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Data reduction methodologies for single fibre fragmentation test: Role of the interface and interphase
Composites: Part A 40 (2009) 449–454
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Composites: Part A journal homepage: www.elsevier.com/locate/compositesa
Data reduction methodologies for single fibre fragmentation test: Role of the interface and interphase Anbu Clemensis Johnson a,b, Simon A. Hayes a, Frank R. Jones a,* a b
Department of Engineering Materials, University of Sheffield, Sir Robert Hadfield Building, Mappin Street, Sheffield S1 3JD, UK Department of Chemical Engineering, Eritrea Institute of Technology, Mai Nefhi, Asmara, P.O. Box 1056, Eritrea
a r t i c l e
i n f o
Article history: Received 8 August 2007 Received in revised form 24 December 2008 Accepted 14 January 2009
Keywords: B. Interface/interphase C. Numerical analysis C. Micro-mechanics Data reduction technique
a b s t r a c t The single fibre fragmentation test (SFFT) is commonly used to characterise the fibre/matrix adhesion. In order to quantify the fibre/matrix adhesion the cumulative stress transfer function (CSTF) methodology was developed so that the elastoplasticity of the matrix could be included in the analysis through the plasticity-effect model [Tripathi D, Chen F, Jones FR. A comprehensive model to predict the stress fields in a single fibre composite. J Comp Mater 1996;30;1514–38., Tripathi D, Jones FR. Measurement of the load-bearing capability of the fibre/matrix interface by single fibre fragmentation. Comp Sci Technol 1997;57:925–35.] The limitations of this technique for the data reduction have been addressed by the use of the Plasticity Model to input the non-linearity of the matrix into methodology for fragmentation of a fibre in a matrix. An improved methodology, known as the revised cumulative stress transfer function (RCSTF) is described. The adhesion of a nanoscale plasma copolymer coated glass/epoxy system has been used to examine this approach to the fragmentation process. This methodology is also extended to account for the presence of an interphase. To validate the three phase model, carbon fibre coated with high and medium modulus epoxy resin were used to simulate fibre/interphase/matrix. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction The single fibre fragmentation test (SFFT) has been widely used to assess the quality of the interface fibre-matrix. Significant effort has been focused on the interface region to enhance the stress transfer capability of the fibres and manufacture durable materials. In the SFFT a single fibre with or without a size or coating is embedded in a resin matrix and a tensile load is applied axially to the fibre. When the applied load exceeds the tensile strength of the fibre, fragmentation is initiated which continues until the fragment lengths are shorter than the critical fibre length (lc) for that system. This occurs because the fragments are too short to be reloaded to fracture. There are several theoretical models in the literature that predict the stress transfer from the polymer matrix to the fibre. However, only a limited number are readily applied to the study of the interface. The most commonly preferred data reduction scheme for SFFT is based on the constant shear model [3]. However, this data reduction scheme has several limitations which have been discussed elsewhere [2,4]. The constant shear model considers that the stress transfer at the interface requires either complete debonding or shear yielding of the matrix. One of the serious limitations in this data reduction technique is insufficient consider* Corresponding author. Tel.: +44 114 222 5477; fax: +44 114 222 5943. E-mail address: f.r.jones@sheffield.ac.uk (F.R. Jones). 1359-835X/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesa.2009.01.008
ation of the various failure events such as fibre/matrix debonding, matrix transverse cracking and shear yielding of the matrix during fragmentation. Lacroix et al. [5] proposed a data reduction scheme which was based on the classical shear lag theory. The cumulative stress transfer function (CSTF) was proposed by Tripathi et al. [1,6] who used the plasticity-effect model to describe the stress profile within a fragment. The plasticity-effect model uses Nairn’s model [7] CSTF to calculate the shear stress profile and hence the point of matrix yield was shown to be effective in differentiating between a range of interfacial behaviour. This data reduction scheme was an improvement over the use of the constant shear model because it included the properties of the matrix and the extent of fibre/matrix debonding. The CSTF technique uses the Nairn’s elastic–elastic variational model to calculate the shear stress profile along a short fibre of given length, which is truncated when the matrix shear strength is matched. Debonding was included by calculating the radial thermal stress from a given value of the friction coefficient in the Coulomb estimation. This shear stress profile is converted into a fragment tensile stress profile to estimate the stress transfer efficiency. Thus, the CSTF value is defined as the average cumulative stress on a fragment. Unfortunately, the CSTF value is strongly dependent on fragment length, so that a poorly bonded fibre which cannot be fragmented further can have a higher CSTF value than a well bonded fibre which has fragmented to a higher degree. Therefore, the stress transfer function was normalised to that for an ideal
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bonded fibre in an elastic matrix which was referred to as stress transfer efficiency (STE) [8]. In this study, we concluded that interphasal properties contribute significantly to the fragmentation process so that any analysis should include the elastic–plastic properties of the interphase, where the shear stress is concentrated. Therefore, the aim of this paper was to examine the revised cumulative stress transfer function (RCSTF) for a study of the stress transfer efficiency in the presence of an interphase, using the plasticity model of Johnson et al. [9]. 2. Materials E-glass fibres (Owens Corning) and HTA carbon fibres (Tenax GmbH) were used in this study. The glass fibres were obtained without water sized film former, silane or other process aids. The glass fibres were coated with 30%, 50%, 90% acrylic acid/70%, 50%, 10% 1,7-octadiene and 100% 1,7-octadiene plasma polymers to form a nanoscale coating. The detailed description of fibre coating using plasma polymerisation (PP) is given elsewhere [10]. The thickness of the coating was shown by X-ray photoelectron spectroscopy to be greater than the analysis depth of 10 nm. Recent measurements using atomic force microscopy (AFM) showed the thickness to be 15.3 nm [11]. The coatings containing 50% and 90% acrylic acid provided a strong interfacial bond whereas the 1,7-octadiene coating provided a poor interface. In order to study the influence of an interphase, HTA carbon fibres (sized) obtained from Cytec Fiberite were coated with a thermoplastic–toughened epoxy resin (high modulus, 3.75 GPa) and medium modulus epoxy resin with modulus 3.02 GPa. The coating process was carried out by moving the glass rod containing the resin very slowly over the fibre to produce a uniform coating. The epoxy resin was cured at 85 °C for 12 h and increased to 180 °C for 2 h. The average thickness of the coating was 1 ± 0.15 lm. Full details of the coating resin system are not available for commercial reasons. These fibres were embedded in a ductile matrix (2.40 GPa) and cured at 80 °C for 4 h, followed by post-curing at 130 °C for 3 h. As shown in Table 1, the ductile matrix used for these experiments was a blend of LY1556 (Vantico, UK) which is a diglycidyl ether of Bisphenol A, and a flexibilising aliphatic epoxy resin Araldite GY298 (Ciba Geigy, UK). The curing agents were Nadic Methylene Anhydride (Stag Polymers and Sealants, UK) and Capcure 3-800 (Henkel-Nopco, UK), a mercaptan terminated polymer. The details of the single fibre fragmentation test are discussed elsewhere [12]. 3. Plasticity model The plasticity model (PLM) was developed by Johnson et al. [9] to calculate the stress in a discontinuous or fractured single fibre in an infinite matrix. The matrix and interphase plastic properties are imposed on the variational model developed by Wu et al. [13]. The variational model is an axisymmetric analysis which determines the elastic stress state around the fibre-breaks and debonded interface of a three-phase composite. The shear stress at the debonded interface was calculated from the coulomb friction law. The mini-
Table 1 Resin formulation used for the ductile 6040 support matrix. Constituent
(pbw*)
LY 1556 GY 298 NMA Capcure 3-800
60 40 63.90 23.93
*
pbw = parts by weight.
misation procedure was applied to both the debonded and the bonded zones simultaneously and the strong interaction of the two zones was correctly described. The tensile stress and the shear stress in the fibre is given by the equations,
rzz ; f ¼ wf wf w0 ; f
ð1Þ
wf nw00 ; f ¼ 2
ð2Þ
srz ; f
Where w is a stress function and wi ; f are stress perturbation functions where the subscript ‘‘i” is the function number. ‘‘f” denotes the fibre, wf is the far field axial stress in the fibre, w0 ; f is the expression for the stresses in the fibre, w00 ; f is the first derivative of the stresses in the fibre. These terms are a complex function of material and geometry and n is the position in the matrix starting from the interface. In the presence of a perfect interface, the plasticity model determines the shear stress at the fibre/matrix interface and the axial stress in the fibre using the variational model by Wu et al. [13]. The shear stress is compared to the shear yield strength of the matrix. If the calculated shear stress is greater than the shear yield strength of the matrix it is replaced by the shear yield strength of the matrix initially and then subsequently with the cold draw strength of the matrix. From the shear stress profile, the tensile stress profile in a fragment was calculated using a force balance.
