Energy criterion for modelling damage evolution in cross-ply composite laminates

Transcription

1 Energy riterion for modelling damage evolution in ross-ply omposite laminates D.T.. Katerelos, J. Varna, C. aliotis To ite this version: D.T.. Katerelos, J. Varna, C. aliotis. Energy riterion for modelling damage evolution in rossply omposite laminates. Composites Siene and Tehnology, Elsevier, 28, 68 (12), pp <1.116/j.ompsiteh >. <hal > HAL Id: hal Submitted on 9 Jul 21 HAL is a multi-disiplinary open aess arhive for the deposit and dissemination of sientifi researh douments, whether they are published or not. The douments may ome from teahing and researh institutions in Frane or abroad, or from publi or private researh enters. L arhive ouverte pluridisiplinaire HAL, est destinée au dépôt et à la diffusion de douments sientifiques de niveau reherhe, publiés ou non, émanant des établissements d enseignement et de reherhe français ou étrangers, des laboratoires publis ou privés.

2 Aepted Manusript Energy riterion for modelling damage evolution in ross-ply omposite laminates D.T.. Katerelos, J. Varna, C. aliotis PII: S (7)367-3 DOI: 1.116/j.ompsiteh Referene: CSTE 3838 To appear in: Composites Siene and Tehnology Reeived Date: 23 April 27 Revised Date: 16 September 27 Aepted Date: 24 September 27 Please ite this artile as: Katerelos, D.T.., Varna, J., aliotis, C., Energy riterion for modelling damage evolution in ross-ply omposite laminates, Composites Siene and Tehnology (27), doi: 1.116/j.ompsiteh This is a PDF file of an unedited manusript that has been aepted for publiation. As a servie to our ustomers we are providing this early version of the manusript. The manusript will undergo opyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the prodution proess errors may be disovered whih ould affet the ontent, and all legal dislaimers that apply to the journal pertain.

3 ENERY CRITERION FOR MODELLIN DAMAE EVOLUTION IN CROSS-PLY COMPOSITE LAMINATES D. T.. Katerelos 1, 2*, J. Varna 1 and C. aliotis 2 1 Division of Polymer Engineering, Department of Applied Physis and Mehanial Engineering, Luleå University of Tehnology, SE , Luleå, Sweden 2 FORTH/ICE-HT, Stadiou str. Platani, Patras, P.O. Box 1414, R-265 4, reee * katerel@pathfinder.gr tel: +46() , fax: +46() Abstrat The energy dissipated in ross-ply laminates during loading-unloading loops is obtained from stress-strain urves for ross-ply laminates and used in an energy based approah to predit the development of matrix raking. The dissipated energy is orrelated to the rak density growth data reorded for a referene laminate. The ritial strain energy release rate, obtained in this way is inreasing with the applied strain. This phenomenon reflets the statistial nature of distribution in the 9-layer: the first raks (lower strain) develop in positions with lower frature toughness. The obtained data are in a good agreement with frature toughness data obtained using LEFM based ompliane alibration model in whih the stiffness hange with inreasing strain is used. Finally the matrix raking development is suessfully simulated using in the LEFM model, the data for ritial strain energy release rate and an earlier derived stiffness-rak density relationship. It has been demonstrated that knowing the laminates geometry and measuring the laminate stiffness redution with strain or (alternatively measuring the dissipated energy) the damage evolution may be simulated, thus reduing the neessity for optial observations to validation only. 1

