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Using DQ method for vibration analysis of a laminated trapezoidal structure with functionally graded faces and damaged core

  • Vanessa Valverde (Facultad de Mecanica, Escuela Superior Politecnica de Chimborazo (ESPOCH)) ;
  • Patrik Viktor (Obuda University, Keleti Karoly Faculty of Business and Management, Obuda University) ;
  • Sherzod Abdullaev (Faculty of Chemical Engineering, New Uzbekistan University) ;
  • Nasrin Bohlooli (Nabi Data Science & Computational Intelligence Research Co.)
  • 투고 : 2023.07.07
  • 심사 : 2023.11.27
  • 발행 : 2024.04.10

초록

This paper has focused on presenting vibration analysis of trapezoidal sandwich plates with a damaged core and FG wavy CNT-reinforced face sheets. A damage model is introduced to provide an analytical description of an irreversible rheological process that causes the decay of the mechanical properties, in terms of engineering constants. An isotropic damage is considered for the core of the sandwich structure. The classical theory concerning the mechanical efficiency of a matrix embedding finite length fibers has been modified by introducing the tube-to-tube random contact, which explicitly accounts for the progressive reduction of the tubes' effective aspect ratio as the filler content increases. The First-order shear deformation theory of plate is utilized to establish governing partial differential equations and boundary conditions for the trapezoidal plate. The governing equations together with related boundary conditions are discretized using a mapping-generalized differential quadrature (GDQ) method in spatial domain. Then natural frequencies of the trapezoidal sandwich plates are obtained using GDQ method. Validity of the current study is evaluated by comparing its numerical results with those available in the literature. After demonstrating the convergence and accuracy of the method, different parametric studies for laminated trapezoidal structure including carbon nanotubes waviness (0≤w≤1), CNT aspect ratio (0≤AR≤4000), face sheet to core thickness ratio (0.1 ≤ ${\frac{h_f}{h_c}}$ ≤ 0.5), trapezoidal side angles (30° ≤ α, β ≤ 90°) and damaged parameter (0 ≤ D < 1) are carried out. It is explicated that the damaged core and weight fraction, carbon nanotubes (CNTs) waviness and CNT aspect ratio can significantly affect the vibrational behavior of the sandwich structure. Results show that by increasing the values of waviness index (w), normalized natural frequency of the structure decreases, and the straight CNT (w=0) gives the highest frequency. For an overall comprehension on vibration of laminated trapezoidal plates, some selected vibration mode shapes were graphically represented in this study.

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참고문헌

  1. Abo-Dahab, S.M., Abouelregal, A.E. and Marin, M. (2020), "Generalized thermoelastic functionally graded on a thin slim strip non-gaussian laser beam", Symmetry, 12(7), https://doi.org/10.3390/sym12071094.
  2. Affdl Halpin, J.C. and Kardos, J.L. (1976), "The Halpin-Tsai equations: A review", Polym. Eng. Sci., 16(5), 344-352. https://doi.org/10.1002/pen.760160512.
  3. Afrookhteh, S.S., Fathi, A., Naghdipour, M. and Alizadeh Sahraei, A. (2016), "An experimental investigation of the effects of weight fractions of reinforcement and timing of hardener addition on the strain sensitivity of carbon nanotube/polymer composites", U.P.B. Sci. Bull., Series B, 78(4), 121-130.
  4. Afrookhteh, S.S., Shakeri, M., Baniassadi, M. and Alizadeh Sahraei, A. (2018), "Microstructure Reconstruction and Characterization of the Porous GDLs for PEMFC Based on Fibers Orientation Distribution", Fuel Cells, 18(2), https://doi.org/10.1002/fuce.201700239.
  5. Al-Fasih, M.Y., Kueh, A.B.H. and Ibrahim, M.H.W. (2020), "Flexural behavior of sandwich beams with novel triaxially woven fabric composite skins", Steel Compos. Struct., 34(2), 299-308, https://doi.org/10.12989/scs.2020.34.2.299.