2p r s dx ¼ p r 2 drf 2 drf ¼ s dx r
ð3Þ ð4Þ
The programme checks whether the axial stress in the fibre with an elasto-plastic matrix has reached 95% of the elastic fibre stress, if not the fragment length was reduced by an infinitely small increment and shear stress at the interface was recalculated and replaced the reduced length with the cold draw strength of the matrix. This process continues until the axial stress in the fibre reaches 95% of the elastic stress of that fragment. The whole procedure is summarised in the flow chart given in Fig. 1. 4. The revised cumulative stress transfer method for the single fibre fragmentation test CSTF methodology was introduced by Tripathi et al. [6] for the estimation of fibre/matrix adhesion. It is based on the assumption that the number and length of the individual fibre fragments in the single fibre fragmentation test specimen depend on the quality of the interface such that a good interface will transfer more stress to the embedded fibre than a poor interface. The calculation requires the properties of the fibre, interphase and matrix including the shear and cold draw strengths, coefficient of friction between the fibre and the matrix or interphase whichever is applicable. The individual fragment lengths with their associated debond lengths at a specific applied strain are experimentally determined and introduced into the calculation. In this methodology, the plasticity model [9] is used to calculate the shear stress for a particular fibre fragment. In the case of a debonded fibre, frictional stress at the interface between the fibre/matrix or fibre/interphase is computed in accordance with Coulomb’s friction law. Thus a shear stress profile is determined; from which the axial stress in the fibre fragment can be computed using the force balance argument. The tensile stress in the fibre fragment is integrated over the length of the fragment to calculate the stress transfer function (STF) and normalised using the fragment length, in order to make STF independent of fragment length. This procedure was followed for all the fragments. The STF value for all individual fragments were summed and normalised with respect to the total number of fragments.
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Input applied stress and mechanical properties of fibre, interphase and matrix Input data Choose one fragment (bonded or de-bonded) from SFFT
[Plasticity model]
Calculate the axial and shear stress at the fibre/matrix interface/interphase region using Wu et al. model [13]
Apply von Mises yield criterion (τ y = σ y / 3 ) and replace all the shear stress value greater than the shear yield strength of the matrix with shear yield strength and cold draw strength
Integrate over the fragment length to calculate the tensile stress in the fibre fragment
Reduce the fragment length, recalculate the shear stress for the fragment, and replace reduced length by cold draw strength
Check if the axial stress in the fibre has reached 95 % of the elastic stress
[RCSTF methodology]
Yes
No
Integrate the tensile stress in the fibre fragment over the length of the fragment divided by fragment length (STF) Repeat the above calculation process for rest of the fragments and add the STF values Cumulative stress transfer function is the total tensile stress transferred to the fibre fragments, which is the sum of STF determined for each fragment divided by total number of fragments
Output data
Fig. 1. Schematic representation of the RCSTF methodology.
This normalised value is known as the cumulative stress transfer function (CSTF). Because of the improvement in the calculation methodology CSTF will be known as Revised CSTF to differentiate the old and new programs. The general strategy for the computation of the RCSTF is given as a flowchart in Fig. 1. FORTRAN 77 was used for computer coding and interfaced with Clear Win+ for graphical interface to view the stress plots directly. The RCSTF is defined as,
Pi¼N R Li RCSTF ¼
i¼1
0
rf ðxÞdx
ð5Þ
Pi¼N
i¼1 Li
Where, rf (x) is the tensile stress in the fibre fragment of length, Li and i = 1, 2, 3. . ..N. The stress transfer efficiency (STE) technique was introduced by Lopattananon et al. [8] by normalising the value of CSTF to one for a perfect interface between an elastic fibre and matrix. Analogously the RSTE can be obtained according to,
STE ¼
½RSTFplastic ½RSTFelastic
100
ð6Þ
Where, [RCSTF]Plastic is the quantity obtained using elasto-plastic nature of the matrix and [RCSTF]Elastic is the quantity obtained assuming a perfect bond and an elastic matrix of the same modulus. 5. Experimental validation of RCSTF methodology 5.1. Validation of RSTF and RSTE methodology The validity of the RCSTF technique was assessed by using a series of given fragment lengths, as shown in Table 2 to examine the effect of fragment length and interphase properties on the computed RSTF and RSTE values. The fibre, matrix and interphase properties used are given in Table 3. The results presented in Table 2 were calculated at a 3% applied strain. It can be seen from Table 2 for fragments 1–3 the RSTF and the RSTE increase with fragment
Table 2 Validation of RSTF and RSTE model using values determined for fibre fragments at an applied strain of 3% except fragment 10 where the applied strain was 6%. Fragment
Bonded / debonded
Applied strain (%)
Interface friction coefficient (l)
Interphase modulus (GPa) / thickness in lm
Fragment length (mm)
Debonded length (mm)
[RSTF] Plastic (MPa)
[RSTF] Elastic (MPa)
RSTE (%)
1 2 3 4 5 6 7 8 9 10
Bonded Bonded Bonded Partially Partially Bonded Partially Bonded Partially Bonded
3 3 3 3 3 3 3 3 3 6
– – – 0.2 0.5 – 0.2 – 0.2 –
– – – – – 3.75/1 lm 3.75/1 lm 3.02/1 lm 3.02/1 lm 3.02/1 lm
0.50 0.75 1 1 1 1 1 1 1 0.5
0 0 0 0.50 0.50 0 0.50 0 0.50 0
619.78 931.20 1220.43 406.31 551.55 5319.71 2218.88 5222.38 2118.27 619.78
1907.60 2034.08 2094.79 2094.38 2094.76 6677.18 6677.34 6637.49 6636.18 3816.38
32.49 45.78 58.26 19.40 26.33 79.67 33.23 78.68 31.92 16.24
Debonded Debonded Debonded Debonded
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Table 3 Mechanical properties of fibre, interphase and matrix used in RCSTF and CSTF calculations.
Table 4 Comparison of average fragment length at saturation and percentage debonding for different plasma coated E-glass fibres.
