Steel and Composite Structures

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Free vibration analysis of a laminated trapezoidal plate with GrF-PMC core and wavy CNT-reinforced face sheets

  • Yingqun Zhang (School of Architecture and Engineering, Weifang Engineering Vocational College) ;
  • Qian Zhao (School of Architecture and Engineering, Weifang Engineering Vocational College) ;
  • Qi Han (School of Architecture and Engineering, Weifang Engineering Vocational College) ;
  • N. Bohlooli (School of Civil Engineering, Urmia University)
  • Received : 2022.11.30
  • Accepted : 2023.07.26
  • Published : 2023.08.10
⟨ Previous Next ⟩

Abstract

This paper has focused on presenting vibration analysis of trapezoidal sandwich plates with 3D-graphene foam reinforced polymer matrix composites (GrF-PMC) core and FG wavy CNT-reinforced face sheets. The porous graphene foam possessing 3D scaffold structures has been introduced into polymers for enhancing the overall stiffness of the composite structure. Also, 3D graphene foams can distribute uniformly or non-uniformly in the plate thickness direction. The effective Young's modulus, mass density and Poisson's ratio are predicted by the rule of mixture. In this study, 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 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. It is explicated that 3D-GrF skeleton type and weight fraction, carbon nanotubes (CNTs) waviness and CNT aspect ratio can significantly affect the vibrational behavior of the sandwich structure. The plate's normalized natural frequency decreased and the straight carbon nanotube (w=0) reached the highest frequency by increasing the values of the waviness index (w).

Keywords

Acknowledgement

Supported by Key Project of Vocational Education Teaching Reform Research in Shandong Province: Research and Practice on the Segmented Training Model of Building Engineering Technology Majors Based on the "Post Course Competition Certificate"(NO.2022050).

References

  1. 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.
  2. 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.
  3. 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.
  4. 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.
  5. 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
  6. 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.
  7. 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.
  8. 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.
  9. Bi, H., Xie, X., Yin, K., Zhou, Y., Wan, S. and 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.
  10. 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.
  11. 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.
  12. 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.
  13. 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.
  14. 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.
  15. Carman G.P. and Reifsnider K.L. (1992), "Micromechanics of short-fibre composites", Compos. Sci. Technol., 43, 137-146. https://doi.org/10.1016/0266-3538(92)90004-M
  16. 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.
  17. 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.
  18. 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
  19. Cox, H.L. (1952), "The elasticity and strength of paper and other fibrous materials", Br. J. Appl. Phys., 3, 72-79. https://doi.org/10.1088/0508-3443/3/3/302
  20. 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.
  21. 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.
  22. 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
  23. 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.
  24. 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.
  25. 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.
  26. 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.
  27. 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.
  28. 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.
  29. Gurses, M., Civalek, O., Ersoy, H. and 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.
  30. 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
  31. Halpin, J.C. and Tsai, S.W. (1969), "Effects of environmental factors on composite materials", AFML-TR-67-423.
  32. 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
  33. 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.
  34. 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.
  35. 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.
  36. Kapidzic, Z. (2013), "Strength analysis and modeling of hybrid composite-aluminum aircraft structures", Linkoping Studies in Science and Technology, Licentiate Thesis No. 1590.characterization of the porous GDLs for PEMFC based on fibers orientation distribution", Fuel Cells, 18(2), https://doi.org/10.1002/fuce.201700239.
  37. 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.
  38. 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.
  39. 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.
  40. 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.
  41. Koizumi, M. (1993), "The concept of FGM", Ceram. Trans. Funct. Grad. Mater., 34, 3-10.
  42. 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.
  43. 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.
  44. 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
  45. 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.
  46. 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.
  47. 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.
  48. Marin, M. and Florea, O. (2014), "On temporal behaviour of solutions in thermoelasticity of porous micropolar bodies", St. Univ. Ovidius Constanta, 22(1), 169-188. https://doi.org/10.2478/auom-2014-0014
  49. Marin, M. (1994), "The Lagrange identity method in thermoelasticity of bodies with microstructure', Int. J. Eng. Sci., 32(8), 1229-1240. https://doi.org/10.1016/0020-7225(94)90034-5.
  50. Marin, M. and Nicaise, S. (2016), "Existence and stability results for thermoelastic dipolar bodies with double porosity", Continuum Mech. Thermodyn., 28(6), 1645-1657. https://doi.org/10.1007/s00161-016-0503-4
  51. Marin, M., Ellahi, R. and Chirila, A. (2017), "On solutions of Saint-Venant's problem for elastic dipolar bodies with voids", Carpathian J. Mathem., 33(2), 219-232. https://doi.org/10.37193/CJM.2017.02.09
  52. Marin, M., Vlase, S., Ellahi, R. and Bhatti, M.M. (2019), "On the partition of energies for the backward in time problem of thermoelastic materials with a dipolar structure", Symmetry, Basel, 11(7), 1-16. https://doi.org/10.