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Fracture toughness of nano- and micro-spherical silica-particle-filled epoxy composites
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Acta Materialia 56 (2008) 2101–2109 www.elsevier.com/locate/actamat
Fracture toughness of nano- and micro-spherical silica-particle-filled epoxy composites Tadaharu Adachi *, Mayuka Osaki 1, Wakako Araki, Soon-Chul Kwon 2 Department of Mechanical Sciences and Engineering, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8552, Japan Received 5 December 2006; received in revised form 1 October 2007; accepted 2 January 2008 Available online 4 March 2008
Abstract The effects of particle size and volume fraction on the mode I fracture toughness of epoxy composites filled with spherical silica particles were investigated through experiments and an analytical model. Spherical silica particles with various particle diameters ranging from 1.56 lm to 240 nm and volume fractions from 0 to 0.35 were added to epoxy resin. We found that the fracture toughness of the composites was approximately linear with respect to the reciprocal of the product of (i) the square root of the mean distance between the particle surfaces and (ii) the normalized mean stress in the matrix given by the equivalent inclusion method. Ó 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Particulate reinforced composites; Nanocomposite; Polymer matrix composites; Fracture; Micromechanical modeling
1. Introduction Polymer materials can be reinforced by filling with several kinds of particles or fibers to make composite materials. The particle size in composites is currently crossing below the micrometer level to form nanocomposites that have superior mechanical properties. It is well known that the size and content of particles strongly affect the mechanical properties of composites [1,2], especially their fracture toughness [3,4]. Several researchers have considered the effects of particle size or volume fraction on the fracture toughness of polymer matrix composites [5–17]. Almost all of them have discussed the effects based on experimental results only. Consequently, general explanations about the effects have
*
Corresponding author. Tel.: +81 3 5734 2156; fax: +81 3 5734 2893. E-mail address: [email protected] (T. Adachi). 1 Present address: Production Engineering Research Laboratory, Hitachi Ltd., 292 Yoshida-cho, Totsuka-ku, Yokohama-shi, Kanagawaken 244-0817, Japan. 2 Present address: Semiconductor Material R&D Center, Samsung Techwin Co. Ltd., San No. 14, Nongseo-Ri, Giheung-Eup, Yongin-Si, Gyeonggi-Do 446-712, Republic of Korea.
not been clear because not only the particle and matrix materials, but also the size and volume fraction of the particles have been very different in these studies. Some researchers have considered these effects on the fracture toughness of composites using analytical models based on crack propagation along the particle surfaces. Taking the interspacing between particles [18] into account, Lange and Radford [19], Evans [20], Green et al. [21], Bower and Ortiz [22,23], Cui et al. [24] and Qiao [25] mathematically considered trapping, pinning and bridging of the crack front at the particles. It can be predicted that a crack in a composite does not always propagate along interfaces between particles and matrix. In particular, fracture phenomena in composites filled with nanometer- or micrometer-sized particles may be different from those in composites filled with larger particles. In this research, the effects of particle size and volume fraction on the mode I fracture toughness of epoxy composites filled with spherical silica particles were investigated. First, composite materials to be used as specimens were prepared by mixing bisphenol A type epoxy resin and spherical silica particles of various particle diameters, ranging from microsized to nanosized – i.e. 1.56 lm to 240 nm – and volume fractions from 0 to 0.35. The fracture
1359-6454/$34.00 Ó 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2008.01.002
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toughness of the composites was measured. The fracture surface of each specimen was observed with a scanning electron microscope (SEM) and a microscopic Raman spectrometer to determine clearly the crack path in the composite. Stress fields near the crack tip were analyzed approximately based on observing the fracture surfaces with the equivalent inclusion method and linear elastic fracture mechanics, taking into account the statistical distribution of particles to obtain the fracture toughness of specimens. Finally, the effects of particle size and volume fraction on the fracture toughness were formulated based on the experimental and analytical results. 2. Specimens Epoxy resin matrix filled with spherical particles of fused silica was used as the material for the specimens. The epoxy resin in the experiment was bisphenol A type epoxide resin (Japan Epoxy Resin, Epikote 828) with methyl-tetrahydro-phthalic anhydride as the curing agent (Hitachi Chem., HN-2200) and 2,4,6-tris (dimethyl aminomethyl) phenol as the accelerator (Daito Curar, DMP-30). The weight ratio of the resin, agent and accelerator was 100:80:0.5. The fillers were spherical silica particles with three diameters, as shown in Fig. 1. The median diameters of the particles Dm were 1.56 lm (Tatsumori, SO-C5), 560 nm (Tatsumori, SO-C2) and 240 nm (Tatsumori, 1-FX). The distributions for the particle diameters normalized by the median diameters are plotted in Fig. 2. These normalized distributions were approximately the same. The mechanical properties of the epoxy resin and silica particles are listed in Table 1 [17,26,27]. The surfaces of the particles were not coated chemically. Hereafter, the median diameter of the particles will be called the particle diameter for short, unless there is the possibility of confusion. After the epoxy resin and the silica particles were combined in a mixing machine for more than 30 min, the mixture was stored in a vacuum vessel to remove voids and was poured into an aluminum mold coated with a Teflon sheet, which was set up in an oven. The mold was 260 mm long, 5 mm wide and 180 mm deep. The volume fractions of the silica particles were within 0 to 0.35. The curing was done in two steps. First, the composite was kept at 353 K for 3 h to gel the matrix resin (pre-curing). The second step, post-curing, which greatly affects the cross-linking reaction of the resin, was done at 413 K for 10 h. The heating rate from pre-curing to post-curing was constant at 72 K h1. After curing, we found that the particles had dispersed adequately without condensation and there were no voids in the composites. The glass transition temperature of each composite was identified from the shift factor of the thermo-viscoelastic properties [17,26] to verify the degree of cross-linking as the cured states of the specimens. Neat epoxy resin and 11 different composites were prepared, as listed in Table 2. We confirmed that the epoxy
Fig. 1. Silica particles: (a) SO-C5 (median diameter Dm = 1.56 lm); (b) SO-C2 (median diameter Dm = 560 nm); and (c) 1-FX (median diameter Dm = 240 nm).
and every composite had been completely cured from the glass transition temperature. For each composite, the distance between the surfaces of the nearest particles was analyzed to obtain the statistical particle dispersion. We assumed that the particles had the same constant diameter with a median diameter of Dm and were distributed with volume fraction / in the matrix resin three-dimensionally and randomly. The mean dis-
T. Adachi et al. / Acta Materialia 56 (2008) 2101–2109
10.0
Table 2 Specimens
100 Cumulative volume
Volume [%]
60 5.0
Volume
40 Dm = 1.56 ∝m
2.5
0
b
20
0
1 2 3 4 5 Normalized particle diameter, D/Dm
15.0
Volume [%]
60 Volume
40 Dm = 560 nm
20
2.5 0
1 2 3 4 5 Normalized particle diameter, D/Dm
10.0
0
100 80
7.5
60 Volume
5.0
40 Dm = 240 nm
2.5
0
20
0
1
2
3
4
5
Cumulative volume [%]
Cumulative volume
Volume [%]
Cumulative volume [%]
10.0
0
0
Normalized particle diameter, D/Dm Fig. 2. Diameter distributions of silica particles: (a) SO-C5 (median diameter Dm = 1.56 lm); (b) SO-C2 (median diameter Dm = 560 nm); and (c) 1-FX (median diameter Dm = 240 nm).