  6. Al-Fasih, M.Y., Kueh, A.B.H. and Ibrahim, M.H.W. (2020), "Failure behavior of sandwich honeycomb composite beam containing crack at the skin", PLOS ONE, https://doi.org/10.1371/journal.pone.0227895.
  7. Al-Fasih, M.Y., Kueh, A.B.H., Abo Sabah, S.H. and Yahya, M.Y. (2017), "Influence of tows waviness and anisotropy on effective Mode I fracture toughness of triaxially woven fabric composites", Eng. Fracture Mech., 182, 521-526. https://doi.org/10.1016/j.engfracmech.2017.03.051.
  8. Al-Fasih, M.Y., Kueh, A.B.H., Abo Sabah, S.H. and Yahya, M.Y. (2018), "Tow waviness and anisotropy effects on Mode II fracture of triaxially woven composite", Steel Compos. Struct., 26(2), 241-253, https://doi.org/10.12989/scs.2018.26.2.241.
  9. Arefi, M. (2015), "Elastic solution of a curved beam made of functionally graded materials with different cross sections', Steel Compos. Struct., 18(3), 659-672. https://doi.org/10.12989/scs.2015.18.3.659.
  10. Bardell, N.S. (1992), "The free vibration of skew plates using the hierarchical finite element method", Comput. Struct., 45, 841-874. https://doi.org/10.1016/0045-7949(92)90044-Z
  11. Barka, M., Benrahou, K.H., Bakora, A. and Tounsi, A. (2016), "Thermal post-buckling behavior of imperfect temperature-dependent sandwich FGM plates resting on Pasternak elastic foundation", Steel Compos. Struct., 22(1), 91-112. https://doi.org/10.12989/scs.2016.22.1.091.
  12. Bennai, R., Ait Atmane, H. and Tounsi, A. (2015), "A new higher-order shear and normal deformation theory for functionally graded sandwich beams", Steel Compos. Struct., 19(3), 521-546. https://doi.org/10.12989/scs.2015.19.3.521.
  13. Bert, C.W. and Malik, M. (1996), "Differential quadrature method in computational mechanics: a review", Appl. Mech. Rev., 49, 1-27. https://doi.org/10.1115/1.3101882.
  14. Bi, H., Xie, X., Yin, K., Zhou, Y., Wan, S., He, L. (2012), "Spongy graphene as a highly efficient and recyclable sorbent for oils and organic solvents", Adv. Funct. Mater., 22, 4421-4425. https://doi.org/10.1002/adfm.201200888.
  15. Bi, H., Yin, K., Xie, X., Zhou, Y., Wan, N. and Xu, F. (2012), "Low temperature casting of graphene with high compressive strength", Adv. Mater., 24, 5124-5129. https://doi.org/10.1002/adma.201201519.
  16. Bouchafa, A., Bouiadjra, M.B., Houari, M.S.A. and Tounsi, A. (2015), "Thermal stresses and deflections of functionally graded sandwich plates using a new refined hyperbolic shear deformation theory", Steel Compos. Struct., 18(6), 1493-1515. https://doi.org/10.12989/scs.2015.18.6.1493.
  17. Bouguenina, O., Belakhdar, K., Tounsi, A. and Bedia, E.A.A. (2015), "Numerical analysis of FGM plates with variable thickness subjected to thermal buckling", Steel Compos. Struct., 19(3), 679-695. https://doi.org/10.12989/scs.2015.19.3.679.
  18. Bouafia, H., Chikh, A., Bousahla,A.A., Bourada, F., Heireche, H., Tounsi, A., Benrahou, K.H., Tounsi,A., Al-Zahrani, M.M. and Hussain, M. (2021), "Natural frequencies of FGM nanoplates embedded in an elastic medium", Adv. Nano Res., 11(3), 239-249. https://doi.org/10.12989/anr.2021.11.3.239.
  19. Boutaleb, S., Benrahou, K.H., Bakora, A., Algarni, A., Bousahla, A.A., Tounsi, A., Tounsi, A. and Mahmoud, S.R. (2019), "Dynamic analysis of nanosize FG rectangular plates based on simple nonlocal quasi 3D HSDT", Adv. Nano Res., 7(3), 191-208. https://doi.org/10.12989/anr.2019.7.3.191.
