gelatin nanocomposite hydrogel: Experimental and numerical validation of hyperelastic model

Materials Science and Engineering C 38 (2014) 299–305

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Graphene oxide/poly(acrylic acid)/gelatin nanocomposite hydrogel: Experimental and numerical validation of hyperelastic model Shahab Faghihi a,⁎, Alireza Karimi a,b, Mahsa Jamadi a, Rana Imani a,c, Reza Salarian d a

Tissue Engineering and Biomaterials Division, National Institute of Genetic Engineering and Biotechnology, Tehran 14965/161, Iran School of Mechanical Engineering, Iran University of Science and Technology, Tehran 16844, Iran Biomedical Engineering Department, Amirkabir University of Technology, Tehran 15875, Iran d Biomedical Engineering Department, Maziar University, Noor, Royan 151, Iran b c

a r t i c l e

i n f o

Article history: Received 18 November 2013 Received in revised form 27 January 2014 Accepted 8 February 2014 Available online 15 February 2014 Keywords: Graphene oxide nanosheet Poly(acrylic acid) Gelatin Constitutive equations Finite element

a b s t r a c t Owing to excellent thermal and mechanical properties, graphene-based nanomaterials have recently attracted intensive attention for a wide range of applications, including biosensors, bioseparation, drug release vehicle, and tissue engineering. In this study, the effects of graphene oxide nanosheet (GONS) content on the linear (tensile strength and strain) and nonlinear (hyperelastic coefficients) mechanical properties of poly(acrylic acid) (PAA)/gelatin (Gel) hydrogels are evaluated. The GONS with different content (0.1, 0.3, and 0.5 wt.%) is added into the prepared PAA/Gel hydrogels and composite hydrogels are subjected to a series of tensile and stress relaxation tests. Hyperelastic strain energy density functions (SEDFs) are calibrated using uniaxial experimental data. The potential ability of different hyperelastic constitutive equations (Neo-Hookean, Yeoh, and Mooney– Rivlin) to define the nonlinear mechanical behavior of hydrogels is verified by finite element (FE) simulations. The results show that the tensile strength (71%) and elongation at break (26%) of composite hydrogels are significantly increased by the addition of GONS (0.3 wt.%). The experimental data is well fitted with those predicted by the FE models. The Yeoh material model accurately defines the nonlinear behavior of hydrogels which can be used for further biomechanical simulations of hydrogels. This finding might have implications not only for the improvement of the mechanical properties of composite hydrogels but also for the fabrication of polymeric substrate materials suitable for tissue engineering applications. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Hydrogels, distinct from solid materials, are composed of a hydrophilic polymer network and a large amount of water. This special softwet structure of hydrogels has enabled them to be used as biocompatible materials for a variety of applications, such as biosensors [1], bioseparation, drug release vehicles [2,3], and tissue engineering scaffolds [4,5]. However, conventional polymeric hydrogels cross-linked by organic cross-linkers are usually mechanically weak due to the irregular distribution of crosslinking which results in a broad distribution of crosslinked chain lengths. In order to extend practical applications of hydrogels at the same time as improving their mechanical performances, graphene has been employed in recent years. While graphene has excellent mechanical properties consists of a tensile modulus of 1.0 TPa and an ultimate strength of 130 GPa [6,7], it is cheap and easy to obtain. It is expected that graphene proceeds as a novel reinforcing agent to support

⁎ Corresponding author at: Tissue Engineering and Biomaterials Division (Director), National Institute of Genetic Engineering and Biotechnology, Tehran 14965/161, Iran. Tel./fax: +98 21 44580386. E-mail addresses: [email protected], [email protected] (S. Faghihi).

http://dx.doi.org/10.1016/j.msec.2014.02.015 0928-4931/© 2014 Elsevier B.V. All rights reserved.

