Micro-Crack Initiation and Propagation in Fiber Reinforced Composites. Andrejs Pupurs

Transcription

1 DOCTORA L T H E S I S Micro-Crac Initiation and Propagation in Fiber Reinorced Coposites Andrejs Pupurs

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3 Micro-crac Initiation and Propagation in Fiber Reinorced Coposites DOCTORAL THESIS by Andrejs Pupurs Division o Materials Science Departent o Engineering Sciences and Matheatics Luleå University o Technology Luleå, Sweden SE Septeber 2012

4 Printed by Universitetstryceriet, Luleå 2012 ISSN: ISBN Luleå

5 Preace This Doctoral thesis was copleted during the period ro January 2008 to August 2012 at the Polyeric coposites group, Division o Materials Science, Luleå University o Technology, Sweden. For the accoplished wor I ust sincerely express y gratitude to y supervisor Proessor Janis Varna, who has been a strong and contributing guide, an advisor, an authority, or, in other words, the best supervisor. I a also very grateul to hi or his hospitality during all y stays in Luleå and or the highly dedicated atosphere, which aes woring in his group a great pleasure and experience. I will also use an opportunity to than the ever-reliable Proessor Roberts Joe or his ruitul suggestions, rears, and assistance in the widest range o probles. The contribution and patience o Proessor Andrejs Krasniovs ro Riga Technical University, Latvia, is also highly regarded. I grateully acnowledge y co-authors Dr. Povl Brøndsted and Dr. Stergios Goutianos ro DTU, Denar, and Pro. Göran Lindbergh ro KTH, Sweden, or their expertize and valuable suggestions. I would also lie to than Pro. Lei Asp and Dr. David Mattsson o Swerea SICOMP or our interesting and productive collaboration. Much o the tie that I spent woring on this thesis was in act borrowed ro y beloved wie Lva. I a incredibly thanul to her or the everyday support and or her understanding o y occupation with this thesis. I hope to be as good support to her, when she deends her Doctoral thesis in a couple o years. Finally, I would lie to than all y dear aily, who have always been a strong inspiration and support to e, all y colleagues and riends at LTU, especially y oice-ate Kostis or all the un we had woring together, and all y Master students, who have always been ind and helpul to e. Luleå, Septeber 2012 Andrejs Pupurs i

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7 Abstract Predicting icro-daage initiation and evolution is one o the ey challenges or sae design o iber reinorced polyer coposites. Micro-scale daage such as, or exaple, single iber brea ay be unnoticeable and negligible during the initial service lie o coposite, however, with any loading and unloading cycles this initially icro-scale daage ay propagate oring acroscopic scale cracs that can signiicantly reduce the service lietie or even lead to unoreseen catastrophic ailure o the coposite structure. The objective o this Doctoral thesis is to develop ethodology or prediction o icro-crac propagation in iber reinorced polyer coposites. Fracture echanics concepts o strain energy release rate are applied or crac growth analysis. Analytical odeling cobined with nuerical FEM calculations are used to obtain the values o the energy release rate. Paraetric analysis is perored to evaluate the signiicance o the applied load and various aterial properties on the icro-crac growth rate. Calculation results are ipleented into power law relation, to predict the crac growth in atigue loading. In Paper I iber/atrix interace debond growth starting ro single iber breas in unidirectional (UD) polyer coposites is studied. Analytical solution or Mode energy release rate G is ound and paraetric analysis is perored in the selsiilar debond crac propagation region. When the iber/atrix interace debond crac is short, the sel-siilarity condition is not valid. Due to interaction with iber brea, G is agniied. In Paper, nuerical FEM odeling is perored to calculate G or short debond cracs. The indings ro G analysis or sel-siilar and short debond cracs are suarized in siple expressions. Siulations o iber/atrix interace debond crac growth in tensiontension atigue using Paris law are perored. In Paper I, debond growth in single iber (SF) coposites subjected to tensiontension atigue is analyzed. Using the sae procedure as or UD coposites, irst, an analytical solution or Mode energy release rate G is ound or sel-siilar crac growth region and then FEM odeling is perored to obtain agniication proiles or short debond cracs. In Paper IV odeling ethodology described in Paper I is advanced urther and the odeling results are copared with experiental data or interace debond crac growth in SF coposites subjected to tension-tension atigue loading. The power law constants are extracted ro the best it between the experiental and odeling results. Validation o results proves that the power law can be ipleented to characterize icro-crac growth in atigue o polyer coposites. In Paper V iber/atrix interace debond growth on the surace o a UD coposite specien subjected to tension-tension atigue is analyzed. 3-D FEM odeling is perored to account or the non-axisyetric stress state due to the edge eect. Siulations o debond growth in tension-tension atigue are perored. Modeling results are copared with SF coposites and with experiental data available in the literature. Finally, in Paper VI racture echanics concepts o energy release rate are used to odel icro-crac initiation and propagation in a carbon iber, which, apart ro the load bearing unction, also serves the purpose o an electrode in a novel lithiu-ion rechargeable battery. When subjected to lithiu ion intercalation, carbon iber experiences a non-unior swelling that leads to developent o high echanical iii

8 stresses. In any cycles o charging-discharging these stresses can introduce daage and reduce the echanical and electrocheical properties o the battery. FEM odeling using theral analogy is perored to solve the transient ion diusion and echanical stress proble. Dierent crac initiation and propagation scenarios are analyzed and copared. iv

9 List o appended papers Paper I A. Pupurs, J. Varna, Energy release rate based iber/atrix debond growth in atigue. Part I: Sel-siilar crac growth, Mechanics o Advanced Materials and Structures, In press, Paper A. Pupurs, A. Krasniovs, J. Varna, Energy release rate based iber/atrix debond growth in atigue. Part : debond growth analysis using Paris law, Mechanics o Advanced Materials and Structures, In press, Paper I A. Pupurs, J. Varna, Fracture echanics analysis o debond growth in a single-iber coposite under cyclic loading, Mechanics o Coposite Materials, vol.47(1), pp , Paper IV A. Pupurs, S. Goutianos, P. Brøndsted, J. Varna, Interace debond crac growth in tension-tension cyclic loading o single iber polyer coposites, Coposites Part A, Accepted, Paper V A. Pupurs, J. Varna, Modeling iber/atrix debond growth in tension-tension cyclic loading o UD coposites, To be subitted to Coposites Part A, Paper VI A. Pupurs, J. Varna, G. Lindbergh, Modeling echanical stress and exoliation daage in carbon iber electrodes subjected to cyclic intercalation/de-intercalation o lithiu ions, To be subitted to Journal o the Mechanics and Physics o Solids, v

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11 Contents Preace... i Abstract... iii List o appended papers... v Contents... vii 1. Introduction Longitudinal tension: ailure echaniss and daage initiation in UD coposites Failure echaniss in tension-tension atigue Fracture echanics analysis o iber/atrix debond growth Paris law or atigue Experiental easureents o debond growth in atigue Objectives o the current wor Suary o appended papers Reerences Paper I Paper Paper I Paper IV Paper V Paper VI vii

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13 1. Introduction For several decades polyer coposite aterials have been considered as aterials o the uture with potential or application in high perorance structures. Wide range o applications in autootive, wind power and aerospace industries has clearly deonstrated the potential and advantages o coposites in the ost deanding environents and loading conditions. One o the reasons o suitability o coposites or high perorance structures is the apparent insensitivity to atigue when the load is applied in iber direction even at stresses close to static racture strength. However, several dierent atigue daage echaniss are nown to exist in coposites [1]. Fatigue daage echaniss in unidirectional (UD) coposites depend on the loading ode (tensile, copressive, etc.), and on whether the loading is parallel or inclined to the iber direction. For UD coposites subjected to axial loading in iber direction icrodaage in or o iber breas or atrix cracs ay develop ro occasional overload or due to anuacturing deects. In the service lie these initially icroscopic laws ay propagate as, or exaple, iber/atrix interace cracs leading to oration o larger scale cracs and eventually resulting in ailure o the structure. In act, it has been estiated that ore than 80% o all service ailures o structures are due to atigue [2]. When loads o the iber axis are applied, atigue o coposites ay be even ore coplex than etallic aterials atigue [1]. Thereore, in order to increase the copetitiveness o coposites it is very iportant to incorporate existing nowledge o coposite daage echaniss in designing atigue-resistant coposite structures and to develop odels that could saely predict the developent o daage propagation and accuulation in cyclic loading. In practice unidirectional coposites are seldo used to eet the requireents to withstand various cobinations o ulti-axial echanical and theral loads dierent coposite lay-ups are designed or dierent applications. However, the atigue lie o the o-axis plies is uch shorter than or the 0 plies, in act, the atigue lie o the coposite is oten indistinguishable ro the atigue lie o the 0 plies since they are the last ones to ail in coposite. It has been shown in any studies [3-5] that the atigue lie o ultidirectional lainates containing also plies with ibers oriented in the load direction (0 ) is controlled by the atigue behavior o the unconstrained 0 plies. This iplies that analyzing the atigue lie o 0 plies is iportant or iproving the atigue perorance o coposites. 1.1 Longitudinal tension: ailure echaniss and daage initiation in UD coposites When long iber coposites (with iber strain to ailure saller than atrix strain to ailure) are loaded to ailure in quasi-static tension along the iber direction, the irst iber breas occur in soewhat rando positions as shown in Fig.1. This is because o the laws and deects in ibers, which cause ibers to brea in a brittle anner at rando stress. However, rando iber ailure stress can be described by statistics. The rando ailure stress o brittle ibers usually resebles Weibull distribution which has been coonly used in ailure characterization o long iber coposites, see, or exaple, [6-8]. In Fig.1 rando iber strength distribution is shown graphically along the iber length. It shows the diagra o ailure stress o the iber ared as F in the picture. Also in Fig.1 axial stress distribution in iber F is shown. Certainly, the axial stress is equal 1

14 to zero at iber breas, but builds up to ar ield stress value (equal to stress in undaaged coposite) ater a certain distance away ro the brea by eans o stress transer echanis through iber/atrix interace. Breas in the iber have an inluence on the stress level and thus the probability o ailure o the neighboring ibers is increased. Stress distribution plot in Fig.1 shows the stress agniication (in iber F ) due to breas in the neighboring ibers. F Fig.1. Axial stress distribution and local iber strength along iber length in a UD coposite with rando iber breas. Figure source taen ro [9]. Further increase o the applied tensile load will cause new iber breas in the coposite in places where local stress versus local strength ratio is the highest. Depending on the properties o ibers, atrix and the iber/atrix interace, there ay be several scenarios o events beore new iber brea occurs. Fig.2 shows the ost coon scenarios occurring in UD polyer coposites. Due to increase o the applied load or due to excess o the energy released during the brea o the brittle iber, the crac ay propagate ro iber into atrix until it is arrested by the neighboring iber (Fig.2a) as observed, e.g., in [10] and [11]. The local stress concentration at the tip o the atrix crac ay then cause the neighboring iber to brea or, depending on aterial properties, the crac ay as well delect as a debond crac growing along the iber/atrix interace. Shear yielding o atrix (Fig.2b) ay occur i the atrix tensile strength is high enough to resist stress concentration at the tip o penny-shaped iber crac. The inluence o shear yielding on the inal ailure o UD coposites has been recently studied in [12]. Another possible scenario is delection o the iber crac into a growing debond crac along iber/atrix interace (Fig.2c). Interace crac ay grow until it eets the other debond crac growing oppositely ro the next brea o the sae iber. The oration o local cracs proceeds until all cracs coalesce into one large crac leading to coplete ailure o the UD coposite. The present thesis is ocused priarily on the interace debond growth, thereore this echanis is described ore in detail. In Fig.3 daage evolution and inal ailure o coposite according to interace debonding scenario are shown. 2

15 As it is shown in the igure, urther increase o load causes iber crac delection in or o interace debonds. Transverse atrix cracs ay also or as a result o stress concentration at debond crac tips when close to iber cracs. As a consequence o growing daage, larger cracs are ored ro sall cracs leading to oration o a ajor crac and the inal ailure o coposite. In Fig.3 inal ailure in or o pull-out o partially broen ibers is shown. a) b) c) Fig.2. Daage developent scenarios ater iber brea oration: a) crac propogation in atrix; b) atrix yielding; c) debonding o iber/atrix interace. rando iber brea debond atrix brea Fiber brea debond inal ailure Fig.3. Scheatic showing o daage events leading to inal ailure o a UD coposite due to increase o the applied tensile load. 1.2 Failure echaniss in tension-tension atigue When UD coposite is subjected to uniaxial tension-tension atigue loading along the iber direction, the daage initiation and its urther evolution strongly depend on the level o the applied stress. Possible daage initiation and evolution echaniss related to coposites have been widely described in [13], [14]. On contrary to quasistatic loading, in atigue loading a stable daage growth and gradual decrease o echanical properties is expected. The our conceptual daage developent scenarios in tension-tension atigue o UD coposites are deonstrated in Fig.4. 3

16 The ost extree developent shown in Fig.4a is coposite ailure at very high applied stress levels which cause the UD coposite to ail within ew load cycles. The stress level or such case exceeds the average coposite tensile strength ( c ) and the ailure taes place in a anner alost identical to static ailure. However, it is nown that carbon and glass ibers are brittle aterials with probabilistic distribution o strength. The weaest ibers ay ail uch earlier than the UD coposite. I a stress level larger than strength o the weaest iber but saller than the coposite strength ( c ) is applied in tension-tension atigue, ibers ay brea in rando positions already during the irst applied load cycle as shown in Fig.4b. As it was ephasized earlier, ibers such as carbon and glass ibers are usually not prone to degradation in atigue. An isolated iber brea causes shear stress concentration at the interace close to the tip o the broen iber. Thereore, when the axial load is constantly repeated, the urther daage evolution ight be in or o growing interace debond cracs (Fig.4b). Debond cracs start to grow at the tips o iber crac due to stress concentration. The rate o debond crac growth with the applied nuber o cycles depends on the shear ailure properties o the iber/atrix interace and on the level o the applied stress. However, the above described is not the only possible daage evolution scenario in high stress tension-tension atigue. Since stress concentration occurs near the debond crac tip, a possibility o transverse crac oration in atrix also exists. a) b) c) d) Fig.4. Possible daage evolution scenarios in tension-tension atigue: a) overload stress ( c ); b) high stress ( c ); c) ediu stress ( FL ); d) low stress ( ). The agniied tensile stress due to iber breaage ay exceed the racture stress o the atrix. Fibers in polyer coposites (glass, carbon ibers) are aterials with high stiness and the aount o energy released during breaing o ibers is oten larger than necessary to cause iber brea only. The excess o the released energy ight result in oration o short debond cracs or cracs propagating in atrix iediately ater the iber brea occurs during the irst cycle. I the atrix crac occurs, then during consecutive tension-tension atigue load cycles it ay propagate until it reaches the neighboring iber. At low and ediu stress atigue loading conditions the growth o the atrix crac will stop at the iber/atrix interace or delect as a debond crac growing along iber/atrix interace. In high stress atigue, however, ater the atrix crac reaches the interace, iber breas ay or as a result o high stresses at the tip o the preceding atrix crac. FL 4

17 In ediu stress tension-tension atigue loading where the axial stress is saller than stress necessary to cause breaage o the weaest ibers during the irst cycle ( FL ), the echanis deonstrated in Fig.4c is typical. While the ibers are, as previously stated, degradation-proo in atigue, the polyer atrix is nown to have a certain atigue liit (FL) above which atrix cracing will occur. Matrix cracs ay develop ater a certain aount o applied load cycles, which thereater ay also delect and continue propagation as debond cracs (Fig.4c). For polyer coposites, the elastic odulus o atrix is usually uch saller than or ibers, thereore it is expected that the energy released due to atrix cracs will also be uch saller. A atrix crac would stop at the iber/atrix interace at lower strains while at higher applied strains the stress at the crac tip ight exceed the racture stress o the ibers leading to ailure o soe ibers in the coposite. In [15] it was shown that the energy release rate or debond cracs initiated ro atrix breas (Fig.4c) decays with the distance ro the priary crac whereas it approaches to an asyptotic value in the case o iber brea being the priary crac (scenario in Fig.4b). In ediu stress tension-tension atigue, the atrix cracs ay or in several places siultaneously which as a result o increasing nuber o applied cycles ay asseble into a large atrix crac bridged by the ibers. The urther scenario ay then be either oration o iber breas leading to inal ailure o coposite or growing debond cracs. Debond growth initiated ro a single atrix or single iber brea can with certain approxiation be considered as an axisyetric proble. The inal possible scenario, shown in Fig.4d, is actually related to ininite atigue lie and it is the opposite o the extree scenario o nearly instant ailure shown in Fig.4a. The applied strain level or the case shown in Fig.4d is below the atigue liit o the polyer atrix ( FL ). Initial daage or deects ay be present as well in this case, however, either the deect size is too sall or the arresting echaniss are very eicient to prevent any propagation. The cyclic axiu strain below which no cracs or only non-propagating cracs ay be initiated in the atrix aterial, deined as the atigue liit o the atrix, ay be obtained by atigue testing o the unconstrained (unreinorced) atrix aterial [1]. The daage evolution scenarios described above are soewhat idealized. In practice, depending on the aterial properties and the level o applied stress, these echaniss ay as well occur siultaneously. Because dierent daage echaniss occur depending on the level o applied stress, analyzing the atigue lie o UD coposites can be coplex. A coprehensive way to deonstrate the atigue lie is to construct a atigue-lie diagra. In Fig.5 typical atigue-lie diagra or UD coposites loaded in iber direction is shown. The atigue-lie diagras were irst introduced by Talreja [1]. Beore that, the socalled Wöhler or S-N diagras were extensively used to show the applied global stress level with respect to the nuber o cycles to ailure. However, since or coposite aterials dierent atigue daage echaniss tae place depending on the applied load level, the S-N diagras are soewhat incoplete and probably isleading representation o atigue perorance o coposites. Thereore, the atigue-lie diagras have shown theselves useul as a raewor to interpret atigue properties o unidirectional coposite aterials. On the vertical axis o a atigue-lie diagra shown in Fig.5 the axiu atigue strain is plotted, while on horizontal axis logarith o the nuber o cycles to ailure is plotted. Strain instead o stress is typically chosen in atigue-lie diagras as both 5

18 ibers and atrix are subjected to the sae strain while stresses in the two phases dier depending on the volue raction and elastic properties o the two phases [1]. ax iber breaage, interacial debonding c atigue liit o atrix atrix cracing, interacial shear ailure log N Fig.5. Fatigue-lie diagra or unidirectional coposites under loading parallel to ibers, redrawn ro [1]. The regions deined in Fig.5 each correspond to daage echaniss previously described and depicted in Fig.4: the region below the atigue liit o atrix in Fig.5 corresponds to no-daage state depicted in Fig.4d. The atrix cracing region corresponds to daage echanis shown in Fig.4c. The iber breaage and interacial debonding region corresponds to Fig.4b. As shown in Fig.5 this region covers the scatter in the iber racture strength. Above the iber breaage and interacial debonding region, ailure o UD coposite will occur within ew load cycles as shown in Fig.4a. The presented echaniss dier substantially ro each other and require dierent approaches or analysis. In the present thesis, the ocus is only on iber breaage and interace debonding echanis (Fig.4b). In Papers I-V included in the present thesis the applied load is parallel to the iber orientation. Fatigue daage echaniss under loading inclined to iber direction (o-axis atigue) including dierent lay-ups o coposites are described in detail in [1,16]. 1.3 Fracture echanics analysis o iber/atrix debond growth Within the current wor, daage evolution scenario according to high stress tensiontension atigue (see Fig.4b) is considered as predoinant over other possible daage echaniss. The sequence o events or such scenario is oration o iber breas during the irst load cycle ater which urther daage evolution is assued to occur in or o growing interace debond cracs Analytical odels In order to analyze debond crac growth along iber/atrix interace, racture echanics concepts can be applied. Energy release rate G related to Mode crac growth is used as a paraeter or debond growth analysis. For UD polyer coposites subjected to tensile loads only Mode crac propagation is relevant because the Poisson s ratio o the atrix is larger than or the ibers. In addition, 6

19 theral expansion coeicient o atrix is also larger than or the ibers eaning that due to cooling down ro anuacturing to roo teperature the radial stresses on the iber/atrix interace will be copressive. Both o the aboveentioned obstacles eliinate the Mode I (opening). In UD long iber coposites with ibers o circular cross-section, debond growth initiated ro a single iber brea can be approxiately considered as a proble with axial syetry. Concentric cylinder assebly (CCA) odel introduced by Hashin [17,18] can be applied in analytical calculations. The UD coposite ay be represented by a CCA odel consisting o three phases broen and partially debonded iber cylinder in the iddle, a atrix cylinder and an eective coposite cylinder, which represents the surrounding undaaged coposite as shown in Fig.6. When considering debond crac growth in a UD coposite, two dierent crac growth conditions have to be separated: 1) i debond cracs are long and the tip o the debond crac is ar away ro the iber brea where it initiated ro, and the oppositely propagating debond crac is also suiciently ar away, the crac will propagate in a sel-siilar anner since there is no interaction. The energy release rate G or sel-siilar (long and non-interactive) debond cracs is thereore a constant value independent on the crac length; 2) i debond cracs are short, the stresses at the tip o debond cracs interact with stresses at the tip o iber crac. Thus debond crac growth related energy release rate is agniied and is larger than in the sel-siilar region. Fig.6. a) UD coposite with a rando iber brea and partial iber/atrix interace debonding b) representation o the proble by CCA odel (F-iber, M-atrix, C- eective coposite). In the sel-siilar region, due to no interaction condition, analytical calculation ethod based on the CCA odel can be conveniently applied. The idea o G calculation ethod is based on strain energy change. In sel-siilar tot tot conditions the dierence between strain energy U b o the bonded and U d o the 7

20 debonded regions o unit length dl d (equal or both regions, see Fig.7) ar away ro the debond crac tip and the iber brea is equal to du (1) tot tot U d U b Energy dissipation due to riction between iber and atrix suraces in this case will be considered as negligible. The bonded and debonded regions are shown in Fig.7. Note that, when the load is applied, sliding o iber with respect to atrix and coposite will occur in the debonded region (Fig.7a), and deorations and strains will thereore be dierent than in the bonded region (Fig.7b.). The strain energy o bonded and debonded regions shown in Fig.7 can be calculated i the average stress state in constituents is nown. The energy release rate is obtained by dividing Eq.(1) by the newly created surace area da. Thus G du uconst (2) da The assuption o constant displaceent ( u const ) is used in Eq(2). In circular iber UD coposites with iber radius r, the newly created surace area related to debond growth by dld is equal to da 2r dl. d a) b) debond Fig.7. Three phase CCA odel o UD coposite: a) debonded region o unit length; b) bonded region o unit length. The detailed procedure to calculate tot U b and tot U d is given in [19], this paper is also included in the present thesis (Paper I). To acilitate the energy release rate calculations the ollowing siple expression is suggested or an arbitrary theroechanical load case in [19]: G z 2 r T r T 2 E r (3) ech th ech th 8

21 E is iber longitudinal odulus, where echanical strain, T r is the iber radius, ech is the applied is the applied teperature dierence,, th and th are coeicients related to echanical, theral and thero-echanical response respectively o the coposite and can be obtained by peroring 3 separate calculations (see [19]). Nairn [20] derived an exact solution or debond growth related energy release rate in single iber coposites, using a constant orce assuption. Using the previous notation, it is as ollows: G 2 Q zr d 1 E z d 2 d r 1, Er E 1 1 Ez r 4 T 2 (4) ech 2 zr, d th E d z 2 r th 2 2 zr d 1, E z z 1 z Q d z (5) 2 Q 1 zr (6) E d where is the theral expansion coeicient o the atrix, z and r are the axial and radial theral expansion coeicients respectively o the iber, zr and r are the in-plane and out-o-plane Poisson s ratios respectively o the iber, E and are the elastic odulus and Poisson s ratio respectively o the atrix Nuerical odeling Linear elastic racture echanics paraeters such as stress intensity actor K and energy release rate G are widely used to analyze daage initiation and propagation in structural aterials. Stress intensity actor K characterizes the stresses, strains and displaceents near the crac tip. Energy release rate G quantiies the net change in potential energy that accopanies an increent o crac extension. While K is a local paraeter, G describes the global behavior. Since the original wor perored by Griith, Inglis and Irwin analytical solutions or K and G have been developed or a wide range o dierent crac geoetries and loading conditions. In previous section analytical odels or UD coposites [19] and single iber coposites [20] were entioned. For cases when analytical solution is not available or too coplicated or practical use, nuerical tools such as inite eleent ethod (FEM), boundary eleent ethod (BEM) and other ethods are convenient tools or calculation o K and G. Considering relatively short debond cracs, the sel-siilarity condition described in previous section is not valid due to interaction between debond crac tip and the iber crac, where the debond initated ro. Due to perturbation o stress, exact analytical solution or energy release rate is not easible. Thereore, or short debonds, the interest o using nuerical odelling arises. For axisyetric probles with singular stresses (at the tip o debond crac) the application o boundary eleent ethod (BEM) is advantageous [21]. However, BEM odeling is liited to isotropic constituents only and thus it is not applicable or carbon ibers (transversally isotropic). Finite eleent ethod (FEM) can be thereore ipleented. Although, z 9

22 when using FEM, the obtained results ay be sensitive to esh reineent at the tip o debond crac, the ethod is not restricted in ters o aterial properties. A three phase axisyetric FEM odel (see Fig.8.) in cobination with virtual crac closure technique (VCCT) [22] can be conveniently applied or calculation o G. Fig.8. Scheatic picture o FEM odel to be used in cobination with virtual crac closure technique. The crac closure technique states that the energy released due to debond crac growth by da is equal to the wor, which is required to close the newly created surace ro size A da bac to size A. As stated previously, or circular cross section iber coposites da 2r dl d. Closing the Mode debond crac by dl d (ro l d dl d to l d ) by applying tangential tractions, points at the debonded surace in the region z l d ; ld dld, which have relative tangential displaceent ld dld ld dld ld dld u ( z) u ( z) u ( z) (7) z are oved bac to coinciding positions. At the end o this procedure the shear stress in point z is equal to l d rz (z), which is the shear stress in ront o the crac with size l d. Then the wor required to close the crac by dl d can be expressed as 10 z ld dld 1 ld dld ld W 2 r u ( z) rz ( z) dz (8) 2 l d This is nown as the crac closure technique. Within the VCCT it is assued that due to sall value o dl d the relative sliding displaceent at the tip o the crac with size l d dl d is the sae as at the tip o the debond crac with size l d : ld dld ld u z) u ( z dl ) (9) ( d The beneit o this assuption is that only one stress state calculation or a given debond length is required. Since the energy release rate is deined by Eq.(2), the ollowing result is obtained using Eqs.(8) and (9)

23 G ( l ) li d dld dl d ld dld ld u ld ld z dl zdz d rz (10) The distribution o the relative tangential displaceent u z behind the debond crac tip and the shear stress rz values ahead o the crac tip can be obtained directly ro the FEM odel as shown in the Fig.8. When using VCCT or G calculations ro FEM results, the size o the area o integration dl d has to be careully analyzed. Since integration o Eq.(10) is perored over a inite distance, the obtained G value will strictly speaing depend on the chosen length o integration. On contrary to condition that dl d 0 in Eq.(10), when using FEM calculations, the accuracy o the result ay be reduced i saller length and thus saller nuber o eleents close to the crac tip is used or the integration. This is because sall aount o eleents at the crac tip cannot accurately represent the actual stress and displaceent proiles close to the crac tip. Soe inal value o the integration length dl d ust be set, by peroring paraetric analysis until acceptable value is ound. The VCCT technique described above, despite its siplicity requires peroring accurate operations with FEM calculation results. To acilitate this, an increasing nuber o dierent autoated techniques and specialized eleents or racture echanics paraeter calculations are incorporated in the latest versions o coercial FEM sotware codes or even ore user-oriented convenience. However, the availability o dierent autoated options requires additional attention towards the calculation results the user has to be aware o what calculations are perored "within the blac box", realize the liitations o each technique and veriy the obtained results by an alternative calculation ethod. Further in this section autoated G calculation routines provided by sotware code ANSYS version 13.0 [23] are analyzed and the results are copared with analytical results and anually obtained nuerical results Coparison o nuerical routines or strain energy release rate calculation Exaple 1 Exaple 2 a) b) Fig.9. a) Exaple 1: Center craced tension (CCT) specien. 2W - specien width, 2a - crac length, - tensile stress. Figure taen ro [24]; b) Exaple 2: Axisyetric CCT specien with penny-shaped crac. Figure taen ro [26]. 11

24 Exaple 1: A siple 2-D proble o center craced tension (CCT) isotropic aterial specien is deonstrated as Exaple 1. Geoetrical paraeters o a inite size center craced tension (CCT) specien are given in Fig.9a. Only Mode I (opening) is relevant or the CCT specien. Exact analytical solutions or K I and G I are available, when the width o the specien is ininite (2W ). When the width o the CCT specien is inite, closed or analytical solution is usually not possible due to edge eects. Several approaches to ind an approxiate solution have been proposed, ro which the ost accurate ones coe ro inite eleent analysis [24]. For plane strain and plane stress states, the closed or analytical solution or stress intensity actor K I or an ininite plate (2W ) is: K I a (11) For the inite size plate the ollowing approxiation is accurate: K I a a a a sec (12) 2W W W This approxiation has less than 2% error copared to inite eleent solution when a /W 0.9 [24]. Exaple 2: The penny-shaped crac in an ininite ediu is another coniguration or which a closed-or analytical solution or K I exists [25] and the stress state in such a case is axisyetric. For axisyetric proble, 2a in Fig.9b is the diaeter o the pennyshaped crac and the solution or K I is: K I 2 a (13) To account or the inite size o an axisyetric CCT specien shown in Fig.9b ( 2 b is the inite width (diaeter) o the saple), the ollowing approxiation can be used [26]: a a b b K I a a 1 b 3 (14) The energy release rate G I can be calculated ro relationship between K I and G I. For linear elastic aterials, K and G are uniquely related. For plane stress state the relation is: 12