Mechanical properties
E-glass fibre
Carbon fibre
Interphase
Plasma copolymer
Avg. fibre length at saturation (mm)
Debonding (%)
Adhesion
–
Good
76
238.7
3.75
0.40 ± 0.02
2.5 ± 5
Good
0.22 – –
0.2 & 0.37 0.36 – 86.00* – –
50% Acrylic acid 50% 1,7-Octadiene 90% Acrylic acid 10% 1,7-Octadiene Unsized glass 30% Acrylic acid 70% 1,7-Octadiene 100% Octadiene
0.38 ± 0.12
Young’s modulus (GPa) Poissons ratio (L & T) Yield strength (MPa) Cold draw strength (MPa) Coefficient of friction Fibre volume fraction (Vf)a R/ra
0.42 ± 0.03 0.46 ± 0.12
– –
Good Intermediate
0.60 ± 0.19
44.24 ± 20
Poor
3.02
2.40
0.36 80.00* –
0.36 46.00 32.30
0.2 0.2 0.000625 –
– –
– –
– –
–
–
–
40
–
L, longitudinal; T, transverse; R/r, unperturbed resin to fibre radius ratio. * Failure stress of resin. a Vf and R/r used in CSTF.
length. However, in an ideal case when two fragments of equal length are considered with one fragment without and the other with debonding as the case in fragments 3 and 4, it can be seen that RSTF and RSTE values decrease due to debonding. From fragments 4 and 5 it can be see that the computed values of RSTF and RSTE increase with coefficient of friction, as expected. The influence of the interphase can be seen from fragments 6 and 8, where the RSTF and RSTE values are higher for high modulus interphase resin compared to the medium modulus interphase resin. Similarly, the high modulus interphase resin was able to transfer more stress to the debonded fibre fragment than the medium modulus interphase resin resulting in a higher value of RSTF and RSTE (fragments 7 and 9). Furthermore, it can be seen that as the applied strain was increased to 6% for fragment 10, the RSTF remained constant. This is because the interphase adjacent to the fibre would have yielded completely resulting in the fibre reaching its maximum stress, therefore a further increase in stress does not result in any further increase in RSTF. However, RSTE decreases because of normalisation of the RSTF with RSTF elastic obtained with elastic matrix (see Section 4). Hence, it can be concluded that the RCSTF value is an exclusive function of fibre fragment lengths in a fragmentation test. While RSTE is a function of fragment length and applied strain. 5.2. Two-phase system For the two-phase system, we have assumed that a 15 nm functional plasma polymer as a coupling system providing adhesion between E-glass fibres and the epoxy resin matrix. By varying the number of functional carboxylic acid groups in the coating, the degree of adhesion can be tuned. To validate the RCSTF methodology the fragmentation data of Marks [14] have been reanalysed. In the fragmentation experiment, the bonded and debonded lengths of fragments at a range of axial applied strains were measured from digital photographs taken directly through a microscope attached to a mini mechanical test frame. The values of RCSTF and RSTE were calculated by summing the calculations for each individual fragment using the mechanical properties of the fibre and resin matrix given in Table 3. Fragment data for fibres coated with 30%, 50%, 90% Acrylic acid/70%, 50%, 10% 1,7-Octadiene, 100% Octadiene plasma polymers and the uncoated fibres were analysed. In order to evaluate the performance of the new model, results from both models are compared. The fragmentation data are summarised in Table 4. Fig. 2 compares the calculated data for RCSTF and CSTF for the well-adhering system, while Fig. 3 gives the analogous data for the poorly adhering fibres. Generally the value of RCSTF is greater
900 30% Acrylic acid
850
RCSTF and CSTF (MPa)
LT1556/GY298 resin matrix
50% Acrylic acid 90% Acrylic acid
800 750
RCSTF
700
CSTF
650 600 550 500 450 400 1
3
5
7
9
11
13
15
Applied strain (%) Fig. 2. Comparison of revised cumulative stress transfer function (RCSTF) and cumulative stress transfer function (CSTF) methods with applied strain for 30%, 50% and 90% acrylic acid coated fragmentation test data.
900
RCSTF and CSTF (MPa)
High Medium modulus modulus
X Unsized glass
100% Octadiene
800
RCSTF CSTF
700 600 500 400 300 1
3
5
7
9
11
13
15
Applied strain (%) Fig. 3. Comparison of revised cumulative stress transfer function (RCSTF) and cumulative stress transfer function (CSTF) methods with applied strain for unsized glass and 100% octadiene coated fragmentation test data.
than that for CSTF for both. Up to an applied strain of 5% the fragmentation process is determined by the statistics of fibre strength. At higher strains the interfacial stress transfer becomes dominant because of the short fragment length and the onset of debonding phenomena. There is a large difference between the values for the poorly adhering octadiene plasma polymer coated fibres. More interestingly the control, uncoated fibres have similar values of RCSTF and CSTF. Fig. 4 presents the stress transfer efficiencies in terms of RSTE and STE. The values of STE tend to be higher than that for RSTE except for the poorly bonded octadiene plasma polymer coated fibres, where the data is coincident with the RSTE. It is also
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RSTE and STE (%)
80 30% Acrylic acid
70
50% Acrylic acid 90% Acrylic acid X Unsized glass 100% Octadiene
60 50
RSTE 40
STE
30 20 10 0
1
3
5
7
9
11
13
15
Applied strain (%) Fig. 4. Comparison of revised stress transfer efficiency (RSTE) and stress transfer efficiency (STE) methods with applied strain for unsized glass and plasma polymer coated fragmentation test data.
electrolytically oxidised fibre. We have coated the fibres with a curable epoxy resin so that the interphase which forms should be significantly different in properties and dimensions compared to the sized fibre control embedded in the matrix. For the coated fibres, a resin of high or medium modulus was applied to the carbon fibres and cured. The actual yield strength of the coatings could not be obtained because the coupons failed in tension before yield was achieved. Therefore, the yield strength and cold draw strength required for the calculation were assumed to be the failure strength of high and medium modulus resin. In reality local matrix yielding will take place at the interphase because of the low dimensional nature of the interphasal resin. The results of the calculations are given in Fig. 6. It can be seen that the values of RCSTF and RSTE are much higher than those for the uncoated fibre. Fig. 7 shows the variation of stress transfer efficiency with average fragment length. The average fragment length decreases with increase in interphase modulus. 6. Discussion
5.3. Three-phase system
6.1. Analysis of two-phase system Adhesion of plasma polymer functionalised E-glass fibres are discussed in detail elsewhere [10]. From the results presented in Figs. 2 and 3 it can be seen that RCSTF values are higher than the CSTF values at lower applied strains (up to 8%) and in close agreement close to the saturation point. On considering Table 4, it can be seen that 50% acrylic acid plasma copolymer coated E-glass fibre has the shortest fragment length at saturation without debonding,
5000
100 RSTE
4000
The RCSTF and RSTE were calculated from the fragmentation data for the coated and non-coated carbon fibres. The non-coated fibre was sized HTA carbon fibre. We recognise that the sizing is a thin epoxy resin coating, which is intended to be soluble, on an
Uncoated carbon fibre
90
High modulus resin coating Medium modulus resin coating
80
3500
70
3000
60
2500
50
2000
40
1500
30
1000
20
500
10
0
RSTE (%)
RCSTF
4500
RCSTF (MPa)
apparent that these calculations do not differentiate between the differing levels of adhesion, given that the functionality of the plasma polymer provides differing levels of adhesion. The reason for this is that the average fragment length differs for each system at a given applied strain. The well-adhering fibres will have a low average fragment length. Thus in Fig. 5 we have plotted the STE and RSTE values against the average fragment length. Now both RSTE and STE tend to converge on an average fragment length which differs for each system. Thus the minimum average fragment length represents the ineffective length of that fibre and resin combination. Both the RSTE and STE can be extrapolated to give the exactly identical value of ineffective length. Thus the 50% acrylic acid plasma copolymer functional coating gives the best stress transfer and hence adhesion. The poorest adhering octadiene plasma polymer coated fibre has the largest ineffective length. Whereas the values of RSTE and STE can be compared at a given fragment length the ineffective length would appear to be a good measure of interfacial stress transfer or adhesion.
