FRACTURE MECHANICS ANALYSIS OF SINGLE FIBER FRAGMENTATION TEST

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1 FRCTUR MCHNICS NLYSIS OF SINGL FIBR FRGMNTTION TST Ramrez F.., ha B.., and Carlsson L.. Department o Mehanal ngneerng, Florda tlant Unversty 777 Glades Road, Boa Raton, FL 3343, US Department o Oean ngneerng, Florda tlant Unversty 0 N. Beah Road, Dana Beah, FL 33004, US BSTRCT smple rature mehans model was developed to quanty the ber/matrx nterae rature toughness usng a moded test proedure or the sngle ber ragmentaton test (SFFT). The approah onsders ntally the event o the rst ber break ollowed by nstantaneous ber/matrx debondng usng the typal SFFT proedure. It s realzed, however, that determnng the nteraal rature toughness rom the nstantaneous F/M debondng event an be too omplex sne ths event s dependent o the energy released by the ber break. In order to separate the ber rature and debond propagaton events, a moded test proedure was developed. The new proedure onssts o unloadng and loadng the SFFT spemen ater the rst ber break and nstantaneous F/M debondng are aheved. In ths way the energy release rate or debond propagaton an be determned ndependently o other rature proesses. The moded SFFT proedure was perormed on -glass/vnylester spemens to llustrate the applablty o the rature mehans model proposed n ths analyss. The rature toughness value obtaned or ths system, G d =6 J/m, s n agreement wth toughness values obtaned by other authors or smlar systems.. INTRODUCTION The strutural ntegrty and letme o polymer ompostes are rtally dependent on the stablty o the ber/matrx (F/M) nterae regon [,]. Thereore, t s extremely mportant to haraterze the F/M nterae to understand the overall perormane o polymer matrx ompostes. ong the test methods to evaluate the bondng between ber and matrx, the sngle ber ragmentaton test (SFFT) method s the smplest n terms o the expermental setup, and thereore ommonly used []. The SFFT onssts o a sngle ber embedded n a resn matrx molded nto a dog-bone spemen. The spemen s tested n tenson and, the spemen s properly desgned, the ber wll break nto several ragments untl ber break saturaton ours. The nal ragmentaton length s alled rtal length, l [3]. Several models have used the rtal length to determne the F/M nterae shear strength [3,4]. Typally, the F/M nterae shear strength has been estmated rom the Kelly-Tyson shear lag analyss [4]. The Kelly-Tyson model, however, s assoated wth nauraes sne t assumes a onstant shear stress along the F/M nterae and does not aount or other alure proesses that are nvolved n the ragmentaton test. For nstane, t has been observed that every ber break s aompaned by a nte amount o F/M debondng (or nstantaneous debondng) [5] and thereore, a rature toughness quantty suh as the energy release rate or propagaton o nteraal debondng may be better lled to quanty the F/M nterae adheson than a shear strength parameter. Currently there are several models based on rature toughness rteron to evaluate the F/M nterae [6,7]. These models, however, are very omplex beause they ntegrate the eets o energy released by a ber rature wth debondng propagaton.