4 Keywords: A. Polymer-matrix omposites (PMCs); B. Matrix raking; C. Damage mehanis; C. Frature mehanis ; D. Raman spetrosopy Introdution Damage Mehanis is the field intensively developing during the last deades and providing tools for studies of omposite materials behaviour after damage initiation. For the damage initiation predition appropriate strength based riteria ould be used. Designing strutural elements for funtionality in the presene of damage, the above sequene has to be aounted for when struture s durability is assessed. Damage in laminated omposite materials develops in three subsequent and mostly interlaed forms: matrix raking (intralaminar damage), delaminations (interlaminar damage) and fibres failure (laminates ollapse). The interfaes between the aforementioned damage modes are not well defined, that is matrix raking may be aompanied with loal delaminations and/or fibres breakage. However, intralaminar, or as it is often alled, transverse matrix raking in plies oriented at angles different than the loading axes is onsidered to appear first. Therefore, during the last thirty years it has been the main subjet of studies for many researhers [1-5]. The rak density and damage entity harateristis in terms of damage vetors and tensors, rak opening displaements et were used to quantify the effet of damage on the overall behaviour of laminates [1-7]. Modelling the initiation and development of matrix raking in a omposite system starts with the evaluation of the stresses distributions within the laminate. In the reviews [1-5] all the main different approahes for the stress distribution determination ould be found. Sine the internal stress state is quantified the intralaminar raking evolution ould be modelled. The models proposed may be ategorized, aording to their basi approahes, into two main groups: strength based models and frature mehanis based models; see 2

5 review [4]. Most of the stress based models aount for the strength distribution within the laminate and the probability of loal failure. Among physial quantities, used to haraterize miro damage development in laminates due to thermo-mehanial loading, the most often used is the ritial strain energy release rate (often erroneously alled frature toughness). It is ommonly onsidered as a material onstant; however, an appropriate test for its determination is still an open question. In stati rak growth problems, the ritial strain energy release rate for the rak growth is either appropriately hosen in order to satisfy experimental damage evolution data [3, 14], or derived from raking data and using the stress state provided analytially [15-2] or numerially (finite elements) [21]. The attempts to use measured laminate Young s modulus hanges or the measured dissipated energy are more rarely desribed in the literature. The strain energy release rate,, is usually alulated analyzing the stress state hange with inreasing rak density or using the hange in the alulated elasti modulus of the laminate. Together with independently obtained it is used to model damage evolution in omposite laminates under both stati and fatigue loading. Monitoring rak growth under fatigue has been presented by several researhers [9-13], while detailed review an be found in [4]. The is used in the Paris power law riterion for individual rak growth during yli loading [27]. As a rak initiation riterion for omposite laminates under stati flexure loading, has been used by Smith and Ogin [22-23]. One example of an approah whih inorporates probabilities in the frature mehanis analysis is the early analytial model proposed by Laws and Dvorak [8]. Statistial Wang et al; see for example [27]. distribution was extensively used by In the present paper the ritial strain energy release rate for two glass fibre reinfored epoxy omposite systems is derived using the dissipated energy obtained from stress - strain urves reorded during an ordinary loading-unloading tensile test with inreasing maximum and 3

6 using the observed rak density relationship with the applied strain for a referene laminate. As an alternative method for obtained determination the stiffness redution with strain is used. The values strongly depend on the maximum strain level in the loading loop whih reflets the statistial nature of the frature toughness distribution along the 9-ply: the values for lower strains orrespond to raks in the weakest loations whereas the new raks appearing at higher strains require more energy. Finally, theoretial preditions of the rak density as a funtion of applied strain are performed using the obtained and an analytial model desribing the longitudinal modulus degradation as a funtion of rak density [26]. Experimental Two ross-ply glass fibres reinfored omposite systems (FRP) were used for the dissipated energy alulations. The first (F1) was manufatured using a modified frame-winding tehnique. Detailed desription of the proedure an be found in the literature [6-7]. A steel frame welded at 9 was used for the glass fibres winding with the inner layer wound firstly. Then, the layer was wound. Tensile oupons were ut from the laminates aording to ASTM D339 standard. A Shell Epiote 828 resin was used and ured with nadi methylanhydride and aelerator K61B in the ratio 1:6:4. The laminate was ured between thik glass plates under 8-1 bar pressure for 3h at 1 C, followed by a post-ure at 15 C for 3h. The fibre volume fration of the thus produed plates was 63 %, the laminate ply thikness was.64-mm for the plies and.62-mm for the 9 plies. In the ase of the seond omposite system (F2) three different laminates with different ply thiknesses were onsidered. They were manufatured using a vauum bag tehnique from unidiretional prepregs with the ommerial ode VICOTEX NVE 913/28%/192/EC9756. The laminates were ured at 9 C for 3 min followed by post uring at 12 C for 6 min under 3-7 bar pressure. Three laminates staking sequenes were 4

