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Effects of adding carbon nanofibers on the reduction of matrix cracking in laminated composites: Experimental and analytical approaches
Polymer Testing xxx (xxxx) xxx
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Effects of adding carbon nanofibers on the reduction of matrix cracking in laminated composites: Experimental and analytical approaches H. Ramezani a, S. Kazemirad b, M.M. Shokrieh a, *, A. Mardanshahi a a
Composites Research Laboratory, Center of Excellence in Experimental Solid Mechanics and Dynamics, School of Mechanical Engineering, Iran University of Science and Technology, Tehran, 16846-13114, Iran b School of Mechanical Engineering, Iran University of Science and Technology, Tehran, 16846-13114, Iran
A R T I C L E I N F O
A B S T R A C T
Keywords: Polymeric composites Carbon nanofiber Matrix cracking Lamb wave Elastic properties Micromechanical models
The main purpose of the present study is to investigate the effect of adding carbon nanofibers (CNFs) in fibrous polymeric composites on the reduction of matrix cracking. A tensile testing machine was used to apply five loading steps and induce different crack densities in [02/906]s composite specimens with and without CNF fillers. After each loading step, the elastic modulus of specimens was measured. The crack density of damaged speci mens was also measured by an optical microscope after each loading step. The Lamb wave propagation method was used to assess the elastic modulus of specimens at different crack densities. It was demonstrated that the addition of CNF fillers into the composite specimens resulted in lower crack densities and less stiffness reduction at certain applied stress levels. Moreover, Hashin and Andersons et al. micromechanical models were assessed by the experimental results for the analysis of matrix cracking in cross-ply laminated composites. For composite specimens with and without CNFs, the stiffness reduction at different crack densities was in good agreement with the predictions of the Hashin model, and the induced crack density at each applied loading step was in good agreement with the predictions of the Andersons et al. model.
1. Introduction Composite structures suffer from different damage mechanisms [1]. Generally, all of the critical damage modes of composites, caused by static and fatigue loadings, are initiated by the transverse cracking of the matrix [2]. The development of methods to postpone the initiation of matrix cracking in laminated composites and to identify its severity may play an important role in improving applications of these materials. The growth of matrix cracking in composites under static and fatigue loadings has been investigated in the past decades [3–5]. Hashin [3] proposed an analytical model based on the variational method to predict the growth of matrix cracking damage in cross-ply multilayered com posites under tensile and shear loadings. It was shown that the Hashin model properly predicts the stiffness reduction at different matrix crack densities as measured by the experiments. Liu and Nairn [6] developed a semi-empirical relation based on the experimental results between the matrix crack density in polymeric composites and the tensile load. They also fitted the obtained experimental results with the energy release rate expression and estimated the microcracking and intralaminar fracture toughness. Tounsi and Amara [7] numerically, analytically, and
experimentally investigated the variation in the longitudinal modulus of the cross-ply laminated composite due to the matrix cracking, by considering a relationship between the stiffness reduction and the variation of the temperature and humidity. Their results showed that the hygrothermal environment had a significant effect on the relative reduction of the longitudinal modulus at higher crack densities. Bouazza et al. [8] presented an analytical model based on the concept of stress turbulence function to predict the effect of transverse matrix cracks on the stiffness reduction of composite laminates. They also showed that the moisture had no significant effect on the longitudinal modulus of the tested composite laminates, while an increase in the temperature caused a reduction in the longitudinal modulus, especially at higher crack densities. Andersons et al. [9] proposed a semi-analytical model based on the shear lag method, in which a relation between the matrix cracking in cross-ply laminated composite and stress in 90-degree layers was given. Deng et al. [10] investigated the mixed effects of three factors of temperature, fiber-matrix debonding, and thermal residual stress on the first matrix cracking stress for the fiber-reinforced ceramic com posites. They also quantitatively analyzed the effect of composite vol ume fraction on the onset of matrix cracking. Barulich et al. [11] used a
* Corresponding author. E-mail address: [email protected] (M.M. Shokrieh). https://doi.org/10.1016/j.polymertesting.2020.106988 Received 31 July 2020; Received in revised form 28 October 2020; Accepted 23 November 2020 Available online 28 November 2020 0142-9418/© 2020 The Authors. Published by Elsevier Ltd. This is an open
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meso-mechanics model for the analysis of matrix cracking, consisted of a continuous three-dimensional finite element model in the mesoscale and a classic thin plates laminate model in the macro-scale for cross-ply laminates. Their results showed that the model with a non-uniform crack density presented a better agreement with experimental data than the model with uniform crack density. Mohammadi and Pakdel [4] developed a damage growth criterion based on the energy release rate that can predict the matrix crack density in laminated composites with different layups under fatigue loading. Based on the analytical and experimental results presented in prior research works, the elastic modulus and ultimate strength of the poly mer may increase by adding certain types of nanofibers or nanoparticles to the resin up to a certain weight fraction and using an appropriate dispersion method [12,13]. The effects of CNFs on failure and fracture behavior of laminated composites were also investigated [14–16]. Furthermore, other properties of the resin such as the heat transfer properties, ultimate strain, and electrical conductivity are improved by adding CNFs to the resin [17–21]. Different optimum weight fractions of CNF were reported in previous studies to obtain the maximum increase in the elastic modulus and ultimate strength of the resin [22–24]. These discrepancies in the optimum weigh fraction may be due to the differ ences of methods used for the dispersion of CNF in the resin, different sizes of CNF used, and the fabrication and testing conditions of the specimens. It was also found in these studies that the effect of CNF on increasing the tensile strength of the resin was more than that on the elastic modulus [25–27]. Shokrieh et al. [28] revealed that the optimum weight fraction of CNF to achieve the best mechanical properties of CNF/epoxy nanocomposites was 0.25 wt%. They observed an increase of 23% in the tensile strength and 10% in the flexural strength for this filler content. Zhou et al. [29] showed that the elastic modulus of epoxy modified with CNF increased with increasing the weight fraction of this nanofiber. Their results revealed that the ultimate tensile strength, fa tigue life, and fracture toughness also increased significantly by increasing the CNF content in the resin, reaching their maximum values at a weight fraction of 2%. Moreover, they observed that the nanofibers caused a random distribution of the matrix cracks. Tabatabaeian et al. [30] investigated the effect of multi-walled carbon nanotubes (MWCNTs) and thickness on the curvature changes and weight loss of laminated composites in different thermal cycles. They observed that the weight loss and curvature reduced in thermal fatigue by adding MWCNTs into the specimens. Ghasemi et al. [31] investigated the effect of MWCNTs on the residual stress in laminated composite with theo retical and experimental approaches. Their results revealed that rein forcing the composite specimens with 1% MWCNTs reduced the residual stresses up to 35%. Material characterization and damage detection methods based on the Lamb wave propagation have been developed for the nondestructive evaluation of multilayered composites [32–39]. For example, Pant et al. [37] proposed an inverse method based on the theory of Lamb wave propagation to monitor the material constants and structural health of aerospace composites and metal structures. This method was used to investigate the effect of the composite layup, stiffness, density, and the Lamb wave propagation direction on the dispersion curve. Mardanshahi et al. [38] proposed a viscoelastic characterization approach based on the Lamb wave propagation method for the identification of matrix cracking in cross-ply laminated composites. In most proposed methods, the fundamental Lamb wave modes were used along with the inverse problem solution techniques. The general criteria for selecting the Lamb wave mode for the nondestructive evaluation of composite structures include limited wave scattering, small wave attenuation, proper fault sensitivity, and simple and feasible wave excitation and detection [40–46]. To the best knowledge of the present authors, the effect of adding CNFs to polymeric composites on the initiation and propagation of the matrix cracking has not been investigated. Moreover, the development and assessment of a non-destructive method based on the Lamb wave
propagation for the evaluation of matrix crack density and stiffness loss in polymeric composites reinforced by CNF may be of great interest. Therefore, the effect of adding CNFs on the matrix cracking behavior of cross-ply polymeric composites was investigated in the present study and assessed by a nondestructive evaluation method based on the Lamb wave propagation technique. In the following sections, two analytical and semi-analytical micromechanical models governing the crack den sity and the resulting stiffness loss of cross-ply composites, and the equations governing the Lamb wave propagation are presented. After ward, the fabrication process of specimens, including the CNF/epoxy nanocomposites and hybrid glass/CNF/epoxy composite specimens is explained. Then, the process of inducing matrix cracks in the specimens, and tensile and Lamb wave propagation tests are described. Finally, the results obtained from the tensile and Lamb wave propagation tests are presented and discussed. It was shown that the matrix crack density and stiffness loss in composite specimens decreased by adding CNF fillers to polymeric composites, and the Lamb wave propagation method may offer a powerful tool for the nondestructive evaluation of such hybrid polymer composite materials. 2. Background theory 2.1. Micromechanical models for matrix cracking The models proposed by Hashin [3] and by Andersons et al. [9] were used in the present study to predict and analyze matrix cracking growth, assess the accuracy of tensile and Lamb wave propagation experiments, and analyze the effect of CNFs on the stiffness and strength of composite specimens. The Hashin model [3] is based on the calculation of the stress field by the variational method, while the Andersons et al. model [9] is based on the calculation of the stress field via the shear lag method. In the Hashin model [3], after calculation of the stress field in a cross-ply multilayered composite specimen under unidirectional tensile loading, a lower limit is proposed for the elastic modulus of the damaged specimen (Ex ) as: 1 1 1 < χ (δ) > ≤ + k2 η(ψ ) Ex Exο ET 2 <δ> where ψ = tt12 , η(ψ ) =