2 The am o ths work s to develop a smple rature mehans model and test proedure or quantyng the nteraal rature toughness. Three undamental rature events are onsdered: ber break, F/M debond ntaton, and debond growth. In order to separate the ber rature and debond propagaton events, a moded sngle ber ragmentaton test proedure s proposed. The new test proedure onssts o unloadng the SFFT spemen ater the rst ber break and nstantaneous F/M debondng are aheved. The SFFT spemen s then loaded agan untl the exstng debond regon grows and n ths way the energy release rate or debond propagaton an be determned ndependently o the energy released rate by the rst ber rature. xperments to llustrate the applablty o the rature mehans model developed n ths analyss are perormed on SFFT spemens onsstng o an -glass ber and vnylester matrx whh are typal materals used n naval applatons.. FRCTUR MCHNICS MODL The undamental rature events n the ragmentaton test are llustrated n Fgure. Fgure : Shemat o rature events and assoated omplane o SFFT spemen. In the rst stage, there s no damage, and the spemen omplane s C o. ter the ber breaks, the energy release rate by the ber rature, G, s determned n terms o the derene n spemen omplane beore, C o, and ater the ber breaks, C. In the seond rature stage, t s onsdered that the exess o released energy by the ber rature may generate nstantaneous F/M debondng. n expresson or G or debondng ollowng a ber break an be derved n terms o omplane derene between C and C (not derved n ths paper). Fnally, n the thrd stage, the SFFT spemen s assumed to be unloaded ater the ber breaks and debonds, and then loaded agan untl the exstng debond starts to grow. The nteraal rature toughness, G d, s determned by the derene n spemen omplane (.e. C 3 -C ) ater an exstng debond length starts to extend. In the rst stage o ths analyss, we wll onsder only the event o rst ber alure assumng that the ber/matrx nterae remans undamaged. The break s assumed to our n the enter o the spemen enablng onsderaton o the hal symmetry seton n Fg.. Ths assumpton s not physally reasonable but should be vald or an arbtrary rature ste.

3 x Fgure. Shemat o ber stress dstrbuton ater a ber break n a symmetry seton o SFFT spemen or the ase o no F/M debondng. The ber stress dstrbuton around a ber break s llustrated n Fgure. ter the ber breaks, t arres no load at the break pont. The regons o the broken ber near the ber break pont are subjet to nteraal shear stress whh wll transer load rom the matrx nto the ber. The zone over whh the ber stress bulds up s dented as the neetve zone. The neetve length, L, s the dstane requred to reover the stress n the ber to the ar-eld stress. The zone where the ber s ully loaded away rom the break pont s onsdered as the eetve zone. The ber tensle stress dstrbuton n the neetve length regon s gven by an axsymetr model developed by Whtney and Drzal [8], σ x x L ( x ) = ( )e ε 0 L () where x s the dstane rom the ber break pont along the ber axs ( 0 x L ), see Fg.. The ar-eld regons o the ber and matrx are assumed to be subjeted to a unorm axal stran ε 0. s a materal property onstant dened as: 4K Gmv = + ( ν vm ) ( K + G ) () where K m = m ( / G v / ) (3),, ν, and ν are the ylndral orthotrop ber axal and transverse modul and Posson s ratos respetvely, and G reers to the shear modulus n the transverse plane o the ber. m and G m are the Young s and shear modulus o the matrx. The Whtney-Drzal model provdes the ollowng expresson or the neetve length:.375( d / ) L = µ (4) where µ = Gm G (5) 4ν m

4 To mplement a rature mehans analyss, the total dsplaement o the SFFT spemen under tensle loadng beore and ater a ber break has to be alulated. Beore a ber break, the total spemen dsplaement, o, s gven by Hooke s law as: PL δ o = (6) where and are the ross seton area and eetve Young s modulus o the SFFT spemen and P s the axal load = ( R) (7) 4 = V + Vmm (8) P = ε o (9) where R s the radus o the (assumed) ylndral SFFT spemen. V and V m are the volume raton o the ber and matrx alulated as: V = and Vm = (0 a,b) where and m are the ross seton areas o ber and matrx, = d and = ( a,b) 4 ter a ber break (see Fg. ), the total dsplaement o the SFFT spemen,, s gven by the sum o the dsplaements o the eetve (e) and neetve () regons, =,e +, () The dsplaement n the eetve zone s gven by: P( L - L ),e = (3) The dsplaement n the neetve zone s gven by: L, = (x)dx 0 (4) Where the stran n the neetve regon s: σ m( x ) ε ( x ) = (5) m m s the matrx stress whh may be obtaned rom equlbrum o ore, P σ ( x) ε ( x) = (6) m Substtuton o q. (6) nto (4) yelds: PL =, mm (7) The total dsplaement o the spemen ater the ber breaks s: P(L L ) PL = + m (8)

