Debon crack growth in atigue along iber in UD composite with broken ibers Johannes Eitzenberger an Janis arna Dept o Applie Physics an Mechanical Engineering Lulea University o Technology, Sween SE 97 87, Lulea, Sween Janis.arna@ltu.se ABSTRACT Assuming that Paris law is applicable or iniviual ebon crack propagation along the iber/matrix interace, the relate strain energy release rate in a uniirectional composite is analyse using FEM an also using simple analytical consierations base on sel-similar ebon crack propagation. Moel with axial symmetry consisting o three concentric cyliners is use: partially ebone broken iber in the mile is surroune by matrix cyliner which is embee in a large block o eective composite with properties calculate using rule o mixtures an Halpin-Tsai expressions. It is shown or pure mechanical loaing that the iber elastic properties have a huge eect on the release energy, whereas iber content in the composite in the consiere realistic range has eect only or short ebons. The interaction between ebons approaching rom both iber ragment ens is investigate an relate to material properties an geometrical parameters. It is shown that the sel-similar ebon propagation moel gives slightly overestimate values o the strain energy release rate which may be relate to interaction eects not inclue in the analytical moel.. INTRODUCTION Since the iber strain to ailure in polymer matrix iber reinorce uniirectional (UD) composites is lower than the matrix strain to ailure the irst ailure event in tensile loaing in these composites is statistical iber ailures. Due to stress transer over the interace the stress in the iber is recovere an with increasing loa multiple iber breaks in are possible. ery oten the iber ailure, which is assume to be a penny-shape crack transverse to the iber axis, is an unstable phenomenon an the energy release uring this event is larger than require. The excess o energy may go to initiation o the iber/matrix ebon at the tip o the iber crack. In other wors the eboning is a creation o ree iber surace growing along the iber in the axial irection. The ebon initiation (transition rom no ebon state to ebon state) is very complex process an ue to lack o relevant inormation it is not suitable or racture mechanics treatment. The eboning can be consiere as an interace crack growth along the iber an racture mechanics may be use or the crack evolution analysis. The stress state at the iber crack an in the ebon tip region is very complex. For long ebons plateau region exists away rom the iber crack an rom the ebon crack tip. Due to urther ebon crack growth the plateau region becomes larger. Long ebon cracks propagate in a sel-similar manner meaning that when the crack grows the local stress proile at the ebon crack ront shits along the iber axis without changes in the shape an in the value. ery long ebons (comparable with the hal-length o the iber ragment) start to interact an the sel-similarity is lost. The energy release rate uring ebon propagation has been previously calculate or ebon along a single iber ragment embee in an ininite matrix in so calle single iber ragmentation (SFF) test. The use methos cover a wie spectrum rom approximate
analytical to numerical base on inite elements (FE) or bounary elements (BE) [-7]. The variational moel base on minimization o the complementary energy [] is probably one o the best analytical solutions but the accuracy is achieve in rather complex calculation routine. The most careul numerical analysis o the local stress state at the ebon crack tip in terms o stress intensity actors an egree o singularity has been perorme in [6] using BE metho. Unortunately this metho at present is limite to isotropic constituents an, hence, not applicable or carbon ibers. enerally speaking, most o the escribe approaches may be aapte or ealing with partially ebone broken iber surroune by matrix in a composite. However a systematic parametric analysis o the energy release rate as aecte by constituent properties, geometrical parameters is not available. The objective o this paper is to perorm the abovementione analysis using FE an to ientiy the most signiicant parameters inluencing the strain energy release rate. A three concentric cyliner assembly moel is consiere. A broken iber in the mile is surroune by a matrix cyliner an the interace is partially ebone, see Fig.. This iber /resin block is embee in outer eective composite cyliner. Figure : Schematic showing the three cyliner geometry with the partially ebone iber in the mile. The stress istribution in the ront o the ebon crack an the isplacement proile behin the crack are inicate.. MODE ENERY RELEASE RATE In the particular case o the ebon crack the raial stresses on the iber surace are compressive. It is ue to larger Poisson s ratio o the matrix an also ue to higher thermal expansion coeicient (i thermal stresses are accounte or). This means that the crack propagation in the analyse problem is in Moe. Eects relate to riction at the interace are neglecte. In atigue Paris law may be applie which requires the change o the strain energy release rate to be calculate. β l = A () N c The unit cell o the composite with a partially ebone iber is shown in Fig.. The raius o the transversally isotropic iber is r. The outer raius o the matrix cyliner r m is relate to the iber content in the composite by = ( r r ) m. To represent the ininite eective composite surrouning the iber/matrix unit, the outer raius o the cyliner assembly R is large.