25 G I 2 K I (15) E where E is elastic odulus. In plane strain the relation is: 2 K I GI (16) E 2 1 where is Poisson's ratio. For axisyetric stress state Eq.(16) is applicable or calculation o G I. Finite eleent calculations o K and G were perored using ANSYS version 13.0 [23]. Taing advantage o the syetry conditions in Fig.9 a and b, only 1/4 o the specien was odeled. Fig.10 shows the FEM odel and the boundary conditions used. a) W P coupling on nodes L syetry b.c. reined esh area b) R i R i a crac tip syetry b.c. Fig.10. FEM odel o a center craced tension specien. a) ¼ odel and boundary conditions b) odel detail at the crac tip. In all calculations hal-width o the saple was W = 1 (or b =1 or Exaple 2), hal-length o the saple was L = 2, the applied load was P = 1 N (regardless o all other paraeters, the applied load P was in all cases equal to 1 N thus representing a load controlled test). Linear elastic aterial properties were: elastic odulus E = 3 GPa, Poisson's ratio = 0.3. Ri is the integration length that was used or crac closure and virtual crac closure techniques. 2-D 8-node structural solid eleents PLANE183 were used in all FEM calculations. This eleent type is recoended or use in racture echanics probles in ANSYS [23]. 3 dierent eshing approaches or crac tip region were used to reveal the signiicance o eshing on calculation results. The irst approach denoted as Case A in urther reerence, is eshing the crac tip region with triangular solid eleents with decreasing eleent size towards the crac tip. Fig.11 shows a ragent o FEM odel and the detail o the eleent esh at the crac tip. 13

26 a) b) crac surace crac tip crac tip Fig.11. a) Eleent esh ragent or Case A, b) detail o eleent esh at the crac tip. As shown in Fig.11 the eleents outside the reined crac tip zone ostly have quadrilateral shape and triangular shape eleents in the crac tip zone are used because it ay be geoetrically diicult to use quadrilateral shape eleents in locations with high reineent or irregular shape. Also, the calculation accuracy (stress distribution) is higher when triangular shape eleents at the crac tip are used. This ind o crac tip eshing approach was used or exaple in [27,28], which are Papers and I respectively included in this thesis. crac surace crac tip Fig.12. Detail o the eleent esh at the crac tip or Case B. The second eshing approach denoted as Case B and deonstrated as a detail in Fig.12 is siilar to Case B with the dierence that no increased eleent size reineent is applied towards the crac tip (spacing is unior). The third eshing approach denoted as Case C corresponds to crac tip region eshing recoendations given in ANSYS user anuals [23]. In this case the crac tip region is eshed with triangular shape singular eleents nown as the crac tip eleents in ANSYS terinology. ANSYS coand KSCON is used to deine the esh at the crac tip. Fig.13 shows the eleent esh and eleent nodes or Case C. 14

27 crac surace crac tip Fig.13. a) Eleent esh at the crac tip or Case C. There are two autoated ethods or calculating the energy release rate in ANSYS The irst ethod is J-integral based ethod. J-integral was presented by Rice [29], he showed the path-independence o this integral and its direct relation to energy release rate. J-integral evaluation in ANSYS 13.0 is based on the doain integral ethod by Shih [30]. The doain integration orulation applies area integration or 2-D probles and volue integration or 3-D probles. Area and volue integrals oer uch better accuracy than contour integral and surace integrals, and are uch easier to ipleent nuerically, as stated in [23]. For a 2-D proble and in the absence o theral strain, path dependent plastic strains, body orces within the integration o area, and pressure on the crac surace, the doain integral representation o the J-integral is given by: J u j ij x1 1 A q W i da x i (17) where q is reerred to as the crac-extension vector, W is the strain energy density, ij and u j are stress and displaceent Cartesian coponents. More details on this ethod are available in [30]. J-integral in ANSYS is calculated using CINT coand with option CINT,TYPE,JINT. In the urther text the energy release rate calculated using J-integral ethod is denoted as G JINT I. The second autoated ethod or calculation o G in ANSYS [23] is based on virtual crac closure technique. VCCT principles or Mode were described earlier (see Eqs.(7-10)). For autoated calculation o G ANSYS uses odiied crac closure ethod based on VCCT and assues that stress states around the crac tip do not change signiicantly when the crac grows by a sall aount a, see Fig.14. For a 2-D crac geoetry shown in Fig.14 with a low-order eleent esh, the energy release rate is deined as: G I 1 2a R v, G 1 Y 2a R Xu (18) 15

28 where G I and G are ode I and energy release rates respectively, u and v are relative displaceents between the top and botto nodes o the crac ace in local coordinates x and y, respectively, R X and R Y are reaction orces at the crac tip node, a is the crac extension as shown in Fig.14. Fig.14. Paraeters or VCCT or 2-D crac geoetry. Figure taen ro [23]. The process o energy release rate calculation in ANSYS involves using CINT coand with option CINT,TYPE,VCCT. This option in ANSYS was irst introduced only in version 13.0 [23]. By using this autoated routine, the integration length is deterined by the progra based on the created esh. In the urther text the energy release rate calculated using VCCT ethod is denoted as G I A.VCCT. To validate the nuerical results obtained by autoated ANSYS ethods described previously, anual calculations were perored. Notation "anual calculations" eans that FEM data o stress and displaceent distributions are still used, but the integration is perored anually. Fig.15. Application o closure stresses which shorten the crac bya. Figure taen ro [24]. In this exaple study o the CCT specien two dierent anual calculation ethods were used: crac closure technique and virtual crac closure technique. The principles o both techniques were already given or Mode crac propagation (see Eq.(7-10)). 16

29 For Mode I the noral stress and displaceent coponents are used as shown in Fig.15. In the urther text, energy release rate calculation results using anual crac closure technique ethod are denoted as G CCT I. The energy release rate calculation results obtained by anual VCCT are denoted as G VCCT I. Table 1 suarizes calculation results or Mode I energy release rate G I. Paraeter R i is the length o integration (see Fig.10), which was used or G CCT VCCT I and G I calculations. Analytical A B Notation Table 1. Calculation results or energy release rate G I. Unit Exaple 1 Plane strain Type o the stress state Exaple 1 Plane stress Exaple 2 Axisyetric L [ ] W [ ] * a [ ] ** R i [ ] B [ ] * P [ N ] 1 1 [ - ] G I G I JINT G I A.VCCT G I VCCT G I CCT G I JINT G I A.VCCT G I VCCT G I CCT G I JINT G I A.VCCT G I VCCT G I CCT G I [ J / 2 ] [ J / 2 ] [ J / 2 ] [ J / 2 ] [ J / 2 ] [ J / 2 ] [ J / 2 ] [ J / 2 ] [ J / 2 ] [ J / 2 ] [ J / 2 ] [ J / 2 ] C [ J / 2 ] [ J / 2 ] * in axisyetric case W B is the outer radius o the CCT specien ** in axisyetric case a is the radius o the penny-shaped crac. Analytical results or the ininite size CCT speciens are denoted as G I and are given in Table 1 as a reerence. The analytical results or inite size CCT speciens (denoted as G I in Table 1) should be considered as target values to which all nuerical results should be copared. These target values are underlined in Table 1. In general the agreeent between the target values and the nuerical values calculated using ANSYS autoated routines - J-integral ethod ( G JINT I ) and VCCT 17

30 ( G IA.VCCT ) - is excellent (except results or axisyetric case). The results calculated by anual VCCT ( G I VCCT ) do not provide as good accuracy although the agreeent with target values is still rather good. Results obtained by crac closure technique ( G I CCT ) underestiates the target values the ost ro all cases. The crac increent has been too large. In the ollowing the inluence o the esh reineent is discussed. As Table 1 shows, the calculation results depend on esh reineent level. Coeicient was introduced to characterize the eleent esh reineent and to peror esh reineent convergence analysis. Well reined esh, especially near the crac tip, is iportant or accurate calculation o racture echanics paraeters, which are based on local stress distributions. On the other hand extreely reined esh is nuerically expensive and ay not provide additional accuracy. The coeicient was introduced in the ANSYS input ile as a ultiplication actor that linearly increases the nuber o eleent divisions on lines. Thus, when the esh reineent coeicient is, or exaple, equal to 3, the nuber o eleents per each line o the FEM odel will be 3 ties bigger than when is equal to 1. Fig.16. Energy release rate G I as a unction o esh reineent coeicient. The total nuber o eleents in the odel is however not exactly 3 ties bigger because the area eshing was perored as "ree eshing" based on the speciied line divisions. Fig.16 shows the dependency on esh reineent actor o energy release rate G I. In Fig.16 results obtained by autoated J -integral routine are denoted as G ANSYS JINT, results obtained by autoated VCCT routine are denoted as G ANSYS VCCT and results obtained by anual VCCT are denoted as VCCT anual. Nuerical results are copared with analytical calculations. In Fig.16 the results are shown or Case C, where crac tip eleents were used (see Fig.13). As Fig.16 shows, the results obtained by two autoated calculation routines (G ANSYS JINT and G ANSYS VCCT) are not sensitive to esh reineent coeicient within the studied range. Furtherore, the agreeent between nuerical results obtained by these two ethods and the analytical results is excellent. On the other hand, results obtained by anual VCCT show signiicant dependence on esh reineent. Judging ro the trend o curve "VCCT anual" in Fig.16, urther esh reineent would provide better agreeent with analytical results, however, it has to be noted that even at esh reineent level o 5 the necessary calculation tie 18

31 increased signiicantly. Thereore, in urther paraetric analysis esh reineent level o 3 was used. As Fig.16 shows, the results obtained by anual VCCT are always below the analytical results. Further, the inluence o integration length is discussed. In the FEM odel geoetry, special area around the crac tip was created to anipulate with the integration length or anual VCCT. It has been shown beore [27] that the calculated energy release rate values depend on the length o integration. A coproise has to be ound between having as short as possible integration length according to deinition in Eq.10 and having a suicient integration length in which stress and displaceent distributions are accurately captured. Fig.17 shows the dependency on integration length R i o energy release rate G I calculated by anual VCCT. The results are copared with analytical results and the results obtained by autoated calculation routines available in ANSYS 13.0, notation is the sae as in Fig.16. The results are shown or Case C (see Fig.13). Curve "VCCT anual" in Fig.17 shows that optial integration length R i exists at which agreeent with analytical results is the best. Agreeent between nuerical and analytical results reduces at very sall integration lengths as well as when the integration length is too large. Fig.17. Energy release rate G I as a unction o integration length R i. Finally, the inluence o crac length is discussed. Fig.18 shows plotted results o energy release rate G I as a unction o crac length a. Lie in Figs.16 and 17, the nuerical results in Fig.18 are copared with analytical calculations. The results are shown or Case C (see Fig.13). As it can be seen in Fig.18, the agreeent between nuerical results obtained by autoated calculation routines (G ANSYS JINT and G ANSYS VCCT) and analytical results is excellent. However, the results obtained by anual VCCT are also in very good agreeent with analytical results. This leads to conclude that even though looing on a narrow vertical scale as in Fig.16 and 17, the dierence between anual VCCT results and analytical results and nuerical results obtained by autoated ANSYS calculation routines is notable, the anual VCCT ethod is still suicient or crac propagation analysis. It was also proved in [27] that although the absolute values o G calculated by anual VCCT are not accurate, the noralized curves o G vs. the crac length correspond very well with accurate calculations perored using BEM [21]. 19

32 Fig.18. Energy release rate G I as a unction o crac length a. In conclusion, it can be noted that the autoated routines or G calculations provide excellent agreeent with analytical results i the crac tip region is properly eshed. Nuerical calculations perored anually, such as crac closure technique and the virtual crac closure technique, are less accurate than the autoated routines although the agreeent with analytical results is still satisying. The previous conclusions were drawn only ro analysis o isotropic aterial and the use o autoated routines has to be validated also or ore coplex cases such as anisotropic aterials and interace cracs. In Papers to V, which are included in this thesis, VCCT was used in a anner described in Eq.(7-10). In Paper VI energy release rate was calculated using autoated VCCT routine available in ANSYS 13.0 (CINT coand with VCCT option). Probably the ost detailed nuerical analysis o the local stress state at the debond crac tip in ters o stress intensity actor and degree o singularity has been perored in [21] using BEM. It is iportant to note that in [21] G was calculated taing into account the presence o rictional stresses along the crac aces. Nuerical results showed that riction opposes the debond growth. For ore details on this [31] can also be suggested or reading. Unortunately, BEM ethod at present is liited to isotropic constituents and, hence, not applicable or carbon ibers. 1.4 Paris law or atigue A broad review on echanics and icro-echaniss in atigue o etals, non-etals and coposite aterials is given by Suresh [32]. Under cyclic loading conditions, the onset o crac growth ro pre-existing law or deect can occur at stress intensity values that are well below the quasi-static racture toughness. For conditions o sallscale yielding, where the nonlinear zone at the crac tip is a ere perturbation in an otherwise elastic aterial, Paris, Goez and Anderson [33] suggested that the increent o atigue crac advance per stress cycle, da /dn, could be related to the range o the stress intensity actors, K, during constant aplitude cyclic loading, by the power law expression. 20

33 da BK (19) dn where N is the nuber o applied cycles, B and are epirical scaling constants. The constants are inluenced by aterial icrostructure, cyclic load requency, waveor, environent, test teperature and load ratio. This approach has since been widely adapted or characterizing the growth o atigue cracs in etals and etal atrix coposites [34], [35] under conditions o sallscale plastic deoration at the crac tip. For coposites, since they are non-hoogeneous aterials, it is ore relevant to use range o energy release rates G instead o stress intensity actor range K. Then the expression Eq.(19) can be rewritten as da BG (20) dn where G G ax G in is the dierence between energy release rates at axial and inial applied strains during the constant aplitude atigue loading. More coplex deinition and physical reasoning o energy release rate range G or anisotropic aterials has been recently discussed in [36]. The ajor appeal o the linear elastic racture echanics approach is that the stress intensity actor range (or energy release rate range), deterined ro reote loading conditions and ro the geoetrical diensions o the craced coponent, uniquelly characterizes the propagation o atigue cracs; this ethod does not require a detailed nowledge o the echaniss o atigue racture [32]. Along with Paris law relation using energy release rate range G (Eq.20), the racture toughness G c can also be used as a paraeter or criteria or crac initiation and propagation analysis [21],[37]. Conceptually, the Paris law (Eq.(20)) could also be ipleented or iber/atrix debond crac growth characterization in atigue. Paper IV included in this thesis and which has recently been accepted or publication [38], deonstrated agreeent between the odeling and experiental results thus proving applicability o the power law or debond growth characterization in atigue. 1.5 Experiental easureents o debond growth in atigue Experiental evidence o iber atrix interace debond growth in atigue has been reported or both unidirectional [39] and odel coposites such as single iber coposites [38]. In [39] growth o the debond length on the surace o carbon iber/epoxy UD coposite was experientally observed. During the anuacturing o the coposite, a well-polished stainless steel plate was placed on top o the release il in order to enhance the surace soothness. The speciens were subjected to load-controlled tension-tension atigue test with the axiu applied strain equal to 0.89%, requency o 10 Hz and the tension-tension load ratio o R=0.1. The debond length was easured using surace replicas ade ro a cellulose acetate il that was pressed on the specien surace. Fig.19 shows the debond lengths plotted with respect to the nuber o the load cycles. 21

34 Fig.19. Debond length increase in atigue loading o carbon iber/epoxy coposite. Data taen ro [39]. The results in Fig.19 show easureents or our dierent debonds, the accuracy o the easureents is 2. The results in Fig.19 deonstrate a large scatter in debond length values. Fig.20. Optical icroscopy iages showing the debond length increase with nuber o cycles in tension-tension cyclic loading: a) N = 1; b) N = 1000; c) N = 5000; d) N = F - iber; M - atrix. Figure ro [38]. More straightorward and easier-to-interpret experiental results have been obtained or single glass iber coposites as reported in [38], which is also one o the papers included in this thesis (Paper IV). In [38], experiental easureents o debond growth in tension-tension atigue were perored or single iber ragentation saples with transparent epoxy atrix. A single iber brea was induced prior to the atigue test and the easureents o debond length were perored using optical icroscopy. The saples were subjected to values o axial strain up to 1.76% 22

35 with requency o 2 Hz and ratio between the inial and axial load in one cycle R=0.1. Fig.20 shows optical icroscopy iages taen or one saple ater a certain nuber o applied load cycles. 2. Objectives o the current wor The ain objective o the present thesis is to develop ethodology or prediction o iber/atrix debond growth in cyclic tension-tension loading o polyer coposites. Fracture echanics concept o energy release rate G is used or debond growth analysis. In order to predict the increase o the debond crac length with the increasing nuber o load cycles the ollowing tass have to be advanced: 1) a coputational tool or calculating the Mode strain energy release rates G ust be developed. As described beore in section 1.3, analytical ethods can be applied or G calculation in sel-siilar crac growth regions, where the debond crac tip is suiciently ar away ro the iber brea. For short debonds, where high interaction with the iber crac is expected, nuerical ethods (e.g., FEM) in cobination with virtual crac closure technique can be applied; 2) When a coputational tool or strain energy release rate calculation is developed, a relation that characterizes the increase o the debond length as a unction o applied load cycles in atigue ust be adapted. Paris law (Eq.20), requently used in atigue daage characterization in etals, is a reasonable candidate. 3) To validate the applicability o Paris law or characterization o iber/atrix debond growth in atigue, independent experiental easureents are required. Once the power law is shown to be applicable, the values o power law paraeters can be deterined ro the best it between odeling and experiental data. The power law paraeters are aterial properties that can be ipleented in debond growth siulations or unidirectional and single iber coposites. The nuerical ethods or calculation o strain energy release rate are not liited to debond crac growth analysis only. In the last paper included in this thesis (Paper VI), daage initiation and propagation in a carbon iber subjected to cyclic charge/discharge with lithiu ions (intercalation) is analyzed. The objective o Paper VI is to analyze the possible daage initiation locations and the copeting propagation echaniss in a carbon iber. 3. Suary o appended papers In Paper I analytical solution or Mode strain energy release rate G was ound or unidirectional (UD) coposites and paraetric analysis perored in the sel-siilar debond crac propagation region only. Three aterials have been studied two odiications o carbon iber/epoxy resin coposites and a glass iber/epoxy coposite. Various iber volue ractions were considered. Concentric cylinder assebly odel was applied or analytical calculation o G. Analytically calculated G values were copared with FEM calculations giving excellent agreeent. A onediensional odel was also proposed or consideration, assuing that during the debond crac propagation, the energy released ro iber is overwheling and the 23

36 energy released ro atrix and the eective coposite is alost negligible. Paraetric analysis o aterial properties was perored and non-iportant paraeters were identiied. G calculations were coprised in siple expressions that can be used or any arbitrary echanical, theral as well as cobined loading cases. These expressions are based on coeicients representing echanical, theral and ixed echanical-theral response o UD coposite. By peroring three calculations o siple loading cases, the coeicients were ound and validated. In Paper G was obtained or short debonds (or UD coposites) by applying FEM calculations in cobination with virtual crac closure technique (VCCT). Total strain energy change ethod was also used or coparison with VCCT results, however, it was observed that the accuracy o VCCT is higher. In order to deterine the appropriate length o the FEM odel, paraetric study was perored prior to doing calculations. The integration length used in VCCT was also studied paraetrically and the optial size was chosen. The sae three aterials as in Paper I with various iber volue ractions were studied. Further in Paper, the calculation results or both short and long debonds were ipleented in power law expression or siulations o debond crac growth with the increased nuber o applied cycles. Since no experiental data on UD coposites were available, trend curves showing signiicance o paraeters such as iber radius, initial teperature change, aterial constants and power law constants were presented and analyzed. In Paper I, debond growth in single iber (SF) coposites subjected to tensiontension atigue is analyzed. Using the sae procedure as or UD coposites in Papers I and, irst, an analytical solution or Mode energy release rate G is ound or sel-siilar crac growth region and then FEM odeling is perored to obtain agniication proiles or short debond cracs. It is shown in Paper I that agniication o the strain energy release rate at short debonds ollows the sae rule or echanical and theral loads. Furtherore, it is shown that agniication curves are in excellent agreeent with accurate BEM calculations [21]. A sipler or o equation is thus derived or calculation o the strain energy release rate. Polynoial type o expression is used or approxiation o agniication coeicients. Debond growth siulations are perored to analyze the inluence on debond growth o iber radius, Paris ( power ) law paraeters and the applied initial teperature change. In Paper IV odeling ethodology described in Paper I is advanced urther and the odeling results are copared with experiental data or interace debond crac growth in SF coposites subjected to tension-tension atigue loading. Experiental easureents o debond length increase in tension-tension atigue were perored or single glass iber/epoxy coposites. Debond length is easured using optical icroscopy. Residual stress and iber pre-stress was included in analysis. The power law constants were extracted ro the best it between the experiental and odeling results. Validation o results proves that power unction o power law is applicable to characterize the debond crac growth in atigue o polyer coposites. The values o the deterined power law paraeters were validated or dierent applied strain levels. 24

37 Also debond growth siulations are perored to reveal the sensitivity o power law paraeters and the eect o the residual stresses on debond growth. In Paper V iber/atrix interace debond growth in close-to-surace region o a UD coposite is analyzed. The UD coposite is subjected to tension-tension atigue. 3-D FEM odeling is perored to account or the non-axisyetric stress state due to the edge eect. Two dierent 3-D FEM odels are advanced or energy release rate calculations depending on the distance between the iber and the coposite specien surace Model 1, where the iber and atrix volues are surrounded by an eective coposite phase and Model 2, where a resin-rich region is assued between the iber and the coposite surace. The calculation results show that closer to the coposite surace the average strain energy release rate G is higher than in the bul o coposite. Shear stress and axial displaceent distributions in ront and behind the debond crac tip were copared to analyze the non-axisyetric stress state. Using the values o power law paraeters deterined in Paper IV, siulations o debond growth in tension-tension atigue were perored or glass iber/epoxy coposite. Siulation results reveal that debond growth rate in vicinity o coposite surace is uch higher than in the bul o the coposite. Thereore, studying debond growth on UD coposite surace is proposed as a ethod or accelerated testing o coposites in atigue. Finally, in Paper VI racture echanics concepts o strain energy release rate are used to odel icro crac initiation and propagation in a carbon iber, which, apart ro the load bearing unction, also serves the purpose o an electrode in a novel lithiu-ion rechargeable battery. When subjected to lithiu ion intercalation, carbon iber experiences a non-unior swelling that leads to developent o high echanical stresses. In any cycles o charging-discharging these stresses can introduce daage and reduce the echanical and electrocheical properties o the battery. FEM odeling using theral analogy is perored to solve the transient ion diusion and echanical stress proble. The irst tas o the paper is to analyze the ost probable daage initiation locations. Stress analysis o an undaaged carbon iber subjected to ion intercalation is perored. According to nuerical odeling results, the irst ode o daage will be in a or o radial cracs developing on the outer layers o the carbon iber due to high tensile hoop stresses. Then the propagation o radial crac is analyzed using racture echanics and calculating the energy release rate G to deterine the crac growth tendency. The results show that radial cracs ay grow in urther cyclic charge/discharge until reaching a certain length at which G becoes zero due to copressive hoop stresses in the core region o the iber. A urther scenario o crac delection is studied. Ater decay o the radial crac growth, the crac ay urther propagate as an arc shaped crac due to tensile radial stresses. The copeting echaniss are analyzed by plotting energy release rate at dierent tie instances and at dierent crac lengths. The crac propagation analysis results deonstrate that exoliation o carbon iber layers ay occur in cyclic charge/discharge thus reducing the echanical and electrocheical perorance o the battery. 25

38 4. Reerences 1. R. Talreja, A continuu echanics characterization o daage in coposite aterials. Proceedings o Royal Society o London, A378, pp , R.H. Dausardt, R.O. Ritchie and B.N. Cox, Fatigue o advanced aterials: Part 1, Advanced Materials and Processes, vol.7, pp , R. Talreja, Fatigue o coposites, In Structure and Properties o Coposites, ed. T.W.Chou, Weinhei:VCH, pp , B. Liu, L.B. Lessard, Fatigue daage-tolerance analysis o coposite lainates: Stiness loss, daage-odelling, and lie prediction, Coposites Science and Technology, vol.51, pp , S.I. Andersen, H. Lilholt, A.A. Lystrup, Properties o coposites with long ibres, In Design o Coposite Structures against Fatigue, ed. R.M.Meyer, Bury St.Edunds: Mechanical Engineering Publications, pp , W.A. Curtin, N. Taeda, Tensile strength o iber-reinorced coposites: I. Model and eects o local iber geoetry, Journal o Coposite Materials, vol.32(22), pp , D.G. Harlow, S.L. Phoenix, Probability distributions or the strength o coposite aterials : A convergent sequence o tight bounds, International Journal o Fracture, vol.17(6), pp R.E. Pitt, S.L. Phoenix, Probability distributions or the strength o coposite aterials IV: Localized load-sharing with tapering, International Journal o Fracture, vol.22(4), pp , D. Hull, T.W. Clyne, An introduction to coposite aterials, Second edition, Cabridge University Press, Cabridge, N. Laws, G.J. Dvora, The eect o iber breas and aligned penny-shaped cracs on the stiness and energy release rates in unidirectional coposites, International Journal o Solids and Structures, vol.23, pp , I.J. Beyerlein, S.L. Phoenix, Stress proiles and energy release rates around iber breas in a laina with propagating zones o atrix yielding and debonding. Coposites Science and Technology, vol.57, pp , S. Behzadi, P.T. Curtis, F.R. Jones, Iproving the prediction o tensile ailure in unidirectional bre coposites by introducing atrix shear yielding, Coposites Science and Technology, vol.69, pp , R. Talreja, Fatigue o coposite aterials, Technoic Publishing Inc., Lancaster, Pennsylvania, B. Harris, editor, Fatigue in coposite aterials, Woodhead Publishing Ltd., Cabridge, A. Pupurs, J. Varna, Unidirectional coposite in echanical atigue: odelling debond growth ro ibre breas, Plastics Rubber and Coposites, vol. 39, pp , R. Talreja, C.V. Singh, Daage and Failure o Coposite Materials, Cabridge Univ Press, Z. Hashin, B.W. Rosen, The elastic oduli o iber-reinorced aterials, Journal o Applied Mechanics, vol. 31, pp , Z. Hashin, Analysis o Coposite Materials a survey, Journal o Applied Mechanics, vol. 50, pp , A. Pupurs, J. Varna, Energy release rate based iber/atrix debond growth in atigue. Part I: Sel-siilar crac growth, Mechanics o Advanced Materials and Structures, In press,

39 20. J.A. Nairn, Y.C. Liu, On the use o energy ethods or interpretation o results o single-iber ragentation experients. Coposite Interaces, vol. 4, pp , E. Graciani, V. Manti, F. París, J. Varna, Nuerical analysis o debond propagation in the Single Fibre Fragentation Test, Coposites Science and Technology, vol. 69(15-16), pp , G.R. Irwin, Fracture, Handbuch der Physi, vol.5, Springer Verlag, Berlin, ANSYS Release 13.0, ANSYS Acadeic Research, ANSYS Inc., Canonsburg, Pennsylvania, T.L. Anderson, Fracture echanics. Fundaentals and Applications, second edition, CRC press, M.L. Willias, On the stress distribution at the base o a stationary crac, Journal o Applied Mechanics, vol. 24, pp , H. Tada, P.C. Paris, G.R. Irwin, The Stress Analysis o Cracs Handboo, Third Edition, ASME Press, New Yor, A. Pupurs, A. Krasniovs, J. Varna, Energy release rate based iber/atrix debond growth in atigue. Part : Debond growth analysis using Paris law. Mechanics o Advanced Materials and Structures, In press, A. Pupurs, J. Varna, Fracture echanics analysis o debond growth in single iber coposite under cyclic loading, Mechanics o Coposite Materials, vol. 47(1), pp , J.R. Rice, A path independent integral and the approxiate analysis o strain concentration by notches and cracs, Journal o Applied Mechanics, vol.35, pp , C.F. Shih, B. Moran, T. Naaura, Energy release rate along a threediensional crac ront in a therally stressed body, International Journal o Fracture, vol.30(2), pp , E. Graciani, Nuerical Analysis o the Single Fiber Fragentation Test Including the Eect o Interacial Friction, Doctoral Thesis, Luleå University o Technology, Luleå, Sweden, S. Suresh, Fatigue o aterials, Cabridge University Press, Cabridge, P.C. Paris, M.P. Goez, W.P. Anderson, A rational analytic theory o atigue, The Trend in Engineering, vol.13, pp. 9-14, Z.H. Xia, W.A. Curtin, Lie prediction o titaniu MMCS under low-cycle atigue, Acta Materialia, vol.49, pp , G. Bao, R.M. McMeeing, Fatigue crac growth in iber-reinorced etalatrix coposites, Acta Metallurgica Et Materialia, vol.42, pp , L.N. McCartney, Energy ethods or atigue daage odelling o lainates, Coposites Science and Technology, vol.68, pp , J. Varna, R. Joe, L.A. Berglund, Interacial toughness evaluation ro the single-iber ragentation test, Coposites Science and Technology, vol.56, pp , A. Pupurs, S. Goutianos, P. Brøndsted, J. Varna, Interace debond crac growth in tension-tension cyclic loading o single iber polyer coposites, Coposites Part A, accepted, E.K. Gastedt, Eects o debonding and iber strength distribution on atigue daage propagation in carbon iber reinorced epoxy. Journal o Applied Polyer Science, vol. 76, pp ,

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41 Paper I A. Pupurs, J. Varna Energy release rate based iber/atrix debond growth in atigue. Part I: Sel-siilar crac growth, Mechanics o Advanced Materials and Structures, In press, 2012.

