Composites: Part A 75 (2015) 89–95
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Characterization of composite materials made from discontinuous carbon fibres within the framework of composite recycling A. Gillet, O. Mantaux ⇑, G. Cazaurang Univ. Bordeaux – I2M, CNRS UMR 5295, F-33400 Talence, France
a r t i c l e
i n f o
Article history: Received 28 January 2014 Received in revised form 16 April 2015 Accepted 2 May 2015 Available online 9 May 2015 Keywords: A. Carbon fibre E. Recycling B. Mechanical properties B. Directional orientation
a b s t r a c t Recycling carbon fibres from waste composite materials would only be efficient if it were possible to separate the fibres and the matrix and to re-use the recycled fibres as new reinforcements. The challenge is to use non-continuous fibres to produce high-strength materials. The formation of defects in ‘‘semi-long’’ fibre composites has not yet been taken into account. In this paper the influence of fibre length and fibre alignment on the strength and the modulus of composite materials is illustrated. It is shown that the presence of defects may be modelled in order to understand what the quality of a second generation composite material would be. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Recycling carbon fibre composites cannot be achievable until it is possible to re-use recycled fibres to produce new composite materials. It is assumed here that the first step involved in recycling a composite material (the fibre/matrix separation step) does not damage the carbon fibre [1,2] when it is performed by solvolysis in supercritical water. As a consequence, apart from the length and the absence of sizing, there is no significant difference between a virgin carbon fibre and a recycled carbon fibre after solvolysis. The fibre/matrix separation process can provide different architectures of the recycled fibres. To be efficient at the core, thermal processes require waste to be ground (thick laminates) [3], the latter generates relatively short fibres (<40 mm). These short recycled fibres can be processed by using injection methods. For example, Wong et al. [4] achieved injection moulded composite parts (30% volume of recycled carbon fibres (RCF)) with a longitudinal modulus of 15 GPa and a strength of 50 MPa. The recycled carbon fibres can also be reused as tapes or mats impregnated with thermoset resins [5] or with thermoplastic resins [6]. Pimenta et al. [5] managed to obtain a modulus of 28 GPa and a tensile strength of 195 MPa with compression moulded mats (30% fibre volume content). Although the results were rather good, the strengths remained far from what is expected from a continuous carbon fibre composite. ⇑ Corresponding author. Tel.: +33 556 84 79 79; fax: +33 556 84 58 43. E-mail address:
[email protected] (O. Mantaux). http://dx.doi.org/10.1016/j.compositesa.2015.05.002 1359-835X/Ó 2015 Elsevier Ltd. All rights reserved.
It seems that the realignment of semi-long recycled carbon fibres is essential for achieving high performance composite materials containing recycled fibres. Indeed, the fibre volume fraction could be high and then, the excellent mechanical properties of the fibres could be exploited optimally. Work has been carried out by different researchers to align the fibres. Akonda et al. [7] achieved a 32 GPa modulus and 160 GPa strength with carbon rovings co-mingled with PP fibres at a 28% fibre volume fraction. Turner et al. [8] studied the elaboration of realigned mats by a hydrodynamic process. Their products present a 80 GPa modulus and a 422 MPa strength with 40% fibre volume content. The capabilities of supercritical water solvolysis for penetrating thick composite waste [9] could be very useful because it is no longer necessary to grind the waste before the fibre/matrix separation step. As a consequence, recovering long carbon fibres would be possible provided that the composite waste is cut into long pieces. Therefore, the I2M lab. (University of Bordeaux) has developed a realignment process for semi-long recycled carbon fibres (50– 200 mm). The machine-prototype is able to transform pieces of fabric (simulating post-solvolysis recycled fibres) into tapes of realigned semi-long fibres. On the assumption that the fibre/matrix separation in supercritical water can recover up to 95% of the mechanical properties of the recycled carbon fibres, the quality of recycled carbon fibres depends on the structural parameters of the semi-product made of recycled fibres: fibre length, the realignment rate and the mass of carbon per unit area. The purpose of this study was to underline the relation between the phenomena involved in the re-shaping of the fibres and the mechanical properties of 2nd generation composites. In this paper, the differences
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between the strength of a composite material with semi-long and imperfectly aligned fibres and the strength of a traditional composite part are explained. This study is divided into two parts: in the first part, an experimental measurement of the longitudinal properties of a virgin staple fibre composite is performed. These properties are compared to those of composites with continuous virgin fibres. In the second part, the decrease in longitudinal properties of the virgin staple fibre composite is interpreted using numerical modelling. The effects of fibre alignment scattering, local fibre mass fraction per unit area and fibre length were investigated.
