Steel and Composite Structures

  • 제20권3호
  • /
  • Pages.623-649
  • /
  • 2016
  • /
  • 1229-9367(pISSN)
  • /
  • 1598-6233(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Using an equivalent continuum model for 3D dynamic analysis of nanocomposite plates

  • Tahouneh, Vahid (Young Researchers and Elite Club, Islamshahr Branch, Islamic Azad University)
  • 투고 : 2014.02.26
  • 심사 : 2015.11.27
  • 발행 : 2016.02.29
⟨ 이전 논문 다음 논문 ⟩

초록

Most of the early studies on plates vibration are focused on two-dimensional theories, these theories reduce the dimensions of problems from three to two by introducing some assumptions in mathematical modeling leading to simpler expressions and derivation of solutions. However, these simplifications inherently bring errors and therefore may lead to unreliable results for relatively thick plates. The main objective of this research paper is to present 3-D elasticity solution for free vibration analysis of continuously graded carbon nanotube-reinforced (CGCNTR) rectangular plates resting on two-parameter elastic foundations. The volume fractions of oriented, straight single-walled carbon nanotubes (SWCNTs) are assumed to be graded in the thickness direction. In this study, an equivalent continuum model based on the Eshelby-Mori-Tanaka approach is employed to estimate the effective constitutive law of the elastic isotropic medium (matrix) with oriented, straight carbon nanotubes (CNTs). The proposed rectangular plates have two opposite edges simply supported, while all possible combinations of free, simply supported and clamped boundary conditions are applied to the other two edges. The formulations are based on the three-dimensional elasticity theory. A semi-analytical approach composed of differential quadrature method (DQM) and series solution is adopted to solve the equations of motion. The fast rate of convergence of the method is demonstrated and comparison studies are carried out to establish its very high accuracy and versatility. The 2-D differential quadrature method as an efficient and accurate numerical tool is used to discretize the governing equations and to implement the boundary conditions. The convergence of the method is demonstrated and to validate the results, comparisons are made between the present results and results reported by well-known references for special cases treated before, have confirmed accuracy and efficiency of the present approach. The novelty of the present work is to exploit Eshelby-Mori-Tanaka approach in order to reveal the impacts of the volume fractions of oriented CNTs, different CNTs distributions, various coefficients of foundation and different combinations of free, simply supported and clamped boundary conditions on the vibrational characteristics of CGCNTR rectangular plates. The new results can be used as benchmark solutions for future researches.

키워드

참고문헌

  1. Aragh, B.S. and Yas, M.H. (2010a), "Static and free vibration analyses of continuously graded fiber-reinforced cylindrical shells using generalized power-law distribution", Acta Mech., 215(1-4), 155-173. https://doi.org/10.1007/s00707-010-0335-4
  2. Aragh, B.S. and Yas, M.H. (2010b), "Three-dimensional analysis of thermal stresses in four-parameter continuous grading fiber reinforced cylindrical panels", Int. J. Mech. Sci., 52(8), 1047-1063. https://doi.org/10.1016/j.ijmecsci.2010.04.006
  3. Batra, R.C. and Jin, J. (2005), "Natural frequencies of a functionally graded anisotropic rectangular plate", J. Sound Vib., 282(1), 509-516. https://doi.org/10.1016/j.jsv.2004.03.068
  4. Benveniste, Y. (1987), "A new approach to the application of Mori-Tanaka's theory in composite materials", Mech. Mater., 6(2), 147-157. https://doi.org/10.1016/0167-6636(87)90005-6
  5. Bert, C.W. and Malik, M. (1996), "Differential quadrature method in computational mechanics: A review", A review; Appl. Mech. Rev., 49(1), 1-28. https://doi.org/10.1115/1.3101882
  6. Bonnet, P., Sireude, D., Garnier, B. and Chauvet, O. (2007), "Thermal properties and percolation in carbon nanotube-polymer composites", J. Appl. Phys., 91(20), 1910.