Table 1 Mechanical properties of epoxy resin and silica particles
Silica particles Epoxy resin
Epoxy Composite
Median diameter Dm
Volume fraction /
– 1.56 lm
0 0.057 0.186 0.301 0.351 0.057 0.184 0.347 0.056 0.118 0.187 0.301
560 nm
100 80
5.0
Particle
240 nm
Cumulative volume
7.5
Specimen
0
12.5
c
Cumulative volume [%]
80
7.5
Density (kg m3)
Young’s modulus (GPa)
Shear modulus (GPa)
Poisson’s ratio
2.2 1.2
72.1 3.8
30.5 1.4
0.18 0.38
Normalized distance between surfaces of nearest particles,/Dm
a
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Glass transition temperature (K)
399 397 391 399 393 397 393 393 393 393 388 399
3
2
Close-packed structure
1
0
0.2
0.4
0.6
0.8
Volume fraction, φ
Fig. 3. Distance between surfaces of nearest particles.
where function C(a, n) is an incomplete gamma function. From Eq. (1), distance hLai is proportional to particle diameter Dm and dependent on volume fraction /. The distribution of particles in the matrix was similar for the same volume fraction. The computational results for Eq. (1) were plotted as far as the maximum volume fraction pffiffiffi for a hexagonal or cubic close-packed structure, i.e. 2p=6 in Fig. 3. Distance hLai at a volume fraction of 0.055 was equal to the p particle diameter Dm and the shortest distance ffiffiffi hLai at / ¼ 2p=6 was 0.134Dm. Because volume fractions / in the experiments ranged from 0.06 to 0.35, hLai was below Dm for every specimen. 3. Experimental procedures 3.1. Measurement of elastic properties
tance between the surfaces of the nearest particles, hLai, was given statistically by Bansal and Ardell [28] as Z 6/ Dm Cð0:5; nÞ expðnÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi hLa i ¼ dn ð1Þ 24/ 0 6/ n
A tensile test at room temperature (298 K) was conducted with a universal testing machine (Instron, 8501). The specimens had a length of 120 mm, a width of 10 mm and a thickness of 5 mm. Longitudinal strain was measured using a strain gage at the midpoint of each
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specimen. The rate of tensile deformation was 2 lm s1. Young’s modulus was evaluated from the strain and tensile load. The average value calculated from five measured results was used as the elastic modulus and the standard deviation was also evaluated. 3.2. Measurement of fracture toughness A three-point bending test on each precracked specimen was conducted to measure the mode I fracture toughness at room temperature (298 K) according to ASTM standard D5045 (Fig. 4). The specimens had a length of 90 mm, a width of 20 mm and a thickness of 5 mm. The lengths of the precrack and span were 10 mm and 80 mm, respectively. The tests were carried out under a constant displacement rate of 2 lm s1 at the loading point with a universal testing machine (Instron, 8501). A brittle fracture occurred in every specimen. The stress intensity factor, KIC, when the specimen broke was defined as the fracture toughness [29]: K IC ¼
SP c f ðnÞ BW 3=2
ð2Þ
where f ðnÞ ¼ n¼
3n1=2 f1:99 nð1 nÞð2:15 3:93n þ 2:7n2 Þg 2ð1 þ 2nÞð1 nÞ
3=2
;
a W
and Pc is the maximum load. S, B, W and a are the span length, the thickness, the width and the precrack length of the specimen, respectively. The average value, excluding the maximum and minimum values from seven experimental results, was used as the fracture toughness and the standard deviation was also evaluated. After the experiments, the fracture surfaces of specimens were observed with an SEM (JEOL, JSM-6301F) and a microscopic Raman spectrometer (JASCO, NRS-1000). The wavelength of the source laser in the Raman spectrometer was 532.3 nm and its measurement region was a circle of diameter 8 lm.
4. Experimental results 4.1. Elastic modulus The relations between Young’s modulus and particle volume fraction are shown in Fig. 5. The triangles and circles denote the averages of measured values. The standard deviations of the Young’s moduli were not plotted in Fig. 5 because the values for all specimens were within 2%. As is well known, Young’s modulus increases with increasing volume fraction. We did not find any effects of particle size on Young’s modulus, but we confirmed that the volume fraction governed it. The modulus at a volume fraction of 0.35 was twice that of neat epoxy resin. Typical mixture laws proposed by Kerner [30], Hashin and Shtrikman [31], Lewis and Nielsen [32] and Nielsen [33], denoted as solid lines in Fig. 5, were compared with the experimental results. Hashin and Shtrikman’s equation was based on the equivalent inclusion method. Lewis and Nielsen’s equation was modified from Kerner’s equation. These theoretical results coincide with our experimental results. Lewis and Nielsen’s equation agreed well with the experimental results because the distribution of particle diameters was considered in the equation. Lewis and Nielsen’s equation can also be applied to the thermo-viscoelastic properties of epoxy composites bi-dispersed by silica particles [28]. Hashin and Shtrikman’s equation, which was the result of the equivalent inclusion method, was also in agreement with the experimental results. 4.2. Fracture toughness The relation between the fracture toughness of composites and the volume fraction of the particles is shown in Fig. 6. The triangles and circles indicate the averages of measured values and the error bars indicate their standard deviations. The solid lines denote fitting for each composite with particles of the same size. The fracture toughness increased as the volume fraction increased. Although the particle diameter had little influence on the fracture toughness for a low volume fraction, the fracture toughness was
Young's modulus, Ec [GPa]
10 Lewis & Nielsen [32, 33] Kerner [30] Hashin & Shtrikman [31]
8 6 4
Particle diameter Dm = 1.56 μm Dm = 560 nm Dm = 240 nm Epoxy
2
0
0.1
0.2
0.3
0.4
Volume fraction, φ Fig. 4. Fracture toughness test (units are millimetres).