. The crack closure technique The energy release rate is calculate using the virtual crack closure technique [8] stating that the energy release ue to the crack surace growth by A is equal to the work require to close this newly create surace rom size A + A back to size A. For the ebon crack Points at the ebone suraces o a crack with size other the relative isplacement being u l + l l + l ( z) = u ( z r = r ) u ( z, r r ) l + l z mz = To close the crack by A = πr l (), z [, + ] l + l are sliing with respect to each l l (3) l we have to apply an increasing tangential traction at each point l + l [ l, l l ] to move it back by u ( z) l l σ ( z)z, where ( z) z + point z equals to zr The work perorme equals to zr l W = u. When it is one the value o the traction in σ is the shear stress in ront o a crack with size l. + l l ( z) σ ( z)z zr (4) In the virtual crack closure technique the assumption is that the sliing isplacement iel at the tip o the crack with size l + l is the same as at the tip o the ebon with sizel u l + l l ( z) = u ( z l ) This assumption is base on the assume sel-similarity o the crack growth between l an l + l which is a goo assumption as long as l is small. The beneit o this assumption is that only one stress state calculation or a given ebon length l is require. From (4) an (5) ollows expression or the work perorme to close the crack by l l + l l l W = πr u ( z l ) σ zr ( z) z (6) l Changing the origin to l by introucing z = z l, equation (6) turns to l l l ( W = π r u z + l l ) σ zr ( z + l ) z (7) W The energy release rate () is eine as = which using (7) gives πr l In numerical calculations l l ( l ) = u ( z + l l ) ( z l ) l lim σ zr + z (8) l l l is usually inite an the calculate energy release rate value epens on the integration istance l. The calculate value is calle energy release rate over istance l. This quantity enote l is analyse in the present paper. (5) 3
. Sel-similar ebon crack growth region I the ebon length is several times larger than the iber raius, the stress state at the iber crack is not interacting with the stress state at the ebon crack tip. Aitionally assuming that the iber ragment is long enough an the interaction with the ebon approaching rom the other en o the iber is negligible one may state that the ebon is growing in a sel similar manner. It means that ue to ebon growth by l the stress perturbation region at the ebon tip is shite in the z-irection by l without any changes in the stress proiles an values. From other han the complex stress state region at the iber crack tip remains unchange. Thinking in terms o the change o the strain energy o the whole system we can observe that the energy change analysis is very straightorwar: a region with the volume πr l which previously ha the strain energy as or long three cyliner assembly with perectly bone interaces is now replace by the same volume where the iber cyliner is separate (ebone) rom the rest o cyliners. Denoting the strain energies or these two states by inexes an an or simplicity neglecting thermal terms we obtain πr l U = U U = ε ( E E ) (9) In (9) E an E is the elastic longituinal moulus o the consiere part o the cyliner assembly in the initial state (perect boning) an in the inal state (ebone iber). The elastic moulus change between two cases may be calculate using FEM but there is also an exact analytical solution available [9,,]. This solution is use to calculate the elastic mouli. Aitional assumption mae is that in the ebone case the raial stresses ue to the presence o the ebone iber insie the assembly may be neglecte an the strain energy o the ebone iber in the zero riction case is equal to zero. The strain energy release rate is calculate as leaing to U = () πr l ε = ε R = ε ( E E ) () 4r As an alternative to the concentric cyliner assembly moel the rule o mixtures can be use to calculate elastic mouli in (). 3. RESULTS AND DISCUSSION Calculations were perorme or carbon iber an or glass ibers in polymeric matrix. The use elastic properties o the matrix are E m =3 Pa ν m =. 4 () The isotropic glass iber has properties E = 7 Pa, ν =. (3) The elastic properties o the transversally isotropic carbon iber are as ollows E L = 5 Pa, E T = 3 Pa, LT = Pa, ν LT =., ν T 3 =. 45 (4) The properties o the eective composite were calculate using the rule o mixtures or longituinal moulus an Poisson s ratio an Halpin-Tsai relationships or transverse moulus an shear moulus. 4