  7. Chang, T. and Gao, H. (2003), "Size-dependent elastic properties of a single-walled carbon nanotube via a molecular mechanics model", J. Mech. Phys. Solids, 51(6), 1059-1074. https://doi.org/10.1016/S0022-5096(03)00006-1
  8. Chen, C.H. and Cheng, C.H. (1996), "Effective elastic moduli of misoriented short-fiber composites", Int. J. Solids Struct., 33(17), 2519-2539. https://doi.org/10.1016/0020-7683(95)00160-3
  9. Cheng, Z.Q. and Batra, R. (2000), "Exact correspondence between eigenvalues of membranes and functionally graded simply supported polygonal plates", J. Sound Vib., 229(4), 879-895. https://doi.org/10.1006/jsvi.1999.2525
  10. Endo, M., Hayashi, T., Kim, Y.A., Terrones, M. and Dresselhaus, M.S. (2004), "Applications of carbon nanotubes in the twenty-first century", Trans. R. Soc. Lond A, Mathematical, Physical and Engineering Sciences, 362(1823), 2223-2238. https://doi.org/10.1098/rsta.2004.1437
  11. Esawi, A.M. and Farag, M.M. (2007), "Carbon nanotube reinforced composites: Potential and current challenges", Mater. Des., 28(9), 2394-2401. https://doi.org/10.1016/j.matdes.2006.09.022
  12. Eshelby, J.D. (1957), "The determination of the elastic field of an ellipsoidal inclusion, and related problems", Proc. R. Soc. A, Mathematical, Physical and Engineering Sciences, 241(1226), 376-396. DOI: 10.1098/rspa.1957.0133
  13. Eshelby, J.D. (1959), "The elastic field outside an ellipsoidal inclusion", Proc. R. Soc. A, Mathematical, Physical and Engineering Sciences, 252(1271), 561-569. DOI: 10.1098/rspa.1959.0173
  14. Fidelus, J.D., Wiesel, E., Gojny, F.H., Schulte, K. and Wagner, H.D. (2005), "Thermo-mechanical properties of randomly oriented carbon/epoxy nanocomposites", Composites Part A, 36(11), 1555-1561. https://doi.org/10.1016/j.compositesa.2005.02.006
  15. Formica, G., Lacarbonara, W. and Alessi, R. (2010), "Vibrations of carbon nanotube-reinforced composites", J. Sound Vib., 329(10), 1875-1889. https://doi.org/10.1016/j.jsv.2009.11.020
  16. Fung, Y.c. and Tong, P. (2001), Classical and Computational Solid Mechanics, World Scientific.
  17. Giordano, S., Palla, P.L. and Colombo, L. (2009), "Nonlinear elasticity of composite materials", Eur. Phys. J. B., 68(1), 89-101. https://doi.org/10.1140/epjb/e2009-00063-1
  18. Griebel, M. and Hamaekers, J. (2004), "Molecular dynamics simulations of the elastic moduli of polymer-carbon nanotube composites", Mech. Eng., 193(17), 1773-1788.
  19. Gupta, U. and Ansari, A. (2002), "Effect of elastic foundation on asymmetric vibration of polar orthotropic linearly tapered circular plates", J. Sound Vib., 254(3), 411-426. https://doi.org/10.1006/jsvi.2000.3518
  20. Gupta, U.S., Lal, R. and Jain, S.K. (1990), "Effect of elastic foundation on axisymmetric vibrations of polar orthotropic circular plates of variable thickness", J. Sound Vib., 139(3), 503-513. https://doi.org/10.1016/0022-460X(90)90679-T
  21. Gupta, U.S., Lal, R. and Sagar, R. (1994), "Effect of an elastic foundation on axisymmetric vibrations of polar orthotropic Mindlin circular plates", Indian J. Pure Appl. Math., 25(12), 1317-1317.