3ψ 2 +12ψ +8 , 60
(1) k2 = σσ20 , t1 and t2 are the half of the
thickness of 0-degree and 90-degree layers, respectively, ET is the stiff ness of 90-degree layers or the transverse stiffness, σ2 is the stress in 90degree layers, and σ0 is the total stress of the laminate. The value of k2 (relative stiffness of 90-degree layers) and E0x (initial laminate stiffness) are obtained from the classical lamination theory of the undamaged composite laminate by having the longitudinal and transverse elastic moduli of the unidirectional composite ply. Also, we have: (2)
δ = a/t2 (
χ (δ) = 2αβ α2 + β2
)
(
cosh 2 αρ − cos 2 βρ α sin 2 βρ + β sinh 2 αρ
) (3)
α = q1/4 cos θ 2
/
(4)
/ β = q1/4 sin θ 2
(5)
tan θ =
√̅̅̅̅̅ 4q − 1 p2
(6)
where a is the half of the distance between two adjacent cracks, and q and p are factors that depend exclusively on the layup and mechanical properties of the composite laminate. If the cracks are equidistant, we have δn = δ = a/t2 , < χ (δ) > = χ (δ), and < δ > = δ. Andersons et al. [9] proposed a mathematical relation between the stress and the crack density by using the Norman equations [47]. For a 2
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cross-ply laminated composite, the shear lag parameter is defined as: √̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ 3(d + b)E0x G12 G23 K= (7) dbEL ET (bG12 + dG23 )
where h is half of the thickness of the plate. Also, γand ζ are defined as: √̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ (14) γ = k2p − k2l
where b and d are the thickness of 0◦ and 90◦ layers of the laminate, respectively. Also, E0x is the elastic modulus of composite laminate, EL and ET are the longitudinal and transverse elastic moduli of each uni directional composite ply, respectively. Moreover, G12 and G23 are inplane and out of plane shear moduli, respectively. In this model the crack density is related to the applied stress in the 90-degree layer based on the following equation [9]: ( ) (ρ )2 σ2 c = cinit − 1 (8) K σinit
ζ=
where kl is the Lamb wavenumber, and ks and kp are shear and compression wavenumbers, respectively. The shear wave velocity is related to the compression velocity through [42]: ( )2 cs 1 − 2ν μ = = (16) cp 2 − 2ν λ + 2μ
where ρc is the crack density, σinit is the in-situ strength of 90-degree layers, σ2 is the stress in 90-degree layers of the multi-layered compos ite specimen, cinit is a material-independent factor that is evaluated by fitting the theoretical master curve given in Eq. (8) to the tensile test data. In this study, cinit was estimated from the experimental data for the composite specimens with and without CNF fillers. The Andersons et al. model [9] was used in the present study to assess the effect of the strength of 90-degree layers on the matrix crack density.
G = ρc2s
(17)
E = 2(1 + ν)G
(18)
3. Materials and methods
2.2. Lamb wave theory
3.1. Materials
Lamb waves are elastic perturbations that propagate in thin solid plates with two stress-free boundaries, in which the particle displace ments of the propagation medium occur in the direction of propagation of the wave and also perpendicular to it [42]. Lamb waves are caused by the interaction of compression waves (p) and shear waves (s) propa gating within the medium. Lamb waves may propagate in two inde pendent symmetric and anti-symmetric modes. In this study, the fundamental anti-symmetric Lamb wave mode (A0) was generated in the composite specimens. The relation between the compression wave ve locity (cp ) and the shear wave velocity (cs ) and the mechanical proper ties of the propagation medium are as: √̅̅̅̅̅̅̅̅̅̅̅̅̅ λ + 2μ (9) cp =
Razeen® LR1100 epoxy resin with a mixing ratio of 13:100 of hardener to resin was used for the fabrication of the pure resin, nano composites, and laminated composite specimens. The CNF used in this study had an outer diameter of 200–600 nm, a length of 5–50 μm, fabricated by US Research Nanomaterials, Inc. Unidirectional E-glass fibers of 220 g/m2 manufactured by P-D Interglas Technologies were used for the fabrication of cross-ply laminated composite specimens.
√̅̅̅
μ ρ
3.2. Fabrication of the specimens Three pure epoxy resin specimens were fabricated based on the dimension requirements of the ASTM D638-03 standard. In the fabri cation of pure resin specimens, the mechanical mixing process was carried out for 5 min with a velocity of 200 rpm. The mixture was subsequently vacuumed for 5 min at a pressure of 1 mbar, after which the molding of the mixture was performed. The specimens were then pre-cured at room temperature for 2 days and were post-cured in the oven at 140 ◦ C for 2 h. The final preparation of the specimens to reach the standard dimensions and to remove the surface bubbles were per formed by sanding and polishing. Three CNF/epoxy specimens were also fabricated to assess the effect of CNFs on the elastic properties of the resin. First, the temperature of the resin was increased to 60 ◦ C to reduce its viscosity and improve the wetting of carbon nanofibers in the fabrication process. A CNF weight fraction of 0.25% was selected for the fabrication of specimens, as it was shown to be the optimum weight fraction for the enhancement of the mechanical properties of CNF/epoxy nanocomposites [28]. The CNF/epoxy mixture was then mechanically stirred for 30 min at a speed of 2000 rpm. The mixture was subsequently heated to 70 ◦ C and then sonicated for 100 min with an ultrasonic instrument (Hielscher Ultra sound Technology, UP400S) with an output power of 75 W and 12 kHz frequency to provide an appropriate dispersion of the CNF fillers within the resin. The vacuum, molding, and following sanding and polishing procedures were conducted similar to those for the pure resin specimens. Three glass/epoxy and three glass/CNF/epoxy unidirectional com posite specimens with [08] and [908] stacking layups, respectively, and three glass/epoxy and three glass/CNF/epoxy cross-ply laminated composite specimens with [02/906]s stacking layup were fabricated with the hand layup technique. The CNF fillers were only added to the epoxy
(10)
where ρ is the mass density of the propagation medium, and μ and λ are Lame constants, defined as: λ=
2ν G 1 − 2ν
μ=G
(11) (12)
where Gis the shear modulus, and ν is the Poisson’s ratio of the material. By solving the Navier equation for elastic materials through the decomposition of the displacement field and defining the potential functions of compression and shear waves, the dispersion equation for the antisymmetric Lamb wave mode is obtained. Due to the low thick ness of the composite specimens fabricated and tested in the present study compared to the Lamb wavelength, it can be assumed that a single wave with similar characteristics propagates over the whole thickness of the laminate. Therefore, the following equation may be appropriately used for Lamb wave propagation within the multilayered composite specimens [42]: tanh (γh) 4k2 γζ = ( 2 l 2) tanh (ζh) 2kl − ks
(15)
The shear wavenumber and the shear wave velocity are obtained through the numerical solution of Eq. (13), having the Lamb wave number measured from experiments and the Poisson’s ratio of speci mens. Then, the shear and elastic moduli of the propagation medium are obtained via:
ρ
cs =
√̅̅̅̅̅̅̅̅̅̅̅̅̅̅ k2s − k2l
(13)
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resin used for the fabrication of 90-degree layers of the glass/CNF/epoxy laminated composite specimens. The overall fabrication process of these specimens was similar to that of the previously mentioned specimens. The fabricated specimens were pressurized with a 200 N force for 48 h during the pre-curing at room temperature to eliminate the bubbles and obtain a uniform thickness and an appropriate volume fraction. After post-curing in the oven, the specimens were cut with the dimensions illustrated in Fig. 1, which were based on the ASTM D3039/D3039M standard for obtaining the unidirectional properties of the composite specimens. The specimen tabs were also fabricated using glass/epoxy composites with [±45]4s layup.
tests was set to 2 mm/min (static loading condition) and an extensom eter (STT-100D) was used to measure the extensions of the specimens during the tests. A similar test procedure was carried out to obtain the elastic modulus and tensile strength of pure epoxy, CNF/epoxy, and unidirectional composite specimens. Fig. 2 shows the static tensile test setup. 3.4. Lamb wave propagation tests A nondestructive Lamb wave propagation test procedure was also performed to obtain the elastic properties of composite specimens. The configuration of the experimental setup used in the present study for the ultrasonic Lamb wave propagation tests is shown in Fig. 3. An ultrasonic actuator connected to a generator was used to induce Lamb waves in the specimens with a frequency of 206 kHz and an excitation voltage of 150 V. A high precision ultrasonic sensor connected to a digital data acqui sition system was used to acquire the propagated wave data at different locations along the propagation direction. A finite impulse response (FIR) digital filter with an equiripple algorithm was used to preprocess the acquired signals and remove possible noises. The Lamb wave phase velocity (cl ) was measured using the phase difference of acquired wave signals at different locations along the propagation direction. The Lamb wavenumber (kl ) was consequently obtained and the elastic modulus of each specimen was estimated. To reduce the effects of the nonhomogeneous dispersion of matrix cracks and increase the accuracy of estimated wave velocity and elastic modulus, the propagated Lamb wave data were acquired for each intact or damaged specimens at three different distances of 5, 6, and 7 cm from the actuator. The average value of the wave velocities for each spec imen, obtained using the data acquired at different locations, was used for the estimation of the Lamb wavenumber and the elastic modulus. This procedure was carried out for the six fabricated composite speci mens (three glass/epoxy and three glass/CNF/epoxy specimens) and each crack density (0, 0.05, 0.1, 0.15, 0.2, and 0.25).