5 The energy release rate, G, or raturng a ber n the SFFT spemen s gven n terms o omplane, P C G = (9) where P s the load requred to break the ber n the SFFT spemen, s the new surae area reated and t s onsdered as the ross seton area o the ber, = π d (0) 4 and C s the derene n the SFFT spemen omplane beore and ater the ber break. The SFFT spemen omplane beore the ber break, C o, s gven by: o L Co = δ = P () and the SFFT spemen omplane ater a ber break, C, s gven by: δ ( L L ) L = = + C P m () Hene, the energy release rate when the ber breaks n the SFFT spemen s gven by: G = P L π d m m (3) 4 s mentoned beore, the exess o released energy by the ber rature wll generate nstantaneous F/M debondng. I debondng between ber and matrx ollows a ber break, three zones are onsdered or F/M debondng analyss as shown n Fgure 3 (.e. the eetve zone, the neetve zone and the debond zone). Fgure 3: Shemat the ber stress dstrbuton n a symmetry seton o SFFT spemen ater a ber break wth F/M debondng. In ths stage o the analyss, an expresson or the nteraal rature toughness, G d, s derved based n the event o debond growth, see Fg. (C 3 C ). In order to separate the nluene o the energy released by the ber rature on debond propagaton, the SFFT spemen s unloaded ater a ber break and nstantaneous debondng are aheved. Then the spemen s loaded agan untl the exstng debond length extends. The energy release rate by debond propagaton, G d, s determned by the derene n omplane, (.e. C 3 -C ), ater an exstng debond length extends.

6 The dsplaement o the spemen ontanng a ber break and nstantaneous debondng s gven by: =,e +, +,d (4) where the dsplaement at the eetve regon,,e, s gven by: P( L - L Ld ),e = The dsplaement at the neetve regon,,, s obtaned rom q. (8) PL = = δ, δ, mm (6) and the dsplaement at the debond regon s gven by: PLd,d = (7) d where the eetve modulus o the debond regon s: = d Vm (8) m The omplane o the SFFT spemen ater nstantaneous debondng s: ( L L Ld ) L Ld C = δ = + + P m (9) d When the SFFT spemen s loaded agan (ater unloaded at ber alure and nstantaneous debondng stage), the exstng debond length wll start to extend at a rtal load, P. I the F/M debondng length extends an nrement d(l d ), the eetve length wll derease by d(l d ) (at eah sde o the ber break pont), see Fg.3. Thereore, the SFFT spemen omplane ater debond propagaton, C 3, s gven by: ( L L Ld dld ) L Ld + dld C 3 = + + m (30) d The derene n omplane s: C = C = 3 C dld d (3) and the nrement n debond area s gven by: = d ( dl d ) (3) The energy release rate by debond propagaton s then, P G d = π d d (33) From the ature rteron, the exstng debond length wll start propagatng when G d s equal to the nteraal rature toughness, G d. Thereore, the nteraal rature toughness s gven by: P C P Gd = = π d d (34) quaton (34) reveals that G d does not depend on the debond length. Hene, the sngle ber ragmentaton test should produe stable F/M debond propagaton. (5)