  22. Han, Y. and Elliott, J. (2007), "Molecular dynamics simulations of the elastic properties of polymer/carbon nanotube composites", Comput. Mater. Sci., 39(2), 315-323. https://doi.org/10.1016/j.commatsci.2006.06.011
  23. Hu, N., Fukunaga, H., Lu, C., Kameyama, M. and Yan, B. (2005), "Prediction of elastic properties of carbon nanotube reinforced composites", Proc. R. Soc. A, Mathematical, Physical and Engineering Sciences, 461(2058), 1685-1910. https://doi.org/10.1098/rspa.2004.1422
  24. Jabbari, M., Bahtui, A. and Eslami, M.R. (2006), "Axisymmetric mechanical and thermal stresses in thick long FGM cylinders", J. Therm. Stresses, 29(7), 643-663. https://doi.org/10.1080/01495730500499118
  25. Jin, Y. and Yuan, F. (2003), "Simulation of elastic properties of single-walled carbon nanotubes", Compos. Sci. Technol., 63(11), 1507-1515. https://doi.org/10.1016/S0266-3538(03)00074-5
  26. Ju, F. and Lee, H.P.K.H. (1995), "Free vibration of plates with stepped variations in thickness on nonhomogeneous elastic foundations", J. Sound Vib., 183(3), 533-545. https://doi.org/10.1006/jsvi.1995.0269
  27. Ke, L.L., Yang, J. and Kitipornchai, S. (2010), "Nonlinear free vibration of functionally graded carbon nanotube-reinforced composite beams", Compos. Struct., 92(3), 676-683. https://doi.org/10.1016/j.compstruct.2009.09.024
  28. Laura, P.A. and Gutierrez, R.H. (1991), "Free vibrations of a solid circular plate of linearly varying thickness and attached to a Winkler foundation", J. Sound Vib., 144(1), 149-161. https://doi.org/10.1016/0022-460X(91)90738-6
  29. Liew, K.M., Han, J.B., Xiao, Z.M. and Du, H. (1996), "Differential quadrature method for Mindlin plates on Winkler foundations", Int. J. Mech. Sci., 38(4), 405-421. https://doi.org/10.1016/0020-7403(95)00062-3
  30. Malekzadeh, P. (2009), "Three-dimensional free vibration analysis of thick functionally graded plates on elastic foundations", J. Compos. Struct., 89(3), 367-373. https://doi.org/10.1016/j.compstruct.2008.08.007
  31. Matsunaga, H. (2000), "Vibration and stability of thick plates on elastic foundations", J. Eng. Mech. ASCE, 126(1), 27-34. https://doi.org/10.1061/(ASCE)0733-9399(2000)126:1(27)
  32. Matsunaga, H. (2008), "Free vibration and stability of functionally graded plates according to a 2-D higherorder deformation theory", Compos. Struct., 82(4), 499-512. https://doi.org/10.1016/j.compstruct.2007.01.030
  33. Moniruzzaman, M. and Winey, K.I. (2006), "Polymer nanocomposites containing carbon nanotubes", Macromolecules, 39(16), 5194-5205. https://doi.org/10.1021/ma060733p
  34. Mori, T. and Tanaka, K. (1973), "Average stress in matrix and average elastic energy of materials with misfitting inclusions", Acta Metall., 21(5), 571-574. https://doi.org/10.1016/0001-6160(73)90064-3
  35. Mura, T. (1982), Micromechanics of Defects in Solids, Springer Science & Business Media.
  36. Odegard, G.M., Gates, T.S, Wise, K.E., Park, C. and Siochi, E.J. (2003), "Constitutive modeling of nanotube-reinforced polymer composites", Compos. Sci. Technol., 63(11), 1671-1687. https://doi.org/10.1016/S0266-3538(03)00063-0
  37. Qian, D., Dickey, E.C., Andrews, R. and Rantell, T. (2000), "Load transfer and deformation mechanisms in carbon nanotube-polystyrene composites", Appl. Phys. Lett., 76(20), 2868-2870. https://doi.org/10.1063/1.126500
  38. Reddy, J. and Cheng, Z.Q. (2001), "Three-dimensional thermomechanical deformations of functionally graded rectangular plates", Eur. J. Mech. A/Solids, 20(5), 841-855.