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43 Energy release rate based iber/atrix debond growth in atigue Part I: Sel-siilar crac growth Andrejs Pupurs 1,2, Janis Varna 1 1 Luleå University o Technology, SE Luleå, Sweden 2 Riga Technical University LV-1048 Riga, Latvia Abstract The strain energy release rate related to debond crac growth along the iber/atrix interace in a unidirectional coposite with a broen iber is analyzed. The UD coposite is represented by a odel with axial syetry consisting o three concentric cylinders: broen and partially debonded iber in the iddle surrounded by atrix which is ebedded in a large bloc o eective coposite. Analytical solution or Mode energy release rate G is ound and paraetric analysis perored in the sel-siilar debond crac propagation region. It is shown that any anisotropic elastic constants o the iber, which are usually not nown, have sall eect on G. Keywords: iber breas, debonding, energy release rate 1. Introduction Loading a unidirectional (UD) iber reinorced polyer coposite in iber direction in quasi-static or in a high stress cyclic tension-tension regie, any iber breas ay occur in rando positions already during the load increase in the irst cycle. This is because iber strain to ailure in UD coposites is lower than the polyer atrix strain to ailure. The described is a echanis or very high strains. The sequence o events at strain levels below the iber breaing liit is not considered in this paper. There the atigue scenario ay be dierent and, or exaple, initiation o sall atrix cracs between ibers could be the irst ode o daage. Dierent positions along the iber have dierent strength values, which are assued to ollow Weibull distribution and the irst iber crac occurs in the weaest position. In result o stress transer over the iber/atrix interace the axial stress level in the iber ater a certain distance ro the iber crac recovers its ar-ield value and with increasing load ultiple iber breas in the sae iber are possible. In cyclic loading with constant aplitude we usually assue that ibers do not experience atigue. Thereore the next step in daage evolution with increasing nuber o cycles ay be developent o interace cracs (debonds) growing along the iber/atrix interace. Probably the ost coplex analysis o the stress perturbation in coposite due to iber breas and interace daage was perored in [1] assuing hexagonal iber pacing o any ibers and using 3-D FEM. In [1] the ocus was on overload in neighboring ibers and a 30º wedge with 9 ibers including the broen one in the iddle was analyzed. Keeping in ind the ain goal o the analysis (stress concentration and ailure o neighboring ibers) authors concluded that ost o the geoetrical details are unnecessary. It was shown that 3-D odel introduced in [2] which is based on shear lag approach and nuerically applied or a hexagon o ibers 31

44 with the broen one in the iddle has suicient accuracy or coposites with high iber contents. These conclusions otivated our choice o concentric cylinder assebly to represent coposite with a broen iber. Oten the iber ailure is an unstable phenoenon and the energy released during this event is larger than required to create a iber brea, which is assued to be a pennyshaped crac transverse to the iber axis. The excess o released energy ay go to initiation o the iber/atrix debond at the tip o the iber brea. The iber debonding can be considered as an interace crac growth along the iber and racture echanics concepts (strain energy release rate) ay be used or the evolution analysis. The debond initiation (transition ro no debond state to debond state) is a very coplex nonlinear process and due to lac o relevant inoration it is not suitable or odeling. Four stress state regions can be distinguished considering a broen iber with a long debond in coposite: a) a very coplex stress state at the iber crac; b) plateau region away ro the iber crac and away ro the debond crac tip; c) the debond tip region with stress singularity; d) a region in ront o the debond tip (ar away ro it) where the iber is perectly bonded to the resin. Due to urther debond crac growth the plateau region b) becoes longer and the region d) correspondingly shorter. Thereore, long debond cracs propagate in a sel-siilar anner eaning that when the crac grows the local stress proile at the crac ront shits along the iber axis without changes in the shape and in the value. Certainly, i iber has ultiple breas, long debonds ro both iber ends start to interact and the selsiilarity is lost. In order to analyze the interace crac growth in ters o racture echanics the strain energy release rate related to the increase o the debond length has to be calculated. It has been done previously or debond growth analysis along a single iber ragent in so-called single iber ragentation (SFF) test. Fig.1. a) Unidirectional coposite with iber brea subjected to echanical ( z ) and theral (T ) loading. b) Concentric cylinder assebly odel (F-iber, M-atrix, C- eective coposite), l d shows the iber/atrix debond length. The used ethods cover a wide spectru ro approxiate analytical to nuerical based on inite eleents (FE) or boundary eleents (BE) [3-5]. The variational odel 32

45 based on iniization o the copleentary energy [4] is probably one o the best analytical solutions available but the accuracy is achieved in rather coplex calculation routine. The ost detailed nuerical analysis o the local stress state at the debond cracs tip in ters o stress intensity actors and degree o singularity has been perored in [5] using BE ethod. Unortunately this ethod at present is liited to isotropic constituents and, hence, not applicable or carbon ibers. Generally speaing, soe o the described approaches ay be adapted or dealing with partially debonded broen iber surrounded by atrix and ebedded in coposite. However results o a systeatic paraetric analysis o the energy release rate due to debond growth in coposite as aected by constituent properties, geoetrical paraeters are not available.the objective o this paper consisting o Part I and Part is to peror the aboveentioned paraetric analysis and use the results to siulate the debond growth according to Paris law in UD coposite. The coposite with a broen and partially debonded iber is represented by three concentric cylinder assebly (CCA) odel as shown in Fig.1. A broen iber in the iddle is surrounded by a atrix cylinder and the interace is debonded over distance l d, see Fig.1b. This iber/resin bloc is ebedded in outer eective coposite cylinder. In Part I an analytical solution or the strain energy release rate G is given in the sel-siilar debond propagation region, the signiicance o dierent paraeters is disclosed and validity o one-diensional odels is discussed. This approach is used also in Part [6] where FEM and the virtual crac closure technique [7] (using nonsingular FEM solution) are used in a paraetric study to calculate G or short debonds where the stress state at the iber brea interacts with the stress state at the crac tip. Methodology or convenient calculation o G or an arbitrary cobined theral and echanical loading is presented. The indings ro these studies are suarized in siple expressions and used in siulation and analysis o the debond growth in tension-tension atigue using Paris law. 2. Analytical odel or energy release rate in sel-siilar debond propagation region 2.1 Calculation ethod Because the loading type relevant to this study is axial tension and the applied teperature is negative, T 0 (aterial cooling ater processing to roo teperature) the radial stress on the iber surace is copressive and the debond crac propagation is in Mode. It is due to larger Poisson s ratio or the atrix and also due to larger theral expansion coeicient o the atrix. An analytical ethod can be applied to energy release rate calculation or a particular case when the tip o the iber/atrix debond crac is ar away ro the iber brea where it was initiated and also ar ro another debond which ay be approaching ro the other end o the iber (typical or short ibers or or iber ragent). In this case the debond crac propagation can be considered as sel-siilar. Matheatically this condition in ters o Fig.2 can be written as ld r ld L (1) 33

46 In Eq.(1) r is the iber radius, L is the odel length and it represents ½ o the iber ragent length. For a odel with single iber brea it ay be interpreted as the distance ro the brea to the load application cross-section. Under conditions o Eq.(1) the debond crac growth by dl d which corresponds to new created surace area da 2r dl d (2) a) shits the debond crac tip (and the corresponding singular stress state in its vicinity) in the z-direction by dl d. b) reduces the bonded region volue in the odel by R 2 dld ( R is the outer radius o the three cylinder assebly in Fig.2) and increases the debonded region volue by the sae aount. debond Fig.2. Regions with length dl d in three phase coposite odel corresponding to bonded and debonded case. Bold line represents the debonded surace in contact where sliding is possible. Obviously the energy change in the syste due to this event can be calculated (neglecting riction and energy dissipation related to it which are urther discussed in Section 3) as the dierence between strain energy U o the bonded and U o the debonded regions o unit length ar away ro the debond crac tip and the iber brea (shadowed regions in Fig.2) tot b du (3) tot tot U d U b tot d Models representing these two regions are given in Fig.3. Note that in the debonded region, Fig.3b sliding o the iber with respect to the atrix is possible and hence its deoration is dierent than the deoration o the rest o the odel. The energy release rate is obtained dividing Eq.(3) by the new created surace area Eq.(2) 34

47 G du uconst (4) da Strain energy or ore coplex case with N cylindrical phases where phase 1 ay be bonded or debonded is given below. In the odel in Fig.3 analyzed in this paper N=3 (iber, atrix, eective coposite). Fro the derivation given in Appendix 1 ollows that the strain energy o the odel a) or b) in Fig.3 can be calculated i the average stress state in constituents is nown U tot V 2 N 1 N V zz z0 z TT V r r (5) In Eq.(5) N=3 and =1,2,3 are iber, atrix and eective coposite respectively. 1 Fig.3. Geoetrical showing o the bonded region (a), which due to crac growth by dl turns to debonded region (b) with length dl. d Due to dierences in boundary and interace conditions the averages stresses in Eq.(5) are obviously dierent in the bonded and in the debonded case leading to two tot tot dierent valuesu b, U d. Model assuptions and calculation expressions are given in Subsections 2.2 and Stress state and strain energy in the bonded region The constant strain in axial direction is denoted by z0 and the teperature dierence is T. In Eqs.(5)-(18) assuption is ade that in the perectly bonded region, which is ar away ro the debond tip and ro the iber brea, the axial deoration in all phases is the sae =1,2,3 (6) z0 z0 d The solution or the -th phase is given in Appendix 2 u 1 r A1 r A2 r (7) 35

48 A 1 and H r and A g H T (8) z 1 z0 3 2 A A r H T (9) r 1 2 z0 2 A A r H T (10) 1 2 z0 A 2 are 2 3 unnown constants yet to be solved and r r,,, H 3 are unctions o the -th phase elastic constants given in Appendix 2. g,, The displaceent and stress in Eqs.(7)-(9) have to satisy continuity conditions on all interaces, i.e. the solutions obtained or each phase separately ust satisy the ollowing 2 3 conditions: (i) Radial displaceent ust be zero on the syetry axis r (ii) Displaceent and radial stress continuity at all interaces u r u r (11) 1 r u r r 1,2,...,3 1 (12) r 1 r r (iii) Zero radial stress at the outer boundary r 1,2,...,3 1 (13) N r r R o the cylinder assebly R 0 (14) For a given z0 the syste o 2 3 equations ollowing ro these conditions allows or deterination o 2 3 constants A 1 and A 2 (their values will depend on the applied strain and teperature). According to the above explanation, in siulations the teperature dierence and the axial strain are applied siultaneously. Unortunately, the experiental procedure (and as a consequence also the procedure used in siulations) is dierent: irst the specien is produced at high teperature and cooled down or processing to roo teperature. Theral stresses occur during this step and the specien has copressive strain th z0 beore any echanical loading. This state is assued as the starting (zero echanical strain) state in tests. Then certain aount o echanical ech tensile strains z0 is applied. The relationship with z0 in above expressions is as ollows ech th z0 z0 z0 (15) Thereore, in order to use in siulations the proper value o z0 corresponding to experiental we irst have to calculate the theral strain th. It can be easily ech z0 z0 36

49 done using expressions in this subsection: the correct value ust result in zero average stress in direction z r r dr 0 av z r z (16) R r 1 Insertion o Eq.(8) in Eq.(16) ater integration yields 3 1 A r r 1 r r g H T z0 3 2 (17) th The proble is conveniently solved by using z0 as a nuerical paraeter. For every value o th z0 the syste o linear equations is solved and Eq.(17) used to chec whether zero average stress (with required accuracy) is obtained. th Certainly, the calculated negative z0 value corresponds to the ree theral expansion strain o the coposite in the given theral conditions, which can be obtained as cop z T provided the coposite longitudinal theral expansion coeicient is nown. Fro Eqs.(8)-(10) the average stress expressions to be used in Eq.(5) can be written as A g H T (18) z 1 z0 3 2 A H T (19) r 1 z0 Substituting Eq.(18) and Eq.(19) and accounting or Eq.(6) in Eq.(5) we obtain U b tot V 2 3 V 1 3 A 1 g z0 H 3 T z0 z T 2TV r A 1 z0 H r T 1 (20) 2.3 Stress state and strain energy in the debonded region In each phase o the debonded region represented by odel in Fig.3b the solutions or displaceents and stresses in Eqs.(7)-(10) are still valid with the only dierence that r 1 0 (21) zo z zo The rest o phases (atrix and the eective coposite which both are perectly bonded together) have the sae deoration with respect to the stress ree state as beore given by Eq.(15) and denoted as previously by z0. The axial deoration o the debonded iber consists o ree theral expansion and strain due to radial interaction with the surrounding atrix (caused by theral isatch and dierences in Poisson s ratios). The axial strain value can be obtained ro condition that axial stress in iber is zero 37

50 z z (22) which according to Eq.(A.2.8) leads to g A T g H z z (23) Substituting Eq.(23) in stress expressions or iber (Eqs.(9)-(10)) and realizing that A due to Eq.(11) we obtain r H r g H T g A (24) 1 1 r (25) The stress expressions or the rest o cylindrical phases reain as given by Eqs.(8)- (10). Radial displaceents in all phases are given by Eq.(7). The interace conditions at phase interaces reain as in the bonded case: radial displaceents and stresses are continuous. The outer radial boundary o the assebly is ree o stresses and Eq.(14) is still valid. Solving or constants i A as described in Section 2.2 (the pure theral case is o no interest now) we can write the expression or the strain energy o the debonded odel, Fig.3b. Since the iber stresses are not coordinate dependent the average stresses in the phase 1 (iber) are given by Eqs.(22), (24) and (25). Finally r r r z r z z z tot d H g H T g A TV V T H A V T T T H g A V V U (26) It has to be ephasized that constants i A =1,2,3 are result o the solution in the bonded and debonded case correspondingly and, hence, they are dierent in Eqs.(20) and (26). 2.4 One-diensional energy release rate odel Exact expressions or energy release rate calculation during sel-siilar debond crac growth are given by Eqs.(3)-(4), (20) and (26). It has to be recognized that they are rather coplex and nuerical MATLAB based codes were developed to obtain nuerical values. Calculations discussed in Section 3 showed that the largest energy change due to debonding is in the iber phase. Thereore it appears interesting to ind out i a one-diensional odel considering the strain energy change in iber only can be used and what is the error introduced by this sipliication.

51 First we consider the bonded odel shown in Fig.3a. In 1-D case or a given applied strain (Eq.(15)) which can be written also as z 0 ech cop 0 z z T (27) The iber has only the axial stress (the rest o stress coponents is zero) ech cop T E E 0 0 T (28) The strain energy o the iber is z z z z z z z z U b 2 z z 2E r dl 2 d Ez 2 r dl 2 d ech cop T 2 z0 z z (29) In the debonded 1-D odel the axial iber stress is zero and the radial interaction is neglect leading to U d z 2E 2 z r 2 dl d 0 (30) Hence, the energy release rate due to debond growth by dl d according to the 1-D odel is G b U 2r dl d Ez 4 r ech cop T 2 z0 z z (31) In 1-D odel the coposite axial theral expansion coeicient is E z z V cop E V z (32) E V E V z More accurate value ay be obtained using CCA odel [8,9]. 2.5 Applicability o engineering ethods or eective coposite properties The thero-elastic properties o the eective coposite (phase 3) were calculated using the Concentric Cylinder Assebly (CCA) odel introduced by Hashin [8,9] or the bonded case. The transverse shear odulus was obtained using the Christensen s sel-consistent odel [10]. The whole set o used expressions is given in [11] where ultiphase orthotropic cylinder syste was analyzed. It has to be noted that or deterination o coposite theral expansion coeicients, axial odulus, Poisson s ratio and bul odulus the Hashin s CCA odel is a particular 2-phase case (N=2) o the bonded odel described in Section 2.2. As an alternative the eective coposite properties can be calculated using engineering expressions [12]: rule o ixtures (RoM) or coposite longitudinal odulus EL and or Poisson s ratio LT, Eq.(32) or axial theral expansion 39

52 coeicient L. Transverse odulus E T and in-plane shear odulus G LT ay be calculated using Halpin-Tsai expressions. The transverse theral expansion coeicient o the coposite T was calculated using Schapery s expression [13], which actually is valid only or isotropic ibers T V V L LT 1 1 (33) For carbon ibers TT and T were used. Using the above approxiate engineering expressions to calculate eective coposite properties we observe artiicial interactions at the atrix/coposite interace even in the bonded case. This is because the iber/atrix cylinder does not represent the coposite with properties ollowing ro engineering expressions. This proble does not exist when CCA odel values or the eective coposite are used. 3. Results and discussion 3.1 Material properties This study was perored on three dierent UD coposites - two carbon iber coposites (CF1/EP and CF2/EP) and one glass iber coposite (GF/EP) all o the with epoxy resin atrix (EP). The properties o the constituents (ibers and atrix) and o the eective coposite are given in Tables 1 and 2 respectively. Elastic odulus in longitudinal and transverse directions ( E L and E T ), shear odulus G LT, Poisson s ratios 12, 23 and theral expansion coeicients L and T are presented in the tables. In nuerical results presented in this Section iber radius r 4. Since the energy release rate G is proportional to the iber radius, recalculation to dierent values o r is very siple. Table 1. Elastic properties o aterials. CF carbon ibers, GF glass ibers, EP epoxy resin. Material E L E T G LT L T [GPa] [GPa] [GPa] - - [1 C ] [1 C ] CF1(CF2) 500 (300) GF EP The elastic properties o the eective coposite were calculated cobining Hashin s and Christensen s odels [8-10]. It has to be ephasized here that during cyclic atigue loading the average specien teperature can increase i the loading requency is too high and cooling is not properly arranged. Locally at the sliding interaces the teperature increase can be even uch higher. The teperature increase will change the teperature related G and also the resin properties (or exaple reduced E-odulus). Thereore, the paraetric analysis presented here includes also changed resin properties. The teperature dependence o G in this paper is expressed in a very general or and i the real specien teperature is available the values ay be easy recalculated. It is 40

53 obvious that at increased teperature the theral stress level is lower and G would be slightly lower. Certainly, the case with non-unior teperature has to be analyzed separately using nuerical ethods. Table 2. Elastic properties o the eective coposite (transversally isotropic). Re. V E cl E ct G clt cl ct [GPa] [GPa] [GPa] - - [1/C10-6 ] [1/C10-6 ] CF1/EP CF1/EP CF1/EP CF2/EP CF2/EP CF2/EP GF/EP GF/EP GF/EP Veriication o the analytical odel The energy change du values calculated according to Eq.(3) and the expressions in Section 2 were irst validated coparing to values obtained ro FE odel (coercial code ANSYS [14]). The FE odel consisted o 3 constituents represented as 3 doains, which are eshed in axisyetric eleents. FEM allows to obtain the strain energy values or each constituent (U, U, U C ) and or the tot whole odel ( U ) in a very siple way. Siilarly to analytical odel, the energy release rate can be obtained ro two siulations irst the energy or bonded odel is calculated, then the energy or ully debonded iber/atrix interace odel is calculated (decoupling nodes in axial direction at r r ), and the energy release rate G is obtained ro the energy dierence divided by the contact surace (interace area). The coparison between analytically and nuerically calculated values was ech perored or an arbitrary chosen thero-echanical loading case z0 0,05%, T 100 C. As expected the results coincide or all three coposites with very high accuracy. 3.3 Calculation o G by the analytical CCA odel Siulating debond crac growth in Part we will need the energy release rate or an arbitrary cobination o echanical and theral loading. To avoid unnecessary and tie consuing calculations every tie the loading changes it is useul to reeber that the strain energy is a quadratic unction o stress state. Stress coponents are linearly dependent on applied echanical strain ech z0 and on the teperature dierence T. Hence we suggest the ollowing or G E r z0 ech 2 r th ech z0 T r th T 2 (34) 41

54 Upper index in Eq.(34) is used to note that the coeicients are or sel-siilar 2 crac growth sei-ininite debond. In Eq.(34) G is in J /, strains are in % and teperature in C. With nown coeicients, th and th Eq.(34) is a very useul tool in siulations where G is used. For any coposite these constants can be ound peroring calculation or three loading cases: a) pure echanical; b) pure theral; c) any cobination o theral and echanical. Tables 3a and 3b show all analytically calculated G values or these cases out o which we obtain coeicients, th and th. Fro the echanical loading ech 1% (let section o Table 3a) we obtain values o coeicient, ro the theral loading case T 100C (right section o Table 3a) we obtain values o ech coeicient th and ro thero-echanical loading z0 0.05%, T 100C (Table 3b) we obtain values o coeicient th. In echanical loading G varies a lot dependent on the iber aterial. Coparing Table 3a with Table 1 it is obvious that G is alost proportional to the axial odulus o the iber, which explains the suggested or o Eq.(34). Table 3a. Calculated G values or echanical and theral loading separately ( J 2 ). Mat. ech z0 1% T 100 C V CF1/EP CF2/EP GF/EP Table 3b. Calculated G values or thero-echanical loading ( J 2 ). ech Mat. z0 0.05%, T 100C V CF1/EP CF2/EP GF/EP Table 4 shows the calculated values o coeicients, th and th. The values are presented or all 3 studied aterials and or 3 volue ractions V 0.4, V 0.5, V 0.6. It can be seen that coeicient which is related to echanical ter o the energy release rate is alost volue raction independent (within the observed range). The other two coeicients however depend on iber volue raction quite signiicantly. Coeicient th is just wealy dependent on the iber properties (about 10% variation in the considered range). Coeicient th is related to pure theral ter in 42

55 the energy release rate and is three orders o agnitude larger than ech practical applications z0 2 (expressed in %) is o order o agnitude Since in ost 0 10 whereas T and the echanical ter contains iber odulus (order o 11 4 agnitude10 ) the echanical ter is about 10 ties larger than the theral and the inluence o the teperature related ter or the considered aterials is insigniicant. Table 4. Calculated coeicients in Eq.(34). Mat. th V CF1/EP CF2/EP GF/EP Table 4 continued. Mat. th V CF1/EP CF2/EP GF/EP The eect o the ixed ter varies ro 1-10% dependent on iber properties and volue raction and the value o the echanical strain. Using data in Table 4 to plot the dependence o th and th on iber content one can see that the relationship or all coposites is rather linear (linear it is better or th which as analyzed above has ore practical signiicance). In other words linear interpolation o these constants in the analyzed volue raction region is possible, thus allowing or, and th estiation or any given iber content. 3.4 Validation o calculated coeicients Knowing coeicients in Eq.(34) one can predict G values or aterial subjected to any arbitrary thero-echanical loading conditions o the described type. Certainly it is realized that the perored calculation or the theral case being absolutely correct is still artiicial and does not correspond to what would happen in a real test with a broen iber in coposite (iber brea aces would close at negative teperature change). Nevertheless in the sel-siilar odel in Fig.3 the calculation is possible and in the inite debond length odel in this case we would allow interpenetration o iber crac suraces (negative iber crac opening). ech Two arbitrary thero-echanical loading cases ( z0 0.01%, T 50C and ech z0 0.05%, T 50C ) were considered in order to validate the G deterination using Eq.(34) and the calculated coeicients, th and th. th 43

56 Table 5. Validation o Eq.(34) coparing with direct calculations (values in ( J 2 )). ech ech z0 0.01%, T 50C z0 0.05%, T 50C V CF1/EP ( ) ( ) ( ) ( ) ( ) ( ) CF2/EP ( ) ( ) ( ) ( ) ( ) ( ) GF/EP ( ) ( ) ( ) ( ) ( ) ( ) The validation was perored coparing Eq.(34) with direct analytical calculations or the ixed thero-echanical loading or all three aterials and throughout 3 volue ractions as presented in Table 5. The predicted values are alost identical with directly calculated values (shown in parentheses). 3.5 Identiication o non-signiicant paraeters Soe o thero-elastic constants o transverse isotropic ibers are usually not nown and are estiated very roughly or even arbitrary assued. With a hope to conclude that the rough approxiation o values is justiied or at least to establish liits or these approxiations, in this Section we peror paraetric analysis varying these possibly non-signiicant constants. They are identiied or echanical and theral loading cases separately. Only one property o aterial is changed at a tie (with the respect to reerence aterial in Table 1) and its aect on G value is evaluated. Both carbon and glass iber coposites are inspected. The results are suarized in Tables 6a to 6e, showing the changed property and its value. Results or reerence aterial (properties given in Table 1) are also presented. As it stands in Tables 6a and 6b, changing iber transverse elastic odulus E T, Poisson s ratios 12 and 23 as well as the shear odulus G 12 does not signiicantly aect energy release rate during echanical loading. 2 Table 6a. G values ( J ) or CF1/EP with dierent iber properties, the odiied ech property is indicated, V 0.5, z0 1% (irst row) and V 0.5, T 100 C (second row). Re. (CF1/EP) Modiied property E T G L 0 T

57 2 Table 6b. G values ( J ) or GF/EP with dierent iber properties, the odiied ech property is indicated,v 0.5. z0 1% (irst row) andv 0.5, T 100C (second row). Re. (GF/EP) Modiied property L T Tables 6a and 6b prove that in theral loading energy release rate is sensitive only to the value o the theral expansion coeicients (both in longitudinal and transverse directions) o the constituents. The eect is uch saller in CF1 case than in GF case. The dependence o the previously presented G or reerence aterial with E 3 GPa on the resin odulus are illustrated in Tables 6c and 6d, where E 3.5 GPa. Coparing with Table 3a and 3b we see negligible change in the echanical loading. The theral ter is increasing with resin odulus whereas the ixed ter is arginally decreasing. The results or higher resin odulus expressed in ters o, and presented in Table 6e show the sae trends. th th Table 6c. Calculated G values or CF1/EP and GF/EP both with atrix odulus E 3.5GPa in echanical and theral loading ( J 2 ). Mat. ech z0 1% T 100 C V CF1/EP GF/EP Table 6d. Calculated G values ( J 2 ) or CF1/EP and GF/EP both with atrix odulus E 3.5GPa in thero-echanical loading. ech Mat. z0 0.05%, T 100C V CF1/EP GF/EP Table 6e. Calculated coeicients in Eq. (34) or CF1/EP and GF/EP both with atrix odulus E 3.5GPa. Mat. th V CF1/EP GF/EP

58 Table 6e continued. Mat. th V CF1/EP GF/EP Applicability o Rule o Mixtures This section suarizes results calculated by 1-D energy release odel given in Section 2. The odel states that during debond crac growth, ost o released energy is ro iber and the energy released ro atrix and coposite can be neglected. According to the 1-D odel, see Eqs.(31)-(32), the released energy depends on the iber content V only through the coposite theral expansion coeicient. Table 7 shows coparison o G values calculated by the CCA odel with values obtained by the 1-D odel. For G CCA calculations cl is obtained ro CCA odel while G 1D or 1-D odel is calculated using cl ro Eq.(32). G * 1D is also calculated or 1-D odel only this tie using cl ro ore accurate CCA odel. Presented nubers relate to CF1/EP coposite with iber content V 0.5 or dierent thero-echanical loading cases. Table 7. Coparison o G calculated by CCA and 1-D odels or CF1/EP V 0.5. G values in J 2. % TC G CCA G 1D * G 1D Engineering odels lie the rule o ixtures (RoM) described in Section 2.5 can also be applied as a siple ethod to calculate the eective properties o the coposite. In ollowing we evaluate, what error we induce by applying these siple odels instead o CCA. The error induced can be shown in a siple coparison o radial theral stress distribution e.g. or CF1/EP V 0.5, subjected to theral loading only ( T 100C). 46

59 F M C Fig.4. Radial stress distribution in perectly bonded three phase odel (by FEM) subjected to T 100C only, using eective properties ro RoM and ro Hashin s CCA odel. I eective properties o the coposite are deterined accurately, then there should be no radial stress in the eective coposite. There should also be no radial interaction (stress) at the interace between atrix and the eective coposite. Fig.4 shows that eective properties ro Hashin s odel [8-10] are representative or the coposite, however properties obtained by RoM are incorrect in a way that there is soe considerable radial interaction between the concentric atrix and coposite cylinders that can thereore introduce errors in G calculations. Nevertheless, G or the case presented is G (J / 2 ) which is the sae as the value in Table 3a where the CCA odel was used to calculate eective constants o the coposite. 4. Conclusions Energy release rate G due to sel-siilar growth o iber/atrix interace debond in unidirectional coposite with broen ibers is analyzed using analytical odel with axial syetry. In the odel, subjected to axial echanical strain and teperature dierence, the broen iber with a partially debonded interace is surrounded by resin cylinder, which is ebedded in the eective coposite. In the sel-siilar propagation region o debond the energy release rate is calculated as a dierence o strain energy in a bonded and debonded syste. Applying rule o ixtures as a tool to deterine coposite elastic properties introduces artiicial interaction between the concentric cylinders in CCA odel as shown by FEM siulations. Cobined Hashin s and Christensens odel is shown to give consistent results and thereore is applied in all calculations. Based on calculations or three eleentary loading cases (echanical, theral, cobined) siple quadratic expression to calculate G or an arbitrary loading case is proposed. Coeicients in this expression which are siple unctions o iber content and constituent properties were obtained in paraetric analysis which also 47

60 proved that eect o the ost o the usually unnown anisotropic thero-elastic constants o the iber have sall or zero eect on the calculated G and thereore rough assuptions can be used. Proved by G calculations or echanical, theral and thero-echanical loading cases it was shown that within the range o ost practical applications or a UD coposite with partially debonded regions the inluence o theral load is inial in coparison to echanical loading. This result is dierent than in case o a single iber in an ininite atrix. 1-D odel or G was proposed assuing the energy is released in the iber only. It appears to be a good assuption or the echanical loading but is rather inaccurate or theral loading. 5. Reerences 1. Z.H. Xia, W.A. Curtin, Shear-lag vs. inite eleent odels or stress transer in iber -reinorced coposites, Coposites Science and Technology, vol. 62, pp , T. Oabe, N. Taeda, Y. Kaoshida, M. Shiizu, W.A. Curtin, A 3D shear-lag odel considering icro-daage and statistical strength prediction o unidirectional, iber-reinorced coposites, Coposites Science and Technology, vol. 61, pp , J.A. Nairn, Y.C. Liu, Stress transer into a ragented, anisotropic iber through an iperect interace, International Journal o Solids and Structures, vol. 34, pp , W.Wu, I.Verpoest, J. Varna, Prediction o energy release rate due to the growth o interace crac by variational analysis, Coposites Science and Technology, vol. 60, pp , E. Graciani, V. Manti, F. París, J. Varna, Single iber ragentation test. A BEM analysis, Collection o Technical Papers - AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynaics and Materials Conerence, Norol (Virginia), United States, vol. 2, pp , A. Pupurs, A. Krasniovs, J. Varna, Energy release rate based iber/atrix debond growth in atigue. Part : debond growth analysis using Paris law, Mechanics o Advanced Materials and Structures, In press, G.R. Irwin, Fracture, Handbuch der Physi, vol.5, Springer Verlag, Berlin, Z. Hashin, B.W. Rosen, The elastic oduli o iber-reinorced aterials, Journal o Applied Mechanics, vol. 31, pp , Z. Hashin, Analysis o Coposite Materials a survey, Journal o Applied Mechanics, vol. 50, pp , R.M. Christensen, K.H. Lo, Solutions or eective shear properties in three phase sphere and cylinder odels, Journal o the Mechanics and Physics o Solids, vol. 27, pp , E. Marlund, J. Varna, R.C. Neagu, E.K. Gastedt, Stiness o aligned wood iber coposites: eect o icrostructure and phase properties, Journal o Coposite Materials, vol. 42, pp , B.D. Agarwal, L.J. Broutan, Analysis and perorance o iber coposites, 2nd ed., John Wiley & Sons inc., USA, R.A. Schapery, Theral Expansion coeicients o coposite aterials based on energy principles, Journal o Coposite Materials, vol. 2, pp ,