Fig. 2. Tape of realigned carbon fibres.
minimize additional misalignment. The preforms were then inserted between two metal plates. Both the preforms and the plates were placed in a vacuum bag to provide compaction through a 0.9 bar depression within the mould. A 24-h curing cycle at room temperature and an 8-h post-curing cycle at 80 °C were carried out. For each length of realigned fibre (50 mm, 100 mm and continuous), 4 composite plates were made. In addition, one single plate was made from commercial UD which was used as a reference for perfectly aligned continuous fibres. In each plate, seven 250 20 mm2 samples were cut in the longitudinal direction of the fibres from each plate, using a diamond disc. ±45° glass–epoxy end tabs were glued at the ends of the samples.
2. Mechanical characterization 2.1. Experimental methodology In a real recycling process of composite materials, the first step is the dissolution of the organic matrix (e.g. in supercritical water). During the second step of the recycling process, the recycled fibres are in the form of pieces of dry fabric (without sizing). Then, dry carbon fabric has to be unwoven and the recovered carbon fibres have to be realigned and reshaped. In this study, only the second step is investigated, virgin fibres (hence with sizing), were used. The goal was to investigate the effects of the morphology of the fibrous architecture on the mechanical properties of the composite.
2.1.3. Material characterization Soficar T300 realigned virgin fibres were used, their modulus is Ef = 230 GPa and rf strength is 3530 MPa. Araldite LY-Aradur 50-52 epoxy resin was used, its modulus is Em = 3 GPa and rm strength is 70 MPa after curing (24 h at ambient T° + 4 h post-curing at 100 °C). The fibre volume fraction (Vf) of the composites manufactured with realigned fibres was evaluated by weighing, on the assumption that there was no porosity. The values of Vf ranged from 39% to 46%.
2.1.1. Morphology of dry and realigned fibre tapes The realignment of the fibres was performed on an experimental machine specifically created for this purpose at I2 M in Bordeaux. The T300 3K fibres come from unwoven G803 fabrics (5H satin, 285 g/m2, warp count 7.2, fill count 7.2). These fibres are composed of 3000 7 lm diameter filaments. Square pieces of fabric were cut in the warp/weft direction in order to control the length of the recovered fibres. Unweaving 260 260 mm2 fabric cuts provides 260 mm fibres (the same length as the samples). As a result, the influence of fibre misalignment and mass per unit area variation on the mechanical properties could be assessed in the case of continuous fibres. ‘‘Continuous fibres’’ means that the fibres extend from one end of the specimen to the other (Fig. 1). Unweaving 100 100 mm2 and 50 50 mm2 fabric cuts provides fibres of 100 mm and 50 mm in length respectively. This operation was performed in order to examine the influence of fibre length on the mechanical properties of the composite material. During the re-shaping step, 50 mm wide and 600 g/m2 density tapes were manufactured with the realigned carbon fibres. These tapes contain imperfectly realigned carbon fibres (Fig. 2). These tapes were then assembled to form 160 mm wide and 260 mm long layers. T300 commercial unidirectional was used as a perfectly aligned reference tape (density 125 g/m2, width 50 mm), the latter provided the reference properties.
2.2. Results from mechanical testing Tensile tests were performed in the longitudinal direction of the carbon fibres. The measurement of the longitudinal deformation was carried out using of an extensometer knife the reference length of which was 25 mm. It should be noted that the gauge length was shorter than the fibre length. As a result, the measurements were sensitive to local heterogeneities of the material. Moreover, the samples exhibit bending due to the fact that fibre distribution across the thickness is neither balanced nor symmetrical. Unfortunately, an extensometer with a higher gauge length was not available. The test set-up was the same for all the samples allowing the differences in material properties to be highlighted. 2.2.1. Strength Fig. 3 shows the effect of the fibre length on the strength of the composite materials. The theoretical strength of a T300-epoxy composite, was evaluated at 1450 MPa from the law of mixtures, using manufacturer data (assuming Vf = 40%). As the maximum strength obtained with commercial samples with continuous fibres is about 900 MPa (40%
2.1.2. Production of composite samples Composite plates were manufactured by compression moulding after manual reinforcement impregnation. Rollers were not used to
(a)
(b)
(c)
Fig. 1. Reinforcement architectures: (a) perfectly aligned continuous, (b) realigned continuous fibres and (c) realigned discontinuous fibres.