  39. Salvetat, D. and Rubio, A. (2002), "Mechanical properties of carbon nanotubes: A fiber digest for beginners", Carbon, 40(10), 1729-1734. https://doi.org/10.1016/S0008-6223(02)00012-X
  40. Shen, H.S. (2009), "Nonlinear bending of functionally graded carbon nanotube-reinforced composite plates in thermal environments", Compos. Struct., 91(1), 9-19. https://doi.org/10.1016/j.compstruct.2009.04.026
  41. Shen, H.S. (2011), "Postbuckling of nanotube-reinforced composite cylindrical shells in thermal environments, Part I: Axially-loaded shells", Compos. Struct., 93(8), 2096-2108. https://doi.org/10.1016/j.compstruct.2011.02.011
  42. Shen, H.S. (2012), "Thermal buckling and postbuckling behavior of functionally graded carbon nanotubereinforced composite cylindrical shells", Compos. Part B, Engineering, 43(3), 1030-1038. https://doi.org/10.1016/j.compositesb.2011.10.004
  43. Shen, H.S. and Zhang, C.L. (2010), "Thermal buckling and postbuckling behavior of functionally graded carbon nanotube-reinforced composite plates", Mater. Des., 31(7), 3403-3411. https://doi.org/10.1016/j.matdes.2010.01.048
  44. Shen, H.S. and Zhu, Z.H. (2010), "Buckling and postbuckling behavior of functionally graded nanotubereinforced composite plates in thermal environments", Comput. Mater. Continua., 18(2), 155-182.
  45. Shi, D.L., Feng, X.Q., Huang, Y.Y., Hwang, K.C. and Gao, H. (2004), "The effect of nanotube waviness and agglomeration on the elastic property of carbon nanotube-reinforced composites", J. Eng. Mater. Technol., 126(3), 250-257. https://doi.org/10.1115/1.1751182
  46. Shu, C. and Wang, C. (1999), "Treatment of mixed and nonuniform boundary conditions in GDQ vibration analysis of rectangular plates", Eng. Struct., 21(2), 125-134. https://doi.org/10.1016/S0141-0296(97)00155-7
  47. Sobhani Aragh, B., Hedayati, H., Borzabadi Farahani, E. and Hedayati, M. (2011), "A novel 2-D sixparameter power-law distribution for free vibration and vibrational displacements of two-dimensional functionally graded fiber-reinforced curved panels", Eur. J. Mech. A/Solids, 30(6), 865-883. https://doi.org/10.1016/j.euromechsol.2011.05.002
  48. Tahouneh, V. (2014a), "Free vibration analysis of thick CGFR annular sector plates resting on elastic foundations", Struct. Eng. Mech., Int. J., 50(6), 773-796. https://doi.org/10.12989/sem.2014.50.6.773
  49. Tahouneh, V. (2014b), "Free vibration analysis of bidirectional functionally graded annular plates resting on elastic foundations using differential quadrature method", Struct. Eng. Mech., Int. J., 52(4), 663-686. https://doi.org/10.12989/sem.2014.52.4.663
  50. Tahouneh, V. and Naei, M.H. (2014), "A novel 2-D six-parameter power-law distribution for threedimensional dynamic analysis of thick multi-directional functionally graded rectangular plates resting on a two-parameter elastic foundation", Meccanica, 49(1), 91-109. https://doi.org/10.1007/s11012-013-9776-x
  51. Tahouneh, V. and Naei, M.H. (2015a), "3D free vibration analysis of elastically supported thick nanocomposite curved panels with finite length and different boundary conditions via 2-D GDQ method", Mech. Adv. Mater. Struct., 1-80. DOI: 10.1080/15376494.2015.1068402
  52. Tahouneh, V. and Naei, M.H. (2015b), "Free vibration and vibrational displacements analysis of thick elastically supported laminated curved panels with power-law distribution functionally graded layers and finite length via 2D GDQ method", J. Sandw. Struct. Mater., 1-31. DOI: 10.1177/1099636215600709
  53. Tahouneh, V. and Yas, M.H. (2012), "3-D free vibration analysis of thick functionally graded annular sector plates on Pasternak elastic foundation via 2-D differential quadrature method", Acta Mech., 223(9), 1879-1897. https://doi.org/10.1007/s00707-012-0648-6
  54. Tahouneh, V. and Yas, M.H. (2013), "Semianalytical solution for three-dimensional vibration analysis of thick multidirectional functionally graded annular sector plates under various boundary conditions", J. Eng. Mech., 140(1), 31-46.
