a Thin Laminate S. Kao-Walter 1a, P. Ståhle 2b, S.H.

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1 Key Engineering Mterils Vl. 3 (006 pp nline t (006 Trns Tech Publictins, Switzerlnd A Finite Element Anlysis f Crck Penetrting r Deflecting int n Interfce in Thin Lminte S. K-Wlter, P. Ståhle b, S.H. Chen 3c Dept. f Mechnicl Engineering, Schl f Engineering, Blekinge Institute f Technlgy, S-37 79, Krlskrn, Sweden. Div. f Mteril Mechnics, Mlmö University, Mlmö, Sweden. 3 LNM, Institute f Mechnics, Chinese Acdemy f Sciences, Being, 00080, Chin. Shrn.k-wlter@bth.se, b per.sthle@ts.mh.se, c chenshhu7@htmil.cm Keywrds: Interfce, Crck tip driving frce, Adhesin, Deflecting, Penetrting. Abstrct. The crck tip driving frce f crck grwing frm pre-crck tht is perpendiculr t nd terminting t n interfce between tw mterils is investigted using liner frcture mechnics thery. The nlysis is perfrmed bth fr crck penetrting the interfce, grwing stright hed, nd fr crck deflecting int the interfce. The results frm finite element clcultins re cmpred with sympttic slutins fr infinitesimlly smll crck extensins. The slutin is fund t be ccurte even fr firly lrge munts f crck grwth. Further, by cmpring the crck tip driving frce f the deflected crck with tht f the penetrting crck, it is shwn hw t cntrl the pth f the crck by chsing the dhesin f the interfce reltive t the mteril tughness. Intrductin Thin luminium fil nd plymer lminte cmpsites hve prved their usefulness nd high ptentil thrugh mny pplictins in vrius fields. In prticulr fr pckging mterils, cmpnents mde frm lminted mterils re frequently designed t withstnd extreme lds, nt the lest during the prductin. A better understnding f the filure mechnisms is useful fr the design prcess nd the prductin prcess f lminte with these cmpnents. An imprtnt tsk in the present wrk is t study in detil the frcture behvir f crck situted in n luminium fil (Al-fil lyer nd terminting t the interfce between this nd lw density plyethylene (LDPE lminte. Mny investigtins f the interctin between crck nd n interfce hve been perfrmed. One f the erliest wrks n the subject ws mde by Zk nd Willims []. Lter, Wng nd Ståhle [] used dislctin simultin t extend the nlysis t finite crcks nd, recently, Chen et l. [3] hve studied similr prblem fr finite bdy. By pplying finite element methd, [4-6] presented different nlyses f stress intensity fctrs nd energy relese rtes fr crcks in different bi-mterils. The present wrk pplies finite element methd t nlyze the crck tip driving frce f precrck tht penetrtes r deflects t fllw the interfce between stiff nd wek mteril. The mteril prperties f n Al-fil/LDPE lminte hve been selected becuse f its industril imprtnce. It is ssumed tht the pre-crck is lcted in the stiffer mteril nd tht the interfce is perfect bnd between the tw mterils. Bth mterils re ssumed t be the hmgeneus. Bsic Equtins Accrding t Zk nd Willims [], the stress distributin in the vicinity f tip f crck tht is perpendiculr t bi-mteril interfce cn be written: σ λ ( / r f ( θ, α, = ξ. ( 0σ β All rights reserved. N prt f cntents f this pper my be reprduced r trnsmitted in ny frm r by ny mens withut the written permissin f the publisher: Trns Tech Publictins Ltd, Switzerlnd, (ID: /0/06,0:4:4

2 74 Frcture f Mterils: Mving Frwrds Here, σ 0 is remtely pplied stress, ξ 0 is nn-dimensinl gemetry dependent cefficient nd λ 0 is btined s the lrgest rt in the intervl 0 λ 0 < f the fllwing generting equtin: (β α csλ π = + β ( λ + α + β β. ( The ngulr functins f re knwn (cf. []. Abve, α nd β re the tw s clled Dundurs prmeters [7]: α = Γ (κ + (κ + Γ (κ + + (κ +, β = Γ (κ (κ Γ (κ + + (κ +, (3 where Γ = µ / µ, κ i = 3 4ν i fr the plne strin cse nd κ i = (3 ν i /(+ν i fr the plne stress cse. The mteril prmeters ν i nd µ i re Pissn s rti nd the sher mdulus f lyer i where i= r. Fr µ / µ 0, the expnded result f Eq. gives (cf. [] λ = 0.88 µ / µ. (4 Accrding t [3], this is s even in finite slid in the regin f 0<r/ <.0 fr σ x nd 0<r/ <0.5 fr σ y. A deflecting crck,, lng the interfce is, ccrding t [8], surrunded by the stress field ( / / + iε r g ( θ, α, σ = η σ, (5 β where g re knwn ngulr functins (cf. [], σ is remtely pplied stress nd η is gemetry dependent cnstnt. If the crck prpgtes shrt distnce,, lng the interfce, vi kink, the stress surrunding the tip f the kink shuld be the stress field given in Eq.. The stresses in the field f Eq. therefre frm bundry lyer tht embedded the crck tip nd the kinking crck. Thus the stresses σ in Eq. replce the remte stresses σ in Eq. 5. Thus, the result is redily btined s: λ ( / ( / / + iε r g ( θ, α, σ = ξ. (6 σ β Here gemetry dependent cefficient ξ replces the cnstnts ξ 0 nd η nd ε = + β ln. (7 β Eq. 6 is vlid s lng s the kink is fully embedded by the stress field relted t Eq.. Similrly, the tip f penetrting crck with the length,, extending int the secnd mteril is surrunded by stress field ( / r / h ( σ = η, (8 σ θ where η is n intrduced gemetry dependent cnstnt. The ngulr functins h re knwn s the ngulr functins f the Willims expnsin. After replcing the stress σ in Eq. 8 with σ defined in Eq., the fllwing expressin is btined fr penetrting crck embedded by the field relted t Eq.. σ = ξ σ ( / λ ( / r / h (θ. (9