3.3. Inducing matrix cracks and tensile tests The STM-150 tensile test machine was used for inducing matrix cracks with different densities in the glass/epoxy and glass/CNF/epoxy laminated composite specimens by applying five loading steps to the specimens. These specific loading steps were determined using the Andersons et al. model to be 65.5, 76.0, 90.0, 107.0, and 130.0 MPa, respectively, to induce the crack densities of 0.05, 0.10, 0.15, 0.20, and 0.25 (1/mm) in the composite specimens. After each loading step, the induced cracks were observed and quantified using a digital optical microscope. Moreover, the mechanical properties of the specimens were obtained using the same tensile machine. The tensile stress of 34 MPa was applied to the specimens for the measurement of elastic modulus of intact and damaged specimens. This applied tensile stress was smaller than the tensile strength of the 90-degree lamina and thus smaller than the initiation load to create matrix cracking in the composite specimens. The specimens also underwent the Lamb wave propagation tests after each loading step, after which the next loading step was applied to the specimens. The tensile strength of intact laminated specimens was obtained using tensile tests, in which the applied stress was increased until the fracture occurred. The crosshead displacement rate during the tensile
Fig. 1. a) Dimensions of the laminated and unidirectional composite specimens (all dimensions are in mm). b) The fabricated composite and nano composite specimens. 4
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Fig. 2. The tensile test setup.
Fig. 3. Nondestructive Lamb wave propagation test setup and equipment. 1) Function generator, 2) Oscilloscope, 3) Ultrasonic actuator, 4) Ultrasonic sensor, 5) Computer system and 6) Composite specimen.
4. Results and discussion
ultimate strength was observed as a result of adding CNF fillers to the pure epoxy. This increase in the elastic modulus and ultimate strength of the CNF/epoxy specimens was due to the addition of nanofibers that bridge the molecular chains of the polymer. Similar results were re ported by Shokrieh et al. [28], where an increase of about 10% and 23% were respectively observed in the elastic modulus and ultimate strength of epoxy resin when CNF fillers with 0.25% weight fraction were added to it. Table 1 presents the elastic modulus and strength of the fabricated specimens obtained from tensile tests. The elastic modulus and strength of the 90-degree layers of the composite specimens, i.e. transverse uni directional composites, were increased by 8.42% and 18.35%, respec tively, by adding CNFs to the matrix. The elastic modulus and strength of the cross-ply laminated composite specimens were only increased by 0.66% and 0.65%, respectively. The main reason for the limited improvement of the mechanical properties of laminated composite specimens was that the 0-degree layers carried most of the applied stresses which were not reinforced by the CNF filler. The microscopic images of induced matrix cracking in a typical glass/CNF/epoxy specimen under different tensile loads are shown in Fig. 5. As shown in this figure, the cracks were formed far apart and at an approximately similar distance from each other, thus the simplifying assumptions of the Hashin model are valid for the cases studied in this research. It was also observed in the microscopic images that no matrix cracks were induced in glass/CNF/epoxy specimens in the first loading
The results obtained from the tensile and Lamb wave propagation tests are presented and discussed in this section. The stress-strain dia grams obtained from the tensile tests of typical pure epoxy and CNF/ epoxy specimens are shown in Fig. 4. As shown in this figure, an average increase of about 11.1% in the elastic modulus and 19.7% in the
Fig. 4. The stress-strain diagrams were obtained from static tensile tests of typical pure epoxy and CNF/epoxy specimens. 5
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Table 1 The mean value of the elastic modulus and strength of the fabricated specimens obtained from tensile tests. Epoxy
Elastic modulus (GPa) Strength (MPa)
Unidirectional composites (longitudinal direction)
Unidirectional composites (transverse direction)
Cross-ply laminated composites
Without CNF
With CNF
Without CNF
Without CNF
With CNF
Without CNF
With CNF
2.70 ± 0.05 66.4 ± 2.6
3.00 ± 0.07 79.5 ± 1.2
30.12 ± 0.2 540.1 ± 5.8
9.50 ± 0.7 38.0 ± 3.7
10.30 ± 0.8 45.0 ± 4.6
15.18 ± 0.03 139.2 ± 2.8
15.28 ± 0.06 140.1 ± 3.5
Fig. 5. Microscopic images of a typical glass/CNF/epoxy specimen under tensile loading. The loading respectively increased from the top image to bottom. The length of each specimen shown in the images is 26 mm.
step, unlike the glass/epoxy ones, as seen in the top image of Fig. 5. It was due to the fact that the strength of the 90-degree layers in the presence of CNF fillers was higher than the stress applied to these layers in the first loading step. As the applied load was increased, the crack density in the specimens gradually increased and eventually reached saturation which led to the fracture of specimens. As seen in Fig. 6, some tiny cracks were developed around the through-width transverse matrix cracks, which had a small effect on the elastic modulus of specimens. These tiny cracks were not taken into account in the calculation of the crack density and thus may cause dis crepancies between the experimental results and the predictions of the Hashin and Andersons et al. models. The stress-strain diagram for a typical glass/CNF/epoxy composite specimen is shown in Fig. 7. As seen in this figure, the specimens suf fered from stiffness loss due to inducing matrix cracks after the appli cation of each loading step. Note that the stiffness reduction caused by each loading step was obtained from the slope difference of the linear part of the stress-strain diagrams between two consecutive loading steps. No stiffness reduction was observed in the first step, as the stress applied to the composite laminate was smaller than the in-situ strength of the 90-degree layers, and thus no matrix cracks were induced. As the tensile load was further increased, the matrix cracks were initiated, and the
crack density increased. Fig. 8 shows an example of Lamb wave signals transmitted and received in one of the glass/CNF/epoxy specimens after the first loading step. As seen in Fig. 8, the wave amplitude decayed with the propagation distance, which was due to the inherent viscoelastic properties of com posite specimens and the scattering of the Lamb wave around the matrix cracks. The mean values of the results obtained for the glass/epoxy and glass/CNF/epoxy specimens from the tensile and wave propagation tests after each loading step are presented in Table 2. It is revealed that the discrepancies between the results obtained from these two tests were smaller than 3%, proving the capability of the proposed Lamb wave propagation method for nondestructive evaluation of composite mate rials and structures reinforce by CNF fillers. It was also observed that by adding the CNF fillers into the composite specimens, smaller crack densities and stiffness reduction was obtained at certain applied stress levels. For example, the stiffness reduction decreased 45% and the crack density reduction decreased 40% for the applied stress of 90.0 MPa. Fig. 9 shows the induced crack densities in the glass/epoxy and glass/CNF/epoxy specimens. The Andersons et al. model (Eq. (8)) was fitted to experimental data and the invariant of the model was estimated as 0.05 and 0.025 for the glass/epoxy and glass/CNF/epoxy specimens,
Fig. 6. Images of the matrix crack with larger zoom to see smaller and irregular cracks generated in the specimens. The width of each specimen shown in the images is 3 mm. 6
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Fig. 7. The stress-strain diagram of different loading steps in a typical cross-ply glass/CNF/epoxy specimen.
Fig. 8. The amplitude of Lamb waves transmitted by the ultrasonic actuator and received by the ultrasonic sensor versus the wave arrival time at different prop agation distances. The first half cycle of the wave is only presented. Table 2 The summary of the mean value of the results obtained from the tensile and wave propagation tests. The percentage of stiffness reduction after each loading step in terms of that of the intact specimens is given in parenthesis. Glass/epoxy laminated composite specimens Applied tensile stress (MPa) Stress in 90-degree layers (MPa) Lamb wave velocity (m/s) Elastic modulus from Lamb wave propagation tests (GPa) Elastic modulus from tensile tests (GPa) Crack density (1/mm)
0 0 1361 15.15 (0%) 15.18 (0%) 0
65.5 39.0 1341 14.08 (7.06%) 14.40 (5.14%) 0.05
76.0 46.5 1334 13.86 (8.51%) 14.06 (7.39%) 0.10
90.0 54.5 1297 12.83 (15.31%) 13.01 (14.30%) 0.15
107.0 64.5 1280 12.38 (18.28%) 12.34 (18.71%) 0.20
130.0 78.5 1254 11.71 (22.71%) 11.90 (21.61%) 0.25
90.0 60.5 1347 14.02 (7.94%) 14.10 (7.74%) 0.09
107.0 74.0 1324 13.50 (11.36%) 13.55 (11.34%) 0.13
130.0 89.0 1302 12.85 (15.63%) 12.91 (15.54%) 0.17
Glass/CNF/epoxy laminated composite specimens Applied tensile stress (MPa) Stress in 90-degree layers (MPa) Lamb wave velocity (m/s) Elastic modulus from Lamb wave propagation tests (GPa) Elastic modulus from tensile tests (GPa) Crack density (1/mm)
0 0 1387 15.23 (0%) 15.28 (0%) 0.00
65.5 44.5 1386 15.20 (0.20%) 15.25 (0.20%) 0.00
respectively. As shown in Fig. 9, the initiation of matrix cracking started at higher stresses in glass/CNF/epoxy specimens, which is due to the increased strength of the 90-degree layers. For example, in the case of the stress level of 130 MPa, the stress in the 90-degree layers was ob tained from the classical lamination theory as 89.0 and 75.5 MPa for the glass/epoxy and glass/CNF/epoxy specimens, respectively. The
76.0 51.0 1362 14.51 (4.72%) 14.63 (4.26%) 0.05
consequently induced crack densities at this stress level were obtained by the microscope as 0.25 and 0.17 for the glass/epoxy and glass/CNF/ epoxy specimens, respectively, revealing a 32% crack density drop. Fig. 10 shows the stiffness reduction versus the crack density for the glass/epoxy and glass/CNF/epoxy specimens obtained from the Lamb wave propagation tests and predicted by the Hashin model. As shown in 7
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By comparing the experimental results and the predictions of the Andersons et al. semi-analytical and the Hashin models, it was concluded that the increase in the ultimate strength of the 90-degree layers due to the addition of CNF fillers played a significant role in the reduction of crack density and the resulting stiffness reduction. Data availability The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study. CRediT authorship contribution statement H. Ramezani: Performing tests, Data curation, Data Collections, Writing - original draft. S. Kazemirad: Conceptualization, Methodol ogy, Supervision, Co-Supervision, Writing - review & editing. M.M. Shokrieh: Conceptualization, Methodology, Supervision, Writing - re view & editing.