The iber raius use in calculations was r = 4µ m. The thickness o the matrix cyliner was calculate rom the iber volume raction in the composite using () an is varying with The thickness o the eective composite cyliner was 5 r = µ m. FEM calculations were perorme on one hal o the iber ragment using the commercial coe ANSYS in an axi-symmetric ormulation. The PLANE8 plane element, which is a -D, secon orer element with relatively high accuracy was use in a non-uniorm mesh consisting o both triangular an rectangular elements. To obtain higher accuracy a reine mesh (o triangular elements) was use in the vicinity o the crack tip an at the en o the ebon zone. Symmetry conition was applie on z =, r [ r, R], where R is the outer raius o the ibermatrix-composite system. The axial symmetry is with respect to the z-axis. Displacement in noes on the sie r = R, z [, L ], are couple in the r-irection. ( L = 9r is the nominal length o the system in the axial irection representing one hal o the istance between two iber cracks which is L.) Constant isplacement is applie in the z-irection at z = L, r [, R]. The applie axial strain to the assembly was ε = %. The sliing isplacement is shown in Fig.. Axial coorinate z = µ m correspons to the ebon tip. It can be seen that the axial isplacement o the iber is almost the same along the whole iber surace in the stuie coorinate interval, whereas, the isplacement o the surace o the matrix at the tip o the ebone zone is twice as large as the isplacement at r / rom the tip o the ebon zone. Accoringly, the strong coorinate epenence o the relative motion o the iber an the matrix (u,z u,mz ) at the iber/matrix interace with the axial coorinate is ue to the isplacement o the surace o matrix...7 Longituinal isplacement (microns).6.5.4.3.. UYiber-UYmatrix UYiber UYmatrix 8 8.5 9 9.5 Axial coorinate (microns) Figure : Sliing isplacements at both ebon aces an the relative sliing given by (3). Carbon/epoxy composite with =.55, L = 8r, l = 5r, riction coeicient k =. The energy release rate was calculate accoring to (8) an its epenence on the length o the integration region l. It was oun that or small values o the ratio l / r the calculate values ecrease ue to insuicient accuracy o the use mesh in the local singular stress state region. For large values the calculate values o not have the meaning o strain energy release rate eine or small crack increments. As a compromise the value corresponing to l / r =. has been use to calculate throughout this paper. 5
In Fig. 3 the energy release rate when composite volume raction =.45 is compare with =.55 or carbon/epoxy an glass/epoxy. It can be seen that or meium to large ebon lengths is about the same inepenently o. For short ebon lengths, close to the iber raius, the strain energy release rate in both materials is larger an the values o or higher iber content are lower. Energy release rate Carbon iber 7 65 = 45% 6 = 55% 55 5 45 4 4 6 8 Normalize ebon length Energy release rate lass iber 9.5 9 = 45% 8.5 = 55% 8 7.5 7 6.5 6 4 6 8 Normalize ebon length (a) (b) Figure 3: Strain energy release rate versus normalize ebon length with =.45 an.55 or carbon/epoxy (a) an glass/epoxy (b). l / r in composites The iber in this calculation was suiciently long ( L = 9r ) insuring that the ebon crack oes not interacting with the symmetrical crack on the other en o the ragment. In Fig. 4 an Fig. 5 the eect o the iber ragment length on the calculate values o the strain energy release rate is presente. The epenence on normalize ebon length l / r is shown or ierent iber lengths or carbon/epoxy respective glass/epoxy. It can be seen in Fig. 5 that in carbon iber composite the ecreases with increasing ebon length. This rather linear tren is observe or all