  20. Chen, S., Bao, P., Huang, X., Sun, B. and Wang, G. (2014), "Hierarchical 3D mesoporous silicon@graphene nanoarchitectures for lithium ion batteries with superior performance", Nano Res., 7, 85-94. https://doi.org/10.1007/s12274-013-0374-y.
  21. Chen, C.S., Liu, F.H. and Chen, W.R. (2017), "vibration and stability of initially stressed sandwich plates with FGM face sheets in thermal environments", Steel Compos. Struct., 23(3), 251-261. https://doi.org/10.12989/scs.2017.23.3.251.
  22. Chen, Z., Ren, W., Gao, L., Liu, B., Pei, S. and Cheng, H.M. (2011), "Three-dimensional flexible and conductive interconnected graphene networks grown by chemical vapour deposition", Nat. Mater., 10, 424-428. https://doi.org/10.1038/nmat3001
  23. Dai, Z., Jiang, Z., Zhang, L. and Habibi, M. (2021), "Frequency characteristics and sensitivity analysis of a size-dependent laminated nanoshell", Adv. Nano Res., 10(2), 175-189. https://doi.org/10.12989/anr.2021.10.2.175.
  24. Ebrahimi, F., Fardshad, R.E. and Mahesh, V. (2019), "Frequency response analysis of curved embedded magneto-electro-viscoelastic functionally graded nanobeams", Adv. Nano Res., 7(6), 391-403. https://doi.org/10.12989/anr.2019.7.6.391.
  25. Embrey, L., Nautiyal, P., Loganathan, A., Idowu, A., Boesl, B. and Agarwal, A. (2017) "Three-dimensional graphene foam induces multifunctionality in epoxy nanocomposites by simultaneous improvement in mechanical, thermal, and electrical properties", ACS Appl. Mater. Interfaces, 9, 39717-39727. https://doi.org/10.1021/acsami.7b14078
  26. Eyvazian, A., Hamouda, A.M., Tarlochan, F., Mohsenizadeh, S. and Ahmadi Dastjerdi, A. (2019), "Damping and vibration response of viscoelastic smart sandwich plate reinforced with non-uniform Graphene platelet with magnetorheological fluid core", Steel Compos. Struct., 33(6), 891-906. https://doi.org/10.12989/scs.2019.33.6.891.
  27. Fantuzzi, N., Tornabene, F., Bacciocchi, M. and Dimitri, R. (2017), "Free vibration analysis of arbitrarily shaped Functionally Graded Carbon Nanotube-reinforced plates", Compos. Part B, 115, 384-408. https://doi.org/10.1016/j.compositesb.2016.09.021.
  28. Finot, M. and Suresh, S. (1996), "Small and large deformation of thick and thin-film multilayers: effect of layer geometry, plasticity and compositional gradients", J. Mech. Phys. Solids, 44(5), 683-721. https://doi.org/10.1016/0022-5096(96)84548-0.
  29. Guan, L.Z., Zhao, L., Wan, Y.J. and Tang, L.C. (2018), "Three-dimensional graphene-based polymer nanocomposites: Preparation, properties and applications", Nanoscale, 10, 14788-14811. https://doi.org/10.1039/C8NR03044H.
  30. Gupta, A.K. and Sharma, S. (2014), "Free transverse vibration of orthotropic thin trapezoidal plate of parabolically varying thickness subjected to linear temperature distribution", Shock Vib., 2014, 1-6. http://dx.doi.org/10.1155/2014/392325.
  31. Gupta, A.K. and Sharma, P. (2016), "Vibration study of nonhomogeneous trapezoidal plates of variable thickness under thermal gradient", J.V.C. Control, 22(5), 1369-1379. https://doi.org/10.1177/1077546314535280.
  32. Gurses, M., Civalek, O., Ersoy, H., Kiracioglu, O. (2009), "Analysis of shear deformable laminated composite trapezoidal plates", Mater. Des., 30, 3030-3035. https://doi.org/10.1016/j.matdes.2008.12.016.