Fig. 9. Induced crack density versus the stress level in the 90-degree layers predicted by the Andersons et al. model [9] and obtained from the Lamb wave propagation tests.
Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This research was supported by Iran National Science Foundation, Grant No. 97024007, and Grant No. 97019282. References [1] C. Soutis, Fibre reinforced composites in aircraft construction, Prog. Aero. Sci. 41 (2005) 143–151. [2] M. Kashtalyan, S. Costas, Analysis of composite laminates with intra- and interlaminar damage, Prog. Aero. Sci. 41 (2005) 152–173. [3] Z. Hashin, Analysis of cracked laminates: a variational approach, Mech. Mater. 4 (1985) 121–136. [4] B. Mohammadi, H. Pakdel, Fatigue driven matrix crack propagation in laminated composites, Mater. Des. 146 (2018) 108–115. [5] J. Llobet, P. Maimí, Y. Essa, F. Martin de la Escalera, Progressive matrix cracking in carbon/epoxy cross-ply laminates under static and fatigue loading, Int. J. Fatig. 119 (2019) 330–337. [6] S. Liu, J.A. Nairn, The formation and propagation of matrix microcracks in crossply laminates during static loading, J. Reinforc. Plast. Compos. 11 (1992) 158–178. [7] A. Tounsi, K.H. Amara, Analysis of transverse cracking and stiffness loss in crossply laminates with hygrothermal conditions, Comput. Mater. Sci. 32 (2005) 167–174. [8] M. Bouazza, A. Tounsi, A. Benzair, E. Adda-Bedia, Effect of transverse cracking on stiffness reduction of hygrothermal aged cross-ply laminates, Mater. Des. 28 (2007) 1116–1123. [9] J. Andersons, E. Sparnicnˇs, O. Rubenis, R. Joffe, Estimation of laminate stiffness reduction due to cracking of a transverse ply by employing crack initiation-and propagation-based master curves, Mech. Compos. Mater. 44 (2008) 141–150. [10] Y. Deng, W. Li, X. Zhang, Y. Li, H. Kou, J. Shao, et al., Modeling on temperaturedependent first matrix cracking stress for fiber reinforced ceramics considering fiber debonding and residual thermal stress, Ceram. Int. 44 (2018) 21666–21674. [11] N.D. Barulich, L.A. Godoy, P.M. Dardati, Evaluation of cross-ply laminate stiffness with a non-uniform distribution of transverse matrix cracks by means of a computational meso-mechanic model, Compos. Struct. 185 (2018) 561–572. [12] L. Sun, Z. Ounaies, X. Gao, C.A. Whalen, Z. Yang, Preparation, characterization, and modeling of carbon nanofiber/epoxy nanocomposites, Nanomaterials 2011 (2011) 8. [13] Y. Zhou, F. Pervin, S. Jeelani, P.K. Mallick, Improvement in mechanical properties of carbon fabric - epoxy composite using carbon nanofibers, Mater Process Technol 8 (2007) 445–453. [14] M. Arai, J. Hirokawa, Y. Hanamura, H. Ito, M. Hojo, M. Quaresimin, Characteristic of mode I fatigue crack propagation of CFRP laminates toughened with CNF interlayer, Compos. B Eng. 65 (2014) 26–33. [15] V. Koissin, L. Warnet, R. Akkerman, Delamination in carbon-fibre composites improved with in situ grown nanofibres, Eng. Fract. Mech. 101 (2013) 140–148. [16] M. Arai, T. Sasaki, S. Hirota, H. Ito, N. Hu, M. Quaresimin, Mixed modes interlaminar fracture toughness of CFRP laminates toughened with CNF interlayer, Acta Mech. Solida Sin. 25 (2012) 321–330. [17] Y. Choi, Mechanical and physical properties of epoxy composites reinforced by vapor grown carbon nanofibers, Carbon N Y 43 (2005) 2199–2208.
Fig. 10. Stiffness reduction in the glass/epoxy and glass/CNF/epoxy specimens with crack density obtained from the Lamb wave propagation tests and pre dicted by the Hashin model.
Fig. 10, the stiffness reduction was not significantly different for the glass/CNF/epoxy and glass/epoxy specimens at similar crack densities. Therefore, the main effect of adding CNFs to the matrix of 90-degree layers was the enhancement of their strength, which caused a smaller crack density at a certain stress level and consequently a smaller stiffness reduction. 5. Conclusion The main purpose of the present study was to reduce the density of matrix cracks induced in cross-ply laminated composites under tensile loading by adding CNF fillers in their 90-degree layers. Cross-ply lami nated composite specimens with the stacking sequence of [02/906]s were fabricated, in which the matrix epoxy of the 90-degree layers was reinforced with the CNF fillers. Different cracks densities were induced in the specimens using a tensile testing machine. The crack density was quantified using an optical microscope and the elastic modulus of the specimens was obtained using the tensile test and Lamb wave propa gation technique. The discrepancies between results of the Lamb wave propagation tests and those of the tensile tests were smaller than 3%, proving the capability of the proposed Lamb wave propagation method for nonde structive evaluation of hybrid composite materials and structures. It was also revealed that the matrix crack density and the resulting stiffness reduction of the specimens significantly decreased at each specific stress level by adding 0.25 wt% CNF into the matrix of composite specimens. 8
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Polymer Testing xxx (xxxx) xxx [32] K. Zheng, Z. Li, Z. Ma, J. Chen, J. Zhou, X. Su, Damage detection method based on Lamb waves for stiffened composite panels, Compos. Struct. 225 (2019) 111137. [33] L. Huang, L. Zeng, J. Lin, Z. Luo, An improved time reversal method for diagnostics of composite plates using Lamb waves, Compos. Struct. 190 (2018) 10–19. [34] N.P. Yelve, M. Mitra, P. Mujumdar, Detection of delamination in composite laminates using Lamb wave based nonlinear method, Compos. Struct. 159 (2017) 257–266. [35] Z. Ma, J. Chen, B. Li, Z. Li, X. Su, Dispersion analysis of Lamb waves in composite laminates based on reverberation-ray matrix method, Compos. Struct. 136 (2016) 419–429. [36] P. Ochˆ oa, V. Infante, J.M. Silva, R.M. Groves, Detection of multiple low-energy impact damage in composite plates using Lamb wave techniques, Compos. B Eng. 80 (2015) 291–298. [37] S. Pant, J. Laliberte, M. Martinez, B. Rocha, D. Ancrum, Effects of composite lamina properties on fundamental Lamb wave mode dispersion characteristics, Compos. Struct. 124 (2015) 236–252. [38] A. Mardanshahi, M. Shokrieh, S. Kazemirad, Identification of matrix cracking in cross-ply laminated composites using Lamb wave propagation, Compos. Struct. 235 (2020) 111790. [39] A. Mardanshahi, V. Nasir, S. Kazemirad, M. Shokrieh, Detection and classification of matrix cracking in laminated composites using guided wave propagation and artificial neural networks, Compos. Struct. (2020) 112403. [40] D.N. Alleyne, P. Cawley, Optimization of lamb wave inspection techniques, NDT E Int. 25 (1992) 11–22. [41] P.D. Wilcox, M. Lowe, P. Cawley, The effect of dispersion on long-range inspection using ultrasonic guided waves, NDT&E Int 34 (2001) 1–9. [42] T. Kundu, Ultrasonic Nondestructive Evaluation: Engineering and Biological Material Characterization, CRC Press, 2003. [43] M. Castaings, B. Hosten, Guided waves propagating in sandwich structures made of anisotropic, viscoelastic, composite materials, J. Acoust. Soc. Am. 113 (2003) 2622–2634. [44] N. Toyama, J. Noda, T. Okabe, Quantitative damage detection in cross-ply laminates using Lamb wave method, Compos. Sci. Technol. 63 (2003) 1473–1479. [45] N. Toyama, J. Takatsubo, Lamb wave method for quick inspection of impactinduced delamination in composite laminates, Compos. Sci. Technol. 64 (2004) 1293–1300. [46] B.S. Ben, B.A. Ben, K.A. Vikram, S.H. Yang, Damage identification in composite materials using ultrasonic based Lamb wave method, Measurement 46 (2013) 904–912. [47] N. Laws, G.J. Dvorak, Progressive transverse cracking in composite laminates, J. Compos. Mater. 22 (1988) 900–916.