Fig. 5. Relation between Young’s modulus and volume fraction.
Fracture toughness, KIC [MPa m1/2]
T. Adachi et al. / Acta Materialia 56 (2008) 2101–2109
3
2
Particle diameter Dm = 1.56 ∝m Dm = 560 nm Dm = 240 nm Epoxy
1
0
0
0.1
0.2
0.3
0.4
Volume fraction, φ Fig. 6. Relation between fracture toughness and volume fraction.
strongly dependent on the particle diameter for a high volume fraction. The fracture toughness for a particle diameter of 1.56 lm increased for a volume fraction of up to 0.19 and saturated for a larger volume fraction. The fracture toughness for a particle diameter of 560 nm increased slightly up to the highest volume fraction. The rate of increase was largest for a particle diameter of 240 nm. The fracture toughness at a volume fraction of 0.309 for
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a particle diameter of 240 nm was 1.7 times that for a particle diameter of 1.56 lm and 2.5 times that for epoxy resin. The fracture toughness depended greatly on both the volume fraction and particle diameters, compared with the elastic moduli in Fig. 5. For composites with a low particle volume fraction (<0.1), particle size had little effect on fracture toughness. The size effect strongly affected the fracture toughness in the high volume fraction region. Fracture surfaces observed with an SEM are shown in Fig. 7. The roughness of the fracture surface decreased with decreasing particle size. The fracture surface became fine for a higher volume fraction and for smaller particles. Raw particles did not appear to be exposed on any surface. However, from Fig. 7, it is difficult to clearly find which cracks propagated in the matrix, through particles, or along the interface between the particles and the polymer matrix. The fracture surfaces were analyzed by microscopic Raman spectroscopy to obtain the route of the crack propagation path. The fracture surfaces of specimens were analyzed more than five times at a distance of approximately 3 mm in front of the initial crack tip of the specimen, where several particles were near the surface within view of a Raman spectrometer with a CCD camera attached. The Raman spectrum for the composite with a particle diameter of
Fig. 7. Fracture surfaces.
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rials. We concluded that silica particles had not been exposed on the fracture surface. Therefore, the cracks in all specimens propagated in the matrix resin. Crack propagation has been explained by several researchers as being due to the mechanisms of trapping, pinning and bridging of the crack front at particles [19– 25]. In Figs. 7 and 8, we can see cracks propagated by avoiding particles. Therefore, it would not be appropriate to apply these mechanisms to composites filled with small particles unless the volume fraction were very high.
Epoxide ring
Intensity [a.u.]
Epoxy Si-O-Si Defect of SiO2
Silica
5. Discussion
Composite
5.1. Concept of analysis
200
400
600
800
1000
Raman shift [cm-1] Fig. 8. Raman spectroscopy of fracture surfaces. Particle diameter Dm = 1.56 lm and volume fraction / = 0.186.
1.56 lm and a volume fraction of 0.19 is shown in Fig. 8. The upper line is the spectrum for epoxy resin and the middle line is that for silica particles. The characteristic peaks for epoxy resin [34] and silica [35] in these spectra are indicated by circles. The lower spectrum in Fig. 8 has the results that characterize the composite. We found only peaks for the epoxy resin in the composite’s spectrum and could not find peaks for silica particles. The same results were obtained for the fracture surfaces of other composite mate-
The effects of both particle size and volume fraction on the fracture toughness of a composite were formulated based on reinforcement of the matrix by filler particle and the particle distribution in the matrix. We considered that the mode I fracture toughness of a composite material, K COMP , could be expressed by the sumIC mation of (i) the fracture toughness of a neat matrix resin, EM K MAT IC , and (ii) the improvement in fracture toughness, K IC , provided by reinforcement by particles, as outlined in Fig. 9. The improvement, K EM IC , is attributed to the effect of stress in the matrix interspace between particles and the particle distribution was obtained in Section 2. The model for improvement is shown in Fig. 9(c). It is called the equivalent matrix model (see Section 5.2). It is composed of only the matrix resin without particles and the stress field in the matrix was considered to be caused by
Fig. 9. Analytical model.
T. Adachi et al. / Acta Materialia 56 (2008) 2101–2109
ð3Þ
5.2. Equivalent matrix model Suppose that external stress, r0, is applied to a composite with a crack that is a in length, as shown in Fig. 9. The stress concentration near a particle is slight because individual particles are much closer even for the low volume fraction of particles in Fig. 3. Assuming that tensile stress rm is uniform near the crack tip, the equivalent matrix model can be considered, as described in Fig. 9(c), because the deformation of composites can be expressed approximately by Hashin and Shtrikman’s equation [30], which is the equivalent inclusion method described in Fig. 5. Mean stress rm in the matrix (Fig. 9(c)) is then expressed by external stress, r0, of the composite according to the equivalent inclusion method [36]: rm ¼ r0 gð/Þ
ð4Þ
where function g(/), given in the Appendix, is the normalized mean stress dependent on the elastic moduli of particles and the matrix and the volume fraction /. The mean stress in the matrix is linear with respect to the volume fraction of the particles without the size effect of particles. The numerical results of Eq. (4) for a composite of epoxy and spherical silica particles are derived from the elastic properties in Table 1 as rm ¼ r0 ð1 0:825/Þ
ð5Þ
The minimum value forpmean stress is 0.39r0 at the maxiffiffiffi mum volume fraction, 2p=6. Then, the stress in the matrix resin is reduced by the reinforcement provided by the particles. Particle distribution is similar because mean distance between the surfaces of the nearest particles, hLa i, is proportional to particle diameter for a constant volume fraction due to Eq. (1). Considering the similarity in particle distributions, stress field rij in the matrix can be expressed as a function of xk/hLai: rij ¼ rij ðxk =hLa iÞ
ð6Þ
where xk is the coordinate in Fig. 9(c). Based on Eqs. (4) and (6), tensile stress r22 in the vicinity of a crack tip applied by mean stress rm in the interspace of the matrix region between particles can be analyzed because cracks were confirmed to propagate in the matrix from the results of Figs. 7 and 8: pffiffiffiffiffiffi rm pa ð7Þ r22 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f 2px1 =hLa i where f is a coefficient independent of hLai. If the stress intensity factor given by Eq. (7) coincides with the fracture toughness of the matrix resin, K MAT IC , then the crack propagates through the matrix:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pahLa if