0 1
2
3
4
5
6
7
Applied strain (%) Fig. 6. RCSTF and RSTE with applied strain for fragmentation test data with and without an interphase.
50 30% Acrylic acid 50% Acrylic acid 90% Acrylic acid X Unsized glass 100% Octadiene
RSTE
90
STE
80
30
70
RSTE (%)
RSTE and STE (%)
40
20
10
60 50 40 30
Uncoated carbon fibre HM resin coated carbon fibre LM resin coated carbon fibre
20
0 0.36
0.46
0.56
0.66
0.76
0.86
0.96
Average fragment length (mm)
10 0.4
0.5
0.6
0.7
0.8
0.9
1
Average fragment length (mm) Fig. 5. Comparison of revised stress transfer efficiency (RSTE) and stress transfer efficiency (STE) methods as a function of average fragment length for different fibre surface treatments.
Fig. 7. RCSTF and RSTE with average fragment length for fragmentation test data with and without an interphase.
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A.C. Johnson et al. / Composites: Part A 40 (2009) 449–454
which denotes good adhesion. However, from Fig. 2 the highest RCSTF value was obtained for 30% acrylic acid plasma copolymer coated E-glass fibre which is considered to have intermediate adhesion. This difference is caused by the effect of the fragment length which masks the interfacial contribution. It is evident from this that fragment length has a strong influence on the RCSTF calculations. In Fig. 4, the STE values are higher than the RSTE value except for the 100% octadiene coated fibre, this is because the elastic stress at the interface predicted by the Wu et al. [13] model is higher compared to the prediction by Nairn’s model [7]. In the case of 100% octadiene coated fibre the bonded region of the fibre which forms the effective length is shorter than the fragment from good adhesion. This is the reason for the coincident STE data with the RSTE. The plot of RSTE with average fragment length segregates the good, intermediate and poor interface on the basis of ineffective length. The stress transfer efficiency of the interface appears to be best quantified by extrapolating RSTE to zero to find the minimum fragment length which cannot be loaded to fracture. This equates to the ineffective length of a fibre for reinforcement and is the most appropriate quantity for incorporation into predictive models for composite strength [15]. 6.2. Analysis of three-phase system The modulus of the interphase was shown to be less than the fibre and greater than the bulk matrix using nano-indentation [16]. Therefore, including the interphase in calculations is necessary and requires distinct mechanical properties. From Fig. 6 it can be seen that the RCSTF and RSTE are higher for the high modulus resin coated carbon fibre resulting in better stress transfer compared to the medium modulus and uncoated fibre. The average fragment length at 6% strain using the high modulus resin as matrix was 0.43 ± 0.04 mm and medium modulus resin 0.48 ± 0.02 mm. Since the failure stress of coating resins used in the study were relatively close the variation on the RCSTF values are minimal. The marked difference in the computed values between the coated and uncoated fibre emphasises the requirement of tailoring the interphase. The stress carrying ability of the fibre was enhanced by 58%. From Fig. 7 it can be seen that the fragment length decreases with increase in stress transfer caused by the increase in modulus of the interphase. Because of the low failure strain of the matrix resin saturation in the fragmentation process was not achieved. The chemical and mechanical properties of the interphase will determine the failure mode, debonding of the fibre from the matrix or cracking of the matrix. In a similar manner to the two-phase system, it appears preferable to relate the interfacial response and the role of the interphase to fragment length which cannot be loaded to further fracture. Thus, the extrapolated ineffective length may be a more appropriate parameter. Furthermore, the current numerical calculation considers only the effect of debonding. However, it is necessary, sometimes, to incorporate the effects of matrix cracking after fibre fracture for more precise evaluation of the fragmentation test data. This is the next stage of developing a data reduction model for fragmentation. 7. Conclusion An improved data reduction scheme using the cumulative stress transfer function methodology was developed and successfully implemented in analysing the single fibre fragmentation test for
two-phase system. It was found that the cumulative stress transfer methodology was not very effective in differentiating the level of adhesion with respect to the plasma polymer coating with functional groups. The calculated RCSTF values were higher at the initial stages of the fragmentation test which converges with the CSTF values close to saturation. Consecutively this technique was modified to take into account the presence of an interphase. The stress transfer by the high modulus interphase was higher than the medium modulus interphase. The stress transfer capability of the fibre was enhanced by 58%. This technique can be effectively used for discerning the effects of coating on fibres with known mechanical properties. The ineffective length of a fibre in a particular resin system can be determined and appears to be a better parameter for assessing stress transfer at an interface and through an interphase. Acknowledgements Dr. A.C. Johnson wishes to acknowledge the UK EPSRC (Engineering and Physical Sciences Research Council) for the financial support. We like to thank Dr’s. H. Xu and D.J. Marks for some experimental data. We also like to thank Prof. R.J. Young from UMIST for useful discussions and the late Dr. D. Newman from Advanced Composites Group Ltd., and Cytec Fiberite for supplying fibres and resins for the research. References [1] Tripathi D, Chen F, Jones FR. A comprehsive model to predict the stress fields in a single fibre composite. J Comp Mater 1996;30:1514–38. [2] Tripathi D, Jones FR. Measurement of the load-bearing capability of the fibre/ matrix interface by single fibre fragmentation. Comp Sci Technol 1997;57:925–35. [3] Kelly A, Tyson WR. Tensile properties of fibre reinforced metals: copper/ tungsten and copper/molybdenum. J Mech Phys Solid 1965;13:329–50. [4] Tripathi D, Chen F, Jones FR. The effect of matrix plasticity on the stress fields in a single filament composite and the value of interfacial shear strength obtained from the fragmentation test. Proc Roy Soc A: Math Phys Sci 1996;452:621–53. [5] Lacroix T, Keunings R, Desaeger M, Verpoest I. A new data reduction scheme for the fragmentation testing of polymer composites. J Mater Sci 1995;30:683–92. [6] Tripathi D, Chen F, Jones FR. A Pseudo-energy based method to predict the fibre-matrix adhesion using a single fibre fragmentation test. In: proceedings of ICCM-10 conference 1995. p. 689–96. [7] Nairn JA. A variational mechanics analysis of the stresses around breaks in embedded fibres. Mech Mater 1992;13:131–54. [8] Lopattananon N, Hayes SA, Jones FR. Stress transfer function for interface assessment in composites with plasma copolymer functionalized carbon fibres. J Adhes 2002;78:313–50. [9] Johnson AC, Hayes SA, Jones FR. An improved model including plasticity for the prediction of the stress in fibres with an interface/interphase region. Composites A 2005;36:263–71. [10] Marks DJ, Jones FR. Plasma polymerised coatings for engineered interfaces for enhanced composite performance. Composites A 2002;33:1293–302. [11] Sugihara H, Jones FR. Plasma polymer coatings for improvement of the adhesion of polymer fibres. In: Polymer surface modification and polymer coating by dry process technologies. e.d. Iwamori Satoru, Research Signpost, Trivandrum, India, 2005 p. 127-156. [12] Hayes SA, Lane R, Jones FR. Fibre/matrix stress transfer through a discrete interphase. Part 1: single fibre model composites. Composites A 2001;32:379–89. [13] Wu W, Jacobs E, Verpoest I, Varna J. Variational approach to the stress transfer problem through partially debonded interfaces in a three-phase composite. Comp Sci Technol 1999;59:519–35. [14] Marks DJ, Scaling up of plasma polymer coating for interfacial control in fibre composites, PhD Thesis, Department of Engineering Materials, University of Sheffield, UK, 2002. [15] Rosen BW. Tensile failure of fibrous composites. AIAAJ 1964;2:1985–91. [16] Hodzic A, Stachurski ZH, Kim JK. Nano-indentation of polymer-glass interfaces Part I. Experimental and mechanical analysis. Polymer 2000;41:6895–905.