[18] F. Pervin, Y. Zhou, V.K. Rangari, S. Jeelani, Testing and evaluation on the thermal and mechanical properties of carbon nano fiber reinforced SC-15 epoxy, Mater Sci Eng A 405 (2005) 246–253. [19] G.G. Tibbetts, M.L. Lake, K.L. Strong, B.P. Rice, A review of the fabrication and properties of vapor-grown carbon nanofiber/polymer composites, Compos. Sci. Technol. 67 (2007) 1709–1718. [20] S. Bal, Experimental study of mechanical and electrical properties of carbon nanofiber/epoxy composites, Mater. Des. 31 (2010) 2406–2413. [21] C.Z. Liao, H.M. Wong, K. Wai, K. Yeung, The development, fabrication, and material characterization of polypropylene composites reinforced with carbon nanofiber and hydroxyapatite nanorod hybrid fillers, Int. J. Nanomed. 9 (2014) 1299–1310. [22] H. Ma, J. Zeng, M. Lynn, S. Kumar, D.A. Schiraldi, Processing, structure, and properties of fibers from polyester/carbon nanofiber composites, Compos. Sci. Technol. 63 (2003) 1617–1628. [23] Y. Iwahori, S. Ishiwata, T. Sumizawa, T. Ishikawa, Mechanical properties improvements in two-phase and three-phase composites using carbon nano-fiber dispersed resin, Compos Part A Appl Sci Manuf 36 (2005) 1430–1439. [24] E. Alishahi, S. Shadlou, S. Doagou-r, M.R. Ayatollahi, Effects of carbon nanoreinforcements of different shapes on the mechanical properties of epoxybased nanocomposites, Macromol Mater 298 (2012) 670–678. [25] M.H. Al-Saleh, U. Sundararaj, A review of the mechanical properties of carbon nanofiber/polymer composites, Compos Part A 42 (2011) 2126–2142. [26] M. S´ anchez, J. Rams, M. Campo, A. Ure˜ na, Characterization of carbon nanofiber/ epoxy nanocomposites by the nanoindentation technique, Compos Part B 42 (2011) 638–644. ˇ ˇ [27] M. Slouf, M. F´ aberov´ a, M. Spírkov´ a, R. Poręba, R. Bureˇs, A. Duszov´ a, et al., Thermoplastic polybutadiene-based polyurethane/carbon nanofiber composites, Compos. B Eng. 67 (2014) 434–440. [28] M.M. Shokrieh, M. Esmkhani, F. Vahedi, H.R. Shahverdi, Improvement of mechanical and electrical properties of epoxy resin with carbon nanofibers, Iran Polym 22 (2013) 721–727. [29] Y. Zhou, S. Jeelani, T. Lacy, Experimental study on the mechanical behavior of carbon/epoxy composites with a carbon nanofiber-modified matrix, J. Compos. Mater. 48 (2014) 3659–3672. [30] A. Tabatabaeian, A.R. Ghasemi, Curvature changes and weight loss of polymeric nano-composite plates with consideration of the thermal cycle fatigue effects and different resin types: an experimental approach, Mech. Mater. 131 (2019) 69–77. [31] A.R. Ghasemi, A. Tabatabaeian, B. Asghari, Application of slitting method to characterize the effects of thermal fatigue, lay-up arrangement and MWCNTs on the residual stresses of laminated composites, Mech. Mater. 134 (2019) 185–192.
9
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Polymer Testing journal homepage: http://www.elsevier.com/locate/polytest
Effects of adding carbon nanofibers on the reduction of matrix cracking in laminated composites: Experimental and analytical approaches H. Ramezani a, S. Kazemirad b, M.M. Shokrieh a, *, A. Mardanshahi a a
Composites Research Laboratory, Center of Excellence in Experimental Solid Mechanics and Dynamics, School of Mechanical Engineering, Iran University of Science and Technology, Tehran, 16846-13114, Iran b School of Mechanical Engineering, Iran University of Science and Technology, Tehran, 16846-13114, Iran
A R T I C L E I N F O
A B S T R A C T
Keywords: Polymeric composites Carbon nanofiber Matrix cracking Lamb wave Elastic properties Micromechanical models
The main purpose of the present study is to investigate the effect of adding carbon nanofibers (CNFs) in fibrous polymeric composites on the reduction of matrix cracking. A tensile testing machine was used to apply five loading steps and induce different crack densities in [02/906]s composite specimens with and without CNF fillers. After each loading step, the elastic modulus of specimens was measured. The crack density of damaged speci mens was also measured by an optical microscope after each loading step. The Lamb wave propagation method was used to assess the elastic modulus of specimens at different crack densities. It was demonstrated that the addition of CNF fillers into the composite specimens resulted in lower crack densities and less stiffness reduction at certain applied stress levels. Moreover, Hashin and Andersons et al. micromechanical models were assessed by the experimental results for the analysis of matrix cracking in cross-ply laminated composites. For composite specimens with and without CNFs, the stiffness reduction at different crack densities was in good agreement with the predictions of the Hashin model, and the induced crack density at each applied loading step was in good agreement with the predictions of the Andersons et al. model.
1. Introduction Composite structures suffer from different damage mechanisms [1]. Generally, all of the critical damage modes of composites, caused by static and fatigue loadings, are initiated by the transverse cracking of the matrix [2]. The development of methods to postpone the initiation of matrix cracking in laminated composites and to identify its severity may play an important role in improving applications of these materials. The growth of matrix cracking in composites under static and fatigue loadings has been investigated in the past decades [3–5]. Hashin [3] proposed an analytical model based on the variational method to predict the growth of matrix cracking damage in cross-ply multilayered com posites under tensile and shear loadings. It was shown that the Hashin model properly predicts the stiffness reduction at different matrix crack densities as measured by the experiments. Liu and Nairn [6] developed a semi-empirical relation based on the experimental results between the matrix crack density in polymeric composites and the tensile load. They also fitted the obtained experimental results with the energy release rate expression and estimated the microcracking and intralaminar fracture toughness. Tounsi and Amara [7] numerically, analytically, and
experimentally investigated the variation in the longitudinal modulus of the cross-ply laminated composite due to the matrix cracking, by considering a relationship between the stiffness reduction and the variation of the temperature and humidity. Their results showed that the hygrothermal environment had a significant effect on the relative reduction of the longitudinal modulus at higher crack densities. Bouazza et al. [8] presented an analytical model based on the concept of stress turbulence function to predict the effect of transverse matrix cracks on the stiffness reduction of composite laminates. They also showed that the moisture had no significant effect on the longitudinal modulus of the tested composite laminates, while an increase in the temperature caused a reduction in the longitudinal modulus, especially at higher crack densities. Andersons et al. [9] proposed a semi-analytical model based on the shear lag method, in which a relation between the matrix cracking in cross-ply laminated composite and stress in 90-degree layers was given. Deng et al. [10] investigated the mixed effects of three factors of temperature, fiber-matrix debonding, and thermal residual stress on the first matrix cracking stress for the fiber-reinforced ceramic com posites. They also quantitatively analyzed the effect of composite vol ume fraction on the onset of matrix cracking. Barulich et al. [11] used a
* Corresponding author. E-mail address: [email protected] (M.M. Shokrieh). https://doi.org/10.1016/j.polymertesting.2020.106988 Received 31 July 2020; Received in revised form 28 October 2020; Accepted 23 November 2020 Available online 28 November 2020 0142-9418/© 2020 The Authors. Published by Elsevier Ltd. This is an open
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H. Ramezani et al.