61 14. ANSYS Release 11.0, ANSYS Acadeic Research, ANSYS Inc., Canonsburg, Pennsylvania, D.H. Allen, C. Yoon, Hoogenization techniques or thero-viscoelastic solids containing cracs, International Journal o Solids and Structures, vol. 35, pp Appendix 1. Strain energy o the syste In thero-echanical loading case with echanical load at boundaries and unior teperature change T the strain energy density o a aterial (one phase in the odel) ay be written in Cartesian coordinates as U * 1 2 ij ij ij T (A1.1) In Eq.(A1.1) repeating indexes ean suation over the. The strain energy U is a volue integral over the phase volue. U 1 ij ij dv T 2 2 V ij ij dv (A1.2) V Using divergence theore the irst integral can be reduced to surace integral (see or exaple [13]) leading to the ollowing expression or strain energy U 1 2 S ij u i n j ds T 2 ij ij dv (A1.3) In Eq.(A1.3) S is the whole surace area o the aterial. In the case o the used odel consisting o N concentric cylinders the expression or the total strain energy is V N U tot 1 ij u i n j ds T S N ij ij dv (A1.4) 1 v In Eq.(A1.4) v is the volue o the -th cylinder and S is the whole surace consisting o side boundaries r r (denoted S r ) and end boundaries z z0 0; dl (denoted S z ) The loading cases considered in the Part I o the paper lead to axisyetric solutions and the radial displaceents and stresses do not depend on the axial coordinate. We will analyze the su with surace integrals (irst ter in Eq.(A1.4)) in ore details separating these two types o boundaries N I 1 I 1z I 1r 1 ij u i n j ds 1 ij u i n j ds (A1.5) S z N 1 S r Ater that we will consider the volue integral in Eq.(A1.4). 49

62 A. Integration over end suraces S z At boundaries z z0 we have n x n y 0, n z 1. In the odel or both bonded and debonded cases the stress coponents xz yz 0. The axial displaceents u z0 0 and u z 0 dld constant in each phase. In the bonded case the elongation is the sae in all phases u dld u z0 u 0 u =1,2 N (A1.6) 0 z0 0 z In the debonded case it is dierent in the iber (the separated iber is in contact with the rest o the aterial but is ree to ove in axial direction). Hence, ro Eq.(A1.5) N I 1z 1 ij u i n j ds 1 zz u z0 dl d u z0 0 ds 1 u 0 zz ds S z N 1 S z N 1 Sz N 1 u 0 S z zz (A1.7) In Eq.(A1.7) zz is the average value o the stress coponent zz in the -th phase. Multiplying and dividing Eq.(A1.7) by the odel length dl we inally obtain N I 1z V V zz 2 (A1.8) z0 In Eq.(A1.8) V is volue raction o the -th phase in the odel, z0 is the average axial strain in the phase deined as 1 V S z dl d V z0 u 0 dl d (A1.9) B. Integration over side suraces S r Beore we analyze the surace integrals in Eq.(A1.5) related to S r we irst will express stresses and displaceents in the Cartesian syste through stresses and displaceents in the cylindrical syste where the solution is obtained. The relevant expressions are xx r cos 2 sin 2 yy r sin 2 cos 2 xy r sin cos (A1.10) Since in probles with axial syetry u 0 50

63 u u x y u cos r u sin r (A1.11) Orientation o the noral vector to the surace r r can be expressed as n cos n sin (A1.12) x y Considering the expression under the sign o integral in substituting there Eqs.(A1.10)-(A1.12) we obtain I 1 r or one phase and u n u (A1.13) ij i j r r Since integration in I 1 r is over suraces r r, the values o the radial stress and radial displaceent on this surace are constant which aes the integration trivial. In result N I 1r 1 2dl d r u r r r r r 1 u r r 1 r r 1 (A1.14) 2 1 At interaces between phases r r a) radial displaceents in both phases and b) radial stresses are equal. This is valid or both bonded interace and debonded interace in contact. Thereore any ters in Eq.(A1.14) are cancelled. This is valid also or debonded open interaces because the radial stress is zero there. In addition the ter at r 0 is zero. Finally 0 I 1r 1 2 2Ru rn R R N r 0 (A1.15) because the radial stress at the outer boundary o the odel is zero (ree surace). C. Volue integrals related to theral eects Next we analyze volue integrals in Eq.(A1.4) related to theral eects. I 2 T 2 N ij ij dv (A1.16) 1 v It can be expressed as Consider the expression isotropic I 2 VT 2 N V ij ij (A1.17) 1 ij ij or one phase. Since all phases are transversally 51

64 0 (A1.18) x y r xy Substituting Eqs.(A1.10) and (A1.17) in the aboveentioned expression we obtain ij ij zz z r r (A1.19) ij ij zz z r r (A1.20) The analysis is by this inished. In the last step the expressions or I 1 (Eqs.(A1.8) and (A1.15)) and or I 2 ( Eq.(A1.20) substituted in Eq.(A1.17)) are used in Eq.(A1.4) which leads to U tot V 2 N 1 N V zz z0 z TT V r r 1 (A1.21) Appendix 2. Stress solution in cylindrical phase The constant strain in axial direction is denoted by z0 and the teperature dierence is T. The explanation in this Appendix ollows [8]. All phases are transversally isotropic and ollow Hooe s law which or noral stresses is written here in vectorial or i C ij j i T (A2.1) C ij is the stiness atrix with usual notation. All shear stresses are zero. Hence the only non-trivial equilibriu equation or a phase is r r r 0 r (A2.2) The strain relationship to radial displaceent u r is given by r u r r u r r z z0 (A2.3) and then by use o Eqs.(A2.1) and (A2.2) we obtain the ollowing ield equation 2 u r r 2 u r rr u r r 2 0 (A2.4) The general solution o Eq.(A2.4) can be written as 1 u r A1 r A2r (A2.5) Expressions or strains calculated according to Eq.(A2.3) are 52

65 r A 1 A r 2 2 A A r (A2.6) Introduce new constants g C 33 2C C C C C C (A2.7) Here 3-is the axial direction, 1 and 2-two directions in the plane o isotropy. The stresses according to Eq.(A2.1) are z A 1 g z0 H 3 T r A 1 A 2 r 2 z0 H r T A 1 A 2 r 2 z0 H r T (A2.8) where H H 3 33 z 2 r C C 32 z C 32 r r (A2.9) 53

66 54

67 Paper A. Pupurs, A. Krasniovs, J. Varna Energy release rate based iber/atrix debond growth in atigue. Part : debond growth analysis using Paris law, Mechanics o Advanced Materials and Structures, In press, 2012.

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69 Energy release rate based iber/atrix debond growth in atigue. Part : Debond growth analysis using Paris law Andrejs Pupurs 1,2, Andrejs Krasniovs 2, Janis Varna 1 1 Luleå University o Technology, SE Luleå, Sweden 2 Riga Technical University LV-1048 Riga, Latvia Abstract The strain energy release rate related to debond crac growth along the iber/atrix interace in a unidirectional coposite with a broen and partially debonded iber is analyzed. The ocus in this paper (Part ) in contrast to the sel-siilar crac growth analysis in Part I [1] is on growth o short debonds near the iber brea. Since selsiilarity condition is not valid or interactive cracs, nuerical FEM siulations were used to calculate agniication o previously described coeicients in the strain energy release rate expression. The indings ro these studies are used in siulation o the debond growth in tension-tension atigue using Paris law. Keywords: iber breas, debonding, energy release rate, atigue 1. Introduction When unidirectional iber reinorced polyer coposites (UD) are loaded in longitudinal high stress tension-tension cyclic loading, ultiple iber breas occur in rando positions already during the irst cycle. Assuing that ibers do not experience atigue and ignoring in this paper the possibility that stress concentration at the iber brea can cause racture o the neighboring iber, the initial daage state with iber breas is ixed. The urther daage evolution with increasing nuber o cycles is in or o debonds growing along the iber/atrix interace or penny shaped iber crac propagating in the atrix. The latter scenario or quasi static loading was analyzed in [2], calculating strain energy release rate or hoogenized solid, and it is not considered in this paper. Ater the UD coposite longitudinal odulus reduction in the irst cycle due to iber breas, the growing debonds in atigue loading urther reduce the average stress in the iber and as a consequence the elastic properties o the UD coposite are reduced. This eect which can be expressed in ters o increasing opening o iber cracs has been analyzed in [3]. A 2-D shear lag odel analysis was used in [4] to analyze the copetitive echaniss o debond initiation ro atrix crac and ro iber brea. A stress based criterion was used or debonding. An advantage is that the used Monte-Carlo siulations allow treating any daage entities. Shear lag odel in 2-D orulation was used also in [5] where the iber brea propagated in atrix reaching the neighboring iber and both debonded as well as plastic yielding zones along the interace have been analyzed. These odels are 2-D, they are rough and they do not include radial interaction and in our opinion they can not give a good representation o the debonding in UD coposite. In this paper we consider the iber/atrix debonding in UD coposite as an interace crac growth starting ro iber brea and peror strain energy release rate based 57

70 analysis. Many studies o this type have been perored or a single iber in an ininite atrix, see or exaple [6-8]. Due to dierent Poisson s ratios o iber and resin and dierent theral expansion coeicients the debond crac aces are always in contact and the debond crac propagation is in Mode. Possible rictional eects which are diicult to evaluate in atigue are ignored in this paper. For onotonic quasistatic loading the approach to account or riction suggested in [9] sees to be proising. In Part I o this paper [1] the strain energy release rate dependence on aterial and geoetrical paraeters was analyzed using analytical odel which is valid or sel-siilar crac propagation region. The UD coposite with a broen and partially debonded iber is odeled as three concentric cylinder assebly. Here in Part the strain energy release rate or relatively short debonds (as copared with the iber radius) is analyzed nuerically or a general thero-echanical loading case. The eective coposite properties are calculated using Hashin s Concentric Cylinder Assebly (CCA) odel [10]. A slightly sipliied FEM based calculations o the energy release rate due to debond growth were presented in [11] calculating the properties o the eective coposite with engineering expressions (rule o ixtures, Halpin-Tsai etc). Since the theroelastic constants calculated in this way are not represented by the resin/iber cylinder assebly, soe artiicial stresses at the interace were observed. Theral stresses were ignored in that study. The eect o the inal length o the iber ragent (interaction o debonds coing ro both iber ends) were also addressed in [11]. It will be shown that the strain energy release rate G is a decreasing unction o the distance ro the iber brea (called debond length) and is expected to be singular when the debond length approaches to zero. This eans that the debond crac will be iediately initiated by the iber brea but it will be arrested at certain distance due to the decreasing value o the G. A urther debond growth requires either a higher applied load (quasistatic tensile loading) or an increasing nuber o cycles in atigue loading. In cyclic loading the analysis o the propagation o cracs in atrix is usually based on assued law expressing the crac area growth rate with the nuber o cycles as a unction o stress intensity actor [12] or strain energy release rate [8,9,13]. The ost coon is the relationship in or o power unction and strain energy release rate is preerred or polyer atrices [13, 14]. The arguent in the power law is the dierence between the axiu and iniu value o strain energy release rate during one cycle. In [13] this type o power law was used to predict interlayer delaination growth initiated by intralainar cracs. The evolution o nuber o intralainar cracs in atigue has also been analyzed by based power law [15, 16]. Since delaination and intralainar cracing are ailure odes doinated by atrix and iber/atrix interace properties, we suggest that the power law could be applicable also or a single debond along the iber. We assue that the debond growth rate along the iber can be expressed in or o Paris law with coeicients to be identiied in tests. In order to use the Paris law, energy release rates have to be nown or each value o the debond length (G as a unction o debond length). For this reason the results o the FEM based paraetric analysis are suarized and the dependences on paraeters presented with siple but rather accurate itting unctions. Possible trends in atigue as dependent on aterial and geoetrical paraeters are analyzed. 58

71 2. Theoretical aspects 2.1 Virtual crac closure technique (VCC) We odel the debond growth in the UD coposite using concentric cylinder assebly presented in Fig.1, where the broen and partially debonded iber is surrounded by resin cylinder which is eected in a thic cylinder o eective coposite. The iber radius is r, the debond length is l d and the atrix radius r ollows ro the iber volue raction in the coposite. The assebly is irst (directly ater anuacturing) subjected to teperature change T and then the debond growth is considered at ixed applied displaceent u z (corresponding to certain zech ). Fig.1. Model or virtual crac closure technique. Concentric cylinder geoetry showing debond crac at the iber/atrix interace. The virtual crac closure technique [17] is a convenient ethod to calculate the when doing FE calculations. The calculated relative tangential displaceent u z distribution behind the debond crac tip and the shear stress rz values ahead o the crac tip can be obtained directly ro the FEM odel as shown in the Fig.1. The crac closure technique states that the energy released due to debond crac growth by da is equal to the wor which is required to close the newly created surace ro size A da bac to size A, where da 2r dl d. Upper index in Eqs.(1)-(6) is used to indicate the length o the debond crac. Closing the debond crac by dl d (ro l d dl d to l d ) by applying tangential tractions, points at the debonded surace in the z l ; l dl which have relative tangential displaceent region d d d G u l d dl d (z) u l d dl d l z (z) u d dl d z (z) (1) are oved bac to coinciding positions. At the end o this procedure the shear stress l in point z is equal to d rz (z), which is the shear stress in ront o the crac with size l d. Then the wor required to close the crac by dl d can be expressed as: 59

72 W 2r 1 2 l d dl d u l d dl d l (z) d rz (z)dz (2) l d Within the virtual crac closure technique it is assued that due to sall value o dl d the relative sliding displaceent at the tip o the crac with size l d dl d is the sae as at the tip o the debond crac with size l d : u l d dl d (z) u l d (z dl d ) (3) The beneit o this assuption is that only one stress state calculation or a given debond length is required. Since the energy release rate is deined by relations: and da 2r dl d (4) G du uconst (5) da the ollowing result is obtained using Eqs.(2) and (3) 1 G (l d ) li dl d 0 2dl d l d dl d l d u l d z dl d l d rz dz z (6) Using FEM based solution the integration is over inite region, which ost probably is larger than the region where the solution is local (singular). Thereore, strictly speaing, the obtained value is not G. It depends on the integration region dld and rather should be called energy release rate over distance dl d. In Part I [1] it was shown that the strain energy release rate is a quadratic unction o the applied echanical strain zech and T G E r z0 ech 2 r th ech z0 T r th T 2 (7) Coeicients, th and th in Eq.(7) are larger or short debonds and approach the values or sel-siilar cracs or longer debonds. Thereore they can be written in or with agniication actors l dn * and l * as ollows: th dn * * * l l l dn th dn th th th dn th (8) Coeicients, th and th or sel-siilar cracs were analyzed in Part I [1]. For exaple, is practically independent on iber proeprties and iber content. Magniication coeicients * * *, th and th will be obtained peroring nuerical siulations or sall debond lengths. 60

73 2.2 Application o Paris law For a UD iber coposite subjected to cyclic tension-tension loading with interace cracs growing along iber/atrix interace, we will apply Paris law in order to describe the increase o the debonded interace area da with the increasing nuber o applied cycles. The or o Paris law which we intend to use is as ollows da dn G B G C (9) G is the strain energy release rate dierence between the value corresponding to axiu and iniu load ax, in. In echanical atigue according to Eq.(7): 2 2, T G T ax in th ax in (10) Paraeters B, and are not nown a priori and have to be deterined in tests, G c is the critical strain energy release rate. Generally speaing these paraeters as well as resin elastic properties ay depend on the teperature during the test which in tension-tension cyclic atigue can be changing and have local distribution. Since at the stage we do not have appropriate data we will investigate in paraetric analysis the iportance and eect o these constants on debond growth. Using Eq.(4) and introducing noralized debond length l l / r (11) dn d Eq.(9) can be written in or dldn dn G B * * G ; 2 C 2r B B (12) * Obviously B is diensionless. For siulation purposes Eq.(12) has been written in an increental or: l ( ) dn * G B G l C ( ) dn N, l l l ( 1) ( ) ( ) dn dn dn In Eq.(13) l (1) dn is the initial debond length at the start o siulation. 3. Results and discussion 3.1 FEM odel ; =1,2,3 (13) The goal with nuerical calculations in this paper is two-old: a) to deine the best ethodology or nuerical G deterination in the non-sel-siilar debond crac 61

74 growth region; b) to analyze the eect o the interaction between the iber crac and the debond crac on the energy release rate due to short debond growth The inite eleent odel is rather siple. It consists o 3 constituents represented as 3 areas, which are eshed in axisyetric eleents. Application o FEM [18] allows us to obtain the strain energy values or each constituent (U, U, U C ) and or the whole odel (U tot ) in a very siple way by using several post-solution coands. Fiber radius r was assued equal to 4, outer radius o the atrix r is easily calculated through iber volue raction V. The outer radius o the eective coposite, which is supposed to be ininite, was deterined through paraetric study as a value at which the outer boundaries do not aect the results. The thicness o the outer cylinder was set equal to dc 5 r. To urther validate the chosen geoetrical coniguration, solution with stress ree outer radial boundary was copared with solution based on constant radial displaceent conditions (coupling the nodes on boundary in the radial direction) without inding any noticeable dierence. Because the loading type relevant to this study is axial tension and the applied teperature is negative, T 0 (aterial cooling ater processing to roo teperature) the radial stress on the iber surace is copressive and the debond crac propagation is in Mode. It is due to larger Poisson s ratio or the atrix and also due to larger theral expansion coeicient o the atrix. Since the inluence o riction is assued negligible, these conclusions beneit the build-up o the inite eleent odel, as the debonded interace can thus be siply odeled by coupling the adjacent iber and atrix eleents in the radial direction r and allowing the to ove reely in the longitudinal direction z. The ain beneit is the ability to avoid the use o contact eleents, which would signiicantly increase the calculation tie. 3.2 Nuerical techniques or G calculation Using FEM based solution the integration in Eq.(6) is over a inite region and the obtained value o G depends on the size o the integration region dl d. Since singular eleents were not used in the FEM analysis the accuracy o techniques to be used have to be evaluated and decisions ade coparing with exact solutions in the sel-siilar debond crac propagation region. Using VCC the wor to close the crac is underestiated i the singular nature o the stress distributions at the debond crac tip is not accounted or or i the integration region is larger than the region in which the local solution is doinant. The calculated G value depends on the size o the integration region dl d. The two questions are: a) what is the ost appropriate integration length?; b) what iber length can be considered as ininite in the sense that the calculated G becoes independent on the iber length L and results can be copared with the sel-siilar solution? FEM based calculations were perored or long debond l deb 20 r and two long ibers L 90 ; 200 r. The G values or echanical loading 0.1% case are shown in Table 1 or aterial CF1/EP with iber content V 0.5. As shown in the table, the values o G (denoted G VCC to identiy the ethod used) strictly depend on the chosen length o area o integration, dl d. Debond length l deb 20 r was assued to have a debond belonging to the sel-siilar crac growth region. The analyticalg values at 1% strain or sel-siilar crac growth region are given in 62

75 Table 3a o Part I [1] and or CF1/EP coposite with iber content V 0.5 the value 2 recalculated to 0.1% strain is 0.499J / Table 1. Energy release rate Method Hal iber length (ragent length) G VCC G. G values ( J / 2 ) obtained by various nuerical techniques. As seen in Table 1, the longer we tae dl d, the closer we get to the correct value. However, according to Eq.(6) G is deined as the liit when dl d 0, thereore dl d should be as sall as possible. The bad agreeent with G 0.499J / 2 or saller dl d values can be explained with the saller aount o used nodes in integration in Eq.(6), when the length dl d is sall, which results in rather approxiate inal value. For larger dl d values, ore nodes are used and the solution in this sense is ore accurate because both shear stress and shear displaceent proiles are ore accurate. However, it is not the local solution as required in G deinition. As a coproise between desired agreeent with sel-siilar debond solution and requireent to have the integration interval as short as possible we decided to use dl d 1 r, when calculating G by VCC technique in the ollowing paraetric analysis. In the analyzed case it gives only 5% dierence ro G cca. Also presented in Table 1 is the eect o the length L, see Fig.1. Calculated G VCL values or two lengths L 200 r and L 90 r are presented and only a slight reduction indicating interaction o debonds ro both iber ragent ends is evident. This allows us to consider the shorter length L 90 r as long enough to satisy sel-siilar crac growth conditions and use it in the ollowing nuerical paraetric analysis. Using shorter L is preerable because it also reduces the size o the inite eleent odel, thus aing calculations ore tie-eicient. Shorter L is preerable also when G due to debond crac propagated by dl d is calculated ro the energy change in all constituents (iber, atrix, coposite), see row with G in Table 1. Using this ethod calculations were perored at EN L L dl d 2 r 1.5 r 1 r 0.5 r 0.25 r 200 r r G VCC G EN L 200 r ld 20 r and ld 19 r. The accuracy o this technique depends on the used esh and decreases with increasing size o the odel: or large odel the sall energy change related to G is calculated as a dierence between two very large nubers. The accuracy o both discussed nuerical techniques appears to be rather siilar. All results presented below are obtained using the VCC techniques. 3.3 Paraetric analysis o atrix properties Two aterials have been studied carbon iber/epoxy resin UD coposite denoted as CF1/EP and glass iber/epoxy resin UD coposite denoted as GF/EP. The notations 63

76 o the two coposites are the sae as previously and echanical properties o constituents and coposites can be ound in Table 1 and Table 2 in Part I [1]. In Part I paraetric analysis o iber properties was perored and it was ound that soe anisotropic iber elastic and theral properties that are usually unnown are non-iportant as they do not aect the strain energy release rate values [1]. We assue that these conclusions are valid also or short debonds. First we observe how atrix properties inluence the energy released due to interace crac growth. Paraetric analysis was perored only concerning the elastic odulus o the isotropic epoxy resin atrix. Coeicients, th and th in Eq.(7) were calculated or two values o elastic odulus o the atrix - E 3(GPa) and E 3.5 (GPa). The values o coeicients, th and were calculated by using the ollowing th units in Eq.(7): percent (%) or echanical strain ech and Celsiu degrees ( C) or the applied teperature. These units in Eq.(7) ust be ollowed in order to calculate the energy release rate G in J 2 or an arbitrary thero-echanical loading case. Coparison o agniication coeicients * * *, th and th deined according to Eq.(8) or these two cases o atrix odulus can be ade using data in Tables 2 and 3. Nuerical values or the longest debond length ld 20 r were used as values or the sel-siilar case. Data or CF1/EP are presented in Table 2 and data or GF/EP are presented in Table 3. Table 2. Calculated values o coeicients *, * * th and th or UD CF1/EP coposites with two dierent atrix oduli (in all casesv 0.5). l d 2 r 4 r 5 r 10 r 20 r * E 3 [GPa] E 3.5 [GPa] * E 3 [GPa] th E 3.5 [GPa] * E 3 [GPa] th E 3.5 [GPa] Table 3. Calculated values o coeicients *, * * th and th or UD GF/EP coposites with two dierent atrix oduli (in all casesv 0.5). l d 2 r 4 r 5 r 10 r 20 r * E 3 [GPa] E 3.5 [GPa] * E 3 [GPa] th E 3.5 [GPa] * E 3 [GPa] th E 3.5 [GPa] As one can see the agniication coeicients or debonds longer than ld 2 r is not large. Shorter debonds certainly would have larger agniication but they were not 64

77 analyzed because the interaction with iber brea is very strong in this region and we do not trust the accuracy o our calculations. The eect o atrix odulus on agniication coeicients in the considered atrix odulus region is very arginal and in any cases ay be neglected. 3.4 G as a unction o debond length in thero-echanical loading Fig.2. shows strain energy release rate or CF1/EP as a unction o noralized interace debond length in purely echanical loading with applied strain ech 1%. Results or three iber volue ractions are shown in Fig.2. Fig.2. Energy release rate G as a unction o noralized debond length l dn ( l dn l d /r ). Mechanical loading ech 1%. Presented values were calculated by virtual crac closure technique using FEM siulations. It can be seen ro the igure that at short debonds, high aounts o energy are released during debond crac growth. When the debond length is suiciently long, the aount o released energy sets into its sel-siilar value. The obtained G values at each debond length shorter than sel-siilar conditions length were copared with values or the sel siilar region and agniication * coeicients, (see Eq.(8)), were ound. In calculations the nuerical value corresponding to the largest debond length was assued as the sel-siilar carc value. The coeicients or echanical loading are presented in Fig.3, or theral loading in Fig.4 and the ixed thero-echanical ters in Fig.5. ln( * 1) ln( a) b (15) l dn 65

78 Fig.3. Values o coeicient * as a unction o noralized debond length l dn : a) or CF1/EP; b) or GF/EP. Fig.4. Values o coeicient * th as a unction o noralized debond length l dn : a) or CF1/EP; b) or GF/EP. Fig.5. Values o coeicient * th as a unction o noralized debond length l dn : a) or CF1/EP; b) or GF/EP. It would be convenient or Paris law expression (see Eqs.(8),(10) and (13)) i all agniication coeicients could be coprised in a single relation in ters o the debond length. It was ound that with high accuracy the agniication coeicients ay be itted with exponential unction o type * 1 a e bl dn (14) 66

79 where a and b are constants, * denotes the agniication coeicient, so that or an ininitely long debond length l d the value o * is equal to 1. Taing the logarith o Eq.(14). We see that, to have a good it, the nuerical agniication coeicient data have to ollow a straight line in log-log axes. The itting paraeters a and b can be easily obtained ro the best linear it in these axes.in Fig.3, Fig.4. and Fig.5 values o coeicients * * *, th and th ro the itting expression (Eq.(14)) are copared with the calculated FEM data showing that the itting is good. For a general case when coposite is subjected to both echanical and theral atigue loading, all three coeicients are necessary. Values o itting paraeters are suarized and presented in Tables 4, 5 and 6. Paraeters have been calculated or both studied aterials (CF1/EP and GF/EP) and or three dierent iber volue raction values. Table 4. Values o itting paraeters a and b or *. Material V a b CF1/EP GF/EP Table 5. Values o itting paraeters a and b or th *. Material V a b CF1/EP GF/EP * Table 6. Values o itting paraeters a and b or th. Material V a b CF1/EP GF/EP

80 3.5 Debond crac growth in echanical atigue In act, only values o itting paraeters a and b or each agniication coeicient are necessary in order to describe energy release rate change during debond crac growth and, consequently, to peror Paris law siulations. Moreover, since in the present study we analyze echanical atigue loading only and T during the test is not changing, only and th have to be ound. Paris law expression (Eqs.(9) or (12)) requires soe epirical constants lie B,. * Since in Eq.(12) we are using the noralized value B o the epirical constant B * then or reasons o convenience and realizing that B change the slope o the debond length curve we can tae B * 1. The proper value without having any experiental data is diicult to choose. In [14] where transverse cracing in atigue was analyzed the value 10 was used. Since transverse cracing is atrix and interace properties governed we decided to use siilar values in the ollowing debonding siulations. Material constant G c is used or noralizing the energy dierence G in order to obtain a diensionless entity. Value o G c 200J 2 is taen ro [7] where iber/atrix debond growth in single iber ragentation test * was siulated. The above values o constants B, and G c were considered as noinal and used or both CF1/EP and GF/EP coposites. In siulations the applied axial echanical strain was 1%, the ratio between the inial and axial strain in one cycle was 0.1. Calculations using the Paris law expression (Eq.(12)) were irst used to estiate the eect o theral stresses on debond growth. In Fig.6 the debond growth with the applied nuber o cycles is shown or CF1/EP and or GF/EP. Fig.6. Debond crac growth with the increasing nuber o load cycles in echanical atigue. Signiicance o T or: a) CF1/EP; b) GF/EP. N * N B *. The eect o the teperature dierence T 100C (ro anuacturing teperature to test teperature) on debond crac growth can not be neglected. The eect sees to be larger or GF/EP coposite but the uch larger nuber o required cycles or this coposite should be noticed (due to lower elastic odulus o GF the energy release rate is proportionally lower). I localized heating during specien rictional sliding in cyclic loading would tae place the predicted curves would change. Since the process becoes history dependent and the local teperature distribution and its eect on resin and interace properties in atigue is not nown a 68

81 detailed analysis is not possible. The ost evident trends are that a) theral stresses are reduced, reducing the second ter in G in Eq.(10). This is a consequence o saller teperature dierence T and the low sensitivity o th to resin properties change; b) the irst ter in G expression is not changed (the eect o atrix odulus on is negligible, see Part 1 [1]). Fig.7. Debond crac growth with the increasing nuber o applied cycles in echanical atigue. Signiicance o paraeter or CF1/EP. N * N B *. Fig.8. Debond crac growth with the increasing nuber o applied cycles in echanical atigue. Signiicance o G c or CF1/EP. N * N B *. G C is in J / 2. 69

82 Fig.9. Debond crac growth with the increasing nuber o applied cycles in echanical atigue. Signiicance o r or CF1/EP. N * N B *. Since the applied Paris law expression (Eq.(12)) is based on soe epirical constants * ( B, and G c ) a paraetric analysis o signiicance o these constants was perored siulating pure echanical loading ( T 0 ). Fig.7. shows the large sensitivity o the debond growth rate with respect to constant. In Fig.8. the atigue crac growth is siulated using three dierent values o aterial constant G c. It has to be noted that variation o G c has siilar eect as variation o the applied strain in the test. Finally, in Fig.9 results are presented or three dierent iber radius r values showing uch aster debond crac propagation in ibers with larger radius. The observations ay be useul designing test and/or coposite or validation o the Paris low or debond growth and or paraeter identiication in Eq.(12). It is preerable to have coposite with larger iber radius and with higher odulus. 4. Conclusions In this paper the strain energy release rate in Mode related to iber/atrix debond growth along the broen and partially debonded iber in unidirectional coposite was analyzed using FEM and virtual crac closure technique. The eect o the debond crac and iber brea interaction was analyzed and G agniication coeicients were introduced. To acilitate the debond growth siulations using Paris type o power law the agniication coeicients were described by siple but accurate itting unctions. Large eect o iber elastic odulus and radius on the debond growth was ound. Signiicance o theral stresses in echanical atigue as well as the role o aterial paraeters in the atigue law was evaluated. These coeicients could be easy deterined i reliable experiental data about debond growth in atigue in coposite would be available. 70

83 5. Reerences 1. A. Pupurs, J. Varna, Energy release rate based iber/atrix debond growth in atigue. Part I: Sel-siilar crac growth, Mechanics o Advanced Materials and Structures, In press, N. Laws, G.J. Dvora, The eect o iber breas and aligned penny-shaped cracs on the stiness and energy release rates in unidirectional coposites, International Journal o Solids and Structures, vol. 23, pp , J. Varna, J. Eitzenberger, Modeling UD coposite stiness reduction due to ultiple iber breas and interace debonding, Proc. COMP07: 6th International Syposiu on Advanced Coposites, Coru, Greece, paper 073, pp. 10, S. Ochiai, M. Tanaa, H. Tanaa, S. Kiura, M. Hojo, A odeling study on residual stress-induced interacial debonding and stress-strain behaviour o wealy bonded UD coposites, Coposites Part A, vol. 33, pp , I.J. Beyerlein, S.L. Phoenix, Stress proiles and energy release rates around iber breas in a laina with propagating zones o atrix yielding and debonding. Coposites Science and Technology, vol. 57, pp , B.E. Ki, J.A. Nairn, Experiental veriication o the eects o riction and residual stress on the analysis o interacial debonding and toughness in single iber coposites. Journal o Materials Science, vol. 37, pp , W. Wu, I. Verpoest, J. Varna, Prediction o energy release rate due to the growth o interace crac by variational analysis, Coposites Science and Technology, vol. 60, pp , E. Graciani, V. Manti, F. París, J. Varna, Nuerical analysis o debond propagation in the Single Fibre Fragentation Test, Coposites Science and Technology, vol.69, pp , C.T. Sun, W. Qian, A treatent o interacial cracs in the presence o riction. International Journal o Fracture, vol. 94, pp , Z. Hashin, Analysis o coposite aterials - a survey, Journal o Applied Mechanics, vol. 50, pp , J. Varna, J. Eitzenberger, Debond crac growth in atigue along iber in UD coposite with broen ibers, Proc. 13th European Conerence on Coposite Materials, Stochol, Sweden, pp. 8, Z.H. Xia, W.A. Curtin, Lie prediction o titaniu MMC under low cycle atigue, Acta aterialia, vol. 49, pp , N. Taeda, S. Ogihara, A. Kobayashi, Microscopic atigue daage progress in CFRP cross-ply lainates, Coposites, vol. 26, pp , A.S.D. Wang, P.C. Chou, S.C. Lei, A stochastic odel or the growth o atrix cracs in coposite lainates, Journal o Coposite Materials, vol. 18, pp , J.A. Nairn, S. Hu, Microechanics o Daage: A Case Study o Matrix Microcracing, In Daage Mechanics o Coposite Materials, ed., Raesh Talreja, Elsevier, Asterda, pp , S.L. Liu, Microcracing o Cross-Ply Lainates Under Static and Fatigue Loads, Ph.D. Thesis, University o Utah, USA, G.R. Irwin, Fracture, Handbuch der Physi, vol.5, Springer Verlag, Berlin, ANSYS Release 11.0, ANSYS Acadeic Research, ANSYS Inc., Canonsburg, Pennsylvania,

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85 Paper I A. Pupurs, J. Varna Fracture echanics analysis o debond growth in a single-iber coposite under cyclic loading, Mechanics o Coposite Materials, vol. 47(1), pp , 2011.