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Longitudinal strength (MPa)
1200
commercial UD panel 1
1000
is to use the slenderness of the fibres. The Halpin–Tsai equation, was used because of its simplicity and the opportunity to evaluate all of the elastic properties.
panel 2
800
panel 3 panel 4
600
with g ¼
400 200 0
P 1 þ ngf ¼ Pm 1 gf
nE11 ¼ 2 continuous
100 mm
50 mm
Fig. 3. Effect of fibre length on the strength of the composite materials (error bars are standard deviations). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
less than the theoretical value), this difference highlights the influence of the moulding process. In comparison to the properties of the reference composite (perfect alignment and uniform density), the properties of the composite with continuous realigned fibres showed a loss of strength of 27% which can be attributed to misalignment and scatter of mass per unit area. Finally, the decrease in longitudinal strength compared to the composite with continuous and realigned fibres was of 8% with 100 mm fibres and 28% with 50 mm fibres. The latter highlights the influence of fibre length on mechanical properties. The average strength and Coefficients of Variation (CoV) for the different plates tested are shown in Table 1.
3. Behaviour prediction of a staple fibre composite with alignment and density defects 3.1. Mechanical model To predict the mechanical properties of the composite made from discontinuous long fibres, laminate analogy described in [10] was used. The set of possible orientations is discretised into a vector T = [90°, 85°, . . . , 85°, 90°]. The thickness of each ply is proportional to the number of fibres that fall between the elements of vector T. The orientation of each ply was taken to be equal to 12 ðT i þ T iþ1 Þ. The equivalent laminate is supposed to be symmetrical and balanced. Laminate failure is supposed to occur at the first ply failure. The fracture criterion used for one ply is the Tsai Hill criterion. Many equations to assess the elastic properties of a staple fibre composite [11–13] exist. The common point of these relationships
Pf 1 Pm
L ; d
Pf þn Pm
nE22 ¼ 2;
nG12 ¼ 1
ð2Þ
ð3Þ
where P is the composite’s modulus, Pf and Pm are the fibre and the matrix moduli respectively. ‘‘n’’ is the aspect ratio in the considered direction and f is the fibre volume fraction. The slenderness of fibres was evaluated by determining the equivalent diameter of a roving of 3000 filaments impregnated with resin at a 40% nominal fibre volume fraction.
sffiffiffiffi N d¼ df f
ð4Þ
where N is the number of filaments and df the diameter of a filament. With 3000 filaments of 7 lm in diameter, an equivalent diameter of 0.61 mm was obtained. Fig. 5 shows the evolution of the modulus of an UD composite as a function of fibre length. In a staple fibre composite, the fibre ends generate stress concentration which reduces the strength of the whole composite material. So it is possible to represent the strength in the following way:
rcu ¼ f 2.2.2. Young’s modulus Fig. 4 shows the effect of the fibre length on the elastic modulus of the composite material. The results in Fig. 4 show that the fibre length has little influence on the elastic modulus of the composite. The theoretical modulus of a composite with a T300 UD-epoxy with 40% fibre volume fraction is 94 GPa. All the measurements were close to this value. However, there was wide dispersion with the 50 mm fibres. Table 2 summarizes the average values and coefficients of variation of the longitudinal moduli measured. The scatter on the modulus was high. It therefore appeared that both the variability of the fibre density, the misalignment and the length of the longitudinal fibres strongly influence the composite’s properties. In the second part of this study, simple analytical models to assess the relative impact of these imperfections were used.