Contents lists available at ScienceDirect
Composites: Part A journal homepage: www.elsevier.com/locate/compositesa
Data reduction methodologies for single fibre fragmentation test: Role of the interface and interphase Anbu Clemensis Johnson a,b, Simon A. Hayes a, Frank R. Jones a,* a b
Department of Engineering Materials, University of Sheffield, Sir Robert Hadfield Building, Mappin Street, Sheffield S1 3JD, UK Department of Chemical Engineering, Eritrea Institute of Technology, Mai Nefhi, Asmara, P.O. Box 1056, Eritrea
a r t i c l e
i n f o
Article history: Received 8 August 2007 Received in revised form 24 December 2008 Accepted 14 January 2009
Keywords: B. Interface/interphase C. Numerical analysis C. Micro-mechanics Data reduction technique
a b s t r a c t The single fibre fragmentation test (SFFT) is commonly used to characterise the fibre/matrix adhesion. In order to quantify the fibre/matrix adhesion the cumulative stress transfer function (CSTF) methodology was developed so that the elastoplasticity of the matrix could be included in the analysis through the plasticity-effect model [Tripathi D, Chen F, Jones FR. A comprehensive model to predict the stress fields in a single fibre composite. J Comp Mater 1996;30;1514–38., Tripathi D, Jones FR. Measurement of the load-bearing capability of the fibre/matrix interface by single fibre fragmentation. Comp Sci Technol 1997;57:925–35.] The limitations of this technique for the data reduction have been addressed by the use of the Plasticity Model to input the non-linearity of the matrix into methodology for fragmentation of a fibre in a matrix. An improved methodology, known as the revised cumulative stress transfer function (RCSTF) is described. The adhesion of a nanoscale plasma copolymer coated glass/epoxy system has been used to examine this approach to the fragmentation process. This methodology is also extended to account for the presence of an interphase. To validate the three phase model, carbon fibre coated with high and medium modulus epoxy resin were used to simulate fibre/interphase/matrix. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction The single fibre fragmentation test (SFFT) has been widely used to assess the quality of the interface fibre-matrix. Significant effort has been focused on the interface region to enhance the stress transfer capability of the fibres and manufacture durable materials. In the SFFT a single fibre with or without a size or coating is embedded in a resin matrix and a tensile load is applied axially to the fibre. When the applied load exceeds the tensile strength of the fibre, fragmentation is initiated which continues until the fragment lengths are shorter than the critical fibre length (lc) for that system. This occurs because the fragments are too short to be reloaded to fracture. There are several theoretical models in the literature that predict the stress transfer from the polymer matrix to the fibre. However, only a limited number are readily applied to the study of the interface. The most commonly preferred data reduction scheme for SFFT is based on the constant shear model [3]. However, this data reduction scheme has several limitations which have been discussed elsewhere [2,4]. The constant shear model considers that the stress transfer at the interface requires either complete debonding or shear yielding of the matrix. One of the serious limitations in this data reduction technique is insufficient consider* Corresponding author. Tel.: +44 114 222 5477; fax: +44 114 222 5943. E-mail address: f.r.jones@sheffield.ac.uk (F.R. Jones). 1359-835X/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesa.2009.01.008
ation of the various failure events such as fibre/matrix debonding, matrix transverse cracking and shear yielding of the matrix during fragmentation. Lacroix et al. [5] proposed a data reduction scheme which was based on the classical shear lag theory. The cumulative stress transfer function (CSTF) was proposed by Tripathi et al. [1,6] who used the plasticity-effect model to describe the stress profile within a fragment. The plasticity-effect model uses Nairn’s model [7] CSTF to calculate the shear stress profile and hence the point of matrix yield was shown to be effective in differentiating between a range of interfacial behaviour. This data reduction scheme was an improvement over the use of the constant shear model because it included the properties of the matrix and the extent of fibre/matrix debonding. The CSTF technique uses the Nairn’s elastic–elastic variational model to calculate the shear stress profile along a short fibre of given length, which is truncated when the matrix shear strength is matched. Debonding was included by calculating the radial thermal stress from a given value of the friction coefficient in the Coulomb estimation. This shear stress profile is converted into a fragment tensile stress profile to estimate the stress transfer efficiency. Thus, the CSTF value is defined as the average cumulative stress on a fragment. Unfortunately, the CSTF value is strongly dependent on fragment length, so that a poorly bonded fibre which cannot be fragmented further can have a higher CSTF value than a well bonded fibre which has fragmented to a higher degree. Therefore, the stress transfer function was normalised to that for an ideal
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bonded fibre in an elastic matrix which was referred to as stress transfer efficiency (STE) [8]. In this study, we concluded that interphasal properties contribute significantly to the fragmentation process so that any analysis should include the elastic–plastic properties of the interphase, where the shear stress is concentrated. Therefore, the aim of this paper was to examine the revised cumulative stress transfer function (RCSTF) for a study of the stress transfer efficiency in the presence of an interphase, using the plasticity model of Johnson et al. [9]. 2. Materials E-glass fibres (Owens Corning) and HTA carbon fibres (Tenax GmbH) were used in this study. The glass fibres were obtained without water sized film former, silane or other process aids. The glass fibres were coated with 30%, 50%, 90% acrylic acid/70%, 50%, 10% 1,7-octadiene and 100% 1,7-octadiene plasma polymers to form a nanoscale coating. The detailed description of fibre coating using plasma polymerisation (PP) is given elsewhere [10]. The thickness of the coating was shown by X-ray photoelectron spectroscopy to be greater than the analysis depth of 10 nm. Recent measurements using atomic force microscopy (AFM) showed the thickness to be 15.3 nm [11]. The coatings containing 50% and 90% acrylic acid provided a strong interfacial bond whereas the 1,7-octadiene coating provided a poor interface. In order to study the influence of an interphase, HTA carbon fibres (sized) obtained from Cytec Fiberite were coated with a thermoplastic–toughened epoxy resin (high modulus, 3.75 GPa) and medium modulus epoxy resin with modulus 3.02 GPa. The coating process was carried out by moving the glass rod containing the resin very slowly over the fibre to produce a uniform coating. The epoxy resin was cured at 85 °C for 12 h and increased to 180 °C for 2 h. The average thickness of the coating was 1 ± 0.15 lm. Full details of the coating resin system are not available for commercial reasons. These fibres were embedded in a ductile matrix (2.40 GPa) and cured at 80 °C for 4 h, followed by post-curing at 130 °C for 3 h. As shown in Table 1, the ductile matrix used for these experiments was a blend of LY1556 (Vantico, UK) which is a diglycidyl ether of Bisphenol A, and a flexibilising aliphatic epoxy resin Araldite GY298 (Ciba Geigy, UK). The curing agents were Nadic Methylene Anhydride (Stag Polymers and Sealants, UK) and Capcure 3-800 (Henkel-Nopco, UK), a mercaptan terminated polymer. The details of the single fibre fragmentation test are discussed elsewhere [12]. 3. Plasticity model The plasticity model (PLM) was developed by Johnson et al. [9] to calculate the stress in a discontinuous or fractured single fibre in an infinite matrix. The matrix and interphase plastic properties are imposed on the variational model developed by Wu et al. [13]. The variational model is an axisymmetric analysis which determines the elastic stress state around the fibre-breaks and debonded interface of a three-phase composite. The shear stress at the debonded interface was calculated from the coulomb friction law. The mini-
Table 1 Resin formulation used for the ductile 6040 support matrix. Constituent
(pbw*)
LY 1556 GY 298 NMA Capcure 3-800
60 40 63.90 23.93
*
pbw = parts by weight.
misation procedure was applied to both the debonded and the bonded zones simultaneously and the strong interaction of the two zones was correctly described. The tensile stress and the shear stress in the fibre is given by the equations,
rzz ; f ¼ wf wf w0 ; f
ð1Þ
wf nw00 ; f ¼ 2
ð2Þ
srz ; f
Where w is a stress function and wi ; f are stress perturbation functions where the subscript ‘‘i” is the function number. ‘‘f” denotes the fibre, wf is the far field axial stress in the fibre, w0 ; f is the expression for the stresses in the fibre, w00 ; f is the first derivative of the stresses in the fibre. These terms are a complex function of material and geometry and n is the position in the matrix starting from the interface. In the presence of a perfect interface, the plasticity model determines the shear stress at the fibre/matrix interface and the axial stress in the fibre using the variational model by Wu et al. [13]. The shear stress is compared to the shear yield strength of the matrix. If the calculated shear stress is greater than the shear yield strength of the matrix it is replaced by the shear yield strength of the matrix initially and then subsequently with the cold draw strength of the matrix. From the shear stress profile, the tensile stress profile in a fragment was calculated using a force balance.