ð8Þ
Fracture toughness provided by reinforcement by particles, K EM IC , can also be evaluated using far field stress r0 as follows: pffiffiffiffiffiffi ð9Þ K EM IC ¼ r0 paF where F is the shape function related to the boundary condition of the specimen. Substituting Eq. (8) into Eq. (9), we obtain K EM IC ¼
1 F pffiffiffiffiffiffiffiffiffi K MAT IC gð/Þ hLa i f
ð10Þ
Finally, the fracture toughness of the composite, K COMP , IC can be derived by Eqs. (3) and (10) as ! a COMP pffiffiffiffiffiffiffiffiffi K MAT K IC ¼ 1þ ð11Þ IC gð/Þ hLa i where a = F/f. This equation shows that the fracture toughness of the composite is proportional to g(/)1 hLai1/2 when a crack propagates in the matrix. For a neat matrix resin, i.e. when hLai is infinite, the fracture toughness of the composite, K COMP , coincides with K MAT IC IC . 5.3. Comparison of experimental and analytical results The analytical results computed with Eq. (11) are compared with the experimental results in Fig. 10. The horizontal and vertical lines denote parameters g(/)1hLai1/2 computed for each composite material and the fracture toughness obtained in the experiment. The triangles and circles indicate the average of measured values and the error bars indicate their standard deviations. The experimental results are linear with respect to parameter g(/)1hLai1/2. The agreement between the analytical results and Eq. (11) was confirmed. Therefore, the effect of particle size on fracture toughness can be expressed as the reciprocal of the square root of the mean distance between the surfaces of the nearest particles and the effect of volume fraction can be expressed as the reciprocal of the normalized mean stress in the matrix given by the 1/2
¼ K MAT þ K EM K COMP IC IC IC
¼ rm K MAT IC
Fracture toughness, KIC [MPa m ]
the effects of both volume fraction and particle size. The fracture toughness of the composite, K COMP , is then given IC mathematically by
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3
2
Particle diameter Dm = 1.56 μm Dm = 560 nm Dm = 240 nm Epoxy
1
0
0
2
4
6
g(φ)-1-1/2 [μm-1/2] Fig. 10. Effect of reinforcement by particles on fracture toughness.
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equivalent inclusion method. From the fitting line for the experimental data in Fig. 10, the coefficient a was obtained as 0.27 lm1/2 in Eq. (11). Coefficient a is directly independent of the particle size and the distance between particle surfaces, as previously mentioned. It is probably a parameter of the matrix material, more specifically the molecular behaviors and the interaction of the particles with the cross-linking structure of the polymer. This needs further consideration using experimental results obtained from other composites filled with particles, though coefficient a is important and interesting, especially for nanocomposites. Therefore, Eq. (11) can be efficiently used to design and tailor particle-reinforced composites having appropriate fracture toughness.
rm ¼ r0 gð/Þ ¼ r0 f1 /C 22ij ðS ijkl ekl eij Þg where g (/) is the normalized mean stress. C22ij represents the elastic moduli in a matrix defined as C 2211 ¼ C 2233 ¼
Gm mm ; 1 2mm
C 2222 ¼
2Gm ð1 mm Þ ; 1 2mm
C 22ij ¼ 0ði 6¼ jÞ where G and m are the shear modulus and Poisson’s ratio, respectively, and superscript m denotes the properties of the matrix. Sijkl denotes Eshelby’s tensor for a sphere, 7 5mm ; S 1122 ¼ S 1133 ¼ S 2211 15ð1 mm Þ 1 þ 5mm ¼ S 3322 ¼ 15ð1 mm Þ
S 1111 ¼ S 2222 ¼ S 3333 ¼ ¼ S 2233 ¼ S 3311
6. Conclusions
The eigen strain, eij , given by The effects of particle diameter and volume fraction on mode I fracture toughness of spherical-silica-particle-filled epoxy composites were investigated through experiments and an analytical model. The particle diameters ranged from 240 nm to 1.56 lm and the volume fraction ranged from 0 to 0.35. In the experiments, we found that the fracture toughness increased drastically as the volume fraction increased and the particle diameters decreased. In composites with a low volume fraction of particles, the volume fraction affected the fracture toughness and in composites with high volume fractions, the particle size affected the fracture toughness more. The stress field in the matrix near the crack tip was approximately analyzed based on the equivalent inclusion method and linear elastic fracture mechanics, taking into account the statistical distribution of particles so that the effects of particle size and volume fraction on the fracture toughness of the composite could be established. The validity of the analytical results was confirmed by comparing them with the experimental results. We concluded that the effects of particle size and volume fraction on the fracture toughness could be expressed as the reciprocal of the product of (i) the square root of the mean distance between the surfaces of the nearest particles and (ii) the normalized mean stress in the matrix. Therefore, particle-filled composites with appropriate mechanical properties could be designed and tailored by using the theoretical result. Acknowledgement This research was supported by a Grant-in-Aid for Scientific Research (Project No. 18560072) made available by the Japan Society for the Promotion of Science. Appendix Mean stress in the matrix, rm, if a composite is subjected to tensile stress r0 is based on the equivalent inclusion method [37] and derived as
A1 A 3 A 2 A4 ; ðA1 þ 2A2 ÞðA1 A2 Þ ðA1 þ A2 ÞA4 2A2 A3 e22 ¼ ðA1 þ 2A2 ÞðA1 A2 Þ
e11 ¼ e33 ¼
where ( ) m 2 1 m ð1 m Þ m m A1 ¼ 5ð1 þ m ÞDk þ 2ð7 5m ÞDG þ 30G 1 2mm 15ð1 mm Þ m m 1 m m ð1 m Þ m m ÞDk 2ð1 5m ÞDG þ 30G A2 ¼ 5ð1 þ m 15ð1 mm Þ 1 2mm 1 1 A3 ¼ m fð1 2mm ÞDk þ 2mm DGg; A4 ¼ m fð1 2mm ÞDk 2DGg E E Dk ¼ kp km ; DG ¼ Gp Gm
and k and E are La´me’s constant and Young’s modulus, respectively. Superscript p denotes the properties of a particle. References [1] Nielsen EL, Landel RF. Mechanical properties of polymers and composites. 2nd ed. New York: Marcel Dekker; 1994. [2] Hashin Z. J Mech Rev 1983;50:481. [3] Moloney AC, Kausch HH, Kaiser T, Beer HR. J Mater Sci 1987;22:381. [4] Rice WR. Mechanical properties of ceramics and composites. New York: Marcel Dekker; 2000. [5] Roulin-Moloney AC, Cantwell WJ, Kausch HH. Polym Compos 1987;8:314. [6] Nakamura Y, Yamaguchi M, Kitayama A, Okubo M, Matsumoto T. Polymer 1991;32:2221. [7] Frounchi M, Burford RP, Chaplin RP. Polym Polym Compos 1994;2:77. [8] Ohta M, Nakamura Y, Hamada H, Maekawa Z. Polym Polym Compos 1994;2:215. [9] Jancar J, Dibenedetto AT. J Mater Sci 1995;30:2438. [10] Hussain M, Nishijima S, Nakahira A, Okada T, Niihara K. Molecular state and mechanical properties of epoxy hybrid composite. Mater Res Soc Symp Proc 1996;435:369. [11] Stricker F, Thomann Y, Mulhaupt R. J Appl Polym Sci 1998;68:1891. [12] Cho K, Yang J, Park CE. Polymer 1998;39:3073. [13] Quazi RT, Bhattacharya SN, Koisor E. J Mater Sci 1999;34:607.