ð1Þ
rfu K
þ ð1 f Þ
Em rfu Ef
ð5Þ
with rcu: strength of the composite, rfu: strength of the fibre, Em: elasticity modulus of the matrix, Ef: elasticity modulus of the fibre and K: coefficient of stress concentration
K¼
rmax r0
ð6Þ
K is the coefficient of stress concentration induced by the discontinuity of the fibre (Fig. 6) with r0: nominal stress in the fibre and rmax local increased stress due to the discontinuity of other fibres in the vicinity of the considered fibre. The variability in fibre strength is supposed to be small in comparison with the variability of fibre local mass per area and fibre alignment. rfu is therefore assumed to be constant. K is evaluated using the Wetherhold’s [14,15], critical-zone analytical model for short non-aligned fibre composites. The overstress rmax was sustained by an ‘‘effective’’ fibre after load transfer from ‘‘non-effective’’ fibres. It was therefore necessary to count the number of effective fibres Neff in each section of the composite. If Neff fibres passed through the zone of length lc completely (containing N fibres), it was considered that only the Neff fibres were subjected to loading, while (N Neff) fibres which were interrupted in this zone were ‘‘non-effective’’ (Fig. 6). The lc parameter is the critical length of load transfer between the fibre and the matrix:
lc ¼
rfu d 2s
ð7Þ
where s is the maximum shear strength in the matrix. The stress field pattern in the considered fibre is illustrated in Fig. 7. In a continuous fibre composite, the stress in the fibres is equal to r0 throughout. In a staple fibre composite, in the vicinity of a discontinuity of a ‘‘non-effective’’ fibre, the stress in effective fibres reached an increased value rmax.
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Table 1 Mechanical properties of the composites produced in this study.
Strength (MPa) CoV (%)
Theoretical UD T300 40% fibres
Continuous virgin fibres (UD tapes)
Continuous realigned fibres
100 mm realigned fibres
50 mm realigned fibres
1450 0
922 11
663 15
577 18
446 21
Longitudinal modulus (GPa)
180
commercial UD
160
panel 1
140
panel 2 panel 3
120
panel 4
100 80 60 40 20 0
continuous
100 mm
50 mm
Fig. 4. Effect of the fibre length of the modulus of the composite materials (error bars are standard deviations). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
This model underestimates the actual value of K, as the load is assumed to be uniformly distributed over the effective fibres. This approximation is only valid if the number of ineffective fibre ends in the section is high [16]. The application of the mechanical model requires the local fibre volume fraction, fibre orientations and effective fibre number in each section of a sample to be determined. These parameters are governed by fibre dispersion, predicted with a method depicted in Section 3.2.
Fig. 5. Evolution of the longitudinal modulus as a function of the slenderness (T300 3 K, vf = 40%). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
lc
effective ibres
Fig. 6. Count of effective fibres. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
3.2. Simulation of the fibre dispersion A Monte Carlo simulation was used to generate tapes that were then put together to form the preform architecture. The number of fibres to be deposited was determined from the mass of carbon used to manufacture each plate. Each fibre had the same length (L). For tape dimensions LTAPE bTAPE, the coordinates of the centroid of each fibre were determined by generating two random numbers: xi and yi. xi was a random position uniformly distributed between 0 and LTAPE. yi was a random position uniformly distributed between 0 and bTAPE. A third random number was generated to define the orientation hi in a normal distribution the average is 0 and standard deviation Dh. Moreover, a correction was performed on this orientation if the fibre was not fully contained within the plate (Fig. 8). Fig. 9 shows the fibre volume fraction of a 200 150 mm2 preform (the contrast has been increased for better readability). The density gradually increases from the edges of the preform to reach
L
σmax σ0
Table 2 Moduli of the composites produced in this study. Theoretical UD T300 40% fibres
Continuous virgin fibres (UD tapes)
Continuous realigned fibres
100 mm realigned fibres
50 mm realigned fibres
94 –
90 11
98.2 16
96.7 19
86.2 35
x Fig. 7. Stress distribution in a staple fibre composite material. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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The ratio f/K is also computed in each section of the sample:
Ri ¼
N eff Neff f loc ¼ f loc : ¼ f K N N
ð9Þ
The local fracture factor Ri, used in Eq. (5) allowed the longitudinal strength of the UD ply to be computed. On each section, the local modulus was determined by computing the properties of the UD ply (Eq. (1)). The local orientations were then processed to compute the properties of the equivalent
Fig. 8. Correction of fibre position within the tape.
laminate. The longitudinal modulus Eix and longitudinal failure strength rif of the equivalent laminate were then calculated. To evaluate the modulus of the sample, the mean value of Ex on a zone centered on the middle of the sample, (the length of which was equal to the gauge length) was calculated (Fig. 9). The failure strength of the sample rf was then defined as the minimum value of the failure strengths from all sections of the sample:
rf ¼ mini rif
ð10Þ
tape width
In order to obtain the statistical distribution of these predictions, this procedure was run several times (20 6 samples). 3.3. Prediction of properties
Fig. 9. Simulated density and definition of samples areas. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Transverse and shear fibre modulus were taken from [17]. Assuming a matrix maximum shear strength of s = 40 MPa and considering a constant charge transfer between the fibre and the matrix, the critical length of load transfer was lc = 26 mm. The samples were divided into 150 sections, which guaranteed a small discretisation length when compared with the load transfer length.