polymers and inorganic and metallic materials [8]. However, the major problem in utilizing graphene in polymer composites is its poor dispersibility in most solvents. Graphene is neither hydrophilic nor lipophilic which leads to difficulties in their processing in solvent blending method and poor compatibility with polymer matrices [9]. In contrast, graphene oxide (GO), precursor of graphene from chemical exfoliation of graphite, is amphiphilic because of the existence of oxygenated groups including hydroxyl, carbonyl, carboxyl, and epoxide [10]. As the result, GO can be dispersed in common protic solvents such as water and alcohol through hydrogen bonding. This would facilitate the processing of GO-based composites by solvent blending and assists molecular dispersion of GO in polymeric matrix [11,12]. There has been few studies that have used GO as a reinforcement agent in polymeric composites [13,14]. It is reported that the addition of GO to polyvinyl alcohol (PVA) can significantly improve its mechanical properties [15,16]. Incorporation of 1 wt.% GO has been shown to increase the tensile strength, Young's modulus, and elongation at break of chitosan biopolymer almost 122%, 64%, and 159%, respectively [17]. The tensile strength and Young's modulus of PVA were reported to be increased by 69% and 129% by addition of 3 wt.% GO [18]. Gelatin (Gel) is a partial derivative of collagen, which is the major component of skin, bone, cartilage, and connective tissues. It is

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commonly used in tissue engineering owing to its easy availability, biocompatibility, and biodegradability [19]. The Gel is non-immunogenic compared with its precursor and can promote cell adhesion, migration, differentiation, and proliferation. Therefore, Gel-based scaffolds in particular largely porous sponges were widely used in tissue engineering and regeneration. However, the strength is often not enough to support the tissue growth and the stability of Gel scaffolds is not satisfactory due to their water-solubility [20]. In this study, the role of graphene oxide nanosheets (GONS) as a reinforcement agent in poly(acrylic acid) (PAA)/Gelatin (Gel) composite hydrogels is investigated. The failure stress and strain as well as hyperelastic coefficients are measured using a series of tensile and relaxation tests. A large strain kinematics of nonlinear hyperelastic models (Neo-Hookean, Yeoh, and Mooney–Rivlin) are used to formulate the behavior of hydrogels. The linear and quasilinear models, mostly relevant in the small strain regime, are unable to fully capture the complexities of the hydrogel response at finite strains. In contrast, some of the large strain kinematics models used in the present study, which integrate nonlinearities within their formulations, enable us to successfully reproduce some specific features of the hydrogel deformation over fairly complex loading histories. 2. Experimental 2.1. Materials Natural graphite powders, potassium permanganate (KMnO4), sodium nitrate (NaNO3), concentrated sulfuric acid, hydrogen peroxide, acrylic acid, N,N,N′,N′ tetramethylethylenediamine (TEMED), ammonium persulfate (APS), and BIS were purchased from Merck (Darmstadt, Germany). Gelatin type B (from bovine skin, average molecular weight 50,000) was purchased from Sigma-Aldrich (St. Louis, Missouri, United States). All reagents were used as received. 2.2. Preparation of graphene oxide nanosheets (GONS) The GONS were prepared based on the modified Hummer's method [21]. In a typical experiment, natural graphite powder (1 g) was suspended in 120 ml of sulfuric acid (98%). The mixture was cooled in ice bath and NaNO3 (500 mg) was added under moderate stirring (200 RPM). KMnO4 (6 g) was added over 60 min. The mixture was then allowed to warm to 35 °C with constant stirring. After 48 h the brownish green solution becomes too viscose to stir. 400 ml double distilled water (DDW) was slowly added to the reaction, keeping the temperature at 70 °C for one hour. Finally, 30% H2O2 (10 ml) was added until the color of mixture changed to brilliant yellow. The mixture was rested for 2 days to allow precipitation of graphene oxide nanosheets. The supernatant was removed and precipitated powder were washed ten times with 0.5 M aqueous HCl to remove metal ions and then washed with DDW to remove the acid residue. To achieve nano-sized mono layer graphene oxide sheet, GO suspension was dispersed by a probe-typed ultrasonic treatment (200 W, 2H). The resultant brown solution was dried under freeze drying to give a fine nanographene oxide powder.