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meso-mechanics model for the analysis of matrix cracking, consisted of a continuous three-dimensional finite element model in the mesoscale and a classic thin plates laminate model in the macro-scale for cross-ply laminates. Their results showed that the model with a non-uniform crack density presented a better agreement with experimental data than the model with uniform crack density. Mohammadi and Pakdel [4] developed a damage growth criterion based on the energy release rate that can predict the matrix crack density in laminated composites with different layups under fatigue loading. Based on the analytical and experimental results presented in prior research works, the elastic modulus and ultimate strength of the poly mer may increase by adding certain types of nanofibers or nanoparticles to the resin up to a certain weight fraction and using an appropriate dispersion method [12,13]. The effects of CNFs on failure and fracture behavior of laminated composites were also investigated [14–16]. Furthermore, other properties of the resin such as the heat transfer properties, ultimate strain, and electrical conductivity are improved by adding CNFs to the resin [17–21]. Different optimum weight fractions of CNF were reported in previous studies to obtain the maximum increase in the elastic modulus and ultimate strength of the resin [22–24]. These discrepancies in the optimum weigh fraction may be due to the differ ences of methods used for the dispersion of CNF in the resin, different sizes of CNF used, and the fabrication and testing conditions of the specimens. It was also found in these studies that the effect of CNF on increasing the tensile strength of the resin was more than that on the elastic modulus [25–27]. Shokrieh et al. [28] revealed that the optimum weight fraction of CNF to achieve the best mechanical properties of CNF/epoxy nanocomposites was 0.25 wt%. They observed an increase of 23% in the tensile strength and 10% in the flexural strength for this filler content. Zhou et al. [29] showed that the elastic modulus of epoxy modified with CNF increased with increasing the weight fraction of this nanofiber. Their results revealed that the ultimate tensile strength, fa tigue life, and fracture toughness also increased significantly by increasing the CNF content in the resin, reaching their maximum values at a weight fraction of 2%. Moreover, they observed that the nanofibers caused a random distribution of the matrix cracks. Tabatabaeian et al. [30] investigated the effect of multi-walled carbon nanotubes (MWCNTs) and thickness on the curvature changes and weight loss of laminated composites in different thermal cycles. They observed that the weight loss and curvature reduced in thermal fatigue by adding MWCNTs into the specimens. Ghasemi et al. [31] investigated the effect of MWCNTs on the residual stress in laminated composite with theo retical and experimental approaches. Their results revealed that rein forcing the composite specimens with 1% MWCNTs reduced the residual stresses up to 35%. Material characterization and damage detection methods based on the Lamb wave propagation have been developed for the nondestructive evaluation of multilayered composites [32–39]. For example, Pant et al. [37] proposed an inverse method based on the theory of Lamb wave propagation to monitor the material constants and structural health of aerospace composites and metal structures. This method was used to investigate the effect of the composite layup, stiffness, density, and the Lamb wave propagation direction on the dispersion curve. Mardanshahi et al. [38] proposed a viscoelastic characterization approach based on the Lamb wave propagation method for the identification of matrix cracking in cross-ply laminated composites. In most proposed methods, the fundamental Lamb wave modes were used along with the inverse problem solution techniques. The general criteria for selecting the Lamb wave mode for the nondestructive evaluation of composite structures include limited wave scattering, small wave attenuation, proper fault sensitivity, and simple and feasible wave excitation and detection [40–46]. To the best knowledge of the present authors, the effect of adding CNFs to polymeric composites on the initiation and propagation of the matrix cracking has not been investigated. Moreover, the development and assessment of a non-destructive method based on the Lamb wave
propagation for the evaluation of matrix crack density and stiffness loss in polymeric composites reinforced by CNF may be of great interest. Therefore, the effect of adding CNFs on the matrix cracking behavior of cross-ply polymeric composites was investigated in the present study and assessed by a nondestructive evaluation method based on the Lamb wave propagation technique. In the following sections, two analytical and semi-analytical micromechanical models governing the crack den sity and the resulting stiffness loss of cross-ply composites, and the equations governing the Lamb wave propagation are presented. After ward, the fabrication process of specimens, including the CNF/epoxy nanocomposites and hybrid glass/CNF/epoxy composite specimens is explained. Then, the process of inducing matrix cracks in the specimens, and tensile and Lamb wave propagation tests are described. Finally, the results obtained from the tensile and Lamb wave propagation tests are presented and discussed. It was shown that the matrix crack density and stiffness loss in composite specimens decreased by adding CNF fillers to polymeric composites, and the Lamb wave propagation method may offer a powerful tool for the nondestructive evaluation of such hybrid polymer composite materials. 2. Background theory 2.1. Micromechanical models for matrix cracking The models proposed by Hashin [3] and by Andersons et al. [9] were used in the present study to predict and analyze matrix cracking growth, assess the accuracy of tensile and Lamb wave propagation experiments, and analyze the effect of CNFs on the stiffness and strength of composite specimens. The Hashin model [3] is based on the calculation of the stress field by the variational method, while the Andersons et al. model [9] is based on the calculation of the stress field via the shear lag method. In the Hashin model [3], after calculation of the stress field in a cross-ply multilayered composite specimen under unidirectional tensile loading, a lower limit is proposed for the elastic modulus of the damaged specimen (Ex ) as: 1 1 1 < χ (δ) > ≤ + k2 η(ψ ) Ex Exο ET 2 <δ> where ψ = tt12 , η(ψ ) =
3ψ 2 +12ψ +8 , 60
(1) k2 = σσ20 , t1 and t2 are the half of the
thickness of 0-degree and 90-degree layers, respectively, ET is the stiff ness of 90-degree layers or the transverse stiffness, σ2 is the stress in 90degree layers, and σ0 is the total stress of the laminate. The value of k2 (relative stiffness of 90-degree layers) and E0x (initial laminate stiffness) are obtained from the classical lamination theory of the undamaged composite laminate by having the longitudinal and transverse elastic moduli of the unidirectional composite ply. Also, we have: (2)
δ = a/t2 (
χ (δ) = 2αβ α2 + β2
)
(
cosh 2 αρ − cos 2 βρ α sin 2 βρ + β sinh 2 αρ
) (3)
α = q1/4 cos θ 2
/
(4)
/ β = q1/4 sin θ 2
(5)
tan θ =
√̅̅̅̅̅ 4q − 1 p2
(6)
where a is the half of the distance between two adjacent cracks, and q and p are factors that depend exclusively on the layup and mechanical properties of the composite laminate. If the cracks are equidistant, we have δn = δ = a/t2 , < χ (δ) > = χ (δ), and < δ > = δ. Andersons et al. [9] proposed a mathematical relation between the stress and the crack density by using the Norman equations [47]. For a 2
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cross-ply laminated composite, the shear lag parameter is defined as: √̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ 3(d + b)E0x G12 G23 K= (7) dbEL ET (bG12 + dG23 )
where h is half of the thickness of the plate. Also, γand ζ are defined as: √̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ (14) γ = k2p − k2l
where b and d are the thickness of 0◦ and 90◦ layers of the laminate, respectively. Also, E0x is the elastic modulus of composite laminate, EL and ET are the longitudinal and transverse elastic moduli of each uni directional composite ply, respectively. Moreover, G12 and G23 are inplane and out of plane shear moduli, respectively. In this model the crack density is related to the applied stress in the 90-degree layer based on the following equation [9]: ( ) (ρ )2 σ2 c = cinit − 1 (8) K σinit
ζ=
where kl is the Lamb wavenumber, and ks and kp are shear and compression wavenumbers, respectively. The shear wave velocity is related to the compression velocity through [42]: ( )2 cs 1 − 2ν μ = = (16) cp 2 − 2ν λ + 2μ
where ρc is the crack density, σinit is the in-situ strength of 90-degree layers, σ2 is the stress in 90-degree layers of the multi-layered compos ite specimen, cinit is a material-independent factor that is evaluated by fitting the theoretical master curve given in Eq. (8) to the tensile test data. In this study, cinit was estimated from the experimental data for the composite specimens with and without CNF fillers. The Andersons et al. model [9] was used in the present study to assess the effect of the strength of 90-degree layers on the matrix crack density.
G = ρc2s
(17)
E = 2(1 + ν)G
(18)
3. Materials and methods
2.2. Lamb wave theory
3.1. Materials
Lamb waves are elastic perturbations that propagate in thin solid plates with two stress-free boundaries, in which the particle displace ments of the propagation medium occur in the direction of propagation of the wave and also perpendicular to it [42]. Lamb waves are caused by the interaction of compression waves (p) and shear waves (s) propa gating within the medium. Lamb waves may propagate in two inde pendent symmetric and anti-symmetric modes. In this study, the fundamental anti-symmetric Lamb wave mode (A0) was generated in the composite specimens. The relation between the compression wave ve locity (cp ) and the shear wave velocity (cs ) and the mechanical proper ties of the propagation medium are as: √̅̅̅̅̅̅̅̅̅̅̅̅̅ λ + 2μ (9) cp =
Razeen® LR1100 epoxy resin with a mixing ratio of 13:100 of hardener to resin was used for the fabrication of the pure resin, nano composites, and laminated composite specimens. The CNF used in this study had an outer diameter of 200–600 nm, a length of 5–50 μm, fabricated by US Research Nanomaterials, Inc. Unidirectional E-glass fibers of 220 g/m2 manufactured by P-D Interglas Technologies were used for the fabrication of cross-ply laminated composite specimens.
√̅̅̅
μ ρ
3.2. Fabrication of the specimens Three pure epoxy resin specimens were fabricated based on the dimension requirements of the ASTM D638-03 standard. In the fabri cation of pure resin specimens, the mechanical mixing process was carried out for 5 min with a velocity of 200 rpm. The mixture was subsequently vacuumed for 5 min at a pressure of 1 mbar, after which the molding of the mixture was performed. The specimens were then pre-cured at room temperature for 2 days and were post-cured in the oven at 140 ◦ C for 2 h. The final preparation of the specimens to reach the standard dimensions and to remove the surface bubbles were per formed by sanding and polishing. Three CNF/epoxy specimens were also fabricated to assess the effect of CNFs on the elastic properties of the resin. First, the temperature of the resin was increased to 60 ◦ C to reduce its viscosity and improve the wetting of carbon nanofibers in the fabrication process. A CNF weight fraction of 0.25% was selected for the fabrication of specimens, as it was shown to be the optimum weight fraction for the enhancement of the mechanical properties of CNF/epoxy nanocomposites [28]. The CNF/epoxy mixture was then mechanically stirred for 30 min at a speed of 2000 rpm. The mixture was subsequently heated to 70 ◦ C and then sonicated for 100 min with an ultrasonic instrument (Hielscher Ultra sound Technology, UP400S) with an output power of 75 W and 12 kHz frequency to provide an appropriate dispersion of the CNF fillers within the resin. The vacuum, molding, and following sanding and polishing procedures were conducted similar to those for the pure resin specimens. Three glass/epoxy and three glass/CNF/epoxy unidirectional com posite specimens with [08] and [908] stacking layups, respectively, and three glass/epoxy and three glass/CNF/epoxy cross-ply laminated composite specimens with [02/906]s stacking layup were fabricated with the hand layup technique. The CNF fillers were only added to the epoxy
(10)
where ρ is the mass density of the propagation medium, and μ and λ are Lame constants, defined as: λ=
2ν G 1 − 2ν
μ=G
(11) (12)
where Gis the shear modulus, and ν is the Poisson’s ratio of the material. By solving the Navier equation for elastic materials through the decomposition of the displacement field and defining the potential functions of compression and shear waves, the dispersion equation for the antisymmetric Lamb wave mode is obtained. Due to the low thick ness of the composite specimens fabricated and tested in the present study compared to the Lamb wavelength, it can be assumed that a single wave with similar characteristics propagates over the whole thickness of the laminate. Therefore, the following equation may be appropriately used for Lamb wave propagation within the multilayered composite specimens [42]: tanh (γh) 4k2 γζ = ( 2 l 2) tanh (ζh) 2kl − ks
(15)
The shear wavenumber and the shear wave velocity are obtained through the numerical solution of Eq. (13), having the Lamb wave number measured from experiments and the Poisson’s ratio of speci mens. Then, the shear and elastic moduli of the propagation medium are obtained via:
ρ
cs =
√̅̅̅̅̅̅̅̅̅̅̅̅̅̅ k2s − k2l
(13)
3
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resin used for the fabrication of 90-degree layers of the glass/CNF/epoxy laminated composite specimens. The overall fabrication process of these specimens was similar to that of the previously mentioned specimens. The fabricated specimens were pressurized with a 200 N force for 48 h during the pre-curing at room temperature to eliminate the bubbles and obtain a uniform thickness and an appropriate volume fraction. After post-curing in the oven, the specimens were cut with the dimensions illustrated in Fig. 1, which were based on the ASTM D3039/D3039M standard for obtaining the unidirectional properties of the composite specimens. The specimen tabs were also fabricated using glass/epoxy composites with [±45]4s layup.