a plateau in the central part. The transition zones from the edges to the plateau have a length of about L/2. A thin dark line separating the three tapes can also be seen. On each preform, areas corresponding to the relevant dimensions of a sample (150 20 mm2) were taken from the middle of the preform and processed to determine the fibre volume fraction and the fracture factor of the samples. The total number of fibres N and the number of effective fibres Neff were counted on a predefined set of straight sections for each sample. The number of straight sections was chosen so that the distance between two sections was smaller than the critical length lc. The orientation of each fibre crossing the section was retrieved in order to define the statistical distribution of orientation. These data were used to define the local fibre volume fraction with the relation:
3.3.1. Influence of fibre length Figs. 10 and 11 respectively show the evolution of the fibre volume fraction f (averaged over a gauge length) with the fibre length and the minimal fracture factor for two misalignment values (Dh). Increased fibre length led to an increased fracture factor. From this simulation, a 80% retention factor on strength was obtained. With perfect alignment, the scatter was moderate, and only slightly affected by fibre length. The mean value of f and R was not affected by fibre misalignment. However, in the case of high misalignment values, increased fibre length led to increased scatter of the fibre volume fraction.
f loc ¼
N f N
ð8Þ
3.3.2. Influence of misalignment Fig. 12 shows the evolution of the tensile strength as a function of misalignment for three fibre lengths.
Fig. 10. Evolution of fibre volume fraction with length (a) Dh = 0° and (b) Dh = 15°(error bars are standard deviations). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Fig. 11. Evolution of fracture factor with length (a) Dh = 0° and (b) Dh = 15° (error bars are standard deviations). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 12. Evolution of failure strength as a function of misalignment (error bars are standard deviations). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
On one hand, misalignment caused a decrease in strength especially when the fibres were short (15% for L = 50 mm, 11% for L = 100 mm, 5% for L = 200 mm). On the other hand, scatter increased with the misalignment. This trend was more pronounced with the use of long fibres. Fig. 13 shows the evolution of the modulus with misalignment. Misalignment caused a decrease in the modulus especially when short fibres were used. The modulus scatter increased with misalignment and this increase was even higher when the fibres were long. The effect of misalignment was generally less pronounced on the modulus than on the strength: for L = 50 mm and Dh = 15°, the model predicts 15% strength reduction and 11% modulus reduction. Figs. 14 and 15 respectively, show the comparison of computed strength and modulus with experimental data.
Fig. 14. Comparison of experimental strength with predictions (error bars are standard deviations). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 15. Comparison of experimental modulus with predictions (error bars are standard deviations). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 13. Evolution of the modulus with misalignment. (a) L = 50 mm, (b) L = 100 mm and (c) L = 200 mm (error bars are standard deviations). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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The model overestimates the strength of the composite, and underestimates the scatter. This discrepancy could be reduced by a larger angular scatter, but also with a more representative fibre deposition simulation, which could induce lower fracture factors. This highlights the need to measure the alignment rate and spatial distribution of carbon mass per unit area. The increase in strength with fibre length predicted with the model is much higher than experimentally observed. If the fibre length clearly impacts the strength, the hypothesis of composite fracture induced by fibre breakage should be discussed. An alternative model based on the assumption of matrix fracture should be examined. The discrepancy between measured and predicted mean modulus was lower than for strengths. Arguably, considering an elliptical fibre section would increase the slenderness and would allow to approach the experimental values. In contrast, the scatter predicted by the model is much lower than the experimental scatter. A higher fibre misalignment could provide more realistic predictions. It may also call into question the assumption of a symmetrical and balanced laminate, in which case the effect of induced bending may explain the high dispersion. 