7 manufatured, [/9 4 ] s (F2_1-4), [/9 2 ] s (F2_1-2) and [ 2 /9 2 ] s, (F2_2-2) with an average thikness of 1.35,.8 and 1.4 mm, respetively. Speimens were ut aording to the ASTM D339 standard in the reinforement diretion of the outer layers. Speimen edges were polished in order to eliminate any defets that may appear during sawing. It has been proved [24] that edge polishing is the ause for rapid rak propagation, thus the development of raks overing the full speimen width is expeted. Details on the F2 material manufaturing an be found in [25]. F1 laminates were subjeted to a repeated mehanial tensile testing for the rak pattern development. The mehanial tensile loading was applied through an MTS 858 Mini Bionix tester [6]. The loading rate was.1 mm/min to ahieve ontrolled and stable rak growth. An eletrial strain gauge has been attahed on the speimen in order to ontrol the resulting strain. The speimen was loaded up to a ertain strain level and unloaded to zero stress level. Then the proedure was repeated for higher strain levels until reahing the so alled rak saturation point. The stress-strain urves thus produed are presented in Fig. 2a. Crak density at eah strain level, see the transverse rak appearane in Fig 1, was monitored using the tehnique of laser Raman spetrosopy [6-7]. The initial Young s modulus of the laminate was alulated from the first stress-strain urve provided by the attahed strain gauge and the MTS load ell. F2 material laminates were tested under quasi-stati stepwise tensile loading with an applied displaement rate of 2mm/min [25]. Loading was periodially interrupted and the speimens were unloaded for measuring the rak density. Crak density was monitored by optial means in transmitted light, whih was possible due to the partial transpareny of the materials [25]. The applied strain in the loading diretion was measured by an extensometer. The Young s modulus at eah loading step was derived from the orresponding unloading stress-strain urve within the.1% to.3% axial strain range [25]. 5

8 The two different omposite systems subjeted to rather different strain rates were used to demonstrate that the suggested methodology is appliable in wide range of materials behaviour. The strain rate for all laminates belonging to same material system was kept onstant. Thus the possible effets of more brittle material behaviour at high strain rates and the reep rupture type of effets at low strain rate are inluded in orresponding values of Due to these possible additional effets no omparison of both materials is made.. Modelling In a displaement ontrolled test the ritial strain energy release rate,, is defined by the well known equation U - = A = onstant ( ) (1) where U is the strain energy of the speimen, while A is the total rak surfae area formed by all raks, that is A = N w h 9 (2) where N is the number of raks formed, w is the speimen width and h 9 is the 9 -layer thikness. The strain energy of the speimen (ignoring thermal stresses whih are not large in a glass fibre omposite laminates) an be written as U = E x 2 2 V (3) E x is the damaged laminate longitudinal Young s modulus, is the applied strain and V is the speimen volume V = L w h (4) L is the speimen length and h its thikness. Substituting eqs (2)-(4) into eq (1) the following expression for an be extrated 6

9 1 h E ( ) 2 x - L = 2 h9 N (5) Crak density,, is defined as the ratio of the total number of raks within a length over the length. Thus for the speimen with length, L, it is N = L (6) Combining eqs (5) and (6) is expressed as a funtion of rak density 1 h E ( ) 2 x - = 2 h9 (7) It has been shown [26] that the dependene of laminate longitudinal Young s modulus, E x, on rak density in the damaged ross-ply laminate is of the form E x = 1 + A E (8) ρ ρ u ( ) where E is undamaged laminate longitudinal Young s modulus, A is a onstant whih value depends on the model used (ex. shear lag) and u() is average rak opening displaement normalized with respet to the far-field stress in the 9-layer. We assume, for the sake of simpliity, that the normalized rak opening does not depend on the distane between raks u ( ρ ) = u (9) where the value of u an be alulated by any existing stress model. Atually u is not a onstant and it dereases at high rak density when the interation between raks is notieable. In this paper the model proposed by Lundmark and Varna [26] is used. Differentiating eq (8) and taking into aount eq (9) the partial derivative of the E x with respet to is alulated. Then eq (7) beomes 1 h E 2 h 1 + A u ( ) 2 A u 2 = 9 ( ) (1) whih an be rewritten using eq (8) as 7