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Acta Materialia 56 (2008) 2101–2109 www.elsevier.com/locate/actamat
Fracture toughness of nano- and micro-spherical silica-particle-filled epoxy composites Tadaharu Adachi *, Mayuka Osaki 1, Wakako Araki, Soon-Chul Kwon 2 Department of Mechanical Sciences and Engineering, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8552, Japan Received 5 December 2006; received in revised form 1 October 2007; accepted 2 January 2008 Available online 4 March 2008
Abstract The effects of particle size and volume fraction on the mode I fracture toughness of epoxy composites filled with spherical silica particles were investigated through experiments and an analytical model. Spherical silica particles with various particle diameters ranging from 1.56 lm to 240 nm and volume fractions from 0 to 0.35 were added to epoxy resin. We found that the fracture toughness of the composites was approximately linear with respect to the reciprocal of the product of (i) the square root of the mean distance between the particle surfaces and (ii) the normalized mean stress in the matrix given by the equivalent inclusion method. Ó 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Particulate reinforced composites; Nanocomposite; Polymer matrix composites; Fracture; Micromechanical modeling
1. Introduction Polymer materials can be reinforced by filling with several kinds of particles or fibers to make composite materials. The particle size in composites is currently crossing below the micrometer level to form nanocomposites that have superior mechanical properties. It is well known that the size and content of particles strongly affect the mechanical properties of composites [1,2], especially their fracture toughness [3,4]. Several researchers have considered the effects of particle size or volume fraction on the fracture toughness of polymer matrix composites [5–17]. Almost all of them have discussed the effects based on experimental results only. Consequently, general explanations about the effects have
*
Corresponding author. Tel.: +81 3 5734 2156; fax: +81 3 5734 2893. E-mail address: [email protected] (T. Adachi). 1 Present address: Production Engineering Research Laboratory, Hitachi Ltd., 292 Yoshida-cho, Totsuka-ku, Yokohama-shi, Kanagawaken 244-0817, Japan. 2 Present address: Semiconductor Material R&D Center, Samsung Techwin Co. Ltd., San No. 14, Nongseo-Ri, Giheung-Eup, Yongin-Si, Gyeonggi-Do 446-712, Republic of Korea.
not been clear because not only the particle and matrix materials, but also the size and volume fraction of the particles have been very different in these studies. Some researchers have considered these effects on the fracture toughness of composites using analytical models based on crack propagation along the particle surfaces. Taking the interspacing between particles [18] into account, Lange and Radford [19], Evans [20], Green et al. [21], Bower and Ortiz [22,23], Cui et al. [24] and Qiao [25] mathematically considered trapping, pinning and bridging of the crack front at the particles. It can be predicted that a crack in a composite does not always propagate along interfaces between particles and matrix. In particular, fracture phenomena in composites filled with nanometer- or micrometer-sized particles may be different from those in composites filled with larger particles. In this research, the effects of particle size and volume fraction on the mode I fracture toughness of epoxy composites filled with spherical silica particles were investigated. First, composite materials to be used as specimens were prepared by mixing bisphenol A type epoxy resin and spherical silica particles of various particle diameters, ranging from microsized to nanosized – i.e. 1.56 lm to 240 nm – and volume fractions from 0 to 0.35. The fracture
1359-6454/$34.00 Ó 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2008.01.002
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toughness of the composites was measured. The fracture surface of each specimen was observed with a scanning electron microscope (SEM) and a microscopic Raman spectrometer to determine clearly the crack path in the composite. Stress fields near the crack tip were analyzed approximately based on observing the fracture surfaces with the equivalent inclusion method and linear elastic fracture mechanics, taking into account the statistical distribution of particles to obtain the fracture toughness of specimens. Finally, the effects of particle size and volume fraction on the fracture toughness were formulated based on the experimental and analytical results. 2. Specimens Epoxy resin matrix filled with spherical particles of fused silica was used as the material for the specimens. The epoxy resin in the experiment was bisphenol A type epoxide resin (Japan Epoxy Resin, Epikote 828) with methyl-tetrahydro-phthalic anhydride as the curing agent (Hitachi Chem., HN-2200) and 2,4,6-tris (dimethyl aminomethyl) phenol as the accelerator (Daito Curar, DMP-30). The weight ratio of the resin, agent and accelerator was 100:80:0.5. The fillers were spherical silica particles with three diameters, as shown in Fig. 1. The median diameters of the particles Dm were 1.56 lm (Tatsumori, SO-C5), 560 nm (Tatsumori, SO-C2) and 240 nm (Tatsumori, 1-FX). The distributions for the particle diameters normalized by the median diameters are plotted in Fig. 2. These normalized distributions were approximately the same. The mechanical properties of the epoxy resin and silica particles are listed in Table 1 [17,26,27]. The surfaces of the particles were not coated chemically. Hereafter, the median diameter of the particles will be called the particle diameter for short, unless there is the possibility of confusion. After the epoxy resin and the silica particles were combined in a mixing machine for more than 30 min, the mixture was stored in a vacuum vessel to remove voids and was poured into an aluminum mold coated with a Teflon sheet, which was set up in an oven. The mold was 260 mm long, 5 mm wide and 180 mm deep. The volume fractions of the silica particles were within 0 to 0.35. The curing was done in two steps. First, the composite was kept at 353 K for 3 h to gel the matrix resin (pre-curing). The second step, post-curing, which greatly affects the cross-linking reaction of the resin, was done at 413 K for 10 h. The heating rate from pre-curing to post-curing was constant at 72 K h1. After curing, we found that the particles had dispersed adequately without condensation and there were no voids in the composites. The glass transition temperature of each composite was identified from the shift factor of the thermo-viscoelastic properties [17,26] to verify the degree of cross-linking as the cured states of the specimens. Neat epoxy resin and 11 different composites were prepared, as listed in Table 2. We confirmed that the epoxy
Fig. 1. Silica particles: (a) SO-C5 (median diameter Dm = 1.56 lm); (b) SO-C2 (median diameter Dm = 560 nm); and (c) 1-FX (median diameter Dm = 240 nm).