  33. Haldar, S. and Manna, M.C. (2003), "A high precision shear deformable element for free vibration of thick/thin composite trapezoidal plates", Steel Compos. Struct., 3(3), 213-229. https://doi.org/10.12989/scs.2003.3.3.213
  34. Halpin, J.C. and Tsai, S.W. (1969), "Effects of environmental factors on composite materials", AFML-TR-67-423.
  35. Hanifehzadeh, M. and Mousavi, M.M.R. (2019), "Prediction the structural performance of sandwich concrete panels subjected to blast load considering dynamic increase factor", J. Civil Eng. Sci. Technol., 10(1), 45-58. https://doi.org/10.33736/jcest.1067.2019.
  36. Houmat, A. (2001), "A sector Fourier p-element applied to free vibration analysis of sectorial plates", J. Sound Vib., 243(2), 269-282. https://doi.org/10.1006/jsvi.2000.3410
  37. Hu, H., Zhao, Z., Wan, W., Gogotsi, Y. and Qiu, J. (2013), "Ultralight and highly compressible graphene aerogels", Adv. Mater., 25, 2219-2223. https://doi.org/10.1002/adma.201204530.
  38. Idowu, A., Boesl, B. and Agarwal, A. (2018), "3D graphene foam-reinforced polymer composites", A review, Carbon, 135, 52-71. https://doi.org/10.1016/j.carbon.2018.04.024.
  39. Kamarian, S., Shakeri, M., Yas, M.H., Bodaghi, M. and Pourasghar, A. (2015), "Free vibration analysis of functionally graded nanocomposite sandwich beams resting on Pasternak foundation by considering the agglomeration effect of CNTs", J. Sandw. Struct. Mater., 1-31. https://doi.org/10.1177/1099636215590280.
  40. Kapidzic, Z. (2013), "Strength analysis and modeling of hybrid composite-aluminum aircraft structures", Linkoping Studies Sci. Technol., Licentiate Thesis No. 1590.
  41. Kettaf, F.Z., Houari, M.S.A., Benguediab, M. and Tounsi, A. (2013), "Thermal buckling of functionally graded sandwich plates using a new hyperbolic shear displacement model", Steel Compos. Struct., 15(4), 399-423. https://doi.org/10.12989/scs.2013.15.4.399.
  42. Khadir, A.I., Daikh, A.A. and Eltaher, M.A. (2021), "Novel four-unknowns quasi 3D theory for bending, buckling and free vibration of functionally graded carbon nanotubes reinforced composite laminated nanoplates", Adv. Nano Res., 11(6), 621-640. https://doi.org/10.12989/anr.2021.11.6.621.
  43. Kim, C.S. and Dickinson, S.M. (1989), "On the free, transverse vibration of annular and circular, thin, sectorial plates subjected to certain complicating effects", J. Sound Vib., 134(3), 407-421. https://doi.org/10.1016/0022-460X(89)90566-X.
  44. Kitipornchai, S., Chen, D. and Yang, J. (2017), "Free vibration and elastic buckling of functionally graded porous beams reinforced by graphene platelets", Mater. Des., 116, 656-665. https://doi.org/10.1016/j.matdes.2016.12.061.
  45. Koizumi, M. (1993), "The concept of FGM", Ceram. Trans. Funct. Grad. Mater., 34, 3-10.
  46. Lemaitre, J. and Chaboche, J.L. (1990), "Mechanics of Solid Materials", Cambridge University Press: New York, NY, USA.
  47. Liew, K.M. and Lam, K.Y. (1993), "On the use of 2-d orthogonal polynomials in the Rayleigh-Ritz method for flexural vibration of annular sector plates of arbitrary shape", Int. J. Mech. Sci., 35(2), 129-139. https://doi.org/10.1016/0020-7403(93)90071-2.
  48. Liew, K.M. and Liu, F.L. (2000), "Differential quadrature method for vibration analysis of shear deformable annular sector plates", J. Sound Vib., 230(2), 335-356. https://doi.org/10.1006/jsvi.1999.2623.