iber lengths an the shorter the iber is the stronger is the epenence. This is because the same ebon length constitutes a larger part o a shorter iber than o a longer iber. For short iber ragments the intact part o the iber is much smaller an the interaction with the ebon approaching rom the other ragment en is larger. Accoring to Fig. 4 there is no plateau region in the strain energy release rate which means that the interaction in carbon iber case starts with very short ebon length even or the longest iber ragment. It can be seen in Fig. 5 that the overall tren is the same or ebon growth in glass/epoxy. The ecreases with increasing ebon length or all iber lengths. The shorter the iber is the larger is the epenence on the ebon length. This means that the ecrease is smaller or glass/epoxy than or carbon/epoxy. In other wors, the interaction between ebons rom both iber ragment ens is smaller in glass iber composite case. To gain a eeper insight in the nature o the interaction leaing to the emonstrate overall tren o ecrease with increasing ebon length the ata rom Fig. 4 an Fig. 5 are presente as unction o iber length or ixe length o the ebon, see Fig. 6 an Fig. 7. 6
Energy Release Rate (J/m) Carbon iber 7 6 5 4 3 4 6 8 Normalize Debon Length L = 44r L = 6r L = 8r L = 4r L = 8r Figure 4: The interaction eect on strain energy release rate in carbon/epoxy composite versus normalize ebon length l / r or ierent iber lengths, =.55. lass iber Energy Release Rate (J/m) 9 8 7 6 5 4 3 4 6 8 Normalize Debon Length L = 44r L = 6r L = 8r L = 4r L = 8r Figure 5: The interaction eect on strain energy release rate in glass/epoxy composite versus normalize ebon length l / r or ierent iber lengths, =.55. Carbon iber Energy Release Rate (J/m) 7 6 5 4 3 l =.5r l = r l = 5r l = r 5 5 Norm Fiber Length Figure 6: Strain energy release rate or ebon growth in carbon/epoxy composite versus normalize iber length L / r or ierent ebon lengths when =.55. It can be seen in Fig. 6 or carbon/epoxy that the ecreases with ecreasing iber length or all ebon lengths. The larger the ebon length is the larger is the epenence on the 7
iber length. The reasons or that are explaine above. The ecrease in going rom iber length 8r to 44r is between an 34% epening on the ebon length. The strain energy release rate in glass/epoxy composite, which can be seen in Fig. 7, ollows the same trens as in carbon/epoxy. However, the epenence on the iber length in glass/epoxy is not as strong as in carbon/epoxy. The ecreases in going rom iber length 8r to 44r is between an 5%. Energy Release Rate (J/m) 9 8 7 6 5 4 3 lass iber 5 5 Norm Fiber Length l =.5r l = r l = 5r l = r Figure 7: Strain energy release rate or ebon growth in glass/epoxy composite versus normalize iber length L / r or ierent ebon lengths when =.55. The act that or ebon growth in glass/epoxy composite has a weaker epenence on the iber length compare to carbon/epoxy is relate to the ierences in the stress istribution (plateau value) in both ibers as shown in Fig. 8 an Fig. 9. It can be seen that the ecrease o the plateau value an the length o this zone with ecreasing iber length is much smaller or glass/epoxy (Fig. 9) than or carbon/epoxy (Fig. 8). The ierence can be explaine by the ierence in elastic moulus. The higher the ratio E z /E m, the longer is the istance neee to reach the plateau value in the axial iber stress. For carbon/epoxy the ratio is 5/3 an or glass/epoxy the ratio is 7/3. Thus, ue to lower moulus the stress recovery is aster in glass iber case which leas to smaller stress perturbation zone which in turn results in the weaker epenence or on the iber length. 6 Carbon iber 6 Carbon iber 5 5 Stress (MPa) 4 3 9R 45R R Stress (MPa) 4 3 9R 45R R.5.5 4 6 8 z/l Axial iber coorinate (a) (b) Figure 8: Axial iber stress istribution when = 55% an l = in carbon iber versus normalize axial coorinate z / L (a) versus axial coorinate (b). 8