tests was set to 2 mm/min (static loading condition) and an extensom eter (STT-100D) was used to measure the extensions of the specimens during the tests. A similar test procedure was carried out to obtain the elastic modulus and tensile strength of pure epoxy, CNF/epoxy, and unidirectional composite specimens. Fig. 2 shows the static tensile test setup. 3.4. Lamb wave propagation tests A nondestructive Lamb wave propagation test procedure was also performed to obtain the elastic properties of composite specimens. The configuration of the experimental setup used in the present study for the ultrasonic Lamb wave propagation tests is shown in Fig. 3. An ultrasonic actuator connected to a generator was used to induce Lamb waves in the specimens with a frequency of 206 kHz and an excitation voltage of 150 V. A high precision ultrasonic sensor connected to a digital data acqui sition system was used to acquire the propagated wave data at different locations along the propagation direction. A finite impulse response (FIR) digital filter with an equiripple algorithm was used to preprocess the acquired signals and remove possible noises. The Lamb wave phase velocity (cl ) was measured using the phase difference of acquired wave signals at different locations along the propagation direction. The Lamb wavenumber (kl ) was consequently obtained and the elastic modulus of each specimen was estimated. To reduce the effects of the nonhomogeneous dispersion of matrix cracks and increase the accuracy of estimated wave velocity and elastic modulus, the propagated Lamb wave data were acquired for each intact or damaged specimens at three different distances of 5, 6, and 7 cm from the actuator. The average value of the wave velocities for each spec imen, obtained using the data acquired at different locations, was used for the estimation of the Lamb wavenumber and the elastic modulus. This procedure was carried out for the six fabricated composite speci mens (three glass/epoxy and three glass/CNF/epoxy specimens) and each crack density (0, 0.05, 0.1, 0.15, 0.2, and 0.25).
3.3. Inducing matrix cracks and tensile tests The STM-150 tensile test machine was used for inducing matrix cracks with different densities in the glass/epoxy and glass/CNF/epoxy laminated composite specimens by applying five loading steps to the specimens. These specific loading steps were determined using the Andersons et al. model to be 65.5, 76.0, 90.0, 107.0, and 130.0 MPa, respectively, to induce the crack densities of 0.05, 0.10, 0.15, 0.20, and 0.25 (1/mm) in the composite specimens. After each loading step, the induced cracks were observed and quantified using a digital optical microscope. Moreover, the mechanical properties of the specimens were obtained using the same tensile machine. The tensile stress of 34 MPa was applied to the specimens for the measurement of elastic modulus of intact and damaged specimens. This applied tensile stress was smaller than the tensile strength of the 90-degree lamina and thus smaller than the initiation load to create matrix cracking in the composite specimens. The specimens also underwent the Lamb wave propagation tests after each loading step, after which the next loading step was applied to the specimens. The tensile strength of intact laminated specimens was obtained using tensile tests, in which the applied stress was increased until the fracture occurred. The crosshead displacement rate during the tensile
Fig. 1. a) Dimensions of the laminated and unidirectional composite specimens (all dimensions are in mm). b) The fabricated composite and nano composite specimens. 4
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Fig. 2. The tensile test setup.
Fig. 3. Nondestructive Lamb wave propagation test setup and equipment. 1) Function generator, 2) Oscilloscope, 3) Ultrasonic actuator, 4) Ultrasonic sensor, 5) Computer system and 6) Composite specimen.
4. Results and discussion
ultimate strength was observed as a result of adding CNF fillers to the pure epoxy. This increase in the elastic modulus and ultimate strength of the CNF/epoxy specimens was due to the addition of nanofibers that bridge the molecular chains of the polymer. Similar results were re ported by Shokrieh et al. [28], where an increase of about 10% and 23% were respectively observed in the elastic modulus and ultimate strength of epoxy resin when CNF fillers with 0.25% weight fraction were added to it. Table 1 presents the elastic modulus and strength of the fabricated specimens obtained from tensile tests. The elastic modulus and strength of the 90-degree layers of the composite specimens, i.e. transverse uni directional composites, were increased by 8.42% and 18.35%, respec tively, by adding CNFs to the matrix. The elastic modulus and strength of the cross-ply laminated composite specimens were only increased by 0.66% and 0.65%, respectively. The main reason for the limited improvement of the mechanical properties of laminated composite specimens was that the 0-degree layers carried most of the applied stresses which were not reinforced by the CNF filler. The microscopic images of induced matrix cracking in a typical glass/CNF/epoxy specimen under different tensile loads are shown in Fig. 5. As shown in this figure, the cracks were formed far apart and at an approximately similar distance from each other, thus the simplifying assumptions of the Hashin model are valid for the cases studied in this research. It was also observed in the microscopic images that no matrix cracks were induced in glass/CNF/epoxy specimens in the first loading
The results obtained from the tensile and Lamb wave propagation tests are presented and discussed in this section. The stress-strain dia grams obtained from the tensile tests of typical pure epoxy and CNF/ epoxy specimens are shown in Fig. 4. As shown in this figure, an average increase of about 11.1% in the elastic modulus and 19.7% in the
Fig. 4. The stress-strain diagrams were obtained from static tensile tests of typical pure epoxy and CNF/epoxy specimens. 5
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Table 1 The mean value of the elastic modulus and strength of the fabricated specimens obtained from tensile tests. Epoxy
Elastic modulus (GPa) Strength (MPa)
Unidirectional composites (longitudinal direction)
Unidirectional composites (transverse direction)
Cross-ply laminated composites
Without CNF
With CNF
Without CNF
Without CNF
With CNF
Without CNF
With CNF
2.70 ± 0.05 66.4 ± 2.6
3.00 ± 0.07 79.5 ± 1.2
30.12 ± 0.2 540.1 ± 5.8
9.50 ± 0.7 38.0 ± 3.7
10.30 ± 0.8 45.0 ± 4.6
15.18 ± 0.03 139.2 ± 2.8
15.28 ± 0.06 140.1 ± 3.5
Fig. 5. Microscopic images of a typical glass/CNF/epoxy specimen under tensile loading. The loading respectively increased from the top image to bottom. The length of each specimen shown in the images is 26 mm.
step, unlike the glass/epoxy ones, as seen in the top image of Fig. 5. It was due to the fact that the strength of the 90-degree layers in the presence of CNF fillers was higher than the stress applied to these layers in the first loading step. As the applied load was increased, the crack density in the specimens gradually increased and eventually reached saturation which led to the fracture of specimens. As seen in Fig. 6, some tiny cracks were developed around the through-width transverse matrix cracks, which had a small effect on the elastic modulus of specimens. These tiny cracks were not taken into account in the calculation of the crack density and thus may cause dis crepancies between the experimental results and the predictions of the Hashin and Andersons et al. models. The stress-strain diagram for a typical glass/CNF/epoxy composite specimen is shown in Fig. 7. As seen in this figure, the specimens suf fered from stiffness loss due to inducing matrix cracks after the appli cation of each loading step. Note that the stiffness reduction caused by each loading step was obtained from the slope difference of the linear part of the stress-strain diagrams between two consecutive loading steps. No stiffness reduction was observed in the first step, as the stress applied to the composite laminate was smaller than the in-situ strength of the 90-degree layers, and thus no matrix cracks were induced. As the tensile load was further increased, the matrix cracks were initiated, and the
crack density increased. Fig. 8 shows an example of Lamb wave signals transmitted and received in one of the glass/CNF/epoxy specimens after the first loading step. As seen in Fig. 8, the wave amplitude decayed with the propagation distance, which was due to the inherent viscoelastic properties of com posite specimens and the scattering of the Lamb wave around the matrix cracks. The mean values of the results obtained for the glass/epoxy and glass/CNF/epoxy specimens from the tensile and wave propagation tests after each loading step are presented in Table 2. It is revealed that the discrepancies between the results obtained from these two tests were smaller than 3%, proving the capability of the proposed Lamb wave propagation method for nondestructive evaluation of composite mate rials and structures reinforce by CNF fillers. It was also observed that by adding the CNF fillers into the composite specimens, smaller crack densities and stiffness reduction was obtained at certain applied stress levels. For example, the stiffness reduction decreased 45% and the crack density reduction decreased 40% for the applied stress of 90.0 MPa. Fig. 9 shows the induced crack densities in the glass/epoxy and glass/CNF/epoxy specimens. The Andersons et al. model (Eq. (8)) was fitted to experimental data and the invariant of the model was estimated as 0.05 and 0.025 for the glass/epoxy and glass/CNF/epoxy specimens,
Fig. 6. Images of the matrix crack with larger zoom to see smaller and irregular cracks generated in the specimens. The width of each specimen shown in the images is 3 mm. 6
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Fig. 7. The stress-strain diagram of different loading steps in a typical cross-ply glass/CNF/epoxy specimen.