7 3. XPRIMNTL 3. Materals and spemens The experment was perormed on the sngle ber ragmentaton test spemen shown n Fg. 4 onsstng o an -glass ber and vnylester Derakane 8084 (V D8084) matrx. The propertes o ber and matrx are lsted n Table. Fgure 4: Sngle ber ragmentaton test (SFFT) spemen [9]. Table : Mehanal propertes o -glass ber and vnylester matrx. Fber xal modulus, (GPa) 73 Transverse modulus, (GPa) 73 xal Posson s rato, v 0. Stran to alure, (%) 8-0 Fber rature toughness, ( J / m ) 0 [0] Dameter, d (m) 4 Matrx Young's modulus, m (GPa).7 Posson's rato, m Test Proedure The SFFT spemen s loaded n tenson usng a small tensle stage. The ber rature and F/M debondng are examned durng SFFT testng usng photoelast patterns observed n optal transmsson mrosopy (Olympus BX4 wth a QICM-FST 394 amera). The regon around a ber break exhbts a olored pattern alled brerngene. Ths phenomenon s aused by shear stresses n the matrx []. Fgure 5 shows a shemat o the brerngene patterns around a ber break L d Fgure 5: Shemat eature o photoelast patterns around a ber break. The rst part o the experment onssts n loadng the SFFT spemen n tenson untl the ber breaks. The load requred, P, to break the ber s reorded. The nstantaneous debond length reated by the energy released n ths event s measured and t s used as a reerene to later study debond propagaton. Then, the SFFT spemen s unloaded, and loaded agan untl the exstng debond length nreases. The load requred to extend the debond, P, s reorded.

8 4. RSULTS ND DISCUSSION Ten replate -glass/vnylester SFFT spemens were tested. The spemens were loaded n tenson untl reahng a ber break. Fgure 6 shows a typal brerngene pattern at the rst ber break whh ourred at.% stran. The stran s alulated rom the appled load, P, usng the expresson: ε = P / ( ). Ths alulaton assumes lnear-elast response. Some o the hgher strans enountered were beyond the proportonal stran lmt o the matrx [], but the devaton rom lnearty was qute small. o =.% Fgure 6: Brerngene pattern o the rst ber break n a SFFT spemen. Fgure 7: Fber break pont at external loadng released state. ter determnng the load and the length o the nstantaneous ber/matrx debond regon, L d, at the ber rature event, the SFFT spemen was unloaded. Fgure 7 llustrates the regon near the ber break pont when the external load was removed. ven though the brerngene patterns partally dsappeared ater unloadng the SFFT spemen, there was stll evdene showng that the ber/matrx nterae was damaged by the energy released at ber rature. The SFFT spemen was loaded agan and the ber/matrx debond length at the rst ber break was measured at derent appled strans. Fgure 8 shows the brerngene eets near the ber break as the externally appled stran was nreased. It s evdent that at ertan rtal stran (load), the exstng debond length start to propagate. o =.7% o =.% o =.3% o =.4% o =.7% Fgure 8: Fber/matrx nterae debond growth. 4. Determnaton o Debond Frature Toughness The load, P, requred to break the -glass ber n the SFFT spemen was determned to be N. The energy release rate, G, orrespondng to the ber alure load s J/m (q. (3)). (Narn reported that the energy released rate durng ragmentaton o T50 arbon bers s around 4600J/m [6] whh s n reasonable proporton to the value reported here or -glass bers). small porton o ths released energy atually

9 goes nto raturng the ber (t has been ound that the toughness o a glass ber s only around 0 J/m [9]), whle part o the remanng energy s responsble or the nstantaneous F/M debondng observed [6]. Fgure 9 depts the length, L d, o the debonded regon near a ber break vs. appled stran or spemens that were prevously unloaded ater rst ber rature. L d (m) Debond growth Fber break Stran (%) Fgure 9: Debond length growth as a unton o appled stran. From Fgures 8 and 9, t an be dedued that, or ths omposte system, the exstng debond length starts to propagate at an external load, P, approxmately equal to the load requred to break the ber. Furthermore, the energy release rate requred to grow an exstng debond (or nteraal rature toughness, G d ) determned rom q. (34) s J/m. The energy requred or debond propagaton has been estmated usng pull-out experments by D Frana et al. [3]. They reported a rature toughness, G d, o 50 J/m or glass/epoxy whh s omparable to the rature toughness (6 J/m ) ound n ths researh or a glass/vnylester system. It must be pont out that ths model does not aount or the presene o rton at the nterae. more rgorous analyss nludng rtonal shear stress atng over the debonded regon s urrently been developed. 5. CONCLUSIONS The objetve o ths paper was to develop a test methodology and a smple rature mehans model or determnng the toughness o the ber/matrx nterae by means o the sngle ber ragmentaton tests. It has been realzed by other authors [6] that the energy onsumed by raturng the ber n the SFFT spemen s muh less than the energy avalable n the SFFT spemen at the pont o ber rature. The remanng energy s onsumed n other proesses suh as nstantaneous F/M debondng. Determnng an nteraal rature toughness rom the nstantaneous F/M debondng event an be too omplex sne ths event s dependent o the energy released by the ber break. Thereore, a moded SFFT proedure was desgned to determne the nteraal rature toughness by separatng the events o ber break and nstantaneous debondng rom the event o debond growth. The new SFFT proedure onssts o unloadng the spemen ater the rst ber break and nstantaneous F/M debondng are aheved. The spemen s then loaded agan untl the exstng debond regon grows. The rtal load requred to ntate debond propagaton s then reorded and the energy