  49. Liew, K.M., Xiang, Y., Kitipomchai, S. and Wang, C.M. (1993), "Vibration of thick skew plates based on Mindlin shear deformation plate theory", J. Sound Vib., 168, 39-69. https://doi.org/10.1006/jsvi.1993.1361
  50. Lv, L., Zhang, P., Cheng, H., Zhao, Y., Zhang, Z. and Shi, G. (2016), "Solution-processed ultraelastic and strong air-bubbled graphene foams", Small, 12, 3229-3234. https://doi.org/10.1002/smll.201600509.
  51. Malekzadeh, P. and Karami, G. (2005), "Polynomial and harmonic differential quadrature methods for free vibration of variable thickness skew plate", Eng. Struct., 27, 1563-1574. https://doi.org/10.1016/j.engstruct.2005.03.017.
  52. Marin, M., Agarwal, R.P. and Mahmoud, S.R. (2013), "Nonsimple material problems addressed by the Lagrange's identity", Bound. Value Probl., 2013(1-14), 135.
  53. Marin, M. and Florea, O. (2014), "On temporal behaviour of solutions in thermoelasticity of porous micropolar bodies", An. St. Univ. Ovidius Constanta, 22(1), 169-188.
  54. Marin, M., Seadawy, A. Vlase, S. and Chirila, A. (2022), "On mixed problem in thermoelasticity of type III for Cosserat media", J. Taibah Univ. Sci., 16(1), 1264-1274. https://doi.org/10.1080/16583655.2022.2160290.
  55. Marin, M. (1995), "On existence and uniqueness in thermoelasticity of micropolar bodies", Comptes rendus de l'Academie des sciences Paris, Serie II, B, 321(12), 475-480.
  56. Martone, A., Faiella, G., Antonucci, V., Giordano, M. and Zarrelli, M. (2011), "The effect of the aspect ratio of carbon nanotubes on their effective reinforcement modulus in an epoxy matrix", Compos. Sci. Technol., 71, 1117-1123. https://doi.org/10.1016/j.compscitech.2011.04.002.
  57. McGee, O.G., Kim, J.W. and Kim, Y.S. (1996), "Corner stress singularity effects on the vibration of rhombic plates with combinations of clamped and simply supported edges", J. Sound Vib., 193(13), 555-580. https://doi.org/10.1006/jsvi.1996.0302
  58. Mukhopadhyay, M. (1979), "A semi-analytic solution for free vibration of annular sector plates", J. Sound Vib., 63(1), 87-95. https://doi.org/10.1016/0022-460X(79)90379-1
  59. Mukhopadhyay, M. (1982), "Free vibration of annular sector plates with edges possessing different degrees of rotational restraints", J. Sound Vib., 80(2), 275-279. https://doi.org/10.1016/0022-460X(82)90196-1.
  60. Ni, Y., Chen, L., Teng, K., Shi, J., Qian, X. and Xu, Z. (2015), "Superior mechanical properties of epoxy composites reinforced by 3D interconnected graphene skeleton", ACS Appl. Mater. Interfaces, 7, 11583-11591. https://doi.org/10.1021/acsami.5b02552
  61. Othman, M.I.A., Fekry, M. and Marin, M. (2020), "Plane waves in generalized magneto-thermo-viscoelastic medium with voids under the effect of initial stress and laser pulse heating", Struct, Eng. Mech, 73(6), 621-629. https://doi.org/10.12989/sem.2020.73.6.621.
  62. Park, W.T., Han, S.C., Jung, W.Y. and Lee, W.H. (2016), "Dynamic instability analysis for S-FGM plates embedded in Pasternak elastic medium using the modified couple stress theory", Steel Compos. Struct., 22(6), 1239-1259. https://doi.org/10.12989/scs.2016.22.6.1239.
  63. Pelletier Jacob, L. and Vel Senthil,S. (2006), "An exact solution for the steady state thermo elastic response of functionally graded orthotropic cylindrical shells", Int. J. Solid Struct., 43, 1131-1158. https://doi.org/10.1016/j.ijsolstr.2005.03.079.