  55. Tahouneh, V. and Yas, M.H. (2014), "Influence of equivalent continuum model based on the Eshelby-Mori-Tanaka scheme on the vibrational response of elastically supported thick continuously graded carbon nanotube-reinforced annular plates", Polym. Composite, 35(8), 1644-1661. https://doi.org/10.1002/pc.22818
  56. Tahouneh, V., Yas, M.H., Tourang, H. and Kabirian, M. (2013), "Semi-analytical solution for threedimensional vibration of thick continuous grading fiber reinforced (CGFR) annular plates on Pasternak elastic foundations with arbitrary boundary conditions on their circular edges", Meccanica, 48(6), 1313-1336. https://doi.org/10.1007/s11012-012-9669-4
  57. Thostenson, E.T., Ren, Z.F. and Chou, T.W. (2001), "Advances in the science and technology of carbon nanotubes and their composites", A Review; Compos. Sci. Technol., 61(13), 1899-1912. https://doi.org/10.1016/S0266-3538(01)00094-X
  58. Valter, B., Ram, M.K. and Nicolini, C. (2002), "Synthesis of multiwalled carbon nanotubes and poly (oanisidine) nanocomposite material: Fabrication and characterization of its Langmuir-Schaefer films", Langmuir, 18(5), 1535-1541. https://doi.org/10.1021/la0104673
  59. Vel, S.S. (2010), "Exact elasticity solution for the vibration of functionally graded anisotropic cylindrical shells", Compos. Struct., 92(11), 2712-2727. https://doi.org/10.1016/j.compstruct.2010.03.012
  60. Vel, S.S. and Batra, R. (2004), "Three-dimensional exact solution for the vibration of functionally graded rectangular plates", J. Sound. Vib., 272(3), 703-730. https://doi.org/10.1016/S0022-460X(03)00412-7
  61. Wang, Z.X. and Shen, H.S. (2011), "Nonlinear vibration of nanotube-reinforced composite plates in thermal environments", Comp. Mater. Sci., 50(8), 2319-2330. https://doi.org/10.1016/j.commatsci.2011.03.005
  62. Wang, C.M., Kitipornchai, S. and Xiang, Y. (1997), "Relationships between buckling loads of Kirchhoff, Mindlin, and Reddy polygonal plates on Pasternak foundation", J. Eng. Mech. ASCE, 123(11), 1134-1137. https://doi.org/10.1061/(ASCE)0733-9399(1997)123:11(1134)
  63. Wernik, J. and Meguid, S. (2011), "Multiscale modeling of the nonlinear response of nano-reinforced polymers", Acta Mech., 217(1-2), 1-16. https://doi.org/10.1007/s00707-010-0377-7
  64. Xiang, Y., Wang, C.M. and Kitipornchai, S. (1994), "Exact vibration solution for initially stressed Mindlin plates on Pasternak foundations", Int. J. Mech. Sci., 36(4), 311-316. https://doi.org/10.1016/0020-7403(94)90037-X
  65. Xiang, Y., Kitipornchai, S. and Liew, K.M. (1996), "Buckling and vibration of thick laminates on Pasternak foundations", J. Eng. Mech. ASCE, 122(1), 54-63. https://doi.org/10.1061/(ASCE)0733-9399(1996)122:1(54)
  66. Yas, M.H. and Sobhani Aragh, B. (2010), "Three-dimensional analysis for thermoelastic response of functionally graded fiber reinforced cylindrical panel", Compos. Struct., 92(10), 2391-2399. https://doi.org/10.1016/j.compstruct.2010.03.008
  67. Yas, M.H. and Tahouneh, V. (2012), "3-D free vibration analysis of thick functionally graded annular plates on Pasternak elastic foundation via differential quadrature method (DQM)", Acta Mech., 223(1), 43-62. https://doi.org/10.1007/s00707-011-0543-6
  68. Yokozeki, T., Iwahori, Y. and Ishiwata, S. (2007), "Matrix cracking behaviors in carbon fiber/epoxy laminates filled with cup-stacked carbon nanotubes (CSCNTs)", Composites Part A: Applied Science and Manufacturing, 38(3), 917-924. https://doi.org/10.1016/j.compositesa.2006.07.005