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87 Fracture echanics analysis o debond growth in single iber coposite under cyclic loading Andrejs Pupurs, Janis Varna Division o Polyer Engineering Luleå University o Technology, SE Luleå, Sweden Abstract A odel is developed to analyze iber/atrix debond growth along the broen iber interace in a single iber coposite subjected to tension-tension atigue. Paris law expressed in ters o debond length growth rate and strain energy release rate is used. Analytical solution or Mode energy release rate G is obtained or long debonds where the interace crac growth is sel-siilar. For short debonds the interace crac interacts with the iber brea and thereore FEM odeling in cobination with the virtual crac closure technique was perored to calculate the agniication o G. Finally, the calculated G dependences are suarized in siple expressions that are used to siulate debond growth in atigue. A paraetric study o the eect o Paris law paraeters on debond growth is perored. Keywords: iber brea, debonding, Mode, atigue, energy release rate, Paris law. 1. Introduction The irst ode o daage in unidirectional (UD) iber reinorced polyer coposites subjected to axial high stress tension-tension cyclic loading is ultiple iber ragentation. Due to statistical iber strength distribution, daage (iber breas) ay appear during the irst cycle and ay persist or have urther developent due to stress concentrations caused by growing secondary daage odes lie iber/atrix interace debonding. The iber/atrix debonding in UD coposite can be analyzed as an interace crac growth starting ro iber brea and the analysis can be perored based on strain energy release rate. Paraetric analysis o the eect o dierent geoetrical and elastic paraeters on the energy release rate in UD coposite due to debond growth was perored in [1,2]. Paris law was eployed to siulate the debond growth in tension-tension atigue [2] using cobined analytical and nuerical approach and the signiicance o dierent paraeters in Paris law was revealed. This study deonstrated that independent experiental inoration regarding the values o these paraeters is necessary to siulate debonding in coposite. One possibility is to use single iber (SF) specien to obtain the necessary inoration. SF specien consists o a long iber ebedded in a large bloc o resin, and is subjected to tensile load in the iber direction. The strain to iber ailure is usually lower than the atrix ailure strain and thereore ragentation o the iber is observed beore the specien ails. So, the ragentation process is related to the statistical iber strength distribution. At the iber brea the iber axial stress is zero. However, the axial stress is transerred to the iber through shear stresses at the iber/atrix interace, thus allowing or the appearance o new ragents. The single 75

88 iber ragentation test (SFFT), introduced by Kelly and Tyson [3], has been extensively used or iber-atrix interace characterization in ters o strength and or statistical iber strength characterization. The SF specien with ragented iber can be subjected to axial atigue loading, observing the debond growth with increasing nuber o cycles. Siilar atigue test on odel coposites with only 2-5 ibers have been reported, or exaple, in [4]. I a proper siulation tool is available to siulate the debonding process, the cobined experiental/siulation approach would allow to identiy the atigue law and its paraeters or the interace in question, which can be urther used to analyze atigue o UD coposites with ore realistic iber contents. Fracture echanics approach has been used or the characterization o the interace ailure in quasi-static loading or exaple, by Nairn [5], Varna et al. [6], Nairn and Liu [7] and Wu et al. [8,9]. The approach used in the present paper assues that the propagation o an individual debond along the iber/atrix interace obeys the Paris law, which states that the debond length growth rate has a power law dependence on the strain energy release rate dierence between the highest and lowest levels in the cyclic loading. The SF coposite with partial iber/atrix debond o length l d ay be represented by a two concentric cylinder odel, shown in Fig.1. z iber brea Fig.1. A single iber coposite with broen and partially debonded iber subjected to echanical ( z ) and theral ( T ) loading. F iber, M atrix, S syetry plane. l d shows the debonded length, L is hal o the iber ragent length. First, the strain energy release rate G in Mode due to debond growth will be analyzed using analytical odel or long debonds, then inite eleent ethod (FEM) in cobination with the virtual crac closure technique will be used or short debonds. When the G dependence on debond length and constituent paraeters will be ound and described by siple expressions, it will be used or debond crac growth siulations using Paris law to reveal the trends to expect (the role o involved paraeters, the signiicance o theral stresses, etc.). 76

89 2. Energy release rate or sel-siilar crac growth 2.1 Calculation ethod Axial tension is applied to the SF specien. In addition, since the specien ater processing is cooled to roo teperature, teperature dierence is negative, T 0. Since the Poisson s ratio and the theral expansion coeicient o the atrix are larger than or the iber (see Table 1), the radial stress on the iber surace is copressive and the debond crac propagation is in Mode. An analytical solution or strain energy release rate can be ound in a particular case when the tip o the iber/atrix debond crac is ar away ro the iber brea where it was initiated and when it is also ar ro another debond, which ay be approaching ro the other end o the iber ragent. In this case the debond crac propagation can be considered as sel-siilar. Matheatically this condition in ters o Fig.1 can be written as ld r L ld (1) In (1) r is the iber radius, L is ½ o the iber ragent length. The debond crac growth by dl d : a) shits the debond crac tip (and the corresponding singular stress state in its vicinity) in the z-direction by dl d ; b) reduces the bonded region volue in the odel by R 2 dld (R is the outer radius o the cylinder assebly in Fig.1) and increases the debonded region volue by the sae aount. Neglecting riction and energy dissipation related to it, the energy change in the syste due to this event can be calculated using the dierence between strain energy tot tot U b o the bonded and U d o the debonded regions o unit length located ar ro the debond crac tip and the iber brea tot d tot b dld Models representing these two regions are given in Fig.2. du U U (2) Fig.2. Geoetrical representation o the bonded region (a), which due to crac growth by dl turns to debonded region (b) with length dl. d In the debonded region, Fig.2b, sliding o the iber with respect to the atrix taes place and deorations and strains are dierent. The strain energy release rate is obtained dividing (2) by the newly created surace area da 2r dl 77 d d

90 G du uconst (3) da Strain energy expressions or ore coplex case with N cylindrical phases, where phase 1 ay be bonded or debonded, were derived in [1]. In the odel shown in Fig.2 N 2 (iber and atrix). Fro [1] ollows that the strain energy o the odel a) or b) in Fig.2 can be calculated i the average stress state in constituents is nown U tot V V zz z0 z TT V r r (4) 2 In (4) =1 is iber, =2 is atrix, V R is volue o the unit length o the assebly, V is volue raction o the phase deined as V r 2 /R 2 or ibers and V (R 2 r 2 )/R 2 or atrix. The average stresses are obviously dierent in the bonded and in the debonded case tot tot resulting in two dierent values U, U. Relevant expressions are given below. 2.2 Strain energy in the bonded region The applied teperature dierence is and atrix b d 1 T and the applied strain is the sae in iber (5) 1 2 z0 z0 z0 The elastic solution or two cylinder assebly is given in Appendix. In the odel, the teperature dierence and the axial strain are applied siultaneously. The experiental procedure (and as a consequence also the procedure to be used in siulations) is dierent: irst the specien which is produced at high teperature is cooled down to roo teperature. Theral stresses develop during th this step and the specien has negative strain z0 beore any echanical loading. It is the ree theral expansion strain o the coposite, cop cop z T, where z is axial theral expansion coeicient o the SF coposite (due to very large atrix outer radius it is practically equal to atrix theral expansion coeicient ). In tests and siulations this state is taen as the starting state (zero echanical strain). ech Then certain aount o echanical tensile strain z0, easured by extensoeter or strain gauge, is applied. The relationship with z0 in expressions in Appendix is as ollows ech th (6) z0 z0 z0 Substituting (A3) in (4) we have the strain energy o bonded iber and atrix cylinder assebly U tot b 2 2 V V A1 g z0 H 3 T z0 z T 2TV r A1 z0 H r T (7) 78

91 Expressions or,, g,, deterination are given in Appendix. 2.3 Strain energy in the debonded region H r and In the debonded region shown in Fig.2b we have in the atrix H 3 as well as equations or A 1, (8) z0 z0 Due to sliding the axial strain o the debonded iber is dierent (9) z0 z0 It consists o ree theral expansion strain o the iber and Poisson s interaction strain due to radial interaction with the surrounding atrix. To ephasize that this is solution or the debonded case we use or unnown constants in the solution in d Appendix additional letter d in the upper index, A i. The axial strain value in the debonded iber can be obtained using condition that axial stress in the iber is zero 0 (10) z Expressing z0 ro the irst equation o (A3), using condition (A6) and substituting in the second equation o (A3) we obtain the ollowing expression or 1 r 1 1 1d 1 H 3 1 r A1 1 1 T 1 H r (11) g1 g1 1 1 r (12) The stress expressions or the atrix reain as in the bonded case. The interace conditions at phase interaces also reain as in the bonded case (see Appendix): radial displaceents and stresses are continuous. The outer radial boundary o the assebly is ree o stresses, see (A9). Ater solving or constants A we can use the in the expression or the strain energy o the debonded region which is as ollows 2 tot R U d r r A1 2d 2 2d 2 A g H T T 2T A H T d z0 1 1 H 3 T g g1 3 z0 1 H r T 1 2 d i 2 z0 r (13) 79

92 3. Virtual crac closure technique or short debonds In case o short debond length, the debond crac tip is close to the iber brea and the stress perturbation regions overlap. It was shown previously using FEM or UD coposites [2] and using boundary eleents or SF speciens [10] that due to stress ield interaction, the strain energy release rate G is agniied. The shorter is the debond length, the higher is the degree o interaction and thus debond crac growth related energy release rate values are higher. In this study, agniication o G was deterined ro FEM calculations. An axisyetric odel consisting o broen and partially debonded iber surrounded by an ininite atrix bloc was generated using inite eleent code ANSYS [11]. Virtual crac closure (VCC) technique [12] was used to obtain G ro FEM calculations in a convenient way. The distribution o relative tangential displaceent u z behind the debond crac tip and the shear stress rz values ahead o the crac tip can be obtained directly ro the FEM odel as shown in Fig.3. iber brea Fig.3. Concentric cylinder odel or G calculations by FEM and virtual crac closure technique. Forally, to calculate the wor that is necessary to close the crac by dl d, the tangential displaceents have to be calculated or a debond with length l d dl d and the shear stresses or a debond with length l d. However, when using VCC technique, it is assued that due to sall value o dl d, the relative sliding displaceent at the tip o the crac with size l d dl d is the sae as at the tip o the debond crac with size l d, i.e., ld dld ld u z) u ( z dl ) (14) ( d In (14) the upper index shows the length o the debond crac. This assuption eans that only one stress state calculation or a given debond length is required. Then, the strain energy release rate G can be calculated using expression 1 G (l d ) li dl d 0 2dl d l d dl d l d u l d z dl d l d rz (z)dz (15) 80

93 4. Paris law or debond growth along the single iber The cyclic loading type considered in this study is echanical tension-tension atigue ( ax 0 and in 0) in presence o theral stresses at ixed teperature. We will apply Paris law expression in order to siulate debond growth along iber/atrix interace as a unction o applied nuber o load cycles N. Paris law expression describes the increase o the debond surace area da as: da dn B G G C (16) where: B and are unnown paraeters, which have to be deterined experientally; G is the energy release rate dierence between values corresponding to ax and in ; the constant G C is used in order to noralize energy release rate units. This particular or with explicit inclusion o G C was used previously in [13] and, certainly, other, slightly dierent ors ay be ound in literature. Debond length, which or circular iber is related to debond surace as l d A 2 r will also be noralized with the respect to the iber radius r l l / r (17) dn d Thus (16) can be written in or dldn dn G B * * G ; 2 C 2r B B (18) * Obviously B is also diensionless. For siulation purposes (18) will be used in an increental or l ( ) dn * G ( l B GC ( ) dn ) N, l l l ( 1) ( ) ( ) dn dn dn ; =1,2,3 (19) In (19) (1) l dn is the initial debond length at the start o siulation. 5. Results and discussion 5.1 Analytical results or sel-siilar crac growth Using expressions in Section 2 the energy release rate or sel-siilar crac growth (denoted as G ) was calculated or SF coposites with properties o constituents given in Table 1. Transversally isotropic carbon iber (CF) and isotropic glass iber (GF) coposites with epoxy resin atrix (EP) were studied. In a SF coposite the atrix radius is ininite and it is very convenient to use it in the analytical solution or sel-siilar crac growth (Section 2). However, short debonds G are analyzed using FEM and there the radial size o the odel is liited. 81

94 Thereore, prior to doing FEM analysis or short debonds, the analytical solution or inite R was used to estiate the proper radial size o the odel to be used in FEM. The radius o atrix cylinder was used as a paraeter: it was increased until a liit was reached when urther increase o atrix cylinder did not aect the calculated G value. Logically, a larger size o atrix cylinder corresponds to the ininity condition better, however, since odel with the sae size has to be used or FEM calculations it is preerable to have odel o saller size but accurate enough in order to save calculation tie and eort. Table 1. Elastic properties o SF coposite constituents. Material E L E T G LT L T [GPa] [GPa] [GPa] [-] [-] [1 C ] [1 C ] CF GF EP Paraetric analysis results or CF/EP and GF/EP SF coposites are suarized in Tables 2 and 3 respectively. Fiber radius in all calculations was r 1. Table 2. Dependence o energy release rate G (J/ 2 ) or CF/EP on atrix cylinder radius. Load case Matrix cylinder outer radius R in [] ech z0 T ºC ºC Table 3. Dependence o energy release rate G (J/ 2 ) or GF/EP on atrix cylinder radius. Load case Matrix cylinder outer radius R in [] ech z0 T ºC ºC Based on results shown in Tables 2 and 3 it was decided that atrix cylinder with radius R 100 r is a good representation o an ininite atrix. Following that, the corresponding FEM calculations or short debonds were also perored using a odel with the sae outer radius R. For analytical energy release rate G calculations or sel-siilar cracs, the length o the debond and the debond length increase dl d is not signiicant and any chosen odel length would give the sae accurate result. 82

95 Analytically calculated Table 4. G values or sel-siilar crac growth are presented in Table 4. Energy release rate G (J/ 2 ) or sel-siilar debond crac growth or CF/EP and GF/EP. Material Load case z T 100C z ; T 100C CF/EP GF/EP In order to sipliy recalculating G whenever the applied load is dierent ro that shown in exaples in Table 4, we used the sae approach as suggested in [1]. The ain idea o the approach is that the energy release rate is a quadratic unction o stress state and stress coponents are linearly related to applied echanical strain ech ( z0 ) and applied teperature dierence ( T ), i.e.: G 1 4 E L r ech 2 ech T 2 2 z0 2 th L z0 th L T (20) where coeicients, th and th are related to purely echanical, cobined echanical-theral and purely theral load respectively. Lie or G, the upper index in coeicients is used to denote correspondence to sel-siilar crac growth region. According to (20) the coeicients, th and th can be ound or any SF coposite by peroring the ollowing 3 calculations: 1) purely echanical loading case ( T 0); 2) purely theral loading case ( ech z0 0); 3) any cobination o echanical and theral loading cases ( ech z0 0; T 0). Coeicients, th and th can thus be calculated ro G values given in Table 4. Values o coeicients or both studied SF coposites are suarized in Table 5. ech Note that strain z0 values in / and applied teperature change T in C were used to deterine these coeicients. Table 5. Values o coeicients Material, th th and th. th CF/EP GF/EP Once coeicients, th and th are deterined, G can be calculated or any arbitrary cobination o echanical and theral loads. 5.2 FEM results or G agniication The quadratic or o G with respect to strain and teperature is valid also or short debonds and can be written as 83

96 G ech 2 ech T 1 EL r z0 2 th L z0 th L T (21) Since G is agniied or short debonds [10], we expect according to (21) that coeicients, th and th will also depend on the debond length and they will be higher or shorter debonds. With increasing debond length they have to approach the values, th and th. The latter is valid assuing that the iber ragent is long enough and interaction with another debond approaching ro the other ragent end is negligible. The increase o coeicients, th and th or short debonds is called agniication in the ollowing text. FEM calculations in cobination with VCC technique were used to calculate agniication o these coeicients. It was entioned previously that using a odel with outer atrix radius R 100 r gives a reasonable accuracy o calculations. For FEM calculations when using the odel shown in Fig.3 the length o iber hal ragent L is also iportant. There are two aspects to consider: a) L has to be suiciently large in order to prevent interaction between growing debond crac and the displaceent application surace which is the iddle o the iber ragent (condition o noninteractive debonds); b) a shorter L in the odel would acilitate FEM calculations by saving calculation tie. A coproise between the aboveentioned rules has to be ound. A paraetric study in [2] or UD coposites showed that using a odel with iber hal ragent length L 90 r is a reasonable representation o noninteractive case i ld 30 r. A siilar paraetric study or SF coposites resulted in considering L 200 r as suicient. Within the VCC technique, G is calculated by integrating the product o nuerically obtained displaceent u and stress rz values according to (15). Using the VCC technique in order to calculate G ro FEM results, attention has to be paid to the size o area o integration, i.e., dl d. Since integration in (15) is perored over a inite distance, the obtained G value will depend on the chosen size o integration. Contrary to (15) which states that G is deined as a liit when dl d 0, in FEM calculations the integration region is inite. I or a given esh the integration region is too short, the accuracy will be reduced because the stress and displaceent proiles very close to the crac tip are not accurately represented. Thereore, soe inal value o the length o integration dl d ust be set. In [2] region o integration or a UD was ound by a paraetric analysis. In the present study, two dierent lengths o integration dl d 1 r and dl d 0.5 r were irst copared. Values o G or debond lengths in the range ro l d 2 r to l d 30 r were calculated or both integration lengths. Results or CF/EP in echanical loading are shown as an exaple in Fig.4. In Fig.4, presented versus noralized debond length l dn l d / r. G values are 84

97 Fig.4. Magniication o G or short debonds. CF/EP single iber coposite with r 1 subjected to echanical loading. Applied strain ech z Siilar agniication proiles were obtained or theral and thero-echanical loading cases. According to Fig.4, using shorter integration length dl d 0.5 r, slightly lower G values are obtained. However, the curves presented in Fig.4 loo have very siilar shape and when noralized with respect to their corresponding values at l dn 30 (corresponding to long debonds, lie in the sel-siilar case), the obtained agniication proiles in both cases are very uch the sae (see Fig.5). Fig.5. Noralized agniication proiles in echanical loading case or CF/EP single iber coposite: results or two dierent integration lengths dl d. The sae independence on integration length was also obtained in purely theral and thero-echanical loading cases. Furtherore, the sae was ound or GF/EP as well. We conclude that or the studied SF coposites urther paraetric analysis o the eect o integration length dl d is not required. The noralized agniication 85

98 proiles can be accurately obtained using any chosen length o dl d within the region 0.5 r dl d 1 r. Using the or o (21) with, th and th in order to represent the agniication o G or short debonds, we can write these coeicients as: * (l dn ) * th th (l dn ) th th th * (l dn ) (22) In other words we introduce agniication actors * (l dn ), * * th (l dn ) and th (l dn ) as unctions o noralized debond length. These actors are equal to 1 or long debonds growing in a sel-siilar anner and are larger than 1 (to be deterined ro FEM + VCC technique) or shorter debonds. The coeicients, th and th in (22) do not depend on the debond length and are calculated using the sel-siilar crac growth odel in Section 2 (as, or exaple, in Table 5). In act they could be obtained also ro the FEM solution i the results would have suicient accuracy. Unortunately, the analytically calculated G value or sel-siilar debond growth in CF/EP single iber coposite subjected to echanical loading only ( ech z0 0.01) J/ 2 (see Table 4) is higher than the calculated by FEM at l dn 30 (see Fig.4). Obviously, FEM results or long debonds are too low. The sae was observed or theral and thero-echanical loading cases. It could be due to inite diensions o the odel but also due to nuerical accuracy. Thereore, we decided to use FEM results just to estiate the degree o agniication in noralized or as a unction o noralized debond length. However, in order to calculate the agniication proiles properly when using FEM, G values or long debonds are needed. Fro Fig.4 we see, that at l dn 30 the value o G has alost reached the plateau region. Although it can be intuitively expected ro Fig.4 that G values will still decrease a little or debond lengths l dn 30, in this study we assue that l dn 30 corresponds to plateau value (onset o sel-siilar crac growth region) and in the ollowing the agniication coeicients are calculated as: * j j (l dn ) ; j ;th; th (23) j (l dn 30) Since we will use Paris law expression given by (18) or debond growth analysis in atigue, it would be convenient i proiles o agniication actors * (l dn ), * th (l dn ) * and th (l dn ), could be each represented by siple atheatical expressions. In [2] a siple exponential unction was suggested as a or to coprise G agniication proiles or UD coposites. However, or SF coposites the agniication o G at short debond lengths is considerably higher and a trial itting o * j (l dn ) with an exponential unction led to insuicient agreeent, especially at shorter debond lengths. Thereore, instead o an exponential unction a hyperbolic unction was introduced in the ollowing or: j * 1 C l dn ; j ;th; th (24) 86

99 where C is itting unction (possibly also a wea unction o l dn ). Fitting unction C can be deterined by a siple procedure. Using FEM calculated coeicients j *, the values o hyperbolic itting unction C can be ound ro (24) as: C ( j * 1) l dn (25) The calculated values o C or CF/EP in echanical loading case ( ech z0 0.01) at speciied debond lengths are shown in Fig.6. Fig.6. Fitting o unction C by 2 nd order polynoial or CF/EP in echanical loading case, ech z Obviously, C deined by (24) strongly depends on debond length. Fitting o C in Fig.6 by a 2 nd order polynoial is very good within the observed range o l dn. However, according to physical eaning o agniication coeicient * j (see (22)), C has to give onotonic decrease o G. Obviously, the chosen polynoial unction i Fig.6 does not ulil this condition ater l dn 30. Thereore the itting unction C is deined dierently in 2 regions: C a 1 a 2 (l dn ) a 3 (l dn ) 2 ; 1 l dn 30 C 0 l dn 30 (26) where a 1, a 2 and a 3 are itting paraeters. The accuracy o the itting unction C in description o the G agniication (24) is deonstrated or the case o echanical loading in Fig.7 (or CF/EP) and Fig.8 (or GF/EP). The agreeent with FEM data is excellent. The sae degree o agreeent was observed or other loading cases (theral, thero-echanical). 87

100 Fig.7. Magniication proile * or CF/EP single iber coposite. Fig.8. Magniication proile * or GF/EP single iber coposite. The values o itting paraeters a 1, a 2 and a 3 in (26) were calculated or both aterials (CF/EP, GF/EP) or each loading case. Analyzing the obtained data or the two studied aterials it was noticed that the values o itting paraeters are rather dierent. However, in ters o one aterial the values o coeicients or dierent thero-echanical loading cases are very close. In act, when plotting agniication proiles * * *, th and th on a single graph as shown in Fig.9, we obtain alost coinciding curves or the dierent loading cases. For CF/EP we observe higher degree o agniication than or GF/EP. Since agniication actors * * (l dn ), th (l dn ) and * th (l dn ) proved to be alost identical, we can describe agniication o G by using just one agniication * unction ( l dn ) instead o three. Preerably FEM is perored or echanical rather than theral loading, as it usually requires less coputer power and calculation tie. 88

101 Fig.9. Coparison o agniication proiles or GF/EP and CF/EP. The coeicients or both studied aterials are given in Table 6. We can rewrite the G Table 6. Values o itting paraeters or (26). or j * ; j ;th; th Material a 1 a 2 a 3 CF/EP GF/EP G expression (21) or any debond length in the ollowing or: ech 2 ech 2 T * ( ldn ) EL r z0 th L z0 th L T 4 (27) This is very signiicant or the ollowing, eaning that or SF coposites subjected to echanical, theral or thero-echanical loading, interace debond energy release rate G can be calculated in the ollowing siple way: 1), th and th are calculated analytically; 2) agniication proile * (l dn ) and paraeters a 1, a 2 and a 3 o the itting unction C are obtained ro the FEM data according to procedure described in this section. Nuerical values o these paraeters or the studied coposites are given in Table 6; 3) G values are calculated by (27) or any debond length and or any echanical, theral or cobined thero-echanical loading case. The described procedure or agniication coeicients is based on nuerical integration over inite length in order to obtain the crac closure wor. To estiate the introduced error we copare our agniication actor with results ro [10] which are based on very accurate BEM calculations in the local stress region at the debond tip. The agniication unctions using our approach and BEM are copared 89

102 in Fig.10. The agreeent is rearably good which gives conidence to our results and conclusions. For exaple, BEM results conir that the agniication actors or echanical and ixed thero-echanical loading coincide. Fig.10. Magniication proile * or GF/EP obtained ro FEM and BEM [10] calculations. The ethod described in this study gives an accurate and siple coputational tool or G calculations or any debond length. It cobines analytical solution or long debonds with agniication actors obtained itting nuerical solution or short debonds. 5.3 Debond growth analysis using Paris law The above presented values o itting paraeters a 1,a, a 2,a and a 3,a were used in debond growth analysis as described in the ollowing section. Paris law expression (18) is based on soe epirical constants lie B, and G c. The proper values o these constants are not nown unless experiental data are available. In Paris law expression (18) G (l d ) is the strain energy release rate dierence between the value corresponding to axiu and iniu load. In echanical atigue with a ixed initial teperature change T according to (27) this dierence is equal to: ech 2 ech 2 ech ech z0,ax z0,in 2th L T z0,ax z0, in * 1 G ( ldn ) ( ldn ) EL r 4 (28) In case when the teperature change is zero ( T 0) prior to echanical atigue loading, the second ter in (28) is zero. For the studied coposites a paraetric study o the eect o Paris law constants, G and iber radius r was perored. The paraeters were odiied one at a tie c 90

103 and siulation results were copared to a reerence case. Siulation paraeters or the reerence case were the sae or both aterials. They are presented in Table 7. Table7. Reerence case debond growth siulation paraeters or CF/EP and GF/EP. r B * B G c ech z0,ax ech z0,in T [] [ - ] [ - ] [ 2 ] [J/ 2 ] [/] [ /] [ ºC ] The trends revealing the signiicance o geoetrical and siulation paraeters are shown in Figs Fig.11. Debond growth with increasing nuber o cycles N or CF/EP. Paraetric analysis o : a) iber radius r ; b) paraeter ; c) constant G c d) teperature change T. Reerence case shown with solid continuous line. 91

104 Fig.12. Debond growth with increasing nuber o cycles N or GF/EP. Paraetric analysis o : a) iber radius r ; b) paraeter ; c) constant G c d) teperature change T. Reerence case shown with solid continuous line. The ain observations ro the trend plots in Figs are: a) the debond growth is uch aster in CF/EP (because the echanical part o the energy release rate is proportional to the iber odulus); b) the debond growth rate is higher in coposites with larger diaeter ibers; c) the siulation result is very sensitive to the exponent ; d) increase o the interacial racture toughness G c draatically slows down the debond growth. Certainly, the two paraeters B and G c can be cobined in one paraeter. Finally, we see that copressive theral stresses, induced by initial teperature change T in the iber, change the debond growth signiicantly. 6. Conclusions The debond growth along a broen iber in a single iber coposite subjected to cyclic echanical tension-tension loading is analyzed using racture echanics approach. Paris law or debond growth rate in ters o energy release rate change is used to siulate the debond length increase with the applied nuber o cycles. The strain energy release rate G due to debond growth is calculated analytically in the sel-siilar debond growth region and nuerically or short debonds using FEM and virtual crac closure technique. For short debonds, where debond crac is interacting with iber brea, G is agniied. Magniication is larger in CF/EP case. Debond length dependent agniication actors were introduced or echanical, 92

105 theral and ixed theral-echanical ters in G expression. It was observed that or studied single iber coposites (CF/EP and GF/EP) these three agniication proiles are alost identical. Magniication proiles were itted by one siple expression to acilitate siulations o cyclic loading case. A paraetric study or CF/EP and GF/EP single iber coposites was perored siulating the debond growth and showing the signiicance o Paris law paraeters, theral stresses and the iber radius r. The debond growth siulation tool was developed and trends analyzed or uture coparison with experiental data or SF specien with ragented iber and subjected to cyclic loading. The applicability o Paris law will be evaluated by coparing the easured and siulated debond length growth with the nuber o cycles. The aterial constants in the Paris law will be estiated itting experiental data. Ater that the odel could be used to analyze debond growth trends in unidirectional coposites with realistic iber contents. 7. Reerences 1. A. Pupurs, J. Varna, Energy release rate based iber/atrix debond growth in atigue. Part I: Sel-siilar crac growth, Mechanics o Advanced Materials and Structures, In press, A. Pupurs, A. Krasniovs, J. Varna, Energy release rate based iber/atrix debond growth in atigue. Part : Debond growth analysis using Paris law, Mechanics o Advanced Materials and Structures, In press, A. Kelly, W.R. Tyson, Tensile properties o ibre-reinorced etals: copper/tungsten and copper/olybdenu, Journal o the Mechanics and Physics o Solids, vol.13, pp , S. Goutianos, T. Peijs, Experiental and nuerical investigation into atigue daage echaniss in ultiibre icrocoposites, Plastics Rubber and Coposites, vol.30, pp , J.A. Nairn, A variational echanics analysis o the stresses around breas in ebedded ibers, Mechanics o Materials, vol.13, pp , J. Varna, R. Joe, L.A. Berglund, Interacial toughness evaluation ro the single-iber ragentation test, Coposites Science and Technology, vol.56, pp , J.A. Nairn, Y.C. Liu, Stress transer into a ragented, anisotropic iber through an iperect interace, International Journal o Solids and Structures, vol.34, pp , W. Wu, I. Verpoest, J. Varna, A novel axisyetric variational analysis o stress transer into ibres through a partially debonded interace, Coposites Science and Technology, vol.58, pp , W. Wu, I. Verpoest, J. Varna, Prediction o energy release rate due to the growth o an interace crac by variational analysis, Coposites Science and Technology, vol.60, pp , E. Graciani, V. Manti, F. París, J.Varna, Nuerical analysis o debond propagation in the Single Fibre Fragentation Test, Coposites Science and Technology, vol.69, pp , ANSYS Release 11.0, ANSYS Acadeic Research, ANSYS Inc., Canonsburg, Pennsylvania, G.R. Irwin, Fracture, Handbuch der Physi, vol.5, Springer Verlag, Berlin,

106 13. A.S.D. Wang, P.C. Chou, S.C. Lei, A stochastic odel or the growth o atrix cracs in coposite lainates, Journal o Coposite Materials, vol.18, pp , Z. Hashin, B.W. Rosen, The elastic oduli o iber-reinorced aterials, Journal o Applied Mechanics, vol.31, pp , Z. Hashin, Analysis o Coposite Materials a survey, Journal o Applied Mechanics, vol.50, pp , Appendix. Stress distribution The two cylinder odel shown in Fig 2 is subjected to constant strain in axial direction z0, resulting ro displaceent u z0 2 applied at both ends, and teperature dierence T. In bonded case axial strain in iber and atrix is the sae, in debonded case only atrix is subjected to z0. The expressions below ollow ro Appendix 2 in [8] and are valid or the bonded as well as the debonded case. The derivation is the sae as in [14,15]. Both constituents are transversally isotropic and ollow Hooe s law, which or noral stresses is written here in vectorial or C T (A1) i ij j where C ij is the stiness atrix with usual notation and j are theral expansion coeicients. Here 3 - is the axial direction, 1 and 2 are two directions in the plane o isotropy. All shear stresses are zero. The elastic solution valid or any transversally isotropic cylindrical phase under this type o loading can be written as j 1 u r A1 r A2r (A2) A g z A A r r z0 H T 2 A A r 2 3 z0 z0 H T r H T r (A3) Here new constants are introduced g C 33 2C32 C32 C22 C12 C12 C22 (A4) H 3 C 33 z 2C 32 r H r C 32 (A5) z r Additional index is assigned, where 1, 2 or iber and atrix respectively, to ar the introduced constants or iber and atrix. The unnown constants A1, A2, =1,2 have to be deterined ro continuity and boundary conditions: 1. Radial displaceent ust be zero on the syetry axis 94

107 u r 1 (r 0) 0 (A6) 2. Displaceent and radial stress continuity at the interace u r 1 (r ) u r 2 (r ) r 1 (r ) r 2 (r ) (A7) (A8) 3. Zero radial stress at the outer boundary r R o the cylinder assebly r 2 (R) 0 (A9) 1 Fro (A6) we obtain A 2 0. For a given z0 and T the syste o 3 equations ollowing ro conditions (A7)-A(9) allows or deterination o 3 constants A 1, A1, A2. Obviously their values will depend on the applied strain z0 and teperature T. 95

108 96

109 Paper IV A. Pupurs, S. Goutianos, P. Brøndsted, J. Varna Interace debond crac growth in tension-tension cyclic loading o single iber polyer coposites, Coposites Part A, Accepted, 2012.