  64. Qiu, L., Huang, B., He, Z., Wang, Y., Tian, Z. and Liu, J.Z. (2017), "Extremely low density and super compressible graphene cellular materials", Adv. Mater., 29, 1-6. https://doi.org/10.1002/adma.201701553.
  65. Rajabi, J. and Mohammadimehr, M. (2019), "Hydro-thermomechanical biaxial buckling analysis of sandwich micro-plate with isotropic/orthotropic cores and piezoelectric/polymeric nanocomposite face sheets based on FSDT on elastic foundations", Steel Compos. Struct., 33(4), 509-523. https://doi.org/10.12989/scs.2019.33.4.509.
  66. Ramaiah, G.K. and Vijayakumar, K. (1974), "Natural frequencies of circumferentially truncated sector plates with simply supported straight edges", J. Sound Vib., 34(1), 53-61. https://doi.org/10.1016/S0022-460X(74)80354-8
  67. Rashad, M. and Yang, T.Y. (2018), "Numerical study of steel sandwich plates with RPF and VR cores materials under free air blast loads", Steel Compos. Struct., 27(6), 717-725. https://doi.org/10.12989/scs.2018.27.6.717.
  68. Reddy J.N. (2013), An Introduction to Continuum Mechanics, Cambridge University Press.
  69. Sahla, M., Saidi, H., Draiche, K., Bousahla, A.A., Bourada, F. and Tounsi, A. (2019), "Free vibration analysis of angle-ply laminated composite and soft core sandwich plates", Steel Compos. Struct., 33(5), 663-679. https://doi.org/10.12989/scs.2019.33.5.663.
  70. Saidi, H., Houari, M.S.A., Tounsi, A. and Bedia, E.A. (2013), "Thermo-mechanical bending response with stretching effect of functionally graded sandwich plates using a novel shear deformation theory", Steel Compos. Struct., 15(2), 221-245. https://doi.org/10.12989/scs.2013.15.2.221.
  71. Salah, F., Boucham, B., Bourada, F. and Benzair, A. (2019), "Investigation of thermal buckling properties of ceramic-metal FGM sandwich plates using 2D integral plate model", Steel Compos. Struct., 33(6), 805-822. https://doi.org/10.12989/scs.2019.33.6.805.
  72. Seok, J.W. and Tiersten, H.F. (2004), "Free vibrations of annular sector cantilever plates part 1: Out-of-plane motion", J. Sound Vib., 271(3-5), 757-772. https://doi.org/10.1016/S0022-460X(03)00414-0.
  73. Setoodeh, A.R. and Shojaee, M. (2016), "Application of TW-DQ method to nonlinear free vibration analysis of FG carbon nanotube-reinforced composite quadrilateral plates", Thin-Wall. Struct., 108, 1-11. http://dx.doi.org/10.1016/j.tws.2016.07.019.
  74. Sha, J., Li, Y., Villegas Salvatierra, R., Wang, T., Dong, P. and Ji, Y. (2017), "Three-dimensional printed graphene foams", ACS Nano, 11(7), 6860-6867. https://doi.org/10.1021/acsnano.7b01987.
  75. Sharma, K. and Marin, M. (2013), "Effect of distinct conductive an thermodynamic temperatures on the reflection of plane waves in micropolar elastic half-space", UPB Sci. Bull., Series A Appl. Mathem. Phys., 75(2), 121-132.
  76. Sharma, A., Sharda, H.B. and Nath, Y. (2005a), "Stability and vibration of Mindlin sector plates: an analytical approach", AIAA J., 43(5), 1109-1116. https://doi.org/10.2514/1.4683.
  77. Sharma, A., Sharda, H.B. and Nath, Y. (2005b), "Stability and vibration of thick laminated composite sector plates", J. Sound Vib., 287(1-2), 1-23. https://doi.org/10.1016/j.jsv.2004.10.030.
  78. Shen, H.S., 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.
  79. Shen, H.S. (2009), "Nonlinear bending of functionally graded carbon nanotube reinforced composite plates in thermal environments", Compos. Struct., 91, 9-19. https://doi.org/10.1016/j.compstruct.2009.04.026.