10 2 2 x 9 ACCEPTED MANUSCRIPT 1 h A u ε E = ( ε ) (11) 2 h E The onstant A as alulated from [26] is given by A = E 2 h h T 9 9 E h h (12) Aording to the studies performed by Lundmark and Varna [26] the rak opening displaement in an internal layer an be represented by the following power law u = A + B m m n m E T E L (13) where the onstants A m, B m and n m are given by following empirial forms A =.52 m h9-2 h B m = h 2 h h 9 9 n m = h 2 h (14) Results - Disussion A stop-and-go proedure was employed in the rak forming experimental investigation on both omposite systems. The speimen was loaded up to a ertain strain level where new raks formed, while the applied strain was ontrolled by the strain gauge attahed on one side of the speimen (F1) or the extensometer along the loading diretion (F2). Following, they were unloaded to zero stress, while the residual strain was reorded and the rak pattern was monitored. In the ase of F1 laminate the rak pattern was mapped indiretly through the use of laser Raman spetrosopy for measuring the strain arising within the lamina due to raking. Details on this method have been referred in previously published works [6-7]. The rak pattern arising in the F2 laminates at all strain levels and for all types of speimens used was monitored using an optial mirosope. This proedure 8

11 was repeated until the rak forming mode of damage was ompleted by reahing the soalled saturation point. The stress-strain urves are presented in Figure 2a-d. The first step in the experimental ritial strain energy release rate determination is the alulation of the dissipated energy, U d, at eah loading-unloading yle. By integrating the loading stress-strain urve the energy, U l, given to the material is alulated, whih inludes both the strain energy and the additional work performed on additional displaements due to inrease of the total rak area. The integration of the unloading urve orresponds to the energy returned, U u. The differene between these two values represents the total dissipated energy, U d, in eah loading yle U d = U l - U u (15) The dissipated energy differs from the mehanial strain energy U introdued in the previous setion. It inludes also visoelasti effets as well as redution of strain energy related to thermal stresses as a onsequene of thermal stress relaxation due to raking. The latter phenomenon results in small permanent tensile strains after eah loading-unloading yle. The additional work performed by the external load related to additional maro-sale displaements due to the raks growth is also inluded. The dissipated energy for both materials systems as a funtion of the maximum applied strain in the yle is presented in Fig. 3. Eah maximum applied strain orresponds to a unique rak density,, value for eah different material and lay-up. The proedure for its determination is desribed in Setion Experimental. Alternatively, the measured stiffness redution with strain and the theoretial model an be used to extrat the orresponding rak density. Using these data and the data presented in Fig. 3 a diagram of the dissipated energy as a funtion of for all experiments an be onstruted. U d versus transverse rak density is presented in Fig. 4. Experimental value is given by the total derivative of the dissipated energy, U d, 9

12 with respet to the total rak surfae formed, A = du d da (17) Polynomial fitting of the experimental data in Fig. 4 has been used in order to determine the aording to eq. 17 for eah different rak density value. The rak density,, is related to A with expression (2) where N is the number of raks formed that is related to by (6). Using equations (2), (6) and (17) the following expression, relating atual 9 -layer volume, V 9, is onstruted with through the 1 = V 9 dud dρ (18) The ritial strain energy release rate as a funtion of is presented in Fig. 5. Data presented in Fig. 5 possibly indiate that due to unstable growth more energy was released than the atual required reating raks in the F2_1-4 material. The differene beomes muh smaller for the other two lay-ups with the same of-axis layer thikness. Inverting the previous strain-rak density proedure, i.e. going from yle maximum applied strain to rak density, the relationship between and the applied strain is extrated. The results are presented in Fig. 6. All the experimental data resulting from the three different F2 laminates are plotted to show that the results ould be onsidered laminate lay-up insensitive, while the dependene is different for F1 and F2 omposites. In other words, is a material harateristi and does not vary with ply thikness. As an alternative approah we use the measured stiffness degradation with strain and eq (11) to alulate orresponding to the given strain level (more exatly for raks developing at this strain level). In derivation of (11) the results of previous works [7, 26] on stiffness degradation with inreasing rak density have been used. The model proposed by Lundmark and Varna [26] has been proved to provide a very aurate desription of the 1