2p r s dx ¼ p r 2 drf 2 drf ¼ s dx r
ð3Þ ð4Þ
The programme checks whether the axial stress in the fibre with an elasto-plastic matrix has reached 95% of the elastic fibre stress, if not the fragment length was reduced by an infinitely small increment and shear stress at the interface was recalculated and replaced the reduced length with the cold draw strength of the matrix. This process continues until the axial stress in the fibre reaches 95% of the elastic stress of that fragment. The whole procedure is summarised in the flow chart given in Fig. 1. 4. The revised cumulative stress transfer method for the single fibre fragmentation test CSTF methodology was introduced by Tripathi et al. [6] for the estimation of fibre/matrix adhesion. It is based on the assumption that the number and length of the individual fibre fragments in the single fibre fragmentation test specimen depend on the quality of the interface such that a good interface will transfer more stress to the embedded fibre than a poor interface. The calculation requires the properties of the fibre, interphase and matrix including the shear and cold draw strengths, coefficient of friction between the fibre and the matrix or interphase whichever is applicable. The individual fragment lengths with their associated debond lengths at a specific applied strain are experimentally determined and introduced into the calculation. In this methodology, the plasticity model [9] is used to calculate the shear stress for a particular fibre fragment. In the case of a debonded fibre, frictional stress at the interface between the fibre/matrix or fibre/interphase is computed in accordance with Coulomb’s friction law. Thus a shear stress profile is determined; from which the axial stress in the fibre fragment can be computed using the force balance argument. The tensile stress in the fibre fragment is integrated over the length of the fragment to calculate the stress transfer function (STF) and normalised using the fragment length, in order to make STF independent of fragment length. This procedure was followed for all the fragments. The STF value for all individual fragments were summed and normalised with respect to the total number of fragments.
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Input applied stress and mechanical properties of fibre, interphase and matrix Input data Choose one fragment (bonded or de-bonded) from SFFT
[Plasticity model]
Calculate the axial and shear stress at the fibre/matrix interface/interphase region using Wu et al. model [13]
Apply von Mises yield criterion (τ y = σ y / 3 ) and replace all the shear stress value greater than the shear yield strength of the matrix with shear yield strength and cold draw strength
Integrate over the fragment length to calculate the tensile stress in the fibre fragment
Reduce the fragment length, recalculate the shear stress for the fragment, and replace reduced length by cold draw strength
Check if the axial stress in the fibre has reached 95 % of the elastic stress
[RCSTF methodology]
Yes
No
Integrate the tensile stress in the fibre fragment over the length of the fragment divided by fragment length (STF) Repeat the above calculation process for rest of the fragments and add the STF values Cumulative stress transfer function is the total tensile stress transferred to the fibre fragments, which is the sum of STF determined for each fragment divided by total number of fragments
Output data
Fig. 1. Schematic representation of the RCSTF methodology.
This normalised value is known as the cumulative stress transfer function (CSTF). Because of the improvement in the calculation methodology CSTF will be known as Revised CSTF to differentiate the old and new programs. The general strategy for the computation of the RCSTF is given as a flowchart in Fig. 1. FORTRAN 77 was used for computer coding and interfaced with Clear Win+ for graphical interface to view the stress plots directly. The RCSTF is defined as,
Pi¼N R Li RCSTF ¼
i¼1
0
rf ðxÞdx
ð5Þ
Pi¼N
i¼1 Li
Where, rf (x) is the tensile stress in the fibre fragment of length, Li and i = 1, 2, 3. . ..N. The stress transfer efficiency (STE) technique was introduced by Lopattananon et al. [8] by normalising the value of CSTF to one for a perfect interface between an elastic fibre and matrix. Analogously the RSTE can be obtained according to,
STE ¼
½RSTFplastic ½RSTFelastic
100
ð6Þ
Where, [RCSTF]Plastic is the quantity obtained using elasto-plastic nature of the matrix and [RCSTF]Elastic is the quantity obtained assuming a perfect bond and an elastic matrix of the same modulus. 5. Experimental validation of RCSTF methodology 5.1. Validation of RSTF and RSTE methodology The validity of the RCSTF technique was assessed by using a series of given fragment lengths, as shown in Table 2 to examine the effect of fragment length and interphase properties on the computed RSTF and RSTE values. The fibre, matrix and interphase properties used are given in Table 3. The results presented in Table 2 were calculated at a 3% applied strain. It can be seen from Table 2 for fragments 1–3 the RSTF and the RSTE increase with fragment
Table 2 Validation of RSTF and RSTE model using values determined for fibre fragments at an applied strain of 3% except fragment 10 where the applied strain was 6%. Fragment
Bonded / debonded
Applied strain (%)
Interface friction coefficient (l)
Interphase modulus (GPa) / thickness in lm
Fragment length (mm)
Debonded length (mm)
[RSTF] Plastic (MPa)
[RSTF] Elastic (MPa)
RSTE (%)
1 2 3 4 5 6 7 8 9 10
Bonded Bonded Bonded Partially Partially Bonded Partially Bonded Partially Bonded
3 3 3 3 3 3 3 3 3 6
– – – 0.2 0.5 – 0.2 – 0.2 –
– – – – – 3.75/1 lm 3.75/1 lm 3.02/1 lm 3.02/1 lm 3.02/1 lm
0.50 0.75 1 1 1 1 1 1 1 0.5
0 0 0 0.50 0.50 0 0.50 0 0.50 0
619.78 931.20 1220.43 406.31 551.55 5319.71 2218.88 5222.38 2118.27 619.78
1907.60 2034.08 2094.79 2094.38 2094.76 6677.18 6677.34 6637.49 6636.18 3816.38
32.49 45.78 58.26 19.40 26.33 79.67 33.23 78.68 31.92 16.24
Debonded Debonded Debonded Debonded
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Table 3 Mechanical properties of fibre, interphase and matrix used in RCSTF and CSTF calculations.
Table 4 Comparison of average fragment length at saturation and percentage debonding for different plasma coated E-glass fibres.