3 Key Engineering Mterils Vl Here ξ is cnstnt. By Eq. 6 nd Eq. 9, the crck tip driving frce, G, my be written s fllws (cf. [8]: ν λ0 λ 0 ξσ µ G =. (0 Here, ξ is cnstnt relted t Dunders prmeters. In the cse when the crck deflects distnce int the interfce ξ = ξ (α, β. While in the cse when the crck penetrtes length int hmgenus mteril ξ = ξ, cf. [8]. One cn ls relte the crck tip driving frce t the elstic energy relese rte thrugh [9]: dp G = B d. ( Here is n pplied cnstnt displcement impsed n the edges f the specimen nd B is ssumed t be mm fr the present nlysis. P is the resulting rectin frce. After integrting ver the full grwth f the kink lng the interfce r eqully ver the full grwth f the stright crck, Eq. cn be written s: P Gd = dp = ( P P 0. ( P Here the grwth is in bth cses cnsidered s strting frm the riginl crck with the length. By using Eq. 0, the left side f the Eq. becmes: ( ν λ ( λ ξσ Gd =. (3 0 µ ( λ Hence, fter using the reltin f Eq. 0, we finlly btin, ( λ G = ( P P. (4 This result is nw bsed n the ssumptin tht the stress fields described by Eq. 6 r lterntively Eq. 9 re vlid. With Eq. 4 we re nw equipped with tl fr evlutin f the sympttic fields with regrd t their pplicbility fr lrger kinks nd fr crcks extending fr int the secnd mteril. The limittin t the pplicbility f the sympttic fields will revel itself s difference f G clculted using Eq. 0 nd Eq. 4 by inserting the clcultin results with finite element methd.

4 76 Frcture f Mterils: Mving Frwrds y E D u h interfce h B d A p d Mteril (LDPE Mteril (Al-fil O u x W Fig.. A crck perpendiculr t the bimteril interfce f finite slid. Here, the penetrting crck nd the duble deflecting crck re illustrted by p nd d. Finite Element Anlysis A generl purpse finite element cde [0] is used fr the clcultins. In ttl 735 eight nde plne elements with reduced integrtin is used. The rti f the liner extent f lrgest versus the smllest element is 86. Due t the symmetry, nly hlf f the gemetry is mdelled. A thrugh thickness pre-crck with the length is plced in the stiffer lyer. The nrmlized gemetry prmeters re W / = 3.0, h / =. 0 nd h / = 8. A remte displcement u ws pplied t the bundry DOC in Fig.. Dundurs prmeters where chsen t α =-0.99 nd β = Fr pln strin, this crrespnds t, e.g., the sher mdulus f Al-fil µ = 5.5 MP nd tht f LDPE µ.= 300 MP. The crrespnding Pissn s rti ν is put 0.3 fr bth mterils. Thus, the expnent λ (cf. [6]. In the cse f penetrtin r deflectin, crck with the length p r d is plced s shwn in Fig.. A series f rectin frces P cn be btined. Thus, the crck tip driving frce, G p r G d, cn be clculted fr different crck lengths by pplying Eq. 4. Results nd Discussins Fig. shws the nrmlized crck tip driving frce G d /G nd G p /G fr pre-crck tht penetrtes r deflects distnce frm the interfce. It is ssumed tht the pre-crck is in the stiffer mteril nd tht the interfce is perfectly bnded. The Dundurs prmeter α = mening tht the stiffness f the stiffer mteril is rund 50 times lrger thn the stiffness f the weker mteril (cf. Fig.. Bth the finite element results nd the sympttic results re shwn in Fig.. The sympttic slutins re fund t be ccurte even fr firly lrge munt f crck grwth. Suppse nw tht the tughness f the interfce is equl t Gic nd the mde I tughness f the mteril is equl t Gc. As it hs been discussed in [], s lng s the kink is fully embedded in the stress field Eq. the crck will deflect int the interfce nd remin there if G ic G < d. (5 G c G p F C