110

111 Interace debond crac growth in tension-tension cyclic loading o single iber polyer coposites Andrejs Pupurs 1, Stergios Goutianos 2, Povl Brøndsted 2, Janis Varna 1 1 Departent o Engineering Sciences and Matheatics, Luleå University o Technology, SE , Luleå, Sweden 2 Departent o Wind Energy, Technical University o Denar, Risø Capus, DK- 4000, Rosilde, Denar Abstract Fiber/atrix interace debond crac growth ro a iber brea is deined as one o the ey echaniss o atigue daage in unidirectional coposites. Considering debond as an interace crac its growth in cyclic loading is analyzed utilizing a power law, where the debond growth rate is a power unction o the change o the strain energy release rate in the cycle. To obtain values o two paraeters in the power law cyclic loading o ragented single iber specien is suggested. Measureents o the debond length increase with the nuber o load cycles in tension-tension atigue are perored or glass iber/epoxy single iber coposites. Analytical ethod in the steady-state growth region and FEM or short debonds are cobined or calculating the strain energy release rate o the growing debond crac. Interace ailure paraeters in atigue are deterined by itting the odeling and experiental results. The deterined paraeters or interace atigue are validated at dierent stress levels. Keywords: Polyer-atrix coposites, debonding, atigue, inite eleent analysis 1. Introduction When unidirectional (UD) iber reinorced polyer coposites are loaded in iber direction in increasing quasi-static tensile loading or in high stress cyclic tensiontension loading, ultiple iber breas occur in rando positions. In cyclic loading with constant aplitude we usually assue that ibers do not experience atigue and all iber breas occur during the irst cycle. With an increasing nuber o cycles growth o debonds along the iber/atrix interace is expected and thus racture echanics concepts (strain energy release rate based criteria) ay be used or the evolution analysis. The described is a echanis or very high strains. At lower applied strains the probability o iber ailure is low and the echanis o debond growing ro a iber brea is not the ost typical. The sequence o events at strain levels below the iber racture liit ay be dierent. For exaple, in cyclic loading the initiation o sall atrix cracs between ibers could be the irst ode o daage, which would be ollowed by iber/atrix debond crac propagation along the iber. More detailed description o the possible ailure scenarios in atigue and their dependence on the load level is given in [1,2]. Experiental debond growth observation in UD coposites is diicult and with optical icroscopy it is possible only on specien surace. To authors nowledge there is only one paper [3] available, where the growth o debond initiated ro a iber brea in a UD coposite was easured experientally as a unction o the nuber o cycles. 99

112 The two possible echaniss (debond initiated ro a iber brea and debond starting ro a atrix crac) were analyzed in [4] showing that the Mode strain energy release rate G or the debond crac growth initiated ro a iber brea is several orders o agnitude larger. The G decays with the distance i the debond is initiated ro a atrix crac but, i it initiates ro a iber brea, the G approaches to a non-zero value and the debond growth becoes sel-siilar. A woring hypothesis used in this paper is that the growth o each individual debond can be characterized by Paris type o law, where the debond growth rate is a power unction o the strain energy release rate change in one cycle. To validate this hypothesis we have to show that: a) experiental data on debond growth in cyclic loading can be described by odel based on power law; b) aterial paraeters in this law ay be deterined ro a itting procedure. Experientally it is uch easier to easure the debond length and its increase in a syste consisting o one (single) iber (SF) ebedded in a large bloc o resin rather than in UD coposite with any thousands o ibers in one specien. The single iber ragentation test (SFF test) since it was irst introduced by Kelly and Tyson [5] has been extensively used or strength distribution deterination and or iber-atrix interace characterization. The SFF specien consists o a long iber ebedded in a large (ininite in analytical odels used in this study) bloc o polyer atrix subjected to tensile load in the iber direction applied as a unior strain at the ends o the specien. I the iber ailure strain is lower than the atrix ailure strain, then, ater reaching a certain load, successive ragentation o the iber is observed relecting the statistical iber strength distribution. The iber stress, which is zero at the iber brea, is transerred to the iber ragent through shear stresses at the iberatrix interace, thus allowing or the appearance o new ragents. Apart ro iber strength Weibull paraeter deterination in this test, any o studies have also been dealing with the interacial shear strength evaluation: since the inal (critical) iber ragent length when the ragentation stops depends on the quality o the stress transer at the interace, the critical length is used to estiate the interace shear strength. The data reduction usually ollows sipliied analytical odels (or exaple, the shear lag odel proposed by Cox [6]) with dierent assuptions [7,8]. Thereore, the interace strength value obtained using the SFF test is strongly dependent on the assuptions ade. Since the shear stress distribution at the interace is highly non-unior (singular at the crac tip in elastic case) the use o a strength criterion is questionable. For this reason, a racture echanics approach or interace crac growth is ore appropriate and it also its our goal to identiy atigue related interace properties. Even i atigue is our ain concern the description starts with strain energy release rate calculations in the SFF test as dependent on the debond length. This part is the sae as or the debond growth analysis in quasi-static loading. The interacial radial theral stress which ors during the cool-down to roo teperature, RT, is copressive due to larger theral expansion coeicient o the atrix. Additional copressive radial stresses are ored during the axial echanical loading due to dierent Poisson s ratios. Consequently, there is always a contact between the debonded iber and the resin and the debond crac growth is in pure Mode. Local stress analysis in the singularity region or this case was perored in [9] showing that in the ront o the debond tip only the shear stress is singular. The application o the calculated G is dierent in quasi-static and in atigue loading. In quasi-static case it is used in ailure criterion, which contains critical value 100

113 o this paraeter G C (aterial constant). Experiental results on the debond length as a unction o the applied strain have been reported in [10,11] and used to deterine G C. Theoretical investigations o G using analytical odels as well as nuerical ethods have been presented by several researchers [9,12-18]. Sei-analytical solutions were obtained or the characterization o the interace ailure in the wors o Nairn et al. [12,14], Wu et al. [15,16]. In [13] exact analytical solution or debond propagation G in the region o sel-siilar growth is presented. This solution is used also in the present paper. It has to be entioned that the solution presented in [13] is not applicable or short debonds (debond crac tip is close to the iber brea).. It does not show any agniication o G approaching to iber brea, thus contradicting to all reported nuerical results [9,17,18]. It could be because o the used incorrect boundary conditions on the iber crac surace in [13]: traction ree condition is satisied in average and not in each point. Nuerical ethods based on inite eleents (FEM) or boundary eleents (BEM) were used in [9,17,18]. Probably the ost detailed nuerical analysis o the local stress state at the debond crac tip in ters o stress intensity actor and degree o singularity has been perored in [9,17] using BEM. Unortunately, this ethod at present is liited to isotropic constituents and, hence, not applicable or carbon ibers. Subjecting a single iber coposite with ragented iber to axial cyclic loading we ay observe the debond length increase with nuber o cycles. Siilar atigue test on odel coposites with only 2-5 ibers has been reported in [19]. A hypothesis to veriy is that the debond growth rate with the nuber o cycles ollows the power law with respect to the strain energy release rate change during one cycle. I it appears to be true the data and siulations could be used to ind paraeters (aterial constants) in the power law. The identiied power law could be used to analyze the debond growth in atigue in real coposite with the sae iber/atrix syste. The objectives o this paper are a) to apply cyclic loading on SF specien with ragented iber to observe and quantiy the debond length increase with the nuber o cycles; b) to deterine G dependence on debond length and to describe it by siple itting unction; c) to evaluate using experiental data and G whether the power law can describe the debond length growth rate dependence on the strain energy release rate change during one cycle in atigue d) to deterine aterial constants in this law, i it is applicable. Friction at the interace is neglected in this paper. There are several attepts published to account or it in quasi-static loading [9,11] showing that riction will reduce G and hence the debond growth rate. At current it is not clear or authors how to include it in cyclic loading where the sliding cyclically changes direction and the riction eect is very dependent on the loading history. 2. Theoretical analysis 2.1 Model deinition SF specien shown in Fig. 1 has irst been subjected to teperature dierence T < 0 resulting ro cooling down ro the stress-ree teperature to roo teperature 101

114 T R. Due to ininite radius o the atrix region the theral shrinage strain o the SF th coposite is equal to ree theral expansion strain o the atrix, z T. At teperature T R the specien is subjected to constant strain in axial direction ech. Peroring a one step thero-echanical stress analysis we can interpret the loading sequence as the applied teperature change T and the axial strain z ech T. The iber is considered as transversally isotropic with thero-elastic constants E z, Er E, zr z, r, Gzr Gz, z, r. The atrix is isotropic with constants E, and. Since in the axial direction the Poisson s ratio and the theral expansion coeicient o the atrix are larger than or the iber, the radial stress on the iber surace is copressive and the debond crac propagation is in Mode. 2.2 Energy release rate or debond propagation in the steady-state region In a particular case, when the tip o the iber/atrix debond crac is ar away ro the iber brea, where it was initiated and when it is also ar ro another debond, which ay be approaching ro the other end o the iber ragent, the debond crac propagates in a sel-siilar anner. Matheatically this condition in ters o Fig. 1 and Fig. 2 can be written as: ld r, ld L (1) Fig.1. Representation o a SF coposite: F - iber, M - atrix, S - syetry surace, B - location o iber brea. In Eq. (1) r is the iber radius, L is hal o the iber ragent length. The debond crac grows by dl d and: a) shits the debond crac tip (and the corresponding singular stress state in its vicinity) in the z-direction by dl d ; b) reduces the bonded region volue in the odel by rdl 2 d ( r is the outer radius o the cylinder assebly in Fig. 2, r ) and increases the debonded region volue by the sae aount. 102

115 Neglecting riction and the energy dissipation related to it, Nairn et al. [13] derived exact analytical solution or strain energy release rate in this region. In notation used in the present paper it is as ollows: G 2 Q zr d 1 E z d 2 d r 1, Er E 1 1 Ez r 4 T 2 (2) ech 2 zr, d th E d z 2 r th 2 2 zr d 1, E z z 1 z Q d z 2 zr 1 2 (3) 2 Q 1 (4) E d As ollows ro Eq. (3) and (4) does not depend on theral constants whereas th does. However, or isotropic ibers th is independent on theral expansion coeicients. 2.3 Energy release rate or short debond lengths I the debond is short, the debond crac tip is close to the iber brea. In such case the stress perturbation regions related to the debond crac tip and the iber brea overlap. Previous nuerical studies on UD coposites [20] and single iber coposites [4,9,17,18] have shown that due to this interaction the strain energy release rate G is agniied. The shorter is the debond length the higher is the agniication o G. As the debond length increases, the agniication o G decreases and at certain debond length the crac growth becoes sel-siilar eaning that G will be constant with respect to the debond length. Because o that the analytical solution or calculation o G given in section 2.2 is not valid or short debonds. In this study FEM in cobination with racture echanics concept o virtual crac closure technique [21] was used to calculate the agniication o G at short debond lengths. The crac closure technique states that the energy released due to the debond crac growth by da is equal to the wor, which is required to close the newly created surace ro size A da bac to size A, where da 2r dl d. Fig. 2 scheatically shows the FEM odel. z Fig.2. Scheatic representation o the axisyetric FEM odel: F - iber, M - atrix. 103

116 To close the debond crac by dl d (ro l d dl d to l d ) tangential tractions along the interace have to be applied. The points at the debonded surace in the region z l ; l dl, which have relative tangential displaceent d d d ld dld ld dld ld dld u ( z) u ( z) u ( z) (5) z are oved bac to coinciding positions. At the end o this procedure the shear stress l in point z is equal to d rz (z), which is the shear stress in ront o the crac with size l d. Then the wor required to close the crac by dl d can be expressed as: z ld dld z 1 ld dld ld W 2 r u z ( z) rz ( z) dz (6) 2 Within the virtual crac closure technique it is assued that due to sall value o dl d the relative sliding displaceent at the tip o the crac with size l d dl d is the sae as at the tip o the debond crac with size l d : ld ld dld ld u z z) u z ( z dl ) (7) ( d The beneit o this assuption is that only one stress state calculation or a given debond length is required. Distributions o the shear stress rz and sliding displaceent u z u z u z can be conveniently obtained ro the FEM solution (see Fig. 2). Then, the strain energy release rate G can be calculated using expression ld dld 1 ld ld G ( ld ) li uz z dld rz ( z) dz dld 0 2dl (8) d Using FEM based solution the integration is over a inite region, which is probably larger than the region where the solution is local (singular). Thereore, strictly speaing, the obtained value is not G. It depends on the integration region dl d and should rather be called energy release rate over distance dl d. 2.4 Paris law in tension-tension atigue o SF coposites ax Tension-tension cyclic loading with load ratio R in ech / ech is considered in this study in presence o constant theral stresses at ixed teperature change T. Power law expression will be applied to describe the debond growth along iber/atrix interace as a unction o applied nuber o load cycles N. We assue that the crac surace area A increases with N is described by: ld da dn G B (9) * G 104

117 where: B and are unnown paraeters, which have to be deterined experientally; G is the energy release rate dierence between the values corresponding to ax ech and in ech ; G * = 1 J/ 2 is a constant used to obtain diensionless expressions. Debond length, which or circular iber is related to debond surace as l d A /(2 r ) will be noralized with respect to the iber radius r : l l / r (10) dn d Thus Eq. (9) can be written in or dldn * G B * dn G (11) * B B 2 r (12) 2 * Obviously B is also diensionless. For siulation purposes Eq. (11) was used in an increental or G () B * (l () dn ) N, G * l dn l l l ( 1) ( ) ( ) dn dn dn ; =1,2,3 (13) In Eq. (13) 3. Experiental (1) l dn is the initial debond length at the start o siulation. Tension-tension atigue experients were perored in this study on ragented single iber coposites and the debond length l dn was easured as a unction o the nuber o cycles N. The anuacturing o test speciens and the atigue tests were perored at Risø National laboratory/dtu (Denar). The test specien consisted o a glass iber (diaeter d = 17, P192 sizing) ebedded in an epoxy atrix (E6 resin, H6 hardener, ratio 100:25). Silicone oulds were used or anuacturing o the saples and the ain diensions o saples are shown in Fig. 3. Thicness o all speciens was equal to 2. Fig.3. Diensions (in ) o the atigue test specien. The saples were cured at roo teperature or 24 hours with post-curing step at 80 C or 4 hours. Due to the dierence in theral expansion coeicients between the iber and the atrix, cooling down ro the post-curing teperature to roo teperature causes copressive stresses in the iber. To copensate or it, the iber 105

118 was pre-stressed prior to ould illing using 7g o weight suspended at each end o the iber. In this way iber brea in the SFF specien was obtained at uch lower strain levels than without pre-load. Lower strain levels are preerable to avoid specien racture or non-linear behavior o the epoxy resin atrix. In the strain energy release rate calculations the pre-stress was accounted or by introducing additional theral load which leads to an equal value o the iber axial stress, see Section 5.1. Each specien was subjected to increasing tensile load and the irst brea o the iber was detected using acoustic eission. The applied strain was deterined using iage analysis - reerence points were drawn on the saple surace and the increase o the distance between these points during loading was easured. Iages were taen with a digital caera attached to an optical icroscope. The average strain at the event o iber brea 1st was deterined and it was equal to 2.2 %. Then tension-tension cyclic loading with requency = 2 Hz and R = 0.1 was perored using Instron testing achine under load control. Two dierent values o axial strain level were studied: 1) 1.76 %; 2) 1.32 %. The entioned strain levels correspond to approxiately 80% and 60 % respectively o 1st, where 1st is the average stress at the occurrence o the irst iber brea. Ater certain nuber o cycles, the specien was reoved ro the testing achine and oved to a icroscope to easure the debond length (polarized-light icroscopy). Debond length (denoted as l d1...l d 4 in Fig. 4) was easured at both sides o the iber brea as shown in Fig. 4. The severely daaged iber region around the iber brea was excluded ro the debond length easureents. Fig.4. Scheatic representation o debond cracs growing ro iber brea: F - iber; M - atrix; DF - daaged iber zone around the iber brea; l d1...l d 4 length o the easured debond cracs. Table 1. Basic elastic properties o the constituents. [GPa] zr [-] z [1/ C] E [GPa] [-] [1/ C] E z The basic aterial properties are listed in Table 1. The properties o the E-glass ibers ( E, and ) are ro iber anuacturer s data while the properties o the epoxy resin atrix were deterined ro standard tensile ( E, ) and teperature-strain tests ( ) in teperature range C. 106

119 4. Siulation results and discussion 4.1 G or sel-siilar debond growth paraetric analysis Shear oduli do not enter the solution in section 2.2. The isotropic iber properties and the theral expansion coeicient o the atrix were as speciied in Table 1, whereas the atrix odulus E and the Poisson s ratio are variables o the paraetric analysis. The calculated paraeters and th in Eq. (2) are presented in Table 2. Table 2. Dependence o, th on Poisson s ratio,, and atrix odulus, E. = 0.3 = 0.35 = 0.4 = 0.45 th th th th E = 3 [GPa] E = 3.5 [GPa] E = 4 [GPa] Table 2 shows that or all relevant cobinations o atrix constants the dependence o paraeters, th on atrix elastic properties is very wea and without losing accuracy these paraeters can be taen equal to one. This eans that the echanical strain and the theral expansion strain have equal signiicance in the strain energy release rate expression (2). For isotropic ibers they do not depend on. These conclusions hold also or the carbon iber case. 4.2 G or short debond growth Using FEM in order to calculate G the odel is inite but its diensions should be such to obtain distributions o stresses rz and displaceents u z, which are representative also or the ininite r case analyzed with the analytical odel. Siilarly, in ters o Fig. 2, a suitable size o the odel length L has to be ound to prevent interaction between the debond crac tip and the other debond tip approaching ro the right-hand side (the load application surace in Fig. 2 is in the iddle o the iber ragent). FEM analysis was perored in [18] or a large debond, increasing in steps the atrix radius to r / r = 500 and showing that starting ro ratio 100 the strain energy release rate practically does not change and diers ro the analytical solution or the ininite case [13] by less than 1%. Details regarding the non-unior esh are given in [18].Since our glass iber/epoxy (GF/EP) single iber coposite is siilar to the 107

120 one used in [18], the inding that the outer atrix radius equal to r 100 r and the odel length equal to L 200 r are optial to represent the non-interactive debond case were also used in this study. Constituent elastic properties o the GF/EP coposite are given in Table 1. In [18] the elastic properties o the epoxy resin were E = 3 GPa and = 0.4. However, coparing the calculated G values ro the present study and ro [18], see Fig. 5, we can conclude that there is alost no dierence. Thereore, the agniication coeicients ro [18] are applicable also or the case studied in this paper. Fig.5. G or GF/EP single iber coposite with constituent properties given in Table 1 and ro [18]. ech = 1 %, T = 0 C. The accuracy o the obtained agniication coeicient values was investigated in [18] coparing the results with values calculated using BEM [9] or the case o a purely echanical loading. The results o both ethods were alost identical. The quadratic or o the energy release rate G with respect to the applied strain ech and the teperature change T (see Eq. (2)) is also valid or short debonds. It can be written as: Ez r G T 2 ech th z (14) 4 where * (l dn ), th * th (l dn ) th (15) As deonstrated by Fig. 5 the energy release rate G is higher or shorter debond lengths and approaches a constant plateau value when the debond length is long and debond growth becoes sel-siilar. The coeicients * (l dn ) and th * (l dn ) in Eq. (15) are deined as the agniication coeicients and they are equal to 1 when the debond growth is sel-siilar. Coeicients and th are calculated analytically using Eq. (3) and (4), whereas agniication coeicients * (l dn ) and th * (l dn ) are deterined ro FEM calculations. For the convenience o urther siulations it would be desirable to use siple atheatical expressions to describe the agniication coeicients * (l dn ) and th * (l dn ) as unctions o the debond length. In [18] it was shown that a siple hyperbolic unction is rather accurate. It was also shown that the agniication coeicients are alost identical or the cases o echanical and theral loading. 108

121 * * Thereore ( ldn ) th ( ldn ) 0( ldn ). The or or 0 (l dn ) used here is the sae as in [18] a1 2 1 a2 a3ldn; 1 ldn 30 0 l (16) dn 1; ldn 30 Table 3 shows the values o the itting paraeters a 1, a 2 and a 3 ound in [18]. Table 3. Values o the itting paraeters in Eq. (16). a 1 a 2 a It was shown in [18] that the calculated agniication o G is in excellent agreeent with BEM calculations [9]. 4.3 Modeling debond growth using power law We showed in Section 4.1 that coeicients and th can be assued equal to 1 without losing the accuracy o calculation o G. Furtherore, in [18] and in Section 4.2 we showed that the agniication coeicients or echanical and theral cases * * are identical, i.e., ( ldn ) th ( ldn ) 0( ldn ). Hence, using Eq. (14) or the case o echanical tension-tension cyclic loading at ixed teperature, the energy release rate range G can be calculated as: 2 r Ez 0 ( ldn ) ax 2 in 2 ax in G ( ldn ) ( ech) ( ech) 2( z ) T ( ech ech) (17) 4 Now, the G Eq. (17) can be ipleented in Eq. (13) and debond growth siulations can be perored. The dependence o the growth rate on the iber radius ollows (r ). Paraetric analysis was perored evaluating the signiicance on the debond growth rate o the power law paraeters, B * and the teperature change T. Table 4 shows the values o paraeters or the reerence case and in Figs. 6 to 9 these paraeters are varied to reveal their signiicance. Table 4. Paraeter values or the reerence case o debond growth siulations. r E B B * ax ech R T [] [GPa] [-] [ 2 ] [-] [%] [-] [ C] Results in Fig. 8 deonstrate that the initial teperature change signiicantly aects the growth o the debond crac. As described beore, copressive stresses appear in the iber when the coposite is cooled down ro the anuacturing (stress-ree) teperature to roo teperature ( T < 0). Because o these copressive stresses, 109

122 G is saller and the debond growth rate is uch slower or the case o T = -50 C than or T = -20 C and T = 0 C. Fig.6. Debond growth siulations with dierent values o paraeter. Fig.7. Debond growth siulations with dierent values o paraeter B *. Fig.8. Debond growth siulations with dierent values o initial teperature change T ( C). 110

123 5. Experiental results and identiication o power law paraeters 5.1 Accounting or the theral and curing stresses and the pre-load It is well nown that the elastic odulus and theral expansion coeicients o the epoxy resin depend on teperature (the orer reduces and the latter increases with the teperature increase) [22]. However, it is not clear whether in the very beginning o the cool-down the stresses develop in an elastic anner or there is a viscoelastic stress relaxation. Siilar uncertainty is with respect to the cheical shrinage o the atrix during the cross-lining: is it lined with introduction o elastic stresses or is it a relatively stress-ree process? Nevertheless, all calculations in this paper were perored using a linear theroelastic odel. As an input the elastic odulus o the atrix easured at roo teperature (RT) and the average theral expansion coeicient obtained using NETZSCH DIL 402 C dilatoeter in the teperature range between RT and 74 C were used. In addition, it is suggested that the stresses introduced by cheical shrinage can be accounted or by increased stress-ree teperature. The use o linear thero-elasticity has to be validated in tests where the eect o the theral and cheical shrinage o the polyer is easured in-situ, easuring directly the strain in the aterial o interest or easuring the overall response o the syste and bac-calculating. An excellent technique using the orer option, which unortunately is not available in our lab, is based on Raan spectroetry: a sensor iber is introduced in the resin and strain developent in this iber is easured [23]. In our study we used the latter approach easuring the curvature o a bi-aterial bea at several teperatures. The irst aterial in the bea was a CF/EP 0-layer with nown theroelastic properties, anuactured separately. The resin layer was cured on the surace o the CF/EP layer. During the curing cheical shrinage in the resin was expected and ater the post-curing and the cooling-down the curvature o the bea was the suary result o the cheical and theral shrinage. Ater that the teperature was increased in steps and the curvature at each teperature was deterined. Increasing the teperature the obtained curvature versus teperature curve was very linear showing that the stress is a linear unction o the teperature change. The slope o the curve was rather siilar to the slope calculated using the RT elastic odulus and the average theral expansion coeicient, validating the application o the linear thero-elasticity as the irst approxiation. The curvature reached zero at teperature about 10 C higher than the post-curing teperature. The non-zero curvature at the post-curing teperature was attributed to the cheical shrinage strains developed during the cross-lining. In the linear calculation routine these strains can be considered as a part o theral strains introducing a new higher stress ree teperature and a higher teperature change ater cool-down, Tcure = -10 C. The obtained value o the stress ree teperature increase is close to the value 15 C given in [24]. It has to be noted that due to unavailability o the original aterial (the theral properties test was designed and perored ater the atigue testing progra was inished) the Araldite LY 5052 / Aradur 5052 epoxy with siilar elastic properties was used in the described test. Thereore the values given above should not be used or quantiication purpose. Based on these experients, three nuerical values were chosen or the paraetric analysis: T cure = 0 C (no curing shrinage), - 10 C and -20 C.To partially copensate or the copressive stresses caused by the above theral and curing shrinage o the resin, a echanical pre-load was applied to 111

124 the iber, by suspending 7g weight at each end o the iber prior to ould illing. Thus or the given iber diaeter ( d = 17 ) the iber pre-stress was equal to = MPa. This stress was accounted or using theral analogy, introducing additional teperature change T which leads to equal axial iber stress ech Tech (18) E ) ( Using the aterial properties given in Table 1, T ech = 50.1 C. During the cyclic loading the teperature dierence between the post-curing and the roo (operating) teperature was not changing. Since the post-curing teperature was T C = 80 C and the operating teperature T R = 22 C, we obtain Tth = -58 C. The resultant teperature change, T used in calculations is T T T T (19) th cure ech The T cases considered in siulations in Fig. 9 were: T = -7.9 C, C and C. Fig.9. Debond growth siulations with dierent values o initial teperature change T ( C). 5.2 Measureents o the debond length The experiental easureents o the debond length increase in cyclic echanical loading were perored or three saples denoted urther in the text as saple A, B and C respectively. Fig. 9 shows that the used variation in T, which relects the unnown variation o cheical shrinage, ay be iportant in debond developent. Since the inoration about cheical shrinage is not available, in the ollowing data ax reduction it is ignored (T cure = 0 C). For saples A and B ech was 1.76 % while or saple C ax ech was 1.32 %. Other loading paraeters or all saples were: R =0.1, T =-7.9 C, =2Hz. The paraeter T shows the suary eect o the teperature change (see Eq. (19) in Section 5.1), is the loading requency. 112

125 Table 5. Values o paraeters and lnb * or saples A and B. Saple lnb * A B Typical optical icroscopy iages o the iber/atrix debonds taen ater a certain nuber o cycles in the case o the applied stress level ax = 80% o 1st are shown in Fig. 10. Fig.10. Optical icroscopy iages showing the debond length increase with nuber o cycles in tension-tension cyclic loading: a) N = 1; b) N = 1000; c) N = 5000; d) N = F - iber; M - atrix. Fig.11. Experiental easureents o debond length in cyclic loading or saples A and B. 113