Mechanical properties
E-glass fibre
Carbon fibre
Interphase
Plasma copolymer
Avg. fibre length at saturation (mm)
Debonding (%)
Adhesion
–
Good
76
238.7
3.75
0.40 ± 0.02
2.5 ± 5
Good
0.22 – –
0.2 & 0.37 0.36 – 86.00* – –
50% Acrylic acid 50% 1,7-Octadiene 90% Acrylic acid 10% 1,7-Octadiene Unsized glass 30% Acrylic acid 70% 1,7-Octadiene 100% Octadiene
0.38 ± 0.12
Young’s modulus (GPa) Poissons ratio (L & T) Yield strength (MPa) Cold draw strength (MPa) Coefficient of friction Fibre volume fraction (Vf)a R/ra
0.42 ± 0.03 0.46 ± 0.12
– –
Good Intermediate
0.60 ± 0.19
44.24 ± 20
Poor
3.02
2.40
0.36 80.00* –
0.36 46.00 32.30
0.2 0.2 0.000625 –
– –
– –
– –
–
–
–
40
–
L, longitudinal; T, transverse; R/r, unperturbed resin to fibre radius ratio. * Failure stress of resin. a Vf and R/r used in CSTF.
length. However, in an ideal case when two fragments of equal length are considered with one fragment without and the other with debonding as the case in fragments 3 and 4, it can be seen that RSTF and RSTE values decrease due to debonding. From fragments 4 and 5 it can be see that the computed values of RSTF and RSTE increase with coefficient of friction, as expected. The influence of the interphase can be seen from fragments 6 and 8, where the RSTF and RSTE values are higher for high modulus interphase resin compared to the medium modulus interphase resin. Similarly, the high modulus interphase resin was able to transfer more stress to the debonded fibre fragment than the medium modulus interphase resin resulting in a higher value of RSTF and RSTE (fragments 7 and 9). Furthermore, it can be seen that as the applied strain was increased to 6% for fragment 10, the RSTF remained constant. This is because the interphase adjacent to the fibre would have yielded completely resulting in the fibre reaching its maximum stress, therefore a further increase in stress does not result in any further increase in RSTF. However, RSTE decreases because of normalisation of the RSTF with RSTF elastic obtained with elastic matrix (see Section 4). Hence, it can be concluded that the RCSTF value is an exclusive function of fibre fragment lengths in a fragmentation test. While RSTE is a function of fragment length and applied strain. 5.2. Two-phase system For the two-phase system, we have assumed that a 15 nm functional plasma polymer as a coupling system providing adhesion between E-glass fibres and the epoxy resin matrix. By varying the number of functional carboxylic acid groups in the coating, the degree of adhesion can be tuned. To validate the RCSTF methodology the fragmentation data of Marks [14] have been reanalysed. In the fragmentation experiment, the bonded and debonded lengths of fragments at a range of axial applied strains were measured from digital photographs taken directly through a microscope attached to a mini mechanical test frame. The values of RCSTF and RSTE were calculated by summing the calculations for each individual fragment using the mechanical properties of the fibre and resin matrix given in Table 3. Fragment data for fibres coated with 30%, 50%, 90% Acrylic acid/70%, 50%, 10% 1,7-Octadiene, 100% Octadiene plasma polymers and the uncoated fibres were analysed. In order to evaluate the performance of the new model, results from both models are compared. The fragmentation data are summarised in Table 4. Fig. 2 compares the calculated data for RCSTF and CSTF for the well-adhering system, while Fig. 3 gives the analogous data for the poorly adhering fibres. Generally the value of RCSTF is greater
900 30% Acrylic acid
850
RCSTF and CSTF (MPa)
LT1556/GY298 resin matrix
50% Acrylic acid 90% Acrylic acid
800 750
RCSTF
700
CSTF
650 600 550 500 450 400 1
3
5
7
9
11
13
15
Applied strain (%) Fig. 2. Comparison of revised cumulative stress transfer function (RCSTF) and cumulative stress transfer function (CSTF) methods with applied strain for 30%, 50% and 90% acrylic acid coated fragmentation test data.
900
RCSTF and CSTF (MPa)
High Medium modulus modulus
X Unsized glass
100% Octadiene
800
RCSTF CSTF
700 600 500 400 300 1
3
5
7
9
11
13
15
Applied strain (%) Fig. 3. Comparison of revised cumulative stress transfer function (RCSTF) and cumulative stress transfer function (CSTF) methods with applied strain for unsized glass and 100% octadiene coated fragmentation test data.
than that for CSTF for both. Up to an applied strain of 5% the fragmentation process is determined by the statistics of fibre strength. At higher strains the interfacial stress transfer becomes dominant because of the short fragment length and the onset of debonding phenomena. There is a large difference between the values for the poorly adhering octadiene plasma polymer coated fibres. More interestingly the control, uncoated fibres have similar values of RCSTF and CSTF. Fig. 4 presents the stress transfer efficiencies in terms of RSTE and STE. The values of STE tend to be higher than that for RSTE except for the poorly bonded octadiene plasma polymer coated fibres, where the data is coincident with the RSTE. It is also
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A.C. Johnson et al. / Composites: Part A 40 (2009) 449–454
RSTE and STE (%)
80 30% Acrylic acid
70
50% Acrylic acid 90% Acrylic acid X Unsized glass 100% Octadiene
60 50
RSTE 40
STE
30 20 10 0
1
3
5
7
9
11
13
15
Applied strain (%) Fig. 4. Comparison of revised stress transfer efficiency (RSTE) and stress transfer efficiency (STE) methods with applied strain for unsized glass and plasma polymer coated fragmentation test data.
electrolytically oxidised fibre. We have coated the fibres with a curable epoxy resin so that the interphase which forms should be significantly different in properties and dimensions compared to the sized fibre control embedded in the matrix. For the coated fibres, a resin of high or medium modulus was applied to the carbon fibres and cured. The actual yield strength of the coatings could not be obtained because the coupons failed in tension before yield was achieved. Therefore, the yield strength and cold draw strength required for the calculation were assumed to be the failure strength of high and medium modulus resin. In reality local matrix yielding will take place at the interphase because of the low dimensional nature of the interphasal resin. The results of the calculations are given in Fig. 6. It can be seen that the values of RCSTF and RSTE are much higher than those for the uncoated fibre. Fig. 7 shows the variation of stress transfer efficiency with average fragment length. The average fragment length decreases with increase in interphase modulus. 6. Discussion
5.3. Three-phase system
6.1. Analysis of two-phase system Adhesion of plasma polymer functionalised E-glass fibres are discussed in detail elsewhere [10]. From the results presented in Figs. 2 and 3 it can be seen that RCSTF values are higher than the CSTF values at lower applied strains (up to 8%) and in close agreement close to the saturation point. On considering Table 4, it can be seen that 50% acrylic acid plasma copolymer coated E-glass fibre has the shortest fragment length at saturation without debonding,
5000
100 RSTE
4000
The RCSTF and RSTE were calculated from the fragmentation data for the coated and non-coated carbon fibres. The non-coated fibre was sized HTA carbon fibre. We recognise that the sizing is a thin epoxy resin coating, which is intended to be soluble, on an
Uncoated carbon fibre
90
High modulus resin coating Medium modulus resin coating
80
3500
70
3000
60
2500
50
2000
40
1500
30
1000
20
500
10
0
RSTE (%)
RCSTF
4500
RCSTF (MPa)
apparent that these calculations do not differentiate between the differing levels of adhesion, given that the functionality of the plasma polymer provides differing levels of adhesion. The reason for this is that the average fragment length differs for each system at a given applied strain. The well-adhering fibres will have a low average fragment length. Thus in Fig. 5 we have plotted the STE and RSTE values against the average fragment length. Now both RSTE and STE tend to converge on an average fragment length which differs for each system. Thus the minimum average fragment length represents the ineffective length of that fibre and resin combination. Both the RSTE and STE can be extrapolated to give the exactly identical value of ineffective length. Thus the 50% acrylic acid plasma copolymer functional coating gives the best stress transfer and hence adhesion. The poorest adhering octadiene plasma polymer coated fibre has the largest ineffective length. Whereas the values of RSTE and STE can be compared at a given fragment length the ineffective length would appear to be a good measure of interfacial stress transfer or adhesion.
