外文原文-碳纖維增強聚合物基復(fù)合材料具有不同編織結(jié)構(gòu)的電磁特性的數(shù)值模擬

Computational Modeling of the Electromagnetic Characteristics of Carbon Fiber-Reinforced Polymer Composites with Different Weave Structures A.M. Hassana, J.F. Douglasb, and E.J. Garboczia aMaterials and Structural Systems Division, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA bMaterials Science and Engineering Division, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA Abstract. Carbon fiber reinforced polymer composites (CFRPC) are of great interest in the aerospace and automotive industries due to their exceptional mechanical properties. Carbon fibers are typically woven and inter-laced perpendicularly in warps and wefts to form a carbon fabric that can be embedded in a binding matrix. The warps and wefts can be interlaced in different patterns called weaving structures. The primary weaving structures are the plain, twill, and satin weaves, which give different mechanical composite properties. The goal of this work is to computationally investigate the dependence of CFRPC microwave and terahertz electromagnetic characteristics on weave structure. These bands are good candidates for the Nondestructive Evaluation (NDE) of CFRPC since their wavelengths are comparable to the main weave features. 3D full wave electromagnetic simulations of several different weave models have been performed using a finite element (FEM) simulator, which is able to accurately model the complex weave structure. The computational experiments demonstrate that the reflection of electromagnetic waves from CFRPC depend sensitively on weave structure. The reflection spectra calculated in this work can be used to identify the optimal frequencies for the NDE of each weave structure. Keywords: Carbon Fiber, Twill, Satin, Plain, Terahertz, Microwave, Screening, COMSOL PACS: 81.05.Ni, 81.05.Lg, 81.05.uj, 81.70.-q, 81.70.Ex INTRODUCTION The global carbon fiber market is expanding rapidly and is expected to almost double, increasing from US $1.8 billion in 2013, to US $3 billion in 2018 1. Carbon fibers are graphite filaments where the carbon atoms are arranged in a crystalline form aligned along the axis of the fiber. Typically, the carbon fibers have diameters ranging from 5 m to 10 m and are grouped into bundles containing thousands of carbon fibers. These bundles are typically woven and inter-laced perpendicularly in warps and wefts to form a carbon fabric that can be embedded in a binding polymer to form carbon fiber-reinforced polymer composites (CFRPC). The weaving structures can be divided into three primary structures termed plain, twill, and satin weaves as shown in Figure 1, as well as many other variations. The weave geometries were generated using the TexGen open source software 2. FIGURE 1. Different weaving structures generated using TexGen 2. PlainTwillSatin warp weft xy 40th Annual Review of Progress in Quantitative Nondestructive Evaluation AIP Conf. Proc. 1581, 1494-1499 (2014); doi: 10.1063/1.4864999 2014 AIP Publishing LLC 978-7354-1211-8/$30.00 1494 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: /termsconditions. Downloaded to IP: 89 On: Fri, 21 Feb 2014 05:42:44 Different weaving structures produce different mechanical properties. For example, a study by Rattan et al. showed that the plain weave performed best followed by the satin weave and then the twill weave in an erosive wear study performed on CFRPC fabricated using the impregnation technique 3. The goal of this present work is to focus on the electromagnetic (EM) properties of CFRPC to supplement the numerous studies on their mechanical properties. Recently, the computational modeling of the electromagnetic properties of woven fabric composites has been receiving increasing interest 4-5. Mirotznik et al. used the Rigorous Coupled Wave (RCW) algorithm to predict the EM properties of woven glass fiber composites at frequencies ranging from DC up to 50 GHz 4. The RCW algorithm was able to predict and explain experimentally detected guided mode resonances from glass fiber composites. The main assumption of the model was that the order did not matter in the regions in Figure 1 where the weft and warp overlap. That is, the region marked as x in Figure 1, where the warp is on top of the weft, was considered the same as the region marked as y in Figure 1, where the weft is on top of the warp 4. Therefore, all the overlap regions were modeled with double the thickness of the non-overlapping regions and with dielectric properties equal to the average of the individual dielectric properties of the warps and wefts 4. Wang et al. used the full wave Finite Difference Time Domain (FDTD) to study the properties of CFRPC in a broadband ranging from DC to 2 THz 5. However, the carbon fibers modeled by Wang et al. were non-interlacing 5. In this work, similar to Wang et al., full wave simulations of CFRPC were performed to investigate their EM properties in a wide range of frequencies, but interlacing carbon fibers were explicitly simulated. Also, the exact order of the warps and wefts in the overlap regions was modeled. The full wave solver employed in this work is the Radio-Frequency module of the commercial Finite Element COMSOL Multiphysics package* 6. CFRPC weaves with different geometrical parameters are investigated to elucidate how these parameters affect the EM response. The EM response of CFRPC in the microwave and terahertz range is rich with information and, therefore, can be used in the Nondestructive Evaluation (NDE) of CFRPC. VALIDATION OF COMSOL SIMULATIONS To validate the accuracy of the simulation for this application, a test case was selected and the COMSOL results were compared to analytical solutions. The test case is composed of an array of infinitely long perfectly conducting wires. The radius of each wire, a, is much smaller than the spacing between the wires d (a = 0.01d). The excitation consists of a plane wave incident normal to the wire array. The configuration of the wire array and the excitation are shown in Figure 2(a). The wires are infinitely long in one direction and repeated periodically, with a step d in the other direction, using periodic boundary conditions (PBC). FIGURE 2. (a) Array of perfectly conducting wires (b) Reflection coefficient from an array of perfectly conducting wires for different modes. Periodic structures can lead to the emergence of Floquet modes at higher frequencies in addition to the dominant mode which