13 longitudinal Young s modulus relationship to the rak density. The was alulated by eq (11) for every pair of points (, E x ). The neessary pair of points an be easily found experimentally, from the orresponding stress-strain urves using the unloading urve as desribed. The results are presented in Fig. 7. An average linear fitting is used for all results from the three different F2 laminates sine is onsidered as a frature harateristi of a material not depending on geometrial harateristis. The rather linear behaviour of with applied strain, whih was observed in the dissipated energy alulations, see Fig. 6, is also presented here for the two different materials examined. The numerial values obtained using these two different tehniques are rather lose. Linear fit to these data was used as the first approximation in the following modelling. Certainly, more refined (bi-linear or S-shape) approximation would aount for more details and would improve the preditions. However the auray of the fitting has to be omparable with experimental differenes between urves and the fat that is onsidered as independent of the laminates lay-up. for the F2_1-4 material alulated using the dissipated energy is used in order to develop an analytial method for simulation of the rak density orresponding to any applied strain level. Eq (1) provides the requested relationship and, thus, it was used to derive theoretially the rak density as a funtion of the applied strain for all materials under onsideration. The results are presented in Fig. 8 in omparison to experimentally derived rak density versus maximum applied strain data for both materials systems under investigation. Sine new damage modes (blunted raks, delta raks, urved raks, loal delaminations et) start to be dominant at strains higher than the so alled saturation state, the desribed model for straight intralaminar raks an not be used anymore. In order to extend the approah to this region these damage modes have to be lassified, quantified and their effet on stiffness has to be modelled. The experimental data, from the three different laminates manufatured using the F2 11

14 material, are onsidered separately. The theoretial preditions appear to be in good agreement with the experimental data in the ase of F1 material. Considering the F2 system three different preditions, with respet to the three different initial Young s moduli of the laminates are presented. The variation between the three different urves is not onsiderable, and all three are in agreement with the experimental data. Hene, in the rak density development the ritial strain energy release rate an be treated as a material property to be alulated using marosopially onstruted stress-strain urves. Conlusions Critial strain energy release rate has been used as a mean for the analytial predition of the rak density arising in ross-ply omposite laminates as a onsequene of the external applied strain level. Two different FRP material systems (F1 and F2) were used. F2 system was used in three different staking sequenes with varying layer thikness ratio. The energy dissipated during all loading-unloading yles was alulated using the reorded stress-strain urves. Together with experimental rak density evolution data for a referene laminate it was used in order to derive the as a funtion of a modified rak density. This dependene represents the statistial nature of the omposite frature toughness,, distribution (first raks are in positions with lower frature toughness). Using the referene lay-up rak density data the strain. A linear relationship between was realulated versus the orresponding tensile applied and the maximum applied strain level was observed for both materials systems. This experimental relationship was used for theoretial predition of the rak density evolution with the applied strain in other laminates of the same material. An analytial model was used whih requires only the material thermomehanial properties and the stiffness degradation law as a funtion of inreasing rak density provided any appropriate model. The theoretial alulations of the rak density as a funtion of the 12