0 1
2
3
4
5
6
7
Applied strain (%) Fig. 6. RCSTF and RSTE with applied strain for fragmentation test data with and without an interphase.
50 30% Acrylic acid 50% Acrylic acid 90% Acrylic acid X Unsized glass 100% Octadiene
RSTE
90
STE
80
30
70
RSTE (%)
RSTE and STE (%)
40
20
10
60 50 40 30
Uncoated carbon fibre HM resin coated carbon fibre LM resin coated carbon fibre
20
0 0.36
0.46
0.56
0.66
0.76
0.86
0.96
Average fragment length (mm)
10 0.4
0.5
0.6
0.7
0.8
0.9
1
Average fragment length (mm) Fig. 5. Comparison of revised stress transfer efficiency (RSTE) and stress transfer efficiency (STE) methods as a function of average fragment length for different fibre surface treatments.
Fig. 7. RCSTF and RSTE with average fragment length for fragmentation test data with and without an interphase.
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which denotes good adhesion. However, from Fig. 2 the highest RCSTF value was obtained for 30% acrylic acid plasma copolymer coated E-glass fibre which is considered to have intermediate adhesion. This difference is caused by the effect of the fragment length which masks the interfacial contribution. It is evident from this that fragment length has a strong influence on the RCSTF calculations. In Fig. 4, the STE values are higher than the RSTE value except for the 100% octadiene coated fibre, this is because the elastic stress at the interface predicted by the Wu et al. [13] model is higher compared to the prediction by Nairn’s model [7]. In the case of 100% octadiene coated fibre the bonded region of the fibre which forms the effective length is shorter than the fragment from good adhesion. This is the reason for the coincident STE data with the RSTE. The plot of RSTE with average fragment length segregates the good, intermediate and poor interface on the basis of ineffective length. The stress transfer efficiency of the interface appears to be best quantified by extrapolating RSTE to zero to find the minimum fragment length which cannot be loaded to fracture. This equates to the ineffective length of a fibre for reinforcement and is the most appropriate quantity for incorporation into predictive models for composite strength [15]. 6.2. Analysis of three-phase system The modulus of the interphase was shown to be less than the fibre and greater than the bulk matrix using nano-indentation [16]. Therefore, including the interphase in calculations is necessary and requires distinct mechanical properties. From Fig. 6 it can be seen that the RCSTF and RSTE are higher for the high modulus resin coated carbon fibre resulting in better stress transfer compared to the medium modulus and uncoated fibre. The average fragment length at 6% strain using the high modulus resin as matrix was 0.43 ± 0.04 mm and medium modulus resin 0.48 ± 0.02 mm. Since the failure stress of coating resins used in the study were relatively close the variation on the RCSTF values are minimal. The marked difference in the computed values between the coated and uncoated fibre emphasises the requirement of tailoring the interphase. The stress carrying ability of the fibre was enhanced by 58%. From Fig. 7 it can be seen that the fragment length decreases with increase in stress transfer caused by the increase in modulus of the interphase. Because of the low failure strain of the matrix resin saturation in the fragmentation process was not achieved. The chemical and mechanical properties of the interphase will determine the failure mode, debonding of the fibre from the matrix or cracking of the matrix. In a similar manner to the two-phase system, it appears preferable to relate the interfacial response and the role of the interphase to fragment length which cannot be loaded to further fracture. Thus, the extrapolated ineffective length may be a more appropriate parameter. Furthermore, the current numerical calculation considers only the effect of debonding. However, it is necessary, sometimes, to incorporate the effects of matrix cracking after fibre fracture for more precise evaluation of the fragmentation test data. This is the next stage of developing a data reduction model for fragmentation. 7. Conclusion An improved data reduction scheme using the cumulative stress transfer function methodology was developed and successfully implemented in analysing the single fibre fragmentation test for
two-phase system. It was found that the cumulative stress transfer methodology was not very effective in differentiating the level of adhesion with respect to the plasma polymer coating with functional groups. The calculated RCSTF values were higher at the initial stages of the fragmentation test which converges with the CSTF values close to saturation. Consecutively this technique was modified to take into account the presence of an interphase. The stress transfer by the high modulus interphase was higher than the medium modulus interphase. The stress transfer capability of the fibre was enhanced by 58%. This technique can be effectively used for discerning the effects of coating on fibres with known mechanical properties. The ineffective length of a fibre in a particular resin system can be determined and appears to be a better parameter for assessing stress transfer at an interface and through an interphase. Acknowledgements Dr. A.C. Johnson wishes to acknowledge the UK EPSRC (Engineering and Physical Sciences Research Council) for the financial support. We like to thank Dr’s. H. Xu and D.J. Marks for some experimental data. We also like to thank Prof. R.J. Young from UMIST for useful discussions and the late Dr. D. Newman from Advanced Composites Group Ltd., and Cytec Fiberite for supplying fibres and resins for the research. References [1] Tripathi D, Chen F, Jones FR. A comprehsive model to predict the stress fields in a single fibre composite. J Comp Mater 1996;30:1514–38. [2] Tripathi D, Jones FR. Measurement of the load-bearing capability of the fibre/ matrix interface by single fibre fragmentation. Comp Sci Technol 1997;57:925–35. [3] Kelly A, Tyson WR. Tensile properties of fibre reinforced metals: copper/ tungsten and copper/molybdenum. J Mech Phys Solid 1965;13:329–50. [4] Tripathi D, Chen F, Jones FR. The effect of matrix plasticity on the stress fields in a single filament composite and the value of interfacial shear strength obtained from the fragmentation test. Proc Roy Soc A: Math Phys Sci 1996;452:621–53. [5] Lacroix T, Keunings R, Desaeger M, Verpoest I. A new data reduction scheme for the fragmentation testing of polymer composites. J Mater Sci 1995;30:683–92. [6] Tripathi D, Chen F, Jones FR. A Pseudo-energy based method to predict the fibre-matrix adhesion using a single fibre fragmentation test. In: proceedings of ICCM-10 conference 1995. p. 689–96. [7] Nairn JA. A variational mechanics analysis of the stresses around breaks in embedded fibres. Mech Mater 1992;13:131–54. [8] Lopattananon N, Hayes SA, Jones FR. Stress transfer function for interface assessment in composites with plasma copolymer functionalized carbon fibres. J Adhes 2002;78:313–50. [9] Johnson AC, Hayes SA, Jones FR. An improved model including plasticity for the prediction of the stress in fibres with an interface/interphase region. Composites A 2005;36:263–71. [10] Marks DJ, Jones FR. Plasma polymerised coatings for engineered interfaces for enhanced composite performance. Composites A 2002;33:1293–302. [11] Sugihara H, Jones FR. Plasma polymer coatings for improvement of the adhesion of polymer fibres. In: Polymer surface modification and polymer coating by dry process technologies. e.d. Iwamori Satoru, Research Signpost, Trivandrum, India, 2005 p. 127-156. [12] Hayes SA, Lane R, Jones FR. Fibre/matrix stress transfer through a discrete interphase. Part 1: single fibre model composites. Composites A 2001;32:379–89. [13] Wu W, Jacobs E, Verpoest I, Varna J. Variational approach to the stress transfer problem through partially debonded interfaces in a three-phase composite. Comp Sci Technol 1999;59:519–35. [14] Marks DJ, Scaling up of plasma polymer coating for interfacial control in fibre composites, PhD Thesis, Department of Engineering Materials, University of Sheffield, UK, 2002. [15] Rosen BW. Tensile failure of fibrous composites. AIAAJ 1964;2:1985–91. [16] Hodzic A, Stachurski ZH, Kim JK. Nano-indentation of polymer-glass interfaces Part I. Experimental and mechanical analysis. Polymer 2000;41:6895–905.