stress (MPa) lass iber 8 7 6 5 4 9R 3 45R R.5.5 z/l stress (MPa) lass iber 8 7 6 5 4 9R 3 45R R 3 4 5 6 7 8 Axial iber coorinate (a) (b) Figure 9: Axial iber stress istribution when = 55% an l = in glass iber versus normalize axial coorinate z / L (a) versus axial coorinate (b). Using the expression () or strain energy release rate which is vali in the region o sel similar ebon crack growth we obtain the ollowing values. Using concentric cyliner assembly moel [] For carbon iber composite For glass iber composite =.45 =5 J/m =. 45 =7 J/m =.55 =. =5 J/m 55 =7 J/m (5). Using Rule o mixtures The results with the use accuracy coincie with results rom cyliner assembly moel. The obtaine values or the sel-similar ebon cracks o not epen on the iber content in the composite. For long ebons this result was also obtaine rom FEM. The numerical values are slightly higher than obtaine by FEM. One explanation or this is that accoring to FEM there always was an interaction between cracks lowering the values. This interaction is not accounte or in the sel-similar crack moel. Another reason or ierences may be that in the concentric cyliner moel we have neglecte the compressive raial pressure rom the ebone iber to the matrix/eective composite system. It has to be note that the calculate values o are proportional to the iber moulus, see (3) an (4). Certainly, the presente results or mechanical loaing case have to be superimpose with results or thermal stresses. 4. CONCLUSIONS The strain energy release rate in Moe relate to iber/matrix interace ebon growth along the iber surace in uniirectional composites is analyse using FEM consiering mechanical stresses only. The parametric analysis perorme to reveal the signiicance o constituent properties, iber volume raction iber ragment length an the ebon length on the lea to ollowing conclusions. is proportional to the iber moulus an is much larger in carbon iber The composite The iber volume raction has no eect on the ebons it is larger or lower volume raction or long ebons whereas or short 9
The interaction between ebons rom both iber ragment ens ecreases the values this eect being stronger in carbon iber composites. The ierence is cause by higher stress recovery rate in glass iber composites ue to lower elastic moulus. The sel-similar ebon propagation moel with strain energy changes calculate using concentric cyliner assembly solution give a goo approximation o the The numerical values are by 5-% higher than obtaine using FEM. The obtaine results will be use to simulate ebon growth uring atigue loaing.. 5. REFERENCES. Wu, W., erpoest, I. & arna, J., A novel axisymmetric variational analysis o the stress transer into ibre through a partially ebone interace, Composites Science an Technology, 58, (998) 863-877.. Nairn J.A. an Liu Y.C., Stress transer into a ragmente, anisotropic iber through an imperect interace. Int. J Solis Structures, 997; 34: 55 3. McCartney L.N., New theoretical moel o stress transer between ibre an matrix in auniaxially ibre- reinorce composite. Proc. R. Soc. Lonon A, 989; 45: 5-44. 4. R. Joe, J. arna an L.A. Berglun, Analysis o Single Fiber Fragmentation Data, 3r Int. Con. on Deormation an Fracture o Composites, 7-9 March, 995, uilor, UK, pp.6-3. 5. J.arna, R. Joe an L.A. Berglun, Interacial toughness evaluation rom the singleibre ragmentation test, Composite Science an Technology, vol.56, 9 (996), 5-. 6. raciani E, Mantič, París F an arna J. Single iber ragmentation test. A BEM analysis. Collection o Technical Papers - AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics an Materials Conerence, :988-997, Norolk (irginia), Unite States, 3. 7. W.Wu, I.erpoest an J. arna, Preiction o energy release rate ue to the growth o interace crack by variational analysis, Composites Sci an Technology, 6, (), 35-36. 8. Irwin.R., Fracture, Hanbuch er Physik, vol.5, Berlin: Springer erlag, 958. p.55. 9. Hashin Z, Rosen BW., 964, The elastic mouli o iber-reinorce materials, Journal o Applie Mechanics, 3(), 3-3.. Hashin Z., 983, Analysis o Composite Materials a Survey, Journal o Applie Mechanics, 5, 48-55.. E. Marklun, J. arna, R. C. Neagu, E. K. amstet, Stiness o aligne woo iber composites: Eect o microstructure an phase properties, Journal o Composite Materials, 8, accepte.