and every composite had been completely cured from the glass transition temperature. For each composite, the distance between the surfaces of the nearest particles was analyzed to obtain the statistical particle dispersion. We assumed that the particles had the same constant diameter with a median diameter of Dm and were distributed with volume fraction / in the matrix resin three-dimensionally and randomly. The mean dis-
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10.0
Table 2 Specimens
100 Cumulative volume
Volume [%]
60 5.0
Volume
40 Dm = 1.56 ∝m
2.5
0
b
20
0
1 2 3 4 5 Normalized particle diameter, D/Dm
15.0
Volume [%]
60 Volume
40 Dm = 560 nm
20
2.5 0
1 2 3 4 5 Normalized particle diameter, D/Dm
10.0
0
100 80
7.5
60 Volume
5.0
40 Dm = 240 nm
2.5
0
20
0
1
2
3
4
5
Cumulative volume [%]
Cumulative volume
Volume [%]
Cumulative volume [%]
10.0
0
0
Normalized particle diameter, D/Dm Fig. 2. Diameter distributions of silica particles: (a) SO-C5 (median diameter Dm = 1.56 lm); (b) SO-C2 (median diameter Dm = 560 nm); and (c) 1-FX (median diameter Dm = 240 nm).
Table 1 Mechanical properties of epoxy resin and silica particles
Silica particles Epoxy resin
Epoxy Composite
Median diameter Dm
Volume fraction /
– 1.56 lm
0 0.057 0.186 0.301 0.351 0.057 0.184 0.347 0.056 0.118 0.187 0.301
560 nm
100 80
5.0
Particle
240 nm
Cumulative volume
7.5
Specimen
0
12.5
c
Cumulative volume [%]
80
7.5
Density (kg m3)
Young’s modulus (GPa)
Shear modulus (GPa)
Poisson’s ratio
2.2 1.2
72.1 3.8
30.5 1.4
0.18 0.38
Normalized distance between surfaces of nearest particles,
a
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Glass transition temperature (K)
399 397 391 399 393 397 393 393 393 393 388 399
3
2
Close-packed structure
1
0
0.2
0.4
0.6
0.8
Volume fraction, φ
Fig. 3. Distance between surfaces of nearest particles.
where function C(a, n) is an incomplete gamma function. From Eq. (1), distance hLai is proportional to particle diameter Dm and dependent on volume fraction /. The distribution of particles in the matrix was similar for the same volume fraction. The computational results for Eq. (1) were plotted as far as the maximum volume fraction pffiffiffi for a hexagonal or cubic close-packed structure, i.e. 2p=6 in Fig. 3. Distance hLai at a volume fraction of 0.055 was equal to the p particle diameter Dm and the shortest distance ffiffiffi hLai at / ¼ 2p=6 was 0.134Dm. Because volume fractions / in the experiments ranged from 0.06 to 0.35, hLai was below Dm for every specimen. 3. Experimental procedures 3.1. Measurement of elastic properties
tance between the surfaces of the nearest particles, hLai, was given statistically by Bansal and Ardell [28] as Z 6/ Dm Cð0:5; nÞ expðnÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi hLa i ¼ dn ð1Þ 24/ 0 6/ n
A tensile test at room temperature (298 K) was conducted with a universal testing machine (Instron, 8501). The specimens had a length of 120 mm, a width of 10 mm and a thickness of 5 mm. Longitudinal strain was measured using a strain gage at the midpoint of each
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specimen. The rate of tensile deformation was 2 lm s1. Young’s modulus was evaluated from the strain and tensile load. The average value calculated from five measured results was used as the elastic modulus and the standard deviation was also evaluated. 3.2. Measurement of fracture toughness A three-point bending test on each precracked specimen was conducted to measure the mode I fracture toughness at room temperature (298 K) according to ASTM standard D5045 (Fig. 4). The specimens had a length of 90 mm, a width of 20 mm and a thickness of 5 mm. The lengths of the precrack and span were 10 mm and 80 mm, respectively. The tests were carried out under a constant displacement rate of 2 lm s1 at the loading point with a universal testing machine (Instron, 8501). A brittle fracture occurred in every specimen. The stress intensity factor, KIC, when the specimen broke was defined as the fracture toughness [29]: K IC ¼
SP c f ðnÞ BW 3=2
ð2Þ
where f ðnÞ ¼ n¼
3n1=2 f1:99 nð1 nÞð2:15 3:93n þ 2:7n2 Þg 2ð1 þ 2nÞð1 nÞ
3=2
;
a W
and Pc is the maximum load. S, B, W and a are the span length, the thickness, the width and the precrack length of the specimen, respectively. The average value, excluding the maximum and minimum values from seven experimental results, was used as the fracture toughness and the standard deviation was also evaluated. After the experiments, the fracture surfaces of specimens were observed with an SEM (JEOL, JSM-6301F) and a microscopic Raman spectrometer (JASCO, NRS-1000). The wavelength of the source laser in the Raman spectrometer was 532.3 nm and its measurement region was a circle of diameter 8 lm.
4. Experimental results 4.1. Elastic modulus The relations between Young’s modulus and particle volume fraction are shown in Fig. 5. The triangles and circles denote the averages of measured values. The standard deviations of the Young’s moduli were not plotted in Fig. 5 because the values for all specimens were within 2%. As is well known, Young’s modulus increases with increasing volume fraction. We did not find any effects of particle size on Young’s modulus, but we confirmed that the volume fraction governed it. The modulus at a volume fraction of 0.35 was twice that of neat epoxy resin. Typical mixture laws proposed by Kerner [30], Hashin and Shtrikman [31], Lewis and Nielsen [32] and Nielsen [33], denoted as solid lines in Fig. 5, were compared with the experimental results. Hashin and Shtrikman’s equation was based on the equivalent inclusion method. Lewis and Nielsen’s equation was modified from Kerner’s equation. These theoretical results coincide with our experimental results. Lewis and Nielsen’s equation agreed well with the experimental results because the distribution of particle diameters was considered in the equation. Lewis and Nielsen’s equation can also be applied to the thermo-viscoelastic properties of epoxy composites bi-dispersed by silica particles [28]. Hashin and Shtrikman’s equation, which was the result of the equivalent inclusion method, was also in agreement with the experimental results. 4.2. Fracture toughness The relation between the fracture toughness of composites and the volume fraction of the particles is shown in Fig. 6. The triangles and circles indicate the averages of measured values and the error bars indicate their standard deviations. The solid lines denote fitting for each composite with particles of the same size. The fracture toughness increased as the volume fraction increased. Although the particle diameter had little influence on the fracture toughness for a low volume fraction, the fracture toughness was
Young's modulus, Ec [GPa]
10 Lewis & Nielsen [32, 33] Kerner [30] Hashin & Shtrikman [31]
8 6 4
Particle diameter Dm = 1.56 μm Dm = 560 nm Dm = 240 nm Epoxy
2
0
0.1
0.2
0.3
0.4
Volume fraction, φ Fig. 4. Fracture toughness test (units are millimetres).