126 Fig. 11 shows experiental easureents or saples A and B. The horizontal axis shows the nuber o load cycles N in logarithic scale. The our easureents or each saple in Fig. 11 correspond to the schee shown in Fig. 4. Copared to saples A and B, the debond growth in saple C, or which the applied stress level was lower, was uch saller. 5.3 Deterination o power law paraeters We will use the power law (11) to analyze the experiental data with two tass in ind: a) to veriy whether the power unction is suitable or the description o the debond growth; b) to ind the paraeters in the power law i it is applicable. Expressing G in J/ 2 (11) can be written as dl dn dn B* G (20) The energy release rate range G in the power law expression (20) was calculated using Eq. (17). In act, Eq. (17) can be written as a product o two parts: the debond length dependent part (applied load-independent) denoted as G 0 : and the applied load (, T ) dependent part denoted as Consequently, 2 r 0 ( ) 0 Ez ldn G (21) 4 g : ax 2 in 2 ax in g ( ) ( ) 2( ) T ( ) (22) ech ech z ech ech G G 0 g (23) First the validity o the power law with respect to the strain energy release rate change in the cycle was veriied. Siilarly as with the Weibull strength distribution analysis the ost convenient way is to plot (20) in the log-log axes dldn * 0 ln ln B ln G dn ln g (24) In order to use (24) the derivative o experiental debond length data with respect to is necessary. In this study the derivation was replaced by inite dierences as: where dl l dn dni dn N l dni l i1 dn l N 1 i N N i 2 i dn (25) (26) 114

127 Using the inite dierences instead o derivatives in the representation o the crac growth rate, the scatter was rather large, see Fig.12 and soe deviation ro linearity 0 at low G (long debonds) was ound. As an alternative the experiental debond length versus N data were itted by a sooth (exponential) unction and then the derivative with respect to N was calculated. Using this ethod the results in log-log axes were siilar to the presented here and thereore this approach is not described in ore details. Based on the results in Fig 12 we conclude that the power unction is a proper unction to use. The paraeters and B * in the power law Eq. (20) are iber/atrix interace properties and their values were deterined ro the best it with experiental easureents. Fro Eq. (24) the paraeter can be easily deterined ro the slope o the linear unction in Fig 12. The ordinate axis intercept o the plotted linear unction is then according to Eq. (24) equal to D ln B * lng and thus paraeter B * can be calculated ro: ln B * D lng (27) The power law paraeters according to the aboveentioned procedure were ound or saples A and B. Since power law paraeters are iber/atrix interace related properties the paraeters deterined or saples A and B are expected to be valid also or saple C, or which the level o applied echanical strain was lower. Fig. 12 shows the plot o ln l dn 0 vs. lng N or the saples A and B. Linear it or each saple and the approxiation equations are also presented. Each experiental data point in Fig. 12 and in the urther igures corresponds to the average o our debond length easureents l d1, l d 2, l d 3 and l d 4 shown in Fig. 4. Fig.12. Linear approxiation o ln l dn vs. lng N 0 or saples A and B. 115

128 Table 5 shows the values o and lnb * or both saples. Table 5. Values o paraeters and lnb * or saples A and B. Saple lnb * A B It is useul to estiate the accuracy o the itting perored in the log-log axes. Fig. 13 shows that the overall agreeent between experiental data and siulations or saples A, B and C is acceptable. It has to be stressed that odeling data or saples A and B are sel-predictive, while odeling data or saple C were obtained using constants ro saple A (Table 5). Siulations show that the debond or saple C does not grow at this load, which conirs the experiental observation. Using constants ro saple B in this siulation leads to the sae result. Fig.13. Average noralized debond length l dn as a unction o applied load cycles N or saples A, B and C. Experiental and odeling results. 6. Conclusions Peroring a cyclic tension-tension loading on specien with single ragented iber ebedded in the resin we easured iber/atrix debond growth with the nuber o cycles when the axiu load in the loading was 80% o the load at which the irst iber brea was observed. Debonds initiated by iber brea did not grow when the axiu load in the cyclic loading was 60% o the load or iber brea. The strain energy release rate in Mode or debond growth was analyzed cobining analytical solution in the steady-state region and FEM solution or short debonds. It was shown that the steady-state paraeters are very insensitive with respect to atrix properties. 116

129 For short debonds the agniication o echanical and theral strain energy release rates ollows the sae rule. Using the quantiied debond length versus nuber o cycles data it was shown that the power law with respect to the strain energy release rate change is applicable or debond growth characterization in tension-tension atigue. Data reduction schee was suggested and paraeters were obtained or use in siulations. Siulations showed that the obtained paraeters give acceptable predictions or cases, when the debond grows as well as when it does not grow. Since these paraeters are aterial properties they do not depend on the applied loading case. Thus, it is expected that the paraeters deterined in this study using single iber coposites can also be applied or the case o UD coposites ade o the sae aterial syste.in such case the energy release rate used in the power law will be calculated using the local icrostructure, or exaple the local iber volue raction and the non-uniority o the iber distribution. 7. Reerences 1. R. Talreja, A continuu echanics characterization o daage in coposite aterials. Proceedings o Royal Society o London, A378, pp , R. Talreja, C.V. Singh, Daage and Failure o Coposite Materials, Cabridge Univ Press, E.K. Gastedt, Eects o debonding and iber strength distribution on atigue daage propagation in carbon iber reinorced epoxy, Journal o Applied Polyer Science, vol. 76, pp , A. Pupurs, J. Varna, Unidirectional coposite in echanical atigue: odelling debond growth ro ibre breas, Plastics Rubber and Copos, vol.39, pp , A. Kelly, W.R. Tyson, Tensile properties o ibre-reinorced etals: copper/tungsten and copper/olybdenu. Journal o the Mechanics and Physics o Solids, vol. 13, pp , H.L. Cox, The elasticity and strength o paper and other ibrous aterials, British Journal o Applied Physics, vol.3, pp , J.M. Whitney, L.T. Drzal, Axisyetric stress distribution around an isolated iber ragent. In: N.J. Johnston, editor. Toughened Coposites, ASTM STP, 937. Philadelphia: Aerican Society or Testing and Materials, pp , I. Verpoest, M. Desaeger, R. Keunings, Critical review o direct icroechanical test ethods or interacial strength easureents in coposites. In: Ishida H, editor. Controlled Interphases in Coposite Materials. Asterda: Elsevier, pp , E. Graciani, V. Manti, F. París, J. Varna, Nuerical analysis o debond propagation in the Single Fibre Fragentation Test, Coposites Science and Technology, vol. 69(15-16), pp , B.W. Ki, J. Nairn, Observations o iber racture and interacial debonding phenoena using the ragentation test in single iber coposites, Journal o Coposite Materials, vol. 36(15), pp , B.W. Ki, J. Nairn, Experiental veriication o the eects o riction and residual stress on the analysis o interacial debonding and toughness in single iber coposites, Journal o Materials Science, vol.37, pp ,

130 12. J.A. Nairn, Y.C. Liu, Stress transer into a ragented, anisotropic iber through an iperect interace, International Journal o Solids and Structures, vol. 34, pp , J.A. Nairn, Y.C. Liu, On the use o energy ethods or interpretation o results o single-iber ragentation experients, Coposite Interaces, vol. 4, pp , J.A. Nairn, Exact and variational theores or racture echanics o coposites with residual stresses, traction loaded cracs, and iperect interaces, International Journal o Fracture, vol. 105, pp , W. Wu, I. Verpoest, J. Varna, A novel axisyetric variational analysis o stress transer into ibres through a partially debonded interace, Coposites Science and Technology, vol. 58, pp , W. Wu, I. Verpoest, J. Varna, Prediction o energy release rate due to the growth o interace crac by variational analysis, Coposites Science and Technology, vol.60, pp , E. Graciani, V. Manti, F. París, J. Varna, Single iber ragentation test. A BEM analysis. In: Collection o Technical Papers - AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynaics and Materials Conerence, 2 nd Edition. Norol (Virginia), pp , A. Pupurs, J. Varna, Fracture echanics analysis o debond growth in single iber coposite under cyclic loading, Mechanics o Coposite Materials, vol.47(1), pp , S. Goutianos, T. Peijs, Experiental and nuerical investigation into atigue daage echaniss in ultiibre icrocoposites, Plastics Rubber and Coposites, vol. 30, pp , A. Pupurs, A. Krasniovs, J. Varna, Energy release rate based iber/atrix debond growth in atigue. Part : Debond growth analysis using Paris law, Mechanics o Advanced Materials and Structures, In Press, G.R. Irwin, Fracture, Handbuch der Physi, vol.5. Berlin: Springer Verlag, L.G. Zhao, N.A. Warrior, A.C. Long, A thero-viscoelastic analysis o processinduced residual stress in ibre-reinorced polyer-atrix coposites, Materials Science and Engineering A, vol , pp , J. Parthenios, D.G. Katerelos, G.C. Psarras, C. Galiotis, Araid ibers; a ultiunctional sensor or onitoring stress/strain ields and daage developent in coposite aterials. Engineering Fracture Mechanics, vol. 69, pp , M.M. Shorieh, S. Masoud Kaali, Theoretical and Experiental Studies on Residual Stresses in Lainated Polyer Coposites, Journal o Coposite Materials, vol. 39, pp ,

131 Paper V A. Pupurs, J. Varna Modeling iber/atrix debond growth in tension-tension cyclic loading o UD coposites, To be subitted to Coposites Part A, 2012.

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133 Modeling iber/atrix debond growth in tension-tension cyclic loading o UD coposites Andrejs Pupurs, Janis Varna Departent o Engineering Sciences and Matheatics, Luleå University o Technology, SE , Luleå, Sweden Abstract The iber/atrix interace crac (debond) growth ro iber brea in unidirectional coposite subjected to high stress tension-tension cyclic loading is analyzed. The debond growth is siulated calculating the strain energy release rate G by FEM in 3D orulation and using power law with respect to the G change to describe the debond growth rate. Two odels were applied. In Model 1 the partially debonded iber/atrix cylindrical unit with a iber brea is surrounded ro all sides by eective coposite. In Model 2 the eective coposite was used around the iber/atrix unit except the region between the unit and the specien surace where neat atrix was placed. Calculations show that G is slightly larger when the analyzed iber is close to the specien surace. The debond growth was siulated using interace atigue paraeters obtained easuring debond length in a specien with single ragented iber. Siulations show that debonds ro iber breas close to the specien surace grow uch aster than ro iber breas inside the coposite. 1. Introduction The atigue perorance o unidirectional (UD) coposites is usually characterized by atigue - lie diagras analyzed very systeatically in [1,2,3]. Three regions in the atigue lie diagra have been distinguished analyzing tension-tension cyclic loading in the iber direction. At high loads ultiple iber racture taes place. In [2] the high strain/stress region where the daage is initiated by iber breas was described as non-progressive. Indeed the scatter band in this region is alost horizontal and a clear dependence o the atigue lie on the load level cannot be observed. The atigue lie can vary between several cycles and several thousand o cycles. Nevertheless, conceptually the dependence on load in this region is obvious. Since the iber strength is a statistical property which, or exaple, ay be described by Weibull distribution [4], the nuber (requency) o iber breas which occur during the irst loading cycle strongly depends on the load level. Multiple iber racture (ragentation) is possible because the load is transerred into the broen iber through the intact interace. In cyclic loading with constant aplitude we usually assue that ibers do not experience atigue and all iber breas occur during the irst cycle. During cycling debonds start to grow along the iber/atrix interace and this is the ain ode o daage progression. Finally, the extensive debonding connects the ultiple iber breas and leads to catastrophic rupture o the UD coposite. Hence, understanding o the 121

134 debond growth process in atigue is crucial or atigue lie analysis in this region. Considering the debond as an interace crac its propagation in UD coposite has been analyzed using racture echanics concepts (strain energy release rate based criteria) [5,6]. The described is a echanis or very high strains. In the interediate load region the probability o iber ailure is low and the echanis o debond growing ro a iber brea ay not be the ost typical. For exaple, in cyclic loading the initiation o sall atrix cracs between ibers oriented transverse to the iber direction could be the irst ode o daage. When these cracs hit the interace they ay urther grow along the iber as interace cracs (debonds). Thus, according to this sequence o echaniss the atigue lie is governed by atrix cracing and interace resistance to shear loading. The two possible echaniss (debond initiated ro a iber brea and debond starting ro a atrix crac) were analyzed in [7]. The interacial radial theral stress which ors during the cool-down to roo teperature is copressive due to larger theral expansion coeicient o the atrix. Additional copressive radial stresses are ored during the axial echanical loading due to dierent Poisson s ratios. Consequently, there is always a contact between the debonded iber and the resin and the debond growth is in pure Mode. It was ound that the Mode strain energy release rate G or the debond crac growth initiated ro a iber brea is several orders o agnitude larger than i it is initiated ro a atrix crac. The decays with the distance i the debond is initiated ro a atrix crac but, i it initiates ro a iber brea, the G approaches to a non-zero asyptotic value and the debond growth becoes sel-siilar. Finally, it is oten assued [2] that below a certain cyclic load, called the atigue liit, the existing deects will not grow and the atigue lie is ininite. Certainly, this is an approxiation. More detailed description o the possible ailure scenarios in atigue o UD coposites and their dependence on the load level is given in [1,3]. The present investigation is ocusing on the high load region and the debond growth ro iber breas during cyclic loading in UD coposite is analyzed. The hypothesis used in this paper is that the growth rate o an individual debond can be characterized by a power law: the rate is a power unction o the strain energy release rate change in one cycle. This hypothesis was validated in [8] analyzing the debond growth in atigue in the Single Fiber Fragentation test (SFF test). It was shown that the experiental debond length data in cyclic loading can be described by odel based on power law. The aterial paraeters in this law were deterined by itting. Experiental observation o debond growth as a unction o cycles in UD coposites is a very coplex tas: using optical icroscope a iber brea on the specien surace has to be ound, the debond growth onitored and the debond length recorded. To authors nowledge there is only one paper [9] available, where the length o a debond initiated ro a iber brea in a UD coposite was easured as a unction o the nuber o cycles. These easureents are easier to peror during cyclic loading o a SFF test specien with ragented iber. Siilar atigue test on odel coposites with only 2-5 ibers has been reported in [10]. We suggest that the cyclic SFF test (or siilar test with a sall nuber o ibers) is used to ind the interace characterizing atigue paraeters in the energy release rate dependent power law. Then these paraeters can be used to siulate debonding in a UD coposite ade o the sae aterial syste. The objectives o this paper are: 122 G

135 1) to calculate the energy release rate or debond growth in UD coposite investigating the dependence on the geoetrical position o the broen iber with respect to the specien surace, the eect o the iber properties etc. The proble does not has axial syetry and the analysis is 3-diensional; 2) to use interace atigue paraeters identiied in [8] to siulate the debond growth in a coposite identiying the ost iportant eatures. The used siple odel consists o three phases: a iber surrounded by atrix and ebedded in the eective coposite with elastic constants calculated using the Hashin s CCA odel and Christensen s generalized sel-consistent schee [11-13]. Since a UD coposite consists o any ibers the theral copressive stresses are uch saller in a UD coposite than in a SFF test. They are not included in the presented nuerical results. The riction at the interace is also neglected in this paper. It has been shown that in quasi-static loading [14,15] the riction reduces G and hence the debond growth rate is lower. Unortunately, it is not clear how to deal with it in a cyclic loading: the sliding changes direction during one cycle and the eect o the riction depends on the loading history. 2. Models or UD coposite with broen and partially debonded iber The coposite with a broen and partially debonded iber is subjected to echanical axial strain z, see Fig.1. Due to dierent theral expansion coeicients o the atrix and the iber and the change in teperature T 0, ater anuacturing the iber in the coposite is under axial and radial copression. The copressive stresses in the undaaged iber are uch saller than in the SFF test, where the iber is surrounded by a large bloc o atrix. This dierence is easy to understand, i we consider a cylindrical unit cell consisting o iber and the atrix which is used to calculate the coposite theral expansion coeicient. In UD coposites the volue o the atrix approxiately equals the volue o the iber and due to uch higher stiness the iber is governing the theral expansion. Based on this consideration and due to the planed use o experiental data or aterial syste cured at roo teperature (RT) we ignore the theral stresses in the ollowing analysis. The UD coposite specien is represented by three phases: a) iber; b) atrix; c) eective coposite with elastic properties calculated using the Hashin s concentric cylinder assebly odel (CCA) [11] and the generalized sel-consistent schee by Christensen [12] (or the out-o-plane shear odulus). In the odel, see Fig.1, the broen iber o radius r is surrounded by a atrix cylinder with an outer radius r corresponding to the iber volue raction V. This unit is ebedded in an eective coposite. I the distance l c ro the iber axis to the specien surace is relatively large, lc 2. 5 r, the iber/atrix unit is surrounded with the eective coposite ro all sides including the region between iber and the specien surace, see Model 1 in Fig.1. I the iber is very close to the specien surace, lc 2. 5 r we use a ore realistic assuption that a neat atrix is covering the unit ro the top instead o the eective coposite, see Model 2 in Fig 1. In the plane, where the iber is broen, the iber surace is traction ree. For the atrix and or the eective coposite we use there syetry condition. The iber hallength is denoted LS and ixed axial displaceent is applied to all points in the crosssection corresponding to the iddle o the iber ragent. 123

136 Model 1 specien surace l d l c z specien surace r l c r r d c iber brea syetry plane d c debond iber atrix eective coposite L s syetry plane Model 2 specien surace l d l c z r specien surace r l c r d c iber brea syetry plane d c debond iber atrix eective coposite L s syetry plane Fig.1. Models representing the broen and partially debonded iber in the UD coposite: Model 1 - the distance o the iber center to the surace l 2. 5 r ; Model 2 - the distance lc 2. 5 r. 124 c

137 The iber/atrix interace is debonded over distance l d, see Fig.1. For siplicity we assue that the geoetry o the debond has axial syetry ( l d ( ) ) even when it is close to the surace. When the analysed iber is ar ro the specien surace, the stress state at the interace is close to axisyetric. Approaching the specien surace the axial syetry is lost and all coponents o the stress/strain state becoe dependent on the local angular coordinate. Thus, according to Fig.1 a 3D FEM odel is required or stress calculations and the strain energy release rate is a unction o. To sipliy the discussion the average o the G over will be calculated and used in discussions. The use o the average G is consistent with the above assuption that the debond length does not depend on the angle. 3. Mechanics o debond propagation 3.1 Power law or debond growth in atigue A SFF test specien with ragented iber was subjected to tension-tension cyclic loading and the debond length growth along the iber with the nuber o cycles was recorded in [8]. Coparing the debond growth rate data with the calculated strain energy release rate a power law dependence was ound. Hence, the ollowing expression describes the rate o the increase o the debond surace area da : da dn B G (1) where: B and are the interace atigue resistance characterizing paraeters deterined in this test by data itting. G is the strain energy release rate dierence between values corresponding to the highest, ax and the lowest strain in in a cycle. The debond length increent or a circular iber is related to debond surace increent as: 3.2 Deterination o the strain energy release rate 2 r 1 dl da (2) d Interace debond growth related energy release rate G can be calculated by various analytical and nuerical ethods one o the being the crac closure technique introduced by Irwin [16]. The crac closure technique states that the energy released due to debond crac growth by da is equal to the wor, which is required to close the newly created surace ro size A da bac to size A. Fig.2 can be used or geoetric representation o the urther stateents. z 0 is the axial coordinate o the debond crac tip. Closing the debond crac by length dl d (ro l d dl d to l d ) by applying to the part o the crac aces tangential tractions in the iber direction, the points at the debonded surace in the region z 0, dl d, which have relative tangential displaceent 125

138 ld dld ld dld ld dld u ( z) u ( z) u ( z) (3) z are oved bac to coinciding positions. Here and in the ollowing the upper index or stress and displaceent shows the length o the considered debond. At the end o this l procedure the shear stress in point z is equal to d xz (z), which is the shear stress in ront o the crac with length l d. For the case, when the iber is ar away ro the specien surace, the stress state around the interace is axisyetric, the wor required to close the crac by dl d can be expressed as: dld 1 ld dld ld dw dld 2 r uz ( z) rz ( z) dz (4) 2 0 In (4) rz is the shear stress coponent relevant to Mode crac propagation. For an angle ( ) dependent 3-D stress state the expression (4) can be rewritten as: 2 dl z d 1 ld dld ld dw dld u ( z, ) (, ) z rz z r d dz (5) Due to the extreely sall value o dl d the distribution o the relative sliding displaceent in the vicinity o the debond crac tip o the size l d dl d is practically the sae as at the tip o the debond crac with size l d. Usually the VCCT (Virtual Crac Closure Technique) is used assuing that z ld dld ld u z) u ( z dl ) (6) z ( z d The useulness o this assuption is that only one stress state calculation or a given debond length is required. The strain energy release rate is deined as: dw G (7) da where da 2r dld. Thus G can be easily calculated ro a single stress state by cobining equations (5), (6) and (7): The the distance 2 dl d G 1 ld ld li u ( z dl, ) (, ) 0 z d rz z r d dz dld 4r dl (8) d 0 0 G deined according to (8) represents the average value over all angles and over dl d (i the latter is inite). 3.3 FEM odel Axisyetric FEM odels or analysis o the debond growth in the bul o the coposite, have been reported in [6,7]. However, or the case, when the iber is in 126

139 vicinity o the specien surace, axisyetric odel is not applicable due to edge (surace) eects. In this study 3-D FEM odels corresponding to geoetrical conigurations shown in Fig.1 were generated using inite eleent code ANSYS [17]. As shown in Fig.1 only one hal o the odel had to be generated, taing advantage o the syetry conditions. The values o the geoetrical paraeters L s and d c (see Fig.1) were selected based on the convergence analysis perored in [6] where it was ound that the odel length L s 90 r and the eective coposite size d c 5 r are suicient to represent a long partially debonded iber in an ininite coposite. In the present study relatively short debond lengths were studied ( l 1.5 r 4 r ), thereore the total length o the odel equal to L S 30 r d was used in calculations. Contact eleents were used on the iber and atrix suraces in the debonded region. The odel was eshed with esh reineents near the tip o the debond crac in order to ensure accuracy o stress and displaceent distributions which are needed or G calculations. Since the VCCT was used or G calculations, the inite eleent esh reineent was optiized or the chosen integration length. Fig.2 shows the detail o the FEM odel, including the length o integration dl d. In all calculations perored in this study the length o integration region was equal to dl d 1 r, which was selected based on analysis in [6,] and the accuracy was veriied coparing with boundary eleents in [14,18]. This value is assued to be suiciently sall (actually it should satisy condition dl d 0) and at the sae tie suiciently large to obtain accurate stress and displaceent distributions. Distribution o displaceent uz and shear stress rz were obtained by peroring post-processing (path operations) in ANSYS. Since non-axisyetric stress state was studied, pre-deined paths were generated along the whole circuerence o the iber/atrix interace. Fig.2. Detail o the FEM odel used or G calculations. F iber, M atrix, C eective coposite. To acilitate the path operations, the inite eleent esh was unior along the circuerence. Code coprising Eq.(8) in a discretized or was generated or convenience o calculations. Although the stress state close to the coposite 127

140 specien surace is not axisyetric, or siplicity we assue that the geoetry o the debond has axial syetry ( l d ( ) ). 4. Material properties Two dierent UD coposites were studied carbon iber/epoxy atrix coposite (CF/EP) and glass iber/epoxy atrix coposite (GF/EP). The properties o the coposite constituents are given in Table 1. Table 1. Elastic properties o constituents. Material E L E T G LT LT 23 [GPa] [GPa] [GPa] [-] [-] CF GF EP The thero-elastic constants o the undaaged CF/EP and GF/EP coposites (eective coposite phase properties) except the transverse shear odulus G 23 were calculated using the Concentric Cylinder Assebly (CCA) odel introduced by Hashin [11]. The transverse shear odulus was obtained ro the out-o-plane shear odulus calculated using Christensen s sel-consistent odel [12]. Only one iber volue raction equal to V 0. 5 was studied in this paper. Fiber radius was or all cases, unless stated dierently, equal to r 4. Debond growth siulations in this paper were perored or GF/EP coposite with interace atigue paraeters * and B taen ro [8]. The elastic properties o glass ibers and epoxy atrix in [8] were very siilar to those in Table Results and discussion Deterination o the stress state and the strain energy release rate using the VCCT was perored as described in Section 3. The presented G is calculated using Eq.(8), which includes integration over the angular coordinate. Thereore the presented G is an average over all angles and not a value corresponding to speciic angle. The usage o the average value is consistent with the used assuption that the crac ront is the circular independent on the debond length. Since the stress state in the used odels does not have axial syetry but we are using the average G it is o interest to present and to analyse data showing the angular dependence o stresses and displaceents in the crac ront region. In all cases unior axial displaceent u z 0.01 L s was applied at the end surace o the FEM odel (see Fig.2). Results in this Section are presented as unctions o noralized debond length l dn deined as l dn l d r Model 1 As described in Section 2, in Model 1, shown in Fig 1, we assue that the iber/resin unit is ro all sides (including the specien s surace direction) surrounded by the 128

141 hoogenized eective coposite. Figs. 3 and 4 show the strain energy release rate, G dependence on debond length or CF/EP and GF/EP coposites. Siilarly as in the axisyetric case analysed earlier in [6, 7] the strain energy release rate G or debond growth is the highest close to the iber brea (short debonds). With increasing debond length l d the G values approach to the value in the sel-siilar propagation region (sei-ininite debond and an ininite iber). In nuerical calculations the selsiilar growth region was not reached due to liited coputational resources balanced by needs o the 3-D proble with contact eleents and stress singularities. The range o debond lengths ro ld 1. 5 r to 4 r was studied. As it can be seen ro the trends in Figs. 3 and 4, this range o debond lengths corresponds to interaction region and a urther increase o debond length will result in saller values o G. Analyzing the dependency o G on iber distance ro the specien s surace, calculations were perored or several values o the iber depth paraeter l c. Calculations or CF/EP coposite were done or three values: l c = 10, 15 and 20 and the results are shown in Fig.3. It can be noted that G is consistently higher, when the iber is closer to the surace o the specien but the dierence is not large (less than 2%, see Fig.3). The asyptotic value sees to be larger or debonded ibers closer to the specien surace. Since the energy release rate is higher, the growth o the interace debond crac will be aster or the ibers, which are closer to the specien surace copared to the ibers in the bul o the coposite. Fro calculation results or CF/EP shown in Fig.3 it can be noted that there is no signiicant dierence between results or cases, when iber distance ro surace is l c = 15 or l c = 20. This distance l c = 15, at which the energy release rate G becoes alost independent o distance ro the UD specien surace indicates the transition ro the close-to-surace region to the bul o coposite region. Fig.3. Energy release rate G as a unction o the noralized debond length l dn or CF/EP according to Model 1. l c values are in. The results in Fig. 4 or GF/EP coposite shown or c l = 10 and 15 conir these trends. 129

142 Fig.4. Energy release rate G as a unction o the noralized debond length l dn or GF/EP according to Model 1. l c values are in. Observing Fig.4 the approaching to the respective asyptotic value sees to be slightly aster, i the distance l c ro the specien surace is saller. This trend is less pronounced in CF/EP case, see Fig.3. Fig.5. Shear stress rz distribution according to Model 1 in ront o the tip o the debond crac or CF/EP. Angular coordinate 180, l c values are in. Debond length l 1. 5 r. dn In the transition region and closer to the specien surace the stress distribution in the top and in the botto points o the interace ay be dierent. To analyze the interacial shear stress rz and axial displaceent u z dependence on the distance ro the coposite surace l c and on the angular coordinate, the results are presented in two dierent ors: a) dependence on the distance l c at =180 (this is the closest point to the specien surace); b) dependence on or ixed value o l c. 130

143 The dependence on l c at =180 is shown in Figs. 5 and 6 or CF/EP and GF/EP respectively. In this location there is a very sall (alost negligible in the presented scale) dependence on the distance l c within the observed range ro l c = 10 to 20. The dierence in stresses in this location does not exceed 1%. The dierence or <180 is even saller. Fig.6. Shear stress rz distribution according to Model 1 in ront o the tip o the debond crac or GF/EP. Angular coordinate 0, l c values are in. Debond length ldn 1. 5 r. To inspect the deviation ro axial syetry as the distance ro the specien surace decreases, the dependency o shear stress rz distribution on the angular coordinate was investigated. Fig.7. Shear stress rz distribution according to Model 1 in ront o the tip o the debond crac or CF/EP. The distance ro the surace l c 10. Debond length l dn r In Figs. 7 and 8 stress distributions at 3 dierent angular directions = 0, 90 and 180 are presented or CF/EP and GF/EP respectively. The results conir that the 131

144 dependence o the shear stress rz distribution on the angular coordinate is very wea, which eans that the axisyetric approxiation in this region still would be acceptable. The sae can be concluded ro analysis o displaceent distributions shown or CF/EP in Fig. 9. These conclusions are consistent with the calculated sall (2-3%) change o the G when l c changes ro 20 to 10 (see Figs.3 and 4). Fig.8. Shear stress rz distribution according to Model 1 in ront o the tip o the debond crac or GF/EP. Distance ro the surace l c 10. Debond length ldn 1. 5 r. Fig.9. Axial displaceent u z distribution According to Model 1 behind the tip o the debond crac or CF/EP. l c values are in. Debond length l 4 r. 5.2 Model 2 When the distance ro the coposite surace l c is too sall to have an additional iber on the top o the analysed debonded iber, the use o the eective coposite on the top o the iber/atrix unit assued in Model 1 is not justiied. It is ore probable that the unit is covered with a neat atrix region and not with the eective coposite. Thus the transition point ro Model 1 to Model 2 ay be estiated as lc r 2 r (neglecting the size o the atrix zone in the unit). According to this estiate the case 132 dn

145 with l c =10 analysed above using Model 1 should as well be analysed using Model 2. Fig.10. Energy release rate G as a unction o the noralized debond length l dn or CF/EP calculated using Model 2. l c values are in. The strain energy release rate G according to Model 2 was calculated in the range o l c ro 6 to 10. The results or CF/EP and GF/EP coposites are presented in Figs. 10 and 11 respectively. The trends are siilar as using Model 1 but it sees that the approaching to the asyptotic value with increasing debond length l dn is slower than according to Model1 (copare with Figs.3 and 4). It could be related to ore eicient stress transer when the debonded iber is surrounded by a stier eective coposite. Fig.11. Energy release rate G as a unction o the noralized debond length l dn or GF/EP calculated using Model 2. l c values are in. For both coposites the G values slightly increase, when the debonded iber is closer to the specien surace. However, siilarly as using Model 1 the dierence is very sall (1-2%). Coparing Figs. 3 and 4 with Figs. 10 and 11 one ay notice that 133