5 Key Engineering Mterils Vl G / G FEM, deflected sympttic, deflected FEM, penetrted sympttic, penetrted / Fig.. Nrmlized crck tip driving frce G/G O fr deflected nd penetrted crck versus nrmlized kink /. Here, G / L µ h ( λ ( + υ. = ( lg(g/g 5.00 Fig. 3 shws the rti f the crck tip driving frce fr deflectin nd fr penetrtin, G d /G p, versus different α vlues. The results re cmpred t tht frm [8]. Gd greement cn be fund when α<0. Fr α = This gives G d /G p =0.488 mening tht the crck tip driving frce fter deflectin frm the stright pth is rund hlf f the vlue fr penetrtin (cmprisn is mde t the sme crck length. In reference [8] different Dunders' prmeter bet, i. e. β = 0, ws used. This my hve cused the differences between the result f [8] nd the present result (see Fig. 3. In the present nlysis rti f length f kink t length f riginl crck / = which my, even thugh smll, be insufficient t recver the sympttic field surrunding the tip f smll kink. Fr = the result is bcked up by the results in Fig. b. But fr is lrger thn, sy, 0.3 the result my be questined. This cmputtin explins tht the prbbility f crck deflectin int the interfce t incresed displcement is lrge nly if the tughness f the interfce is less thn hlf f the tughness f the weker lyer. A full understnding f the finl frcture f n Al-fil / LDPE lminte requires cnsidertin f plstic defrmtin becuse f the high ductility f the LDPE lyer. The tughness f ductile mteril tested using lrge-scle gemetry cnnt esily be trnsferred t very smll specimen lg(/ FEM Ref.[8] 3 G d /G p 0-0 α Fig. 3. The rti f the crck tip driving frce f deflected crck t duble penetrted crck t the sme crck length fr different mteril cmbintins.

6 78 Frcture f Mterils: Mving Frwrds Here, interctin with the Al-fil is expected since the liner extent f the plstic zne is f the sme rder f mgnitude r lrger thn the thickness f the lminte lyers (cf. []. The crck tip driving frce fr very lng crck in the interfce hs lwer limit which equls the elstic energy stred in the tw lyers f the lminte. Fr lwer limiting tughness f the interfce the crck will, therefre, just spntneusly grw in the interfce t given ld. Cnclusins The frcture f lyered lminte is studied s crss-sectin crck prpgtin. A finite element methd is used t clculte driving frces fr the crck tip depending n whether the crck will grw int the interfce r will just cntinue stright hed. The lyer tughness shuld be cnfined between lwer limit nd n upper limit t ensure delmintin. The results give guidnce fr the selectin f the rti f mteril stiffnesses nd the rti f frcture tughnesses f the undmged lyer nd the interfce t mximize the tughness f the entire structure. Here mteril prperties f n Al-fil nd n LDPE plymer lminte ws chsen. Fr these mterils, the cnclusin is tht the tughness f the interfce shuld be t lest hlf f the tughness f the LDPE lyer if benefit frm the tughening delmintin is desired.lrge-scle plstic effects were nt cnsidered in this nlysis. It is nticipted tht such effects wuld limit the relibility f the results fr ccurte quntittive predictins f ductile mterils (e.g. Al-fil / LDPE. Acknwledgements This wrk ws grnted by Tetr Pk R&D AB, Tetr Pk Crtn Ambient AB nd the Print Reserch Prgrm (TF. The uthrs wuld like t thnk the cmputer- nd prgrm supprt t Blekinge Institute f Technlgy nd FEM-tech AB. Thnks ls t Mr. Chng Li t Mlmö University fr ssisting during the erly stges f the finite element clcultin. References [] A.R. Zk nd M.L.Willims: J. Appl. Mech. Vl. 30 (963, p. 4. [] T.C. Wng nd P. Ståhle: Act Mech. Sinic Vl-4 (998, p. 7. [3] S.H. Chen, T.C. Wng nd S. K-Wlter: Int. J. Slids nd Structure Vl. 40 (003, p. 73. [4] P. Ståhle, nd C.F. Shih: Mteril Reserch Sciety Sympsium Prceeding Vl.39 (99, Bstn. [5] P. Delfin, J. Gunnrs nd P. Ståhle: Ftigue Frct. Engng Mter. Struct. Vl.8 (995, p. 0. [6] S. K-Wlter: On the Frcture f Thin Lmintes (Dctrl thesis, Blekinge Inst. f Tech., Sweden, ISSN: , 005 [7] J. Dundurs: In Mthemtics f Dislctin (edited by T. Mur, ASME, New Yrk, 969, p. 70. [8] M.Y. He, A. Evns nd J.W. Hutchinsn: Int. J. Slids nd Structure Vl.3 (994, p [9] T.L. Andersn: Frcture Mechnics, Fundmentls nd Applictins (CRC Press LLC, ISBN: , Flrid, USA, 995. [0] ABAQUS, User s mnul, (versin 6.4, Hibbit, Krlssn nd Srensn, ABAQUS, Inc. Printed in U.S.A., 003. [] M.Y. He nd J.W. Hutchinsn: Int. J. Slids nd Structure Vl.5(989, p. 053.