10 release rate or debond growth an be determned ndependently o the energy released rate by the ber rature. Ths proedure was perormed on -glass/vnylester spemens to llustrate the applablty o the model. The rature toughness value determned n ths paper, G d = 6 J/m, s n reasonable agreement wth results obtaned or glass/epoxy by other nvestgators [3]. CKNOWLDGMNTS The authors aknowledge support or ths work at Florda tlant Unversty rom ONR grant No. N managed by Dr. Yapa Rajapakse. Thanks are due to Shawn Pennell or the art work. RFRNCS. Ray, B.C., Temperature et Durng Humd geng on Interaes o Glass and Carbon Fbers Renored poxy Compostes Journal o Collod and Interae Sene, 006, 98: -7.. Rao, V., Herra-Frano, P.J., Ozzelo,.D., Drzal, L.T. Dret Comparson o the Fragmentaton Test and the Mroond Pull-out Test or Determnng the Interaal Shear Strength The Journal o dheson, 99,34: Drzal, L.T, Rh, M.J., Campng, J.D., Park, W.J., Interaal Shear Strength and Falure Mehansms n Graphte Fber Compostes 35 th nnual Renored Plasts/Compostes Conerene, Paper 0C, Kelly.., Tyson, W.R., Tensle Propertes o Fber-Renored Metals: Copper/Tungsten and Copper/Molybdenum Journal o the Meh. and Phys. o Solds, 965, 3, , 5. Km B.W., Narn J.., xpermental Veraton o the ets o Frton and Resdual Stress on the nalyss o Interaal ddbondng and Toughness n Sngle Fber Compostes, Journal o materal Sene, 00, 37: Narn J.., Lu Y.C., On the Use o nergy Methods or Interpretaton o Results o Sngle-Fber Fragmentaton xperments, Composte Interaes, 996, 4: Wagner H. D, Narn J.. and Deteasss, Toughness o Interaes rom Intal Fber-Matrx Debondng n a Sngle Fber Composte Fragmentaton ppled Composte Materals, 995, : Whtney J.M., Drzal L.T., xsymmetr Stress Dstrbuton round n Isolated Fber Fragment, STM Speal Tehnal Publaton, 987, Feh, S., Wonsyld, K., Mnzar, D., Westmann, P. and Lholt, H., Testng Proedure or the Sngle Fber Fragmentaton Test, Rso Natonal Laboratory, Rso-R- 483(N), Rosklde, Denmark, Harrs, B., Morley, J., Phllps, D.C., Frature Mehansm n Glass-Renored Plasts, Journal o Materal Sene, 975, 0:050. Tmoshenko, S., Theory o lastty MGraw-Hll, Ramrez, F.., Carlsson, L.., ha, B.., valuaton o nvronmental ets on Polymer Matrx Compostes by Mromehanal and Maromehanal Tests, 3 th uropean Conerene on Composte Materals, Frana, C.D., Ward, T.C., Claus, R.O., The Sngle Fbre Pull-Out Test : Revew and Interpretaton, Compostes: Part, 996, 7:59-6