Fig. 5. Relation between Young’s modulus and volume fraction.
Fracture toughness, KIC [MPa m1/2]
T. Adachi et al. / Acta Materialia 56 (2008) 2101–2109
3
2
Particle diameter Dm = 1.56 ∝m Dm = 560 nm Dm = 240 nm Epoxy
1
0
0
0.1
0.2
0.3
0.4
Volume fraction, φ Fig. 6. Relation between fracture toughness and volume fraction.
strongly dependent on the particle diameter for a high volume fraction. The fracture toughness for a particle diameter of 1.56 lm increased for a volume fraction of up to 0.19 and saturated for a larger volume fraction. The fracture toughness for a particle diameter of 560 nm increased slightly up to the highest volume fraction. The rate of increase was largest for a particle diameter of 240 nm. The fracture toughness at a volume fraction of 0.309 for
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a particle diameter of 240 nm was 1.7 times that for a particle diameter of 1.56 lm and 2.5 times that for epoxy resin. The fracture toughness depended greatly on both the volume fraction and particle diameters, compared with the elastic moduli in Fig. 5. For composites with a low particle volume fraction (<0.1), particle size had little effect on fracture toughness. The size effect strongly affected the fracture toughness in the high volume fraction region. Fracture surfaces observed with an SEM are shown in Fig. 7. The roughness of the fracture surface decreased with decreasing particle size. The fracture surface became fine for a higher volume fraction and for smaller particles. Raw particles did not appear to be exposed on any surface. However, from Fig. 7, it is difficult to clearly find which cracks propagated in the matrix, through particles, or along the interface between the particles and the polymer matrix. The fracture surfaces were analyzed by microscopic Raman spectroscopy to obtain the route of the crack propagation path. The fracture surfaces of specimens were analyzed more than five times at a distance of approximately 3 mm in front of the initial crack tip of the specimen, where several particles were near the surface within view of a Raman spectrometer with a CCD camera attached. The Raman spectrum for the composite with a particle diameter of
Fig. 7. Fracture surfaces.
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rials. We concluded that silica particles had not been exposed on the fracture surface. Therefore, the cracks in all specimens propagated in the matrix resin. Crack propagation has been explained by several researchers as being due to the mechanisms of trapping, pinning and bridging of the crack front at particles [19– 25]. In Figs. 7 and 8, we can see cracks propagated by avoiding particles. Therefore, it would not be appropriate to apply these mechanisms to composites filled with small particles unless the volume fraction were very high.
Epoxide ring
Intensity [a.u.]
Epoxy Si-O-Si Defect of SiO2
Silica
5. Discussion
Composite
5.1. Concept of analysis
200
400
600
800
1000
Raman shift [cm-1] Fig. 8. Raman spectroscopy of fracture surfaces. Particle diameter Dm = 1.56 lm and volume fraction / = 0.186.
1.56 lm and a volume fraction of 0.19 is shown in Fig. 8. The upper line is the spectrum for epoxy resin and the middle line is that for silica particles. The characteristic peaks for epoxy resin [34] and silica [35] in these spectra are indicated by circles. The lower spectrum in Fig. 8 has the results that characterize the composite. We found only peaks for the epoxy resin in the composite’s spectrum and could not find peaks for silica particles. The same results were obtained for the fracture surfaces of other composite mate-
The effects of both particle size and volume fraction on the fracture toughness of a composite were formulated based on reinforcement of the matrix by filler particle and the particle distribution in the matrix. We considered that the mode I fracture toughness of a composite material, K COMP , could be expressed by the sumIC mation of (i) the fracture toughness of a neat matrix resin, EM K MAT IC , and (ii) the improvement in fracture toughness, K IC , provided by reinforcement by particles, as outlined in Fig. 9. The improvement, K EM IC , is attributed to the effect of stress in the matrix interspace between particles and the particle distribution was obtained in Section 2. The model for improvement is shown in Fig. 9(c). It is called the equivalent matrix model (see Section 5.2). It is composed of only the matrix resin without particles and the stress field in the matrix was considered to be caused by
Fig. 9. Analytical model.