15 applied strain were ompared to experimental data and a good agreement was found. It has been demonstrated that the energy dissipated during a loading unloading yle and the measured stiffness redution an be used to determine material property and used in theoretial damage evolution models. whih an be treated as statistial Aknowledgments Dr S. L. Ogin and Mr R. D. Whattingham of the University of Surrey are aknowledged for their ontribution in the speimen manufaturing proess as well as in the development of the whole projet. Mr E. Sprniš and Dr R. Joffe are aknowledged for onduting tensile experiments. D. T.. Katerelos would like to aknowledge the Swedish Institute for granting a uest post-dotoral sholarship during whih this paper was developed. Referenes 1. Talreja R. Damage haraterization by internal variables. In: Pipes RB, Talreja R, editors. Composite Materials series, vol. 9. Damage Mehanis of Composite Materials. Amsterdam: Elsevier, Varna J, Joffe R, Akshantala NV, Talreja R. Damage in omposite laminates with off-axis plies. Comp Si Teh 1999; 59(14): Nairn J. Matrix miroraking in omposites. In: Kelly A, Zweben C, Talreja R, Manson J-A, editors. Comprehensive Composite Materials, vol. 2. Polymer Matrix Composites. Amsterdam: Elsevier Berthelot J-M. Transverse raking and Delamination in Cross-Ply lass-fiber and Carbon-Fiber Reinfored Plasti Laminates: Stati and Fatigue Loading. Appl Meh Rev 23; 56(1): Kashtalyan M, Soutis C. Analysis of Composite Laminates with Intra- and Interlaminar 13

16 Damage. Progr Aerosp Si 25; 41(2): Katerelos D, MCartney LN, aliotis C. Loal Strain re-distribution and Stiffness Degradation in Cross-Ply Polymer Composites under Tension. Ata Mater 25; 53(12): Katerelos DT, Lundmark P, Varna J, aliotis C. Analysis of Matrix Craking in FRP Laminates using Raman Spetrosopy. Comp Si Teh 26; 67(9): Laws N, Dvorak J. Progressive Transverse Craking in Composite Laminates. Jrnl Comp Mater 1988; 22(1): Lafarie-Frenot MC, Hénaff-ardin C. Formation and rowth of 9 Ply Fatigue Craks in Carbon/Epoxy Laminates. Comp Si Teh 1991; 4(3): Tong J, uild FJ, Ogin SL, Smith PA. Off-Axis Fatigue Crak rowth and the Assoiated Energy Release Rate in Composite Laminates. Appl Comp Mater 1997; 4 (6): Hénaff-ardin C, Lafarie-Frenot MC. The Use of a Charateristi Damage Variable in the Study of Transverse Craking Development under Fatigue Loading in Cross-Ply Laminates. Int Jrnl Fatigue 22; 24(2-4): Yokozeki T, Aoki T, Ishikawa T. Fatigue rowth of Matrix Craks in the Transverse Diretion of CFRP Laminates. Comp Si Teh 22; 62(9): Li C, Ellyin F, Wharmby A. On Matrix Crak Saturation in Composite Laminates. Comp B Eng 23; 34(5): Caiazzo AA, Costanzo F. Modeling the Constitutive Behavior of Layered Composites with Evolving Craks. Int Jrnl Sol Str 21; 38(2): Rebière JL, amby D. Analytial and Numerial Analyses of Transverse Craking in a Cross-Ply Laminate Influene of the Constraining Effet. Comp Str 1992; 2(2): Ogihara S, Takeda N, Kobayashi A. Experimental Charaterization of Mirosopi Failure Proess under Quasi Stati-Tension in Interleaved and Toughness-Improved CFRP 14

17 Cross-Ply Laminates. Comp Si Teh 1997; 57(3): Anderssen R, radin PA, ustafson C. Predition of the Stiffness Degradation in Cross- Ply Laminates due to Transverse Matrix-Craking: An Energy Method Approah. Adv Comp Mater 1998; 7(4): Ji FS, Dharani LR, Mall S. Analysis of Transverse Craking in Cross-Ply Composite Laminates. Adv Comp Mater 1998; 7(1): Adolfsson E, udmundson P. Matrix Crak Initiation and Progression in Composite Laminates subjeted to Bending and Extension. Int Jrnl Sol Str 1999; 36(21): Ladevèze P, Lubineau. On a Damage Mesomodel for Laminates: Miro-Meso Relationships, Possibilities and Limits. Comp Si Teh 21; 61(15): Yokozeki T, Aoki T, Ishikawa T. Conseutive Matrix Craking in Contiguous Plies of Composite Laminates. Int Jrnl Sol Str 25; 42(9-1): Smith PA, Ogin SL. On Transverse Matrix Craking in Cross-Ply Laminates Loaded in Simple Bending. Comp A 1999; 3(8): Smith PA, Ogin SL. Charaterization and Modelling of Matrix Craking in a (/9) 2s FRP Laminate Loaded in Flexure. Pro Royal So London Series A 2; 456(23): Croker LE, Ogin SL, Smith PA, Hill PS. Intra-Laminar Frature in Angle-Ply Laminates. Comp A 1997; 28(9-1): Rubenis O, Sprniš E, Andersons J, Joffe R. The Effet of Crak Spaing Distribution on Stiffness Redution of Cross-Ply Laminates. Appl Comp Mater 27; 14(1): Lundmark P, Varna J. Constitutive Relationships for Laminates with Ply Craks in In- Plane Loading. Int Jrnl Dam Meh 25; 14(3): Wang A.S.D., Chou P.C., A stohasti model for the growth of matrix raks in omposite materials, J. Compos. Mater., 1984; 18:

18 Figure Captions FIURE 1. A typial rak pattern in ross-ply laminates FIURE 2. Typial stress-strain urves for the two materials examined, (a) F1, (b) F2_1-4, () F2_1-2 and (d) F2_2-2. FIURE 3. Dissipated Energy as a funtion of yle maximum applied strain for all the materials tested FIURE 4. The Dissipated Energy plotted against the modified rak density orresponding to eah applied strain level for both materials systems FIURE 5. Critial strain energy release rate as a funtion of the modified rak density FIURE 6. Critial strain energy release rate as a funtion of applied strain. The data of the three different F2 laminates are fitted by an average linear fitting urve FIURE 7. Theoretial evaluation of the data are onsidered as one data-set as a funtion of maximum applied strain. F2 FIURE 8. Theoretial predition of the rak density orresponding to applied strain level in omparison with experimentally derived data. The results are presented for both materials systems used 16

19 Figure 1 Stress (MPa) 16 Stress Strain (a) Stress (MPa) Stress Strain (b) Stress (MPa) Stress Strain () Stress (MPa) Stress Strain (d) Figure 2 17

20 Dissipated Energy (J) F1 U d (J) F2_1-4 U d (J) F2_1-2 U d (J) F2_2-2 U d (J) Linear Fit of F1 data Linear Fit of F2_1-4 data Linear Fit of F2_1-2 data Linear Fit of F2_2-2 data Figure Applied Strain (%) Dissipated Energy (J) Figure 4 F1 U d (J) F2_1-4 U d (J) F2_1-2 U d (J) F2_2-2 U d (J) Polynomial Fit of F1 data Polynomial Fit of F2_1-4 data Polynomial Fit of F2_1-2 data Polynomial Fit of F2_2-2 data Crak Density, (r/mm) 18

21 (J/m 2 ) Figure 5 F1 (J/m 2 ) F2_1-4 (J/m 2 ) F2_1-2 (J/m 2 ) F2_2-2 (J/m 2 ) 75 Linear fit of F1 data 5 Linear fit of F2_1-4 data 25 Linear fit of F2_1-2 data Linear fit of F2_2-2 data Crak Density, (r/mm) 19

22 (J/m 2 ) F1 (J/m 2 ) F2_1-4 (J/m 2 ) F2_1-2 (J/m 2 ) F2_2-2 (J/m 2 ) Linear Fit of F1 Linear Fit of F2_1-4 Linear Fit of F2_1-2 Linear Fit of F2_ Applied Strain (%) Figure 6 2

23 (J/m 2 ) F1 (J/m 2 ) Linear Fit of F1 data F2_1-4 (J/m 2 ) F2_2-2 (J/m 2 ) F2_1-2 (J/m 2 ) Linear Fit of F2 data Applied Strain % Figure 7 21

24 Crak Density (r/mm) (F1) (F2_1-4) (F2_1-2) (F2_2-2) Theoretial (F1) (r/mm) Theoretial (F2_1-4) (r/mm) Theoretial (F2_1-2) (r/mm) Theoretial (F2_2-2) (r/mm) Figure 8 Applied Strain % 22