4. Conclusion The Influence of the fibre length on the properties of a composite with discontinuous imperfectly realigned virgin fibres was investigated. From these tests it was possible to obtain composites with a stiffness equivalent to a commercial fibre product, and strength of about 50% of the strength of an ideal UD composite. The composite properties however showed high scatter. The causes of this scatter and its effect on the decreased strength and on the modulus were investigated using a simple model which highlighted the following points: – A higher fibre length increases the strength of the laminate (using Toray T300 3 K fibres). The numerical model predicts a possible recovery of the UD with the use of very long (>200 mm) fibres. – A decrease in the modulus was observed only when short fibres were used. Nevertheless, the opportunity to obtain the same modulus as perfect UD with fibre length over 100 mm was numerically predicted. – Fibre misalignment is the main cause of scatter, as it induces both scatter on fibre volume fraction and a decrease in/of the number of loaded fibres. The numerical model does not correctly predict the properties of the composite. Thus, improvements are required (i) on the knowledge of the input data, such as the slenderness of the fibres, and (ii) on the dispersion of the fibre orientation and fibre mass per unit area. Moreover, the hypothesis of fibre breakage inducing
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composite failure appears to lead to overly optimistic estimates, and taking into account matrix fracture near fibre ends should be studied. Some improvements to the realignment process are required for recovering longer fibres reducing misalignment. It was shown in this study that the use of semi-long fibres would be the best option to manufacture composites with high mechanical properties but it also led to increased scatter. Therefore, preference should be given to any fibre/matrix separation technique which allows healthy and decimetric fibres to be recovered. References [1] Yang Y, Boom R, Irion B, van Heerden DJ, Kuiper P, de Wit H. Recycling of composite materials. Chem Eng Process 2012;51:53–68. [2] Pimenta S, Pinho ST. Recycling carbon fibre reinforced polymers for structural applications: technology review and market outlook. Waste Manage 2011;31:378–92. [3] Turner TA, Pickering SJ, Warrior NA. Development of recycled carbon fibre moulding compounds – preparation of waste composites. Compos B 2011;42:517–25. [4] Wong KH, Syed Mohammed D, Pickering SJ, Brooks R. Effect of coupling agents on reinforcing potential of recycled carbon fibre for polypropylene composite. Compos Sci Technol 2012;72:835–44. [5] Pimenta S, Pinho ST, Robinson P, Wong KH, Pickering SJ. Mechanical analysis and toughening mechanisms of a multiphase recycled CFRP. Compos Sci Technol 2010;70:1713–25. [6] Giannadakis K, Szpieg M, Varna J. Mechanical performance of a recycled carbon fibre/PP composite. Exp Mech 2011;51:767–77. [7] Akonda MH, Lawrence CA, Weager BM. Recycled carbon fibre-reinforced polypropylene thermoplastic composites. Composite Part A 2012;43:79–86. [8] Turner TA, Warrior NA, Pickering SJ. Developement of high value moulding compounds from recycled carbon fibres. Plast, Rubber Compos 2010;39(3– 5):151–6. [9] Morin C, Loppinet-Serani A, Cansell F, Aymonier C. Near-and supercritical solvolysis of carbon fibre reinforced polymers (CFRPs) for recycling carbon fibres as a valuable resource: state of the art. J Supercrit Fluids 2012;66:232–40. [10] Feraboli P, Cleveland T, Stickler P, Halpin J. Stochastic laminate analogy for simulating the variability in modulus of discontinuous composite materials. Composite Part A 2010;41:557–70. [11] Hsueh CH. Young’s modulus of unidirectional discontinuous-fibre composites. Compos Sci Technol 2000;60:2671–80. [12] Fu SY, Lauke B. The elastic modulus of misaligned short-fibre reinforced polymers. Compos Sci Technol 1998;58:389–400. [13] Fu SY, Yue CY, Hu X, Mai YW. On the elastic stress transfer and longitudinal modulus of unidirectional multi-short-fiber composites. Compos Sci Technol 2000;60:3001–12. [14] Wetherhold RC. Probabilistic aspects of the strength of fiber-dominated shortfiber composites I – aligned fiber. Mater Sci Eng 1987;91:7–12. [15] Wetherhold RC. Probabilistic aspects of the strength of fiber-dominated shortfiber composites II – biased fiber distribution. Mater Sci Eng 1987;91:13–8. [16] Beyerlein IJ, Landis CM. Shear-lag model for failure simulations of unidirectional fiber composites including matrix stiffness. Mech Mater 1999;31:331–50. [17] Liakus J, Wang B, Cipra R, Siegmund T. Processing–microstructure-property predictions for short fiber reinforced composite structures based on a spray deposition process. Compos Struct 2003;61:363–74.