10 min and then pipetted between two glass slides, which were separated by 1.5 mm divider. The system was then heated in an oven at 60 °C for 3 h. 2.4. Measurement and characterization 2.4.1. Experimental setup The experimental setup and stress–strain analysis were described comprehensively in our previous studies [22–24]. Briefly, a low strain rate of 5 mm/min which is typical for surgical procedures and provides more insight into hydrogels behavior was employed by the action of an axial servo motor [25,26]. In order to guarantee a firm fixation of samples between the jaws of the machine, a small tensile pre-load of 0.03 N was applied to each hydrogel strip. The sample's length was measured after the application of the pre-load. This also helped minimize the bending effect caused by the weight of each specimen. All tests were performed at room temperature (25 °C) and each sample was tested only once. Force was applied to each hydrogel strip and the maximum stress and strain of each sample were measured before the incidence of failure. 2.4.2. Constitutive equations–strain energy density functions The hyperelastic material constitutive model is used to define the response of hydrogel materials fabricated in this study. The hyperelastic materials use stress–strain relationship that is derived from strain energy density functions (SEDFs) [27,28]. The hyperelastic constitutive model is used to describe materials that are able to undergo large, recoverable elastic strain, such as rubber­like polymers and soft biological tissues. An isotropic material has the same mechanical response regardless of loading direction. For an isotropic hyperelastic material the strain energy density function W is a scalar function of the right Cauchy–Green deformation tensor, C. The scalar function is composed of either the principal invariants or the principal stretches of the deformation, both of which are derived from the right Cauchy–Green deformation tensor. The experimental data was used to calibrate an isotropic hyperelastic SEDF in order to generate stress–strain relations that can be used in finite element simulation of hydrogels. Under the assumption that the hydrogel is an incompressible and isotropic material [24,29], it was possible to fit a general polynomial isotropic SEDF form (Eq. (1)). W¼

XN

C iþ j¼1 ij



I1 −3

i 

j I2 −3

ð1Þ

XN 1 2i þ i¼1 ð J−1Þ ⋮i; j ¼ 0; 1; …; N⋮i þ j ¼ 1; 2; …; N: Di In general, an isotropic hyperelastic incompressible material is characterized by a strain-energy density function W which is a function of two principal strain invariants: W = W(I1,I2), where I1 and I2 are defined as [30], 2

2

2

I1 ¼ λ1 þ λ2 þ λ3

2 2

2 2

ð2Þ

2 2

I2 ¼ λ1 λ2 þ λ1 λ3 þ λ2 λ3 :

ð3Þ

2.3. Preparation of composite hydrogels Semi-interpenetrated network (semi-IPN) hydrogel composites were prepared using the thermal initiated redox polymerization method with APS and TEMED used as reaction catalysts. A given amount of gelatin and deionized water were stirred at 40 °C for an hour to obtain a gelatin solution with a concentration of 10 wt.%. The gelatin solution was divided equally into four beakers and the desired amount of GONS was added to each beaker to give the four GO/Gel mixtures with GONS contents of 0, 0.1 wt.%, 0.3 wt.%, and 0.5 wt.%. Then AA (5 ml), BIS (0.01 g), APS (0.1 g), and TEMED were added to the GO/Gel mixtures stirring at 0 °C. The mixture was sonicated for

Here, λ21, λ22, and λ23 are the squares of the principal stretch ratios, linked by the relationship of λ1 λ2 λ3 = 1, due to incompressibility. An effort was made to achieve approximately uniform contraction of the hydrogel with a specimen thickness of 1.0 ± 0.05 mm (mean ± SD) up to 30% strain. Then the Eulerian and Lagrangian principal axes of strain and stress are aligned with the direction of tension, X1, say, and with any two orthogonal axes (lateral) X2 and X3, say. Due to symmetry and incompressibility, the stretch ratios are: 1 λ1 ¼ λ and λ2 ¼ λ3 ¼ pffiffiffi λ