  80. Shokrollahi, S. and Shafaghat, S. (2016), "A global Ritz formulation for the free vibration analysis of hybrid metal-composite thick trapezoidal plates", Scientia Iranica Transactions B: Mech. Eng., 23(1), 249-259.
  81. Shu, C. (2012), Differential Quadrature and its Application in Engineering, Springer Science & Business Media.
  82. Sobhani Aragh, B., Nasrollah Barati, A.H. and Hedayati, H. (2012), "Eshelby-Mori-Tanaka approach for vibrational behavior of continuously graded carbon nanotube-reinforced cylindrical panels", Compos. B Eng., 43(4), 1943-1954. https://doi.org/10.1016/j.compositesb.2012.01.004.
  83. Strek, W., Tomala, R., Lukaszewicz, M. and Cichy, B., Gerasymchuk, Y. and Gluchowski, P. (2017), "Laser induced white lighting of graphene foam", Sci. Rep., 7.
  84. Tahouneh, V. (2016), "Using an equivalent continuum model for 3D dynamic analysis of nanocomposite plates", Steel Compos. Struct., 20(3), 623-649. https://doi.org/10.12989/scs.2016.20.3.623.
  85. Tahouneh, V. (2017), "The effect of carbon nanotubes agglomeration on vibrational response of thick functionally graded sandwich plates", Steel Compos. Struct., 24(6), 711-726. https://doi.org/10.12989/scs.2017.24.6.711.
  86. Tahouneh, V., Naei, M.H. and Mosavi Mashhadi, M. (2018), "The effects of temperature and vacancy defect on the severity of the SLGS becoming anisotropic", Steel Compos. Struct., 29(5), 647-657, https://doi.org/10.12989/scs.2018.29.5.647.
  87. Tahouneh, V., Naei, M.H. and Mosavi Mashhadi, M. (2020), "Influence of vacancy defects on vibration analysis of graphene sheets applying isogeometric method: Molecular and continuum approaches", Steel Compos. Struct., 34(2), 261-277, https://doi.org/10.12989/scs.2020.34.2.261.
  88. Tahouneh, V., Naei, M.H. and Mosavi Mashhadi, M. (2019), "Using IGA and trimming approaches for vibrational analysis of L-shape graphene sheets via nonlocal elasticity theory", Steel Compos. Struct., 33(5), 717-727, https://doi.org/10.12989/scs.2019.33.5.717.
  89. Torabi, K. and Afshari, H. (2017), "Vibration analysis of a cantilevered trapezoidal moderately thick plate with variable thickness", Eng. Solid Mech., 30(8), 71-92. https://doi.org/10.5267/j.esm.2016.7.001.
  90. Tornabene, F., Fantuzzi, N., Bacciocchi, M. and Viola, E. (2016), "Effect of agglomeration on the natural frequencies of functionally graded carbon nanotube-reinforced laminated composite doubly-curved shells", Compos. Part B, 89, 187-218. https://doi.org/10.1016/j.compositesb.2015.11.016.
  91. Tornabene, F., Fantuzzi, N. and Bacciocchi, M. (2017), "Linear static response of nanocomposite plates and shells reinforced by agglomerated carbon nanotubes", Compos. Part B., 115, 449-476. https://doi.org/10.1016/j.compositesb.2016.07.011.
  92. Tornabene, F., Fantuzzi, N., Ubertini, F. and Viola, E. (2015), "Strong formulation finite element method based on differential quadrature: A survey", Appl. Mech. Rev., 67(2), 1-55. https://doi.org/10.1115/1.4028859.
  93. Tornabene, F., Fantuzzi, N. and Bacciocchi, M. (2019), "Refined shear deformation theories for laminated composite arches and beams with variable thickness: Natural frequency analysis", Eng. Anal. Bound. Elem., 100, 24-47. https://doi.org/10.1016/j.enganabound.2017.07.029.
  94. Tornabene, F., Fantuzzi, N. and Bacciocchi, M. (2017), "Foam core composite sandwich plates and shells with variable stiffness: Effect of the curvilinear fiber path on the modal response", J. Sandw. Struct. Mater., 21(1), 320-365. https://doi.org/10.1177/1099636217693623.