operates at all frequencies 4. The Floquet modes will be designated with an index m and the dominant mode will correspond to an index m=0. Figure 2(b) shows the reflection coefficient of the different modes from the wire array in Figure 2(a). The reflection coefficient is plotted versus d/ where is the wavelength of the incident wave in free space. Therefore, the parameter d/ is proportional to the frequency of the incident plane wave. The * Certain commercial equipment and/or materials are identified in this report in order to adequately specify the experimental procedure. In no case does such identification imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the equipment and/or materials used are necessarily the best available for the purpose. Perfectly cond. wire Normal Incident plane wave d (a)(b) 1495 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: /termsconditions. Downloaded to IP: 89 On: Fri, 21 Feb 2014 05:42:44 magnitude of the reflection coefficient of the dominant mode is shown in blue (circles) in Figure 2(b). The higher order modes, m=1 and m=-1, start propagating at 2. FIGURE 3. (a) Top view of the non-interlacing carbon fiber weave, (b) side view of the non-interlacing carbon fiber weave, and (c) the total reflected power from the non-interlacing carbon fiber weave for different heights between the warp and the weft. Side View d/12 E Normal Incident plane wave E Normal Incident plane wave X PBC PBC PBC PBC d d PBC PBC h (a) (b)(c) Carbon fiber x y zx y z 1496 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: /termsconditions. Downloaded to IP: 89 On: Fri, 21 Feb 2014 05:42:44 Figure 3c compares different heights, h, between the weft and the warp to the case where h=0 and the warp and weft merge in the same plane as shown in Figure 4. The cases h=d/2, d/4, and 0 are shown in solid blue, dashed red, and dash-dot black, respectively, in Figure 3(c). As the height h decreases, the reflected power converges slowly to the limiting case of h=0. Typical values of d for CFRPC are 0.2 mm 5. Therefore, the frequency range in Figure 3(c) corresponds to a frequency range of 0.75 THz to 4.5 THz, which is within the range of operation of current commercial THz systems 8. Moreover, once the carbon fibers are embedded in a polymer, the features shown in Figure 3(c) will occur at frequencies that are lower by a factor equal to the refractive index of the polymer. FIGURE 4. Limiting case when the warp and weft merge and are in the same plane. FIGURE 5. (a) Top view of the unit cell of the interlacing carbon fiber plain weave, (b) side view of the unit cell of the interlacing carbon fiber plain weave, and (c) Three dimensional unit cell of the interlacing carbon fiber plain weave. Figure 5(a) shows the top view, Figure 5(b) shows the side view, and Figure 5(c) shows the COMSOL three dimensional (3D) unit cell of the carbon fibers arranged in an interlacing plain weave. The interlacing case in Figure 5 has exactly the same dimensions as the non-interlacing cases in Figure 3 and Figure 4. The effect of different heights between the warp and the weft were investigated as shown in Figure 6. FIGURE 6. Total reflected power from the interlacing carbon fiber plain weave for different heights between the warp and the weft. d/4 d/4 d/4 d/4 d/4 d/4 h d/12 d/4d/4 d/4 d/12 h (a)(b)(c) 1497 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: /termsconditions. Downloaded to IP: 89 On: Fri, 21 Feb 2014 05:42:44 Comparing Figure 6 with Figure 3(c), it is clear that there are significant differences between the total reflected power from the non-interlacing and the interlacing weaves. Therefore, the exact geometry of the weave needs to be simulated for an accurate prediction of the total reflected power. Moreover, at low frequencies d the total reflected power in Figure 6 is lower than the total reflected power in Figure 3(c). This can be explained by the fact that the interlacing introduces gaps where the field can penetrate, which reduces the power reflected back to the source direction. Finally, as the height between the warp and weft gets smaller, the total reflected power converges slowly to the case where h=0 shown in Figure 4. The tightness of the weave can be increased by reducing the spacing between the fibers. Figure 7(a) shows the same interlacing plain weave unit cell with the same carbon fiber dimensions but with all the spacing between the fibers reduced to 0.15d instead of 0.25d in Figure 5(a). The total reflected power from the interlacing plain weave with the reduced spacing is shown in Figure 7(b). Figure 7(b) shows that the tighter the weave the higher the total reflected power and the higher the shielding effectiveness. FIGURE 7. (a) Top view of the unit cell of the interlacing carbon fiber plain weave with reduced spacing, (b) Total reflected power from the interlacing carbon fiber plain weave for different spacing. CONCLUSIONS AND FUTURE WORK Extensive computational experiments were performed to study how different weave geometries affect the total reflected power from the carbon fiber fabric. The results of these experiments show that different weave structures exhibit complex frequency selective responses which need to be known before the NDE of CFRPC can be carried out. At low frequencies, the shielding effectiveness of non-interlaced weaves is higher than interlaced weaves. Reducing the height between the warp and the weft and reducing the spacing between the fibers increases the shielding effectiveness. Future work will involve the study of additional weave structures such as satin and twill weaves, modeling the carbon fibers with accurate conductivity values instead of perfect conductors, and embedding the carbon fibers in a polymer. Moreover, the geometry of the weaves will be made more accurate by choosing different cross sections for the fibers instead of the rectangular cross section used in this work, and accurately modeling the actual curvature of the weave instead of simply assuming a square geometry (Fig. 5c). Finally, in this work the warps were assumed to be perfectly straight whereas in true CFRPC they might experience bulging, which will also be studied. ACKNOWLEDGMENTS This work is funded under NIST project, “Carbon Nanocomposite Manufacturing: Processing, Properties, Performance,” which is a large project dedicated to developing microwave in-line quality control sensor for the manufacture of polymer nanocomposites. Spacing=0.25d S