ð4Þ

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301

Fig. 1. The hydrogels composites including (a) GO (0.0 wt.%)/PAA/Gel, (b) GO (0.1 wt.%)/PAA/Gel, (c) GO (0.3 wt.%)/PAA/Gel, and (d) GO (0.5 wt.%)/PAA/Gel.

where λ ≥ 1 is the stretch ratio in the direction of tension. As a result the Eqs. (2) and (3) become: 2

I 1 ¼ λ þ 2λ

−1

and I2 ¼ λ

−2

þ 2λ

ð5Þ

so the W is now a function of λ. During the experimental tension tests, the principal stretch ratio λ was calculated from the measure of elongation e using equation: λ = 1 + e. The nominal/Lagrange stress component along the direction of tension S11 was evaluated as S11 ≡ F/A, where F is the tension force measured in Newtons by the load cell and A is the cross sectional area of the sample in its undeformed state. The experimentally measured nominal stress was then compared to the prediction of the hyperelastic models from the relation [30]. S11 ¼

  f dW f ðλÞ ≡ W λ2 þ 2λ−1 ; λ−2 þ 2λ : ; where W dλ

ð6Þ

Three different isotropic SEDF models were examined: Yeoh (Eq. (7)), Mooney–Rivlin (Eq. (8)), and Neo-Hookean (Eq. (9)). The Yeoh, Mooney–Rivlin [31], and Neo-Hookean [23,26] models are special cases of the polynomial SEDFs of the stretch ratios as their variables

instead of invariants. The polynomial along with the other specialized forms of the SEDFs can be written as: W¼

X3

C i¼1 i0



i XN 1 2i I1 −3 þ ð J−1Þ i¼1 D i

ð7Þ

    1 2 ð J−1Þ W ¼ C 10 I1 −3 þ C 01 I2 −3 þ D1

ð8Þ

  1 2 ð J−1Þ : W ¼ C 10 I1 −3 þ D1

ð9Þ

Where J = det(F) and F is the deformation gradient. The terms I 1 and I2 are the first and second invariants of the left Cauchy–Green strain ten1 sor (B), respectively. For a normalized deformation gradient F ¼ J −3 F T the Cauchy–Green strain tensor assumes the form: B ¼ F F . The principle stretch, λi is the eigenvalue of μi. The polynomial coefficients, μi, αi, and Cij are material constants that were fit from the experimental

Fig. 2. The hydrogels during uniaxial tension test (a) GO (0.0 wt.%)/PAA/Gel, (b) GO (0.3 wt.%)/PAA/Gel.

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Fig. 3. The stress–strain diagrams of hydrogels under uniaxial tensile loading.

data. However, in the present study owing to an incompressibility assumption J equals to 1. 2.4.3. Numerical verification-finite element modeling Finite element models have been generated to simulate three different energy density functions versus their finite element modeling. Modeling was performed to simulate the tests' geometry, dimension, and boundary conditions as well as mechanical response in the forms of displacement, deformation, and stress. An axisymmetric model was made for the unconfined testing model having a length and width of 20 and 10 mm, respectively. In order to examine the calibrated SEDFs as a representative candidate for the hydrogels, a FE simulation was performed and the results were compared to the tensile test measurements. The aim was to verify the ability of the SEDFs to simulate material behavior under different and general 3D stress–strain states. The FE analysis was consisted of a 2D brick element for the hydrogels. Stress–strain curves were generated based on the FE modeling and compared to the results obtained from the tensile experiments.

Fig. 5. Histograms presenting (a) tensile strength and (b) tensile strain of composite hydrogels having different GONS content.

2.5. Statistical analysis Data were first analyzed by the analysis of variance (ANOVA); when statistical differences were detected, student's t-test for comparisons between groups was performed using the SPSS software version 16.0 (SPSS Inc., Chicago, IL, USA). Data are reported as mean ± std at a significance level of p b 0.05. 3. Results and discussion This study aims to characterize the linear and nonlinear mechanical behavior of GO/PAA/Gel composite hydrogels having different GONS content in order to gather accurate information from their hyperelastic properties for incorporation in further scaffold design and fabrication. The prepared hydrogel strips including GONS (0.0 wt.%)/PAA/Gel, GONS (0.1 wt.%)/PAA/Gel, GONS (0.3 wt.%)/PAA/Gel, and GONS

Fig. 4. The stress relaxation diagrams of hydrogel specimens.