T. Adachi et al. / Acta Materialia 56 (2008) 2101–2109
ð3Þ
5.2. Equivalent matrix model Suppose that external stress, r0, is applied to a composite with a crack that is a in length, as shown in Fig. 9. The stress concentration near a particle is slight because individual particles are much closer even for the low volume fraction of particles in Fig. 3. Assuming that tensile stress rm is uniform near the crack tip, the equivalent matrix model can be considered, as described in Fig. 9(c), because the deformation of composites can be expressed approximately by Hashin and Shtrikman’s equation [30], which is the equivalent inclusion method described in Fig. 5. Mean stress rm in the matrix (Fig. 9(c)) is then expressed by external stress, r0, of the composite according to the equivalent inclusion method [36]: rm ¼ r0 gð/Þ
ð4Þ
where function g(/), given in the Appendix, is the normalized mean stress dependent on the elastic moduli of particles and the matrix and the volume fraction /. The mean stress in the matrix is linear with respect to the volume fraction of the particles without the size effect of particles. The numerical results of Eq. (4) for a composite of epoxy and spherical silica particles are derived from the elastic properties in Table 1 as rm ¼ r0 ð1 0:825/Þ
ð5Þ
The minimum value forpmean stress is 0.39r0 at the maxiffiffiffi mum volume fraction, 2p=6. Then, the stress in the matrix resin is reduced by the reinforcement provided by the particles. Particle distribution is similar because mean distance between the surfaces of the nearest particles, hLa i, is proportional to particle diameter for a constant volume fraction due to Eq. (1). Considering the similarity in particle distributions, stress field rij in the matrix can be expressed as a function of xk/hLai: rij ¼ rij ðxk =hLa iÞ
ð6Þ
where xk is the coordinate in Fig. 9(c). Based on Eqs. (4) and (6), tensile stress r22 in the vicinity of a crack tip applied by mean stress rm in the interspace of the matrix region between particles can be analyzed because cracks were confirmed to propagate in the matrix from the results of Figs. 7 and 8: pffiffiffiffiffiffi rm pa ð7Þ r22 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f 2px1 =hLa i where f is a coefficient independent of hLai. If the stress intensity factor given by Eq. (7) coincides with the fracture toughness of the matrix resin, K MAT IC , then the crack propagates through the matrix:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pahLa if
ð8Þ
Fracture toughness provided by reinforcement by particles, K EM IC , can also be evaluated using far field stress r0 as follows: pffiffiffiffiffiffi ð9Þ K EM IC ¼ r0 paF where F is the shape function related to the boundary condition of the specimen. Substituting Eq. (8) into Eq. (9), we obtain K EM IC ¼
1 F pffiffiffiffiffiffiffiffiffi K MAT IC gð/Þ hLa i f
ð10Þ
Finally, the fracture toughness of the composite, K COMP , IC can be derived by Eqs. (3) and (10) as ! a COMP pffiffiffiffiffiffiffiffiffi K MAT K IC ¼ 1þ ð11Þ IC gð/Þ hLa i where a = F/f. This equation shows that the fracture toughness of the composite is proportional to g(/)1 hLai1/2 when a crack propagates in the matrix. For a neat matrix resin, i.e. when hLai is infinite, the fracture toughness of the composite, K COMP , coincides with K MAT IC IC . 5.3. Comparison of experimental and analytical results The analytical results computed with Eq. (11) are compared with the experimental results in Fig. 10. The horizontal and vertical lines denote parameters g(/)1hLai1/2 computed for each composite material and the fracture toughness obtained in the experiment. The triangles and circles indicate the average of measured values and the error bars indicate their standard deviations. The experimental results are linear with respect to parameter g(/)1hLai1/2. The agreement between the analytical results and Eq. (11) was confirmed. Therefore, the effect of particle size on fracture toughness can be expressed as the reciprocal of the square root of the mean distance between the surfaces of the nearest particles and the effect of volume fraction can be expressed as the reciprocal of the normalized mean stress in the matrix given by the 1/2
¼ K MAT þ K EM K COMP IC IC IC
¼ rm K MAT IC
Fracture toughness, KIC [MPa m ]
the effects of both volume fraction and particle size. The fracture toughness of the composite, K COMP , is then given IC mathematically by
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3
2
Particle diameter Dm = 1.56 μm Dm = 560 nm Dm = 240 nm Epoxy
1
0
0
2
4
6
g(φ)-1
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equivalent inclusion method. From the fitting line for the experimental data in Fig. 10, the coefficient a was obtained as 0.27 lm1/2 in Eq. (11). Coefficient a is directly independent of the particle size and the distance between particle surfaces, as previously mentioned. It is probably a parameter of the matrix material, more specifically the molecular behaviors and the interaction of the particles with the cross-linking structure of the polymer. This needs further consideration using experimental results obtained from other composites filled with particles, though coefficient a is important and interesting, especially for nanocomposites. Therefore, Eq. (11) can be efficiently used to design and tailor particle-reinforced composites having appropriate fracture toughness.
rm ¼ r0 gð/Þ ¼ r0 f1 /C 22ij ðS ijkl ekl eij Þg where g (/) is the normalized mean stress. C22ij represents the elastic moduli in a matrix defined as C 2211 ¼ C 2233 ¼
Gm mm ; 1 2mm
C 2222 ¼
2Gm ð1 mm Þ ; 1 2mm
C 22ij ¼ 0ði 6¼ jÞ where G and m are the shear modulus and Poisson’s ratio, respectively, and superscript m denotes the properties of the matrix. Sijkl denotes Eshelby’s tensor for a sphere, 7 5mm ; S 1122 ¼ S 1133 ¼ S 2211 15ð1 mm Þ 1 þ 5mm ¼ S 3322 ¼ 15ð1 mm Þ
S 1111 ¼ S 2222 ¼ S 3333 ¼ ¼ S 2233 ¼ S 3311
6. Conclusions
The eigen strain, eij , given by The effects of particle diameter and volume fraction on mode I fracture toughness of spherical-silica-particle-filled epoxy composites were investigated through experiments and an analytical model. The particle diameters ranged from 240 nm to 1.56 lm and the volume fraction ranged from 0 to 0.35. In the experiments, we found that the fracture toughness increased drastically as the volume fraction increased and the particle diameters decreased. In composites with a low volume fraction of particles, the volume fraction affected the fracture toughness and in composites with high volume fractions, the particle size affected the fracture toughness more. The stress field in the matrix near the crack tip was approximately analyzed based on the equivalent inclusion method and linear elastic fracture mechanics, taking into account the statistical distribution of particles so that the effects of particle size and volume fraction on the fracture toughness of the composite could be established. The validity of the analytical results was confirmed by comparing them with the experimental results. We concluded that the effects of particle size and volume fraction on the fracture toughness could be expressed as the reciprocal of the product of (i) the square root of the mean distance between the surfaces of the nearest particles and (ii) the normalized mean stress in the matrix. Therefore, particle-filled composites with appropriate mechanical properties could be designed and tailored by using the theoretical result. Acknowledgement This research was supported by a Grant-in-Aid for Scientific Research (Project No. 18560072) made available by the Japan Society for the Promotion of Science. Appendix Mean stress in the matrix, rm, if a composite is subjected to tensile stress r0 is based on the equivalent inclusion method [37] and derived as
A1 A 3 A 2 A4 ; ðA1 þ 2A2 ÞðA1 A2 Þ ðA1 þ A2 ÞA4 2A2 A3 e22 ¼ ðA1 þ 2A2 ÞðA1 A2 Þ
e11 ¼ e33 ¼
where ( ) m 2 1 m ð1 m Þ m m A1 ¼ 5ð1 þ m ÞDk þ 2ð7 5m ÞDG þ 30G 1 2mm 15ð1 mm Þ m m 1 m m ð1 m Þ m m ÞDk 2ð1 5m ÞDG þ 30G A2 ¼ 5ð1 þ m 15ð1 mm Þ 1 2mm 1 1 A3 ¼ m fð1 2mm ÞDk þ 2mm DGg; A4 ¼ m fð1 2mm ÞDk 2DGg E E Dk ¼ kp km ; DG ¼ Gp Gm
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