  69. Zhou, D., Cheung, Y.K., Lo, S.H. and Au, F.T.K. (2004), "Three-dimensional vibration analysis of rectangular thick plates on Pasternak foundation", Int. J. Numer. Method. Eng., 59(10), 1313-1334. https://doi.org/10.1002/nme.915

피인용 문헌

  1. Three-dimensional vibration of a ring with a noncircular cross-section on an elastic foundation 2017, https://doi.org/10.1177/0954406217720823
  2. Nonlinear free and forced vibration analysis of FG-CNTRC annular sector plates pp.02728397, 2019, https://doi.org/10.1002/pc.24998
  3. Vibration and Mode Shape Analysis of Functionally Graded Nanocomposite Plates Reinforced by Aggregated Carbon Nanotube vol.378, pp.1757-899X, 2018, https://doi.org/10.1088/1757-899X/378/1/012011
  4. Vibrational analysis of sandwich sectorial plates with functionally graded sheets reinforced by aggregated carbon nanotube pp.1530-7972, 2020, https://doi.org/10.1177/1099636218785972
  5. Free vibration analysis of elastically restrained functionally graded curved beams based on the Mori–Tanaka scheme pp.1537-6532, 2019, https://doi.org/10.1080/15376494.2018.1452318
  6. Comparison of nonlinear Von Karman and Cosserat theories in very large deformation of skew plates vol.10, pp.1, 2018, https://doi.org/10.1007/s40091-018-0184-2
  7. Using modified Halpin-Tsai approach for vibrational analysis of thick functionally graded multi-walled carbon nanotube plates vol.23, pp.6, 2016, https://doi.org/10.12989/scs.2017.23.6.657
  8. Comparison of different cylindrical shell theories for stability of nanocomposite piezoelectric separators containing rotating fluid considering structural damping vol.23, pp.6, 2016, https://doi.org/10.12989/scs.2017.23.6.691
  9. The effect of carbon nanotubes agglomeration on vibrational response of thick functionally graded sandwich plates vol.24, pp.6, 2016, https://doi.org/10.12989/scs.2017.24.6.711
  10. Vibration and mode shape analysis of sandwich panel with MWCNTs FG-reinforcement core vol.25, pp.3, 2017, https://doi.org/10.12989/scs.2017.25.3.347
  11. Effects of CNTs waviness and aspect ratio on vibrational response of FG-sector plate vol.25, pp.6, 2016, https://doi.org/10.12989/scs.2017.25.6.649
  12. Vibration analysis of sandwich sectorial plates considering FG wavy CNT-reinforced face sheets vol.28, pp.5, 2016, https://doi.org/10.12989/scs.2018.28.5.541
  13. Vibration analysis of FG porous rectangular plates reinforced by graphene platelets vol.34, pp.2, 2020, https://doi.org/10.12989/scs.2020.34.2.215
  14. Influence of porosity distribution on vibration analysis of GPLs-reinforcement sectorial plate vol.35, pp.1, 2016, https://doi.org/10.12989/scs.2020.35.1.111
  15. Vibrational characteristic of FG porous conical shells using Donnell's shell theory vol.35, pp.2, 2016, https://doi.org/10.12989/scs.2020.35.2.249
  16. Influence of internal pores and graphene platelets on vibration of non-uniform functionally graded columns vol.35, pp.2, 2016, https://doi.org/10.12989/scs.2020.35.2.295
  17. Vibration behavior of functionally graded sandwich beam with porous core and nanocomposite layers vol.36, pp.1, 2020, https://doi.org/10.12989/scs.2020.36.1.001
  18. Vibration behavior of trapezoidal sandwich plate with functionally graded-porous core and graphene platelet-reinforced layers vol.36, pp.1, 2016, https://doi.org/10.12989/scs.2020.36.1.047
  19. Vibration analysis of sandwich sector plate with porous core and functionally graded wavy carbon nanotube-reinforced layers vol.37, pp.6, 2016, https://doi.org/10.12989/scs.2020.37.6.711
  20. Vibration analysis of damaged core laminated curved panels with functionally graded sheets and finite length vol.38, pp.5, 2016, https://doi.org/10.12989/scs.2021.38.5.477