(0.5 wt.%)/PAA/Gel are shown in Fig. 1. Fig. 2 presents the hydrogel strips with 0.0 wt.% and 0.5 wt.% of GONS content during the tension test. The stress–strain and stress relaxation diagrams of hydrogel samples are illustrated in Figs. 3 and 4, respectively. It can be seen that there is no holding in the stress relaxation diagram which indicates that regardless of the GONS content, the hydrogels showed a near linear mechanical behavior (Figs. 3 and 4). In fact, hydrogels are known to exhibit elastic behavior [32], however, the simple linear behavior fails to

Table 1 Hyperelastic coefficients of hydrogel specimens. Material coefficients (MPa) C10

C01

C20

C30

GO (0.0 wt.%)/PAA/gel Neo-Hookean Yeoh Mooney–Rivlin

0.057 0.031 0.229

– – −0.157

– 0.693 –

– −1.869 –

GO (0.1 wt.%)/PAA/gel Neo-Hookean Yeoh Mooney–Rivlin

0.066 0.044 0.240

– – −0.196

– 0.176 –

– −0.193 –

GO (0.3 wt.%)/PAA/gel Neo-Hookean Yeoh Mooney–Rivlin

0.075 0.050 0.231

– – −0.181

– 0.126 –

– −0.089 –

GO (0.5 wt.%)/PAA/gel Neo-Hookean Yeoh Mooney–Rivlin

0.042 0.027 0.149

– – −0.119

– 0.075 –

– −0.043 –

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Fig. 6. The comparative stress–strain diagrams of GO (0.0 wt.%)/PAA/Gel and GO (0.3 wt.%)/PAA/Gel hydrogels along with the finite element hyperelastic modeling (a, d) Neo-Hookean, (b, e) Yeoh, and (c, f) Mooney–Rivlin.

define the mechanical properties of hydrogels and the hyperelastic properties provide a more accurate estimate of their mechanical performances especially when the loading rate is low [33]. The results indicated that the hydrogels with relatively lower content of GONS (b 0.3 wt.%)

have enhanced mechanical properties compared with those with higher content of GONS (N0.3 wt.%). The tensile strength and strain of GO (0.0 wt.%)/PAA/Gel have increased 71% and 83% after the addition of 0.3 wt.% GONS. This is believed to be related to the hydrogen and

Fig. 7. The finite element modeling of GO (0.0 wt.%)/PAA/Gel using (a) Neo-Hookean, (b) Yeoh, and (c) Mooney–Rivlin hyperelastic models.

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possible covalent bonds between GONS and polymer chains. The relatively lower content of GONS (0.3 wt.%) can be well dispersed in the polymer matrix and enhanced the intermolecular interactions between the components effectively. An excessive amount of GONS (0.5 wt.%) may form large agglomerates that weakened the interfacial interactions in polymeric matrix and results in micro-phase separation between the components (Figs. 3 and 5). This is in agreement with previous reports that suggest application of b 0.3 wt.% of GONS content would enhance the mechanical properties of hydrogel composites [34]. Three independent hyperelastic models, including Neo-Hookean, Yeoh, and Mooney–Rivlin, were implemented to define the nonlinear response of hydrogel strips. The experimental results obtained from the tensile tests were used to calibrate the SEDF candidates used for axial constitutive modeling. Hyperelastic material coefficients for each SEDF candidate are listed in Table 1. The validation of the curve-fitting algorithm has been performed using a FE simulation of the uniaxial tensile test. The hyperelastic models are calibrated from the experimental test results and used to predict the mechanical response of the hydrogels under general stress states. The calibrated models are implemented in a FE analysis that considered general hydrogel geometries and loading conditions. Fig. 6 shows a comparison of the experimental stress–strain diagrams of the GO (0.0 wt.%)/PAA/Gel and GO (0.3 wt.%)/PAA/Gel hydrogels along with the three examined SEDFs consists of Neo-Hookean (a, d), Yeoh (b, e), and Mooney–Rivlin (c, f) hyperelastic models. The results present the average values generated from all samples at their last loading cycles. The ability of the hyperelastic material models to match the test results is examined by their functions. For example, the lack of the second-variable in the Neo-Hookean SEDF has impaired it to lower bound and larger error. The Neo-Hookean function failed to capture the nonlinear behavior of hydrogels under uniaxial stress states. The Yeoh and Mooney–Rivlin SEDFs showed almost similar behavior and adequate ability to predict the different hydrogel behavior for the majority of experimental data range. However, the Yeoh SEDF showed excellent ability to model the entire range of experimental data especially at the failure point. In order to verify the hyperelastic material models in terms of materials coefficients, a finite element model was made to compare the Cauchy stress with stress in loading direction (S11) (Fig. 7). The S11 in constitutive equations is equal to S22 in FE models. The Neo-Hookean model showed a linear stress–strain behavior and diverges from the experimental data toward the end of the loading. The Yeoh SEDF model showed the best adaptation with the entire range of tensile test results, as indicated previously by Kaster et al. that Yeoh hyperelastic model is the most suitable model to define the mechanical behavior of hydrogels [35]. The FE results indicated a good consistency with the experimental results and constitutive modeling as the consequence of the hydrogels' incompressibility. There were less than 5% and 7% difference between the experimental and FE results of Yeoh and Mooney–Rivlin models, respectively (Fig. 7). It was the intention of this study to select a constitutive hyperelastic model that most suitably defines the behavior of composite hydrogels. Therefore, the Yeoh and Mooney–Rivlin functions seem to be the most logical selection for the representation of hydrogels where incompressible behavior is expected for higher stretch values. The addition of GONS successfully reinforced the PAA/Gel hydrogels. It is shown that the mechanical properties of the GO/ PAA/Gel composite hydrogels can be tuned by controlling the amount of GONS incorporated, making it suitable for various tissue engineering applications. However, a linear-elastic model may not give acceptable results as most hydrogels in tissue engineering are subjected to a large-deformation loading. Moreover, many of hydrogel materials are deformed considerably under an applied force and their behavior is often non-linear. The hyperelastic nonlinear mechanical properties of hydrogels are therefore very important to define the nonlinear mechanical behavior of hydrogels. In addition,

hyperelastic material model combined with FE results can be used to predict whether the examined hydrogel is suitable for specific tissue engineering purposes. 4. Conclusions This study investigated the effect of GONS content on the mechanical performance of PAA/Gel composite hydrogels through the measurements of linear and nonlinear mechanical properties. The results indicated that the hydrogels with relatively lower content (b 0.3 wt.%) of GONS have more suitable mechanical properties compared with those with higher content of GONS (N0.3 wt.%). The Yeoh hyperelastic material model showed the most similarity with the experimental data, so it was chosen as the best model that closely defined the nonlinear mechanical behavior of hydrogels in tension. The results could be employed for the design and fabrication of scaffolding materials with desirable mechanical properties intended for tissue engineering applications. Acknowledgments We gratefully acknowledge the financial support of this work by Grant from Maziar University, (Grant No: 921007-IV-478). The authors wish to thank Tissue Engineering and Biomaterials Division, National Institute of Genetic Engineering and Biotechnology (NIGEB), and Tissue Engineering and Biological Systems Research Lab, School of Mechanical Engineering, Iran University of Science and Technology. References [1] R. Yoshida, Y. Uesusuki, Biomacromolecules 6 (2005) 2923–2926. [2] C. Alvarez-Lorenzo, A. Concheiro, A.S. Dubovik, N.V. Grinberg, T.V. Burova, V.Y. Grinberg, J. Control. Release 102 (2005) 629–641. [3] S. Liang, J. Xu, L. Weng, H. Dai, X. Zhang, L. Zhang, J. Control. Release 115 (2006) 189–196. [4] R.M. Coleman, N.D. Case, R.E. Guldberg, Biomaterials 28 (2007) 2077–2086. [5] K. Liu, Y. Li, F. Xu, Y. Zuo, L. Zhang, H. Wang, J. Liao, Mater. Sci. Eng. C 29 (2009) 261–266. [6] C. Lee, X.D. Wei, J.W. Kysar, J. Hone, Science 321 (2008) 385–388. [7] J. Shen, B. Yan, T. Li, Y. Long, N. Li, M. Ye, Compos. A: Appl. Sci. Manuf. 43 (2012) 1476–1481. [8] W. Wang, Z. Wang, Y. Liu, N. Li, W. Wang, J. Gao, Mater. Res. Bull. 47 (2012) 2245–2251. [9] X. Huang, X. Qi, F. Boey, H. Zhang, Chem. Soc. Rev. 41 (2012) 666–686. [10] J. Kim, L.J. Cote, F. Kim, W. Yuan, K.R. Shull, J. Huang, J. Am. Chem. Soc. 132 (2010) 8180–8186. [11] C.W. Yung, L.Q. Wu, J.A. Tullman, G.F. Payne, W.E. Bentley, T.A. Barbari, J. Biomed. Mater. Res. A 83 (2007) 1039–1046. [12] M.A. Rafiee, J. Rafiee, Z. Wang, H. Song, Y.Y. Yu, N. Koratkar, ACS Nano 3 (2009) 3884–3890. [13] A.K. Geim, K.S. Novoselov, Nat. Mater. 6 (2007) 183–191. [14] N. Mohanty, V. Berry, Nano Lett. 8 (2008) 4469–4476. [15] Y. Matsuo, T. Niwa, Y. Sugie, Carbon 37 (1999) 897–901. [16] J. Liang, Y. Huang, L. Zhang, Y. Wang, Y. Ma, T. Guo, Y. Chen, Adv. Funct. Mater. 19 (2009) 2297–2302. [17] X. Yang, Y. Tu, L. Li, S. Shang, X.M. Tao, ACS Appl. Mater. Interfaces 2 (2010) 1707–1713. [18] Y. Xu, W. Hong, H. Bai, C. Li, G. Shi, Carbon 47 (2009) 3538–3543. [19] R.E. Kania, A. Meunier, M. Hamadouche, L. Sedel, H. Petite, J. Biomed. Mater. Res. 43 (1998) 38–45. [20] R. Deepachitra, M. Chamundeeswari, B. Santhosh kumar, G. Krithiga, P. Prabu, M. Pandima Devi, T.P. Sastry, Carbon 56 (2013) 64–76. [21] W. Hummers, R. Offeman, J. Am. Chem. Soc. 80 (1958) 1339–1345. [22] A. Karimi, M. Navidbakhsh, A. Shojaei, S. Faghihi, Mater. Sci. Eng. C 33 (2013) 2550–2554. [23] A. Karimi, M. Navidbakhsh, S. Faghihi, A. Shojaei, K. Hassani, Proc. IMechE H J. Eng. Med. 227 (2013) 148–161. [24] A. Karimi, M. Navidbakhsh, S. Faghihi, Perfusion (2014), http://dx.doi.org/10.1177/ 0267659113513823 (in press. Published online 20 November 2013). [25] A. Karimi, M. Navidbakhsh, A. Motevalli Haghi, S. Faghihi, Proc. IMechE H J. Eng. Med. 227 (2013) 609–614. [26] A. Karimi, M. Navidbakhsh, B. Beigzadeh, S. Faghihi, Int. J. Damage Mech. (2014), http://dx.doi.org/10.1177/1056789513514072 (in press. Published online 27 November 2013). [27] M.A. Bhatti, Advanced Topics in Finite Element Analysis of Structures, John Wiley & Sons, Inc., New York, 2006.

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