146 or the l c =10 case the range o G change according to both odels is dierent. It is larger using Model 1. This dierence is uch larger in CF/EP coposite case, where the eective coposite has uch larger longitudinal odulus than in the GF/EP coposite. The asyptotic G value or CF/EP coposite is uch lower, when Model 2 is used. For GF/EP coposite this dierence is uch saller. Fig.12. Shear stress rz distribution in ront o the debond crac tip or GF/EP calculated using Model 2. Debond length l 1. 5 r, l 6. The described changes are related to the change o the odel. Now we consider how according to Model 2 the distance ro the specien surace (thicness o the resin layer on the top o the iber/resin unit) aects the strain energy release rate due to the debond growth. dn c Fig.13. Displaceent distribution in ront o the debond crac tip or GF/EP calculated using Model 2. Debond length l 1. 5 r, l 6. According to Fig. 11 the G values are higher when the debonded iber is closer to the specien surace. In a uch saller extent this trend is observed also in Fig.10 or short debonds in CF/EP coposite. This trend is siilar to the observed using Model dn c

147 To analyze the deviations ro the axial syetry the shear stress distributions at the interace along the axial coordinate are presented in Fig.12 or the GF/EP coposite. The stress rz is signiicantly lower on the upper part o the interace which is closer to the specien surace. It is interesting to note that or angles up to 90 the angular dependence is negligible. The trend or the displaceent gap uz, shown in Fig.13, is siilar: the sliding is saller on the upper part o the interace and it is alost angle independent on the lower part o the interace. Fig.14. Shear stress rz distributions or GF/EP calculated using Model 2. 1) corresponds to l c =10, 2) corresponds to l c =6. Debond length ldn 1. 5 r. Fig.15. Displaceent distributions or GF/EP calculated using Model 2. 1) corresponds to l c =10, 2) corresponds to l c =6. Debond length l 1. 5 r. The angular dependence is increasing with approaching to the specien surace. It is deonstrated in Fig.14 or the shear stress and in Fig.15 or the sliding displaceent presenting distributions or GF/EP at l c =10 and or l c =6. It is iportant to notice that whereas the values on the upper part o the interace or l c =6 have dn 135

148 decreased, the values in the botto part are slightly higher. This explains why the average G in Fig.11 is slightly higher. Finally we use the calculated strain energy release rate presented in this Section to siulate the debond growth ro the iber brea in tension-tension atigue o UD coposite using expressions in Section 3.1 and the interace properties given in Section 4 or GF/EP coposite. In siulations the polynoial itting was applied to the calculated G data (the itting curves and the polynoials are shown in Fig. 4 or Model 1 and in Fig.11 or Model 2). ax Fig.16. Debond growth siulations or GF/EP coposite. r =8.5, z =1.76%, R=0.1. Initial debond o length equal to ldn 1. 5 r was introduced. In order to copare with SF siulations and experiental data ro [8], debond growth siulations in this paper were perored or UD coposite with radius equal to r =8.5, applied axial strain level ax z =1.76 % and with a load ratio between inial and axial applied strain in one cycle equal to R=0.1. It is nown that energy release rate G is proportional to the square o axial strain z. Also, G is proportional to the iber radius r. Thereore, the polynoial unctions shown in Figs.4 and 11 can be recalculated or any applied strain level and any iber radius. The siulated debond length dependence on the nuber o cycles is shown in Fig. 16. The data points ro SFF test and their it with identiied constants is also shown. The growth o the debond or the iber ost distant ro the surace ( l c =15 ) is the slowest. It was calculated using Model 1. For the debonded iber at the distance ro surace l c =10 the siulation was perored using Model 1, where the eective coposite is between surace, and the iber/atrix unit and also using Model 2, where the eective coposite is replaced by the atrix layer. Using Model 1 the 136

149 growth in this case is aster than using Model 2 (actually it gives the highest growth rate ro all considered). In the case o l c =10 the debond growth or short debonds is with the sae rate according to Model 1 and Model 2. With increasing debond length the growth slows down aster according to Model 2. The rate there or long debonds is governed by the asyptotic G value, which is lower, when the atrix covers the debonded iber (Model 2). According to Model 2 the debond around the iber closest to the specien surace ( l c =6 ) grows aster than the debond at l c =10. It is iportant to note that a) the debond growth rate is uch aster in the single iber ragentation test specien subjected to siilar loading than in the UD coposite; b) the slightly higher strain energy release rate or debonded ibers close to the surace leads to signiicantly higher debond growth rate. The presented data show that the SFF specien or siilar conigurations (or exaple broen ibers close to the UD coposite specien surace) subjected to cyclic loading can be suggested or accelerated testing and characterization o interaces in atigue. 6. Conclusions The iber/atrix interace crac (debond) growth ro distributed iber breas in high stress tension-tension cyclic loading o unidirectional coposites is the analyzed progressive icroechanis in this paper. This process is siulated by calculating the strain energy release rate nuerically in 3-D orulation and using a power law with respect to the strain energy release rate G change to describe the debond growth rate. Two odels were applied. In Model 1 the partially debonded iber/atrix cylindrical unit with a iber brea is surrounded ro all sides by an eective coposite with properties calculated with the concentric cylinder assebly odel and the Christensen s sel-consistent schee. This odel sees to be adequate, when the considered iber is not very close to the specien surace. In Model 2 the eective coposite was used around the iber/atrix unit except the region between the unit and the specien surace, where neat atrix was placed. This odel sees to be ore applicable or ibers very close to the surace. Calculations show that G is slightly higher, when the analyzed iber is closer to the specien surace. However, starting with a distance ro the surace equal to two iber diaeters the eect becoes negligible. It was ound coparing Model 1 and Model 2 that or short debonds G ro both odels is rather siilar. For longer debonds Model 1, in which the iber/atrix unit is separated ro the surace by atrix layer, gives lower values o the strain energy release rate. The debond growth with the nuber o cycles was siulated using the data or interace crac atigue resistance obtained in [8] using single iber ragentation tests specien in cyclic loading. Siulations show that debonds ro iber breas close to the specien surace grow uch aster than ro iber breas inside the coposite: the slightly higher G close to the surace leads to signiicantly higher debond growth rate. 137

150 Under siilar loading the debond growth rate in the single iber ragentation test specien is uch aster than in the UD coposite. This iplies that the SFF specien or siilar odel coposite or, or exaple, broen iber on the UD coposite specien surace subjected to cyclic loading can be suggested or accelerated testing and characterization o interaces in atigue. 7. Reerences 1. R. Talreja, A continuu echanics characterization o daage in coposite aterials. Proceedings o Royal Society o London 1981;A378: R. Talreja, Fatigue o Coposite Materials, Technoic Publishing Co, pp. 181, R. Talreja, C.V. Singh, Daage and Failure o Coposite Materials, Cabridge Univ Press, Z. Chi, T.-W. Chou, G. Shen, Deterination o single ibre strength distribution ro ibre bundle testings, Journal o Materials Science, vol.19(10), pp , A. Pupurs, J. Varna, Energy release rate based iber/atrix debond growth in atigue. Part I: Sel-siilar crac growth, Mechanics o Advanced Materials and Structures, In press, A. Pupurs, A. Krasniovs, J. Varna, Energy release rate based iber/atrix debond growth in atigue. Part : Debond growth analysis using Paris law, Mechanics o Advanced Materials and Structures, In press, A. Pupurs, J. Varna, Unidirectional coposite in echanical atigue: odelling debond growth ro ibre breas, Plastics Rubber and Coposites, vol.39, pp , A. Pupurs, S. Goutianos, P. Brøndsted, J. Varna, Interace debond crac growth in tension-tension cyclic loading o single iber polyer coposites, Coposites Part A, accepted, E.K. Gastedt, Eects o debonding and iber strength distribution on atigue daage propagation in carbon iber reinorced epoxy, Journal o Applied Polyer Science, vol. 76, pp , S. Goutianos, T. Peijs, Experiental and nuerical investigation into atigue daage echaniss in ultiibre icrocoposites, Plastics Rubber and Coposites, vol. 30, pp , Z. Hashin, Analysis o Coposite Materials a survey, Journal o Applied Mechanics, vol.50(3), pp , R.M. Christensen, K.H. Lo, Solutions or eective shear properties in three phase sphere and cylinder odels, Journal o the Mechanics and Physics o Solids, vol. 27(4), pp , E. Marlund, J. Varna, R.C. Neagu, E.K. Gastedt, Stiness o aligned wood iber coposites: eect o icrostructure and phase properties, Journal o Coposite Materials, vol. 42(22), pp , E. Graciani, V. Manti, F. París, J. Varna, Nuerical analysis o debond propagation in the Single Fibre Fragentation Test. Coposites Science and Technology, vol. 69(15-16), pp , B.W. Ki, J. Nairn, Experiental veriication o the eects o riction and residual stress on the analysis o interacial debonding and toughness in single iber coposites, Journal o Materials Science, vol. 37, pp ,

151 16. G.R. Irwin, Fracture, Handbuch der Physi, Berlin, Springer Verlag, vol.5, pp. 551, ANSYS Release 13.0, ANSYS Acadeic Research, ANSYS Inc., Canonsburg, Pennsylvania, A. Pupurs, J. Varna, Fracture Mechanics analysis o debond growth in single iber coposite under cyclic loading, Mechanics o Coposite Materials, vol.47(1), pp ,

152 140

153 Paper VI A. Pupurs, J. Varna, G. Lindbergh Modeling echanical stress and exoliation daage in carbon iber electrodes subjected to cyclic intercalation de-intercalation o lithiu ions, To be subitted to Journal o the Mechanics and Physics o Solids, 2012.

154

155 Modeling echanical stress and exoliation daage in carbon iber electrodes subjected to cyclic intercalation de-intercalation o lithiu ions. Andrejs Pupurs 1, Janis Varna 1, Göran Lindbergh 2 1 Departent o Engineering Sciences and Matheatics, Luleå University o Technology, SE , Luleå, Sweden 2 Departent o Cheical Engineering and Technology, Royal Institute o Technology, SE , Stochol, Sweden Abstract Gradients in lithiu ion concentration distribution in a carbon iber are accopanied by non-unior iber swelling leading to developent o echanical stresses. During lithiu deintercalation these stresses ay lead to initiation and growth o radial cracs in the iber. The ollowing cycle o intercalation ay result in arc-shaped cracs deviating ro the tip o the radial cracs. These phenoena decrease the echanical properties o ibers i used in structural batteries and reduce the charging properties o the battery by decreased diusivity o lithiu ions and by exoliating layers on the iber surace. The crac propagation and possible daage evolution scenarios are analyzed using linear elastic racture echanics. The crac geoetry dependent ion concentration distributions and the elastic stress distributions were ound using inite eleent sotware ANSYS. 1. Introduction One o the aterials with a potential or use as electrodes in lithiu-ion batteries is carbon iber. In uture structural batteries these carbon ibers will also have a load bearing unction. Fiber degradation ay also aect the ion diusivity and the nuber o charge-discharge cycles with high energy eiciency. To ensure the echanical durability o this type o batteries the echanical degradation echaniss in ibers during service lie have to be analyzed to develop guidelines or aterial selection. During the intercalation process lithiu ions enter the iber via the interace by diusion process, see Fig. 1a. In the beginning the ion concentration has a gradient with high concentration at the iber surace. The increase o ion concentration in the aterial leads to anisotropic voluetric changes (swelling) in the transversally isotropic carbon iber characterized by two swelling coeicients 3 and 1 in axial and radial directions respectively. Due to the concentration gradient the outer region o the iber would have larger ree swelling strains than the inner region. Since displaceent continuity has to be satisied, internal stresses appear in the iber, see the 1 st row in Fig.1b: radial stresses, r, are positive (the outer region by tepting to expand ore applies radial tractions to the inner part) whereas hoop stresses,, are negative in the outer region and positive in the inner region (attept o the outer region to expand in direction is constrained by the inner region which due to lower concentration o ions does not expand as uch). The only possible (but not 143

156 very realistic as ollows ro analysis presented urther) daage ode in this step is arc crac oration governed by the radial strength o the iber. At the end o the diusion process the concentration gradient decays and so do all echanical stresses. The radial diusion process in the iber as well as in spherical particles has been analyzed previously, or exaple in [1-3] using series expansion obtaining concentration distribution as a unction o the radial coordinate. The stress distribution proble due to described echanis was solved analytically in a closed or. a) b) 1) 2) 3) Fig.1. Lithiu ion diusion in carbon iber: a) scheatic geoetry; b) possible odes o echanical daage. During the deintercalation, see the 2 nd row in Fig. 1b), the outer region loses the ions irst and would shrin which is constrained by the inner region which is still in the swelled state. Equilibriu is reached with outer regions being under tensile hoop stresses (copressive in the inner regions) and copressive radial stresses. Assuing a ully reversible charge, at the end o the deintercalation the lithiu ion concentration goes to zero and the iber is stress ree. The presented scheatic description is correct under assuption that the surrounding edius have zero elastic properties and do not apply any constraint to the swelling and shrining o the iber. This ay be true in case o a iber surrounded by liquid electrolyte but ay becoe rather dierent, i the electrolyte is solid and this unit is ebedded in other solid aterials. Using the calculated stress distributions, strength based ailure criteria have been applied in [1,2] to analyze possible sites, tie instants and echaniss o ailure. These criteria are applicable to identiy the possible daage initiation but the daage developent has to be approached by racture echanics ethods: sall crac has to be introduced and its growth analyzed by evaluating the available and the required energies or creation o new crac surace. In repeated charging and discharging cycles we ay expect radial crac initiation and growth during discharging (deintercalation) as shown on the 2 nd row o Fig. 1b). It can start in quasi-static anner ro the surace, being arrested when approaching the central part o the iber. The radial growth can continue also in a atigue anner during deintercalation in repeating cycles. 144

157 In the irst approxiation we ay assue that radial cracs do not inluence the ion diusion because it is in radial direction only. Higher order approxiation ay consider the radial crac as a pathway or the electrolyte iniltration eaning that the ions could also intercalate the iber ro the crac surace. Such possibility is negligible, i the electrolyte is solid or i the viscosity o the electrolyte is high because the radial crac opening is extreely sall. In the initial stage o each intercalation, when the radial tensile stress is high, arc cracs can initiate ro the tip o the radial crac and propagate in the circuerential direction as shown in the 3 rd row in Fig 1b. Arc cracs aect the diusion, shielding the ion oveent by their open suraces. In such case the diusion process does not have the axial syetry and cannot be ound analytically by series expansion. Obviously, the stress distributions also becoe dependent on the hoop coordinate and can only be ound nuerically. The diusion process can be very coplex, generally speaing, the syste to analyze contains an electrolyte, where the ion concentration can change and have a gradient and the iber/electrolyte syste ay be surrounded by other solid aterials with certain echanical properties and their own ion diusion paraeters. The objective o the presented paper is to analyze the lithiu ion diusion in the carbon iber by nuerically solving the tie and coordinate dependent diusion proble. The static elasticity proble was solved at selected tie instants corresponding to certain concentration distribution, the stresses are analyzed. Radial and arc cracs were introduced and trends in their growth were analyzed using Virtual Crac Closure Technique (VCTT) well nown in Fracture Mechanics [4]. In this paper we sipliy the proble by considering a single iber in an ininite source o ions. The ocus is on internal stresses in the iber due to concentration gradients and not to the echanical constraint o surrounding aterials which are parts o the structure o the battery. We assue a very low transer resistance o the electrolyte. Hence, the ion concentration in the electrolyte does not change during intercalation or de-intercalation. In such case theral analogy can be used replacing the diusion proble by heat expansion proble. Both theral and corresponding echanical probles were solved using FEM code ANSYS [5]. 2. Theoretical bacground 2.1 Ion concentration in the carbon iber We consider an ininite electrolyte with uniorly distributed carbon ibers characterized by iber content V. This syste can be represented by a cylindrical unit cell with a long iber surrounded by an electrolyte which ay have also nonzero elastic and swelling constants. During the intercalation lithiu ions diuse into the iber and their distribution along the radial coordinate can be described by concentration distribution, which ollows diusion equation c DC t (1) In (1) C is the relative ion concentration in the iber with respect to available sites, is Laplace operator and D is diusion coeicient. In the particular case o a long 145

158 iber the diusion is in-plane and the concentration is a unction o r and Thereore r (2) 2 2 r r r r I the iber is undaaged or i cracs are in radial direction, the diusion is in radial direction only and the concentration does not depend on and the second ter in (2) is zero. In such case the ion concentration dependence on the radial coordinate has a closed or analytical solution. Eq.(1) is a particular case o a ore general equation [6,7]. The or (1) is obtained under assuption that lithiu-lithiu interactions (eect o intercalate activity coeicients 1 ) inside the iber ay be neglected leading to approxiation d(ln 1 ) 0 d(ln C) (3) At the iber surace r r we require that the ion lux N is related to the local current density in the electrolyte i N (4) F The lux at the iber surace according to [3,7] under assuption (3) is N Dc0 C (5) In (5) is the gradient operator, which at the iber surace has only the radial coponent r (6) r In (6) c 0 is the total site concentration in the host aterial (the axiu concentration o lithiu ions inside the iber, i all sites would be occupied). For the current low we use expression ro [1] where i F Ce ( 1 ) ( ) ( 1 ) ( V U V V U V ) C e (7) 1 (8) c a In (8) c and a are cathodic and anodic rate constants, is the syetry actor, 146

159 F / RgT with F being Faraday s constant, Rg - universal gas constant and T - teperature. These expressions are valid under assuption that the electrolyte is well ixed and the transport resistance in the electrolyte is zero (no lithiu ion concentration gradients in the electrolyte). Substituting (5) and (7) in (4) we obtain [1] boundary condition at the iber surace ' ' Dc C ( )( C C ) 0 at r r (9) Here new constants are introduced as ollows a c R ' ' ( V U ) c, c e CR (10) ' ' ' (1 ) ( V U ) a e c a Paraeter C R has a eaning o saturation concentration o ions in the iber. Introducing noralized coordinate and tie r td x, (11) 2 r the diusion Eq.(1) and the boundary condition (9) can be written in diensionless or r C x C (12) ' ' c a xc B( C CR ) 0 at r r B r (13) Dc Operators with index x are deined according to (2), (6) replacing r by the noralized radial coordinate x. By this procedure all unnown paraeters listed above are reduced to one unnown paraeter B (Biot constant). Varying this paraeter ro zero to ininity we can ' ' cover all possible cobinations o paraeters, c, a, D, c 0 and r. As an initial condition or intercalation we assue that at t 0 the concentration is zero Since the diusion proble is linear, we can assue C R 1 when calculating the concentration distribution during intercalation. Concentration distribution or other C R values is obtained by siple ultiplication. During deintercalation the electrocheical paraeters change thus aecting the C R and B values. As an extree case we assue in calculations C R 0 and vary the Biot constant B in a large region. The initial condition in this case is C ( r, t 0) 0 (14)

160 C ( r, t 0) 1 (15) 2.2 Stress distribution in the carbon iber The equilibriu equations or this loading case with all shear stress coponents equal to zero is r r 1 r 0 r r r 1 r r r r 2 r 0 (16) (17) u r u r, r 1 u u u 1 ur, r (18) r r r r r r const (19) z z 0 Plane strain solution is obtained, i z0 0, generalized plane strain solution corresponds to the case, when const obtained ro requireent that the axial orce is equal to zero. z Stress strain relationships or transversally isotropic iber (index 1,2 and 3 correspond to r, and z directions) are r 1c0C z r (20) E E E c0C z r (21) E E E z 3c0C z r (22) E E E In (20)-(22) i, i r,, z are swelling coeicients o the iber in the ain directions. Obviously the diusion proble and the elastic proble are decoupled. We can irst ind the concentration distribution using (12), (13). The concentration distribution does not depend on the elastic stress state. The concentration distribution C ( r,, ) is used as an input in the elastic proble, which can be solved at any arbitrary instant o. The equations (16)-(22) together with proper boundary conditions on the iber surace have to be used to ind the stress-strain state. I the electrolyte does not constrain the deoration o the iber (the electrolyte elastic odulus is very low) the traction ree boundary conditions ay be applied at r r (or x 1 using the noralized coordinates). I the elastic odulus o the electrolyte is high enough to constrain the iber deoration, we have to solve a two body proble with stress and displaceent continuity conditions between both doains

161 2.3 Thero-echanical analogy As described above, when solving the diusion proble we can select C R 1. In result the ion concentration in the iber will vary between 0 and 1. The concentration proiles or other values o C R are proportional to calculated and can be obtained by siple ultiplication. Siilar conclusions apply to the elastic stress proble: elastic stresses or cases when C R 1 can be obtained by ultiplying the obtained stress state by C R. Peroring paraetric analysis we do not need to consider separately paraeter C R and paraeters i, i r,, z. In the stress expressions the ion concentration C is always ultiplied by swelling coeicients i, i r,, z and by c 0. The atheatical description o the concentration distribution (12), (13) is the sae as or heat conduction proble with convection boundary conditions: paraeter B has the eaning o the heat transer coeicient, C is analogous to the teperature distribution and C R is the value o teperature in the surrounding ediu, see [8]. This analogy can be used to eploy coercial inite eleent (FE) codes (ANSYS [5] in the presented paper) to ind concentration and stress distribution. In ANSYS the theral-diusion proble analogy can be used only in probles with single doain or i the condition in or (13) is applied to the outer boundary. In ultiple doain probles the interace conditions between doains or teperature cannot be orulated in ters o (13): teperature in the contact points o two bodies is with zero teperature dierence. This liitation does not aect the current investigation where the diusion proble is solved in a single doain consisting o a iber with conditions (13) applied on the iber surace. 2.4 Fracture echanics approach By stress analysis we can identiy the ost critical tie instants and positions, where the iber daage ay initiate as aected by paraeters and loading cases. To ollow the growth o a certain daage entity (crac) we have to introduce it in the odel as a geoetrical surace with certain length and orientation. Ater that the energy based criteria can be used to analyze the growth conditions quantitatively, see Fig. 2 at the end o this Section or ore details. In Fig.3a a part o the iber with a radial crac o length l r is shown ( l r 0,1 ). As described above during deintercalation the crac is open due to positive hoop stresses and its urther growth in the radial direction by dl r is possible. The propagation is in pure Mode I (opening ode) and it is governed by Mode I energy release rate G I (x). In Fig.3b an arc crac o length l has initiated ro the tip o the radial crac o length l r and conditions or its urther growth in the hoop direction by dl have to be analyzed. In this case the axial syetry is lost and the crac propagation is under ixed ode conditions (both shear stresses and radial stresses are singular at the crac tip and relative crac ace displaceent has opening as well as sliding coponents). In quasi-static conditions the used racture echanics criterion or crac propagation is 149

162 G G c (23) In (23) G is the total energy release rate and G c is its critical value, which is aterial constant. It has to be noted that: a) G ay be ode ixity dependent; b) due to anisotropic iber structure G ay be dierent or crac growth in the radial and in the hoop directions (without regarding the stress state dierences). In atigue loading one can assue that the crac growth ay be governed by a power law siilar to Paris law in atigue o etals dl dn A(G) (24) In (24) A, are aterial constants, G G ax Gin is the change o the strain energy release rate between state where it is the highest and the state where it has iniu. The strain energy release rate contains two coponents G I and G corresponding to the opening and sliding odes respectively G G I G (25) Each o these coponents can be calculated using the VCCT, which is based on the assuption that the energy required to create a unit o a new crac surace, is equal to the wor to close this unit surace. Using the Crac Closure technique (CCT) the wor is calculated using stresses in the ront o a crac o size l and the relative crac ace displaceents are taen ro crac o length l dl. Fig.2 scheatically shows the distribution o stresses and displaceents in the case o opening ode. To avoid necessity to peror calculations or two slightly dierent crac lengths (one or stresses and one with slightly longer crac or displaceents) in the VCCT it is assued that the displaceent distribution at the tip o the crac which is ininitesially longer is the sae as or the initial crac size. The calculation expression or Mode I is ldl 1 G I ( l) li ut ( y dl) T ( y) dy (26) dl 0 2dl In (26) index T is used or the direction transverse to the crac surace, coordinate y is along the crac propagation direction (see Fig.2). l Fig.2. Scheatic representation o stress and displaceent distributions or Mode I propagation. 150

163 3. FEM odel 3.1 Model and the calculation procedure Based on the theral analogy described in Section 2.3, FEM odels were created to calculate transient ion concentration and echanical stress distributions in an undaaged and in a daaged carbon iber electrode. FEM sotware code ANSYS version 13.0 [5] was used to peror all calculations. As deonstrated in Fig.3 to Fig.5 a two phase (iber and atrix electrolyte) odel was generated to account or the possible echanical interaction between the iber and the surrounding electrolyte. The odels shown in Fig.3 to Fig.5 are 2-D transversal representation o the proble. For undaaged iber case an equal alternative to the transversal odel described here would be a 2-D axisyetric odel (with iber axis being the center axis). However, in this study we also account or the daage (cracs) in the iber due to which the axial syetry o the proble is lost. The 2-D eleents were assigned to plane strain behavior. The ion concentration-echanical stress proble was solved in two steps. First the transient ion concentration proble was solved by generating the carbon iber phase and applying the convection boundary condition (Eq (13)) directly on the outer iber surace. In other words, solving the diusion proble the eect o the electrolyte is presented by boundary condition (13). Application o this convection boundary condition iplies that the constant B, and the saturation concentration in the iber C R are speciied (Eq.(13)). As explained beore, the paraetric analysis in this study is reduced to one paraeter, B, thereore the values o other paraeters necessary to peror the theral analysis such as speciic heat and density were set equal to 1. According to the theoretical expressions in Section 2, the results depend only on one paraeter, that is the Biot nuber B. I we also assue that the cathodic and anodic rate constants ' c and ' a reain unchanged, the change in B according to (13) is caused by dierent values o diusivity D and the tie scales, see (11) is dierent (the diusion proble is solved using the noralized tie ). This becoes iportant i we want to copare results between several cases at distinct tie instants. According to Eq.(11) the relation between noralized ties or cases with D 1 and D 2 is 1 ( D2 D1 ) 2. This relationship will be used to recalculate all results to the sae noralized tie. The real tie t can be calculated only i the real value o diusion coeicient D is nown. The initial conditions on all nodes o the iber were speciied by (14) or (15) or intercalation or deintercalation respectively. For each case, the ion concentration distribution at selected tie instants was saved in the results database in ANSYS. In the second step the echanical proble was solved. The atrix phase was added to the iber phase and the elastic properties o both phases were deined. Stress distributions were ound corresponding to selected instants o tie with given concentration distribution. 3.2 Model without daage Taing the advantage o the syetry o the proble, only ¼ o the total transverse FEM odel is needed to peror calculations. 151

164 Fig.3. FEM odel. Two cases were analyzed: Case 1 with the Biot constant B =500 and Case 2 with B =5. The relation between diusion coeicients or Cases 1 and 2 is then the ollowing: D D FEM odel with radial cracs According to the discussion in Introduction, the irst daage in the iber ay be in a or o radial cracs oring at the surace o the iber. Fig.4 shows the geoetry o the FEM odel with radial crac o length l r. Dependent on the stiness o the surrounding electrolyte the crac at the iber surace can be open (surace crac) or closed (the case o a crac closed at the iber/electrolyte interace is shown in Fig. 4b). The opening o the radial crac close to the interace between the iber and atrix ( r r ) depends on the stiness o the atrix, E. It approaches to the opening o a surace crac when atrix odulus approaches to zero. The FEM odel was used to calculate the energy release rate related to the growth o this crac towards the center o iber. The crac ace position corresponds to the angular coordinate o / 4. Due to the syetry conditions on the horizontal and on the vertical axis the iber in this odel has our cracs. Interaction between the is not investigated in this paper. Appropriate esh reineent was used in vicinity o the crac tip as shown in the detail in Fig.4. The representation o the crac geoetry in Fig.4b is scheatic and it illustrates the deored state. 152

165 Fig.4. a) FEM odel or calculating the radial crac growth related energy release rate b) scheatical representation o a radial crac. Since the ion diusion is in radial direction, it is not aected by the presence o the radial crac. When solving the elastic proble contact eleents were generated on the crac suraces to prevent echanical interpenetration. The energy release rate was calculated using the VCCT based routine ipleented in ANSYS version 13.0 [5]. For 2-D geoetries, this routine allows calculation o Mode I and Mode energy release rate. The routine requires deinition o a local coordinate syste at the tip o the crac speciying the crac extension direction and the crac plane noral. Since the radial crac shape in the FEM odel is straight, a Cartesian syste was deined as the local crac tip coordinate syste. 3.4 FEM odel with arc cracs Following the scenario previously described in the Introduction, arc cracs ay delect ro the radial cracs. FEM odel that was used or calculating the energy release rate related to growth o the arc cracs is shown in Fig.5. As shown in the detail in Fig.5b the length o the previously ored radial crac is l r, its angular coordinate is / 4, while l is the arc length o the arc crac, which is assued to grow in circuerential anner along the arc with radius ( r lr ), syetrically with respect to the radial crac. Due to applied syetry conditions the odel corresponds to our radial cracs with arc branches. As in Fig.4b, also in Fig 5b the representation o the crac geoetry is scheatic and illustrates the deored state. Contact eleents were generated on all crac suraces including the initial radial crac surace to prevent interpenetration. As in the case o radial cracs (see Section 3.3.), the calculation o the energy release rate or arc cracs was also perored using the VCCT based routine ipleented in ANSYS. In this case, since the crac is in the shape o an arc, a cylindrical coordinate syste was deined as the local crac tip coordinate syste. 153

166 Fig.5. a) FEM odel or calculating arc crac growth related energy release rate, b) scheatic representation o an arc crac. 4.Results and discussion 4.1 Input paraeters The input values o elastic and swelling paraeters or the reerence case are shown in Table 1. Table 1. Elastic properties o constituents or the reerence case calculation. c 0 C c 0 C E E 3 E 1 G R R [GPa] [GPa] [GPa] [-] [-] [-] [-] [MPa] [-] Subscript in Table 1 denotes iber properties. Apart ro the reerence case, the stress distributions were also calculated or a case with atrix elastic odulus equal to E = 100 MPa. The axial ree swelling strain in Table 1 was obtained by extrapolating the data ro [9]. In [9] the swelling strain 3 c 0 CR o approxiately was experientally deterined and it corresponded to easured capacity o 135Ah/g at 1 hour charge. In [10] it was shown that or PAN based IMS65 carbon iber electrodes the capacity can be increased by up to 3 ties when charging slower, eaning that ore lithiu ions can be intercalated leading to higher swelling. Thus, based on these values, the longitudinal swelling coeicient was assued 3 c 0 CR = to account or 3 ties higher axial swelling than easured in [9]. Swelling in transversal direction was assued 1 c 0 CR =0.01. Although atoic scale experiental data or expansion o interlayer distance in carbon ibers due to lithiu ion intercalation are available (see or exaple [11] using X-Ray diraction), the acroscopic transverse swelling coeicient or carbon ibers has not yet been presented in the literature.