  95. Wang, C., Zhang, C. and Chen, S. (2016), "The microscopic deformation mechanism of 3D graphene foam materials under uniaxial compression", Carbon, 109, 666-672. https://doi.org/10.1016/j.carbon.2016.08.084.
  96. Woo, K.S., Hong, C.H., Basu, P.K. and Seo, C.G. (2003), "Free vibration of skew Mindlin plates by p-version of F.E.M.", J. Sound Vib., 268, 637-656. https://doi.org/10.1016/S0022-460X(02)01536-5
  97. Wu, C.P. and Liu, Y.C. (2016), "A state space meshless method for the 3D analysis of FGM axisymmetric circular plates", Steel Compos. Struct., 22(1), 161-182. https://doi.org/10.12989/scs.2016.22.1.161.
  98. Wu, Y., Yi, N., Huang, L., Zhang, T., Fang, S. and Chang, H. (2015), "Three-dimensionally bonded spongy graphene material with super compressive elasticity and near-zero Poisson's ratio", Nat. Commun., 6, https://doi.org/10.1038/ncomms7141.
  99. Wang, Y.Q. and Zhang Z.Y. (2019), "Bending and buckling of three-dimensional graphene foam plates", Results Phys., 13, https://doi.org/10.1016/j.rinp.2019.02.072.
  100. Xu, X., Li, H., Zhang, Q., Hu, H., Zhao, Z. and Li, J. (2015), "Self-sensing, ultralight, and conductive 3D graphene/iron oxide aerogel elastomer Deformable in a Magnetic Field", ACS Nano, 9, 3969-3977. https://doi.org/10.1021/nn507426u.
  101. Xu, Y., Sheng, K., Li, C. and Shi, G. (2010), "Self-assembled graphene hydrogel via a one-step hydrothermal process", ACS Nano, 4, 4324-4330.
  102. Yavari, F., Chen, Z., Thomas, A.V., Ren, W., Cheng, H.M. and Koratkar, N. (2011), "High sensitivity gas detection using a macroscopic three-dimensional graphene foam network", Sci. Rep., 1, 1-5. https://doi.org/10.1038/srep00166.
  103. Zamani, M., Fallah, A. and Aghdam, M.M. (2012), "Free vibration analysis of moderately thick trapezoidal symmetrically laminated plates with various combinations of boundary conditions", Europ. J. Mech. A/Solids, 36(2012), 204-212. https://doi.org/10.1016/j.euromechsol.2012.03.004.
  104. Zhao, Z., Feng, C., Dong, Y., Wang, Y. and Yang, J. (2019), "Geometrically nonlinear bending of functionally graded nanocomposite trapezoidal plates reinforced with graphene platelets (GPLs)", Int. J. Mech. Mater. Des., 15(4). https://doi.org/10.1007/s10999-019-09442-4.
  105. Zhao, Z., Feng, C., Wang, Y. and Yang, J. (2017), "Bending and vibration analysis of functionally graded trapezoidal nanocomposite plates reinforced with graphene nanoplatelets (GPLs)", Compos. Struct., 180, https://doi.org/10.1016/j.compstruct.2017.08.044.
  106. Zhang, Q., Xu, X., Li, H., Xiong, G., Hu, H. and Fisher, T.S. (2015), "Mechanically robust honeycomb graphene aerogel multifunctional polymer composites", Carbon, 93, 659-670. https://doi.org/10.1016/j.carbon.2015.05.102.
  107. Zhu, C., Han, T.Y.J., Duoss, E.B., Golobic, A.M., Kuntz, J.D. and Spadaccini, C.M. (2015), "Highly compressible 3D periodic graphene aerogel microlattices", Nat. Commun., 6, 1-8. https://doi.org/10.1038/ncomms7962.
  108. Zhu, X.H. and Meng, Z.Y. (1995), "Operational principle fabrication and displacement characteristics of a functionally gradient piezoelectricceramic actuator", Sens. Actuators, 48(3), 169-176. https://doi.org/10.1016/0924-4247(95)00996-5.