Fig. 8. The amplitude of Lamb waves transmitted by the ultrasonic actuator and received by the ultrasonic sensor versus the wave arrival time at different prop agation distances. The first half cycle of the wave is only presented. Table 2 The summary of the mean value of the results obtained from the tensile and wave propagation tests. The percentage of stiffness reduction after each loading step in terms of that of the intact specimens is given in parenthesis. Glass/epoxy laminated composite specimens Applied tensile stress (MPa) Stress in 90-degree layers (MPa) Lamb wave velocity (m/s) Elastic modulus from Lamb wave propagation tests (GPa) Elastic modulus from tensile tests (GPa) Crack density (1/mm)
0 0 1361 15.15 (0%) 15.18 (0%) 0
65.5 39.0 1341 14.08 (7.06%) 14.40 (5.14%) 0.05
76.0 46.5 1334 13.86 (8.51%) 14.06 (7.39%) 0.10
90.0 54.5 1297 12.83 (15.31%) 13.01 (14.30%) 0.15
107.0 64.5 1280 12.38 (18.28%) 12.34 (18.71%) 0.20
130.0 78.5 1254 11.71 (22.71%) 11.90 (21.61%) 0.25
90.0 60.5 1347 14.02 (7.94%) 14.10 (7.74%) 0.09
107.0 74.0 1324 13.50 (11.36%) 13.55 (11.34%) 0.13
130.0 89.0 1302 12.85 (15.63%) 12.91 (15.54%) 0.17
Glass/CNF/epoxy laminated composite specimens Applied tensile stress (MPa) Stress in 90-degree layers (MPa) Lamb wave velocity (m/s) Elastic modulus from Lamb wave propagation tests (GPa) Elastic modulus from tensile tests (GPa) Crack density (1/mm)
0 0 1387 15.23 (0%) 15.28 (0%) 0.00
65.5 44.5 1386 15.20 (0.20%) 15.25 (0.20%) 0.00
respectively. As shown in Fig. 9, the initiation of matrix cracking started at higher stresses in glass/CNF/epoxy specimens, which is due to the increased strength of the 90-degree layers. For example, in the case of the stress level of 130 MPa, the stress in the 90-degree layers was ob tained from the classical lamination theory as 89.0 and 75.5 MPa for the glass/epoxy and glass/CNF/epoxy specimens, respectively. The
76.0 51.0 1362 14.51 (4.72%) 14.63 (4.26%) 0.05
consequently induced crack densities at this stress level were obtained by the microscope as 0.25 and 0.17 for the glass/epoxy and glass/CNF/ epoxy specimens, respectively, revealing a 32% crack density drop. Fig. 10 shows the stiffness reduction versus the crack density for the glass/epoxy and glass/CNF/epoxy specimens obtained from the Lamb wave propagation tests and predicted by the Hashin model. As shown in 7
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By comparing the experimental results and the predictions of the Andersons et al. semi-analytical and the Hashin models, it was concluded that the increase in the ultimate strength of the 90-degree layers due to the addition of CNF fillers played a significant role in the reduction of crack density and the resulting stiffness reduction. Data availability The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study. CRediT authorship contribution statement H. Ramezani: Performing tests, Data curation, Data Collections, Writing - original draft. S. Kazemirad: Conceptualization, Methodol ogy, Supervision, Co-Supervision, Writing - review & editing. M.M. Shokrieh: Conceptualization, Methodology, Supervision, Writing - re view & editing.
Fig. 9. Induced crack density versus the stress level in the 90-degree layers predicted by the Andersons et al. model [9] and obtained from the Lamb wave propagation tests.
Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This research was supported by Iran National Science Foundation, Grant No. 97024007, and Grant No. 97019282. References [1] C. Soutis, Fibre reinforced composites in aircraft construction, Prog. Aero. Sci. 41 (2005) 143–151. [2] M. Kashtalyan, S. Costas, Analysis of composite laminates with intra- and interlaminar damage, Prog. Aero. Sci. 41 (2005) 152–173. [3] Z. Hashin, Analysis of cracked laminates: a variational approach, Mech. Mater. 4 (1985) 121–136. [4] B. Mohammadi, H. Pakdel, Fatigue driven matrix crack propagation in laminated composites, Mater. Des. 146 (2018) 108–115. [5] J. Llobet, P. Maimí, Y. Essa, F. Martin de la Escalera, Progressive matrix cracking in carbon/epoxy cross-ply laminates under static and fatigue loading, Int. J. Fatig. 119 (2019) 330–337. [6] S. Liu, J.A. Nairn, The formation and propagation of matrix microcracks in crossply laminates during static loading, J. Reinforc. Plast. Compos. 11 (1992) 158–178. [7] A. Tounsi, K.H. Amara, Analysis of transverse cracking and stiffness loss in crossply laminates with hygrothermal conditions, Comput. Mater. Sci. 32 (2005) 167–174. [8] M. Bouazza, A. Tounsi, A. Benzair, E. Adda-Bedia, Effect of transverse cracking on stiffness reduction of hygrothermal aged cross-ply laminates, Mater. Des. 28 (2007) 1116–1123. [9] J. Andersons, E. Sparnicnˇs, O. Rubenis, R. Joffe, Estimation of laminate stiffness reduction due to cracking of a transverse ply by employing crack initiation-and propagation-based master curves, Mech. Compos. Mater. 44 (2008) 141–150. [10] Y. Deng, W. Li, X. Zhang, Y. Li, H. Kou, J. Shao, et al., Modeling on temperaturedependent first matrix cracking stress for fiber reinforced ceramics considering fiber debonding and residual thermal stress, Ceram. Int. 44 (2018) 21666–21674. [11] N.D. Barulich, L.A. Godoy, P.M. Dardati, Evaluation of cross-ply laminate stiffness with a non-uniform distribution of transverse matrix cracks by means of a computational meso-mechanic model, Compos. Struct. 185 (2018) 561–572. [12] L. Sun, Z. Ounaies, X. Gao, C.A. Whalen, Z. Yang, Preparation, characterization, and modeling of carbon nanofiber/epoxy nanocomposites, Nanomaterials 2011 (2011) 8. [13] Y. Zhou, F. Pervin, S. Jeelani, P.K. Mallick, Improvement in mechanical properties of carbon fabric - epoxy composite using carbon nanofibers, Mater Process Technol 8 (2007) 445–453. [14] M. Arai, J. Hirokawa, Y. Hanamura, H. Ito, M. Hojo, M. Quaresimin, Characteristic of mode I fatigue crack propagation of CFRP laminates toughened with CNF interlayer, Compos. B Eng. 65 (2014) 26–33. [15] V. Koissin, L. Warnet, R. Akkerman, Delamination in carbon-fibre composites improved with in situ grown nanofibres, Eng. Fract. Mech. 101 (2013) 140–148. [16] M. Arai, T. Sasaki, S. Hirota, H. Ito, N. Hu, M. Quaresimin, Mixed modes interlaminar fracture toughness of CFRP laminates toughened with CNF interlayer, Acta Mech. Solida Sin. 25 (2012) 321–330. [17] Y. Choi, Mechanical and physical properties of epoxy composites reinforced by vapor grown carbon nanofibers, Carbon N Y 43 (2005) 2199–2208.
Fig. 10. Stiffness reduction in the glass/epoxy and glass/CNF/epoxy specimens with crack density obtained from the Lamb wave propagation tests and pre dicted by the Hashin model.
Fig. 10, the stiffness reduction was not significantly different for the glass/CNF/epoxy and glass/epoxy specimens at similar crack densities. Therefore, the main effect of adding CNFs to the matrix of 90-degree layers was the enhancement of their strength, which caused a smaller crack density at a certain stress level and consequently a smaller stiffness reduction. 5. Conclusion The main purpose of the present study was to reduce the density of matrix cracks induced in cross-ply laminated composites under tensile loading by adding CNF fillers in their 90-degree layers. Cross-ply lami nated composite specimens with the stacking sequence of [02/906]s were fabricated, in which the matrix epoxy of the 90-degree layers was reinforced with the CNF fillers. Different cracks densities were induced in the specimens using a tensile testing machine. The crack density was quantified using an optical microscope and the elastic modulus of the specimens was obtained using the tensile test and Lamb wave propa gation technique. The discrepancies between results of the Lamb wave propagation tests and those of the tensile tests were smaller than 3%, proving the capability of the proposed Lamb wave propagation method for nonde structive evaluation of hybrid composite materials and structures. It was also revealed that the matrix crack density and the resulting stiffness reduction of the specimens significantly decreased at each specific stress level by adding 0.25 wt% CNF into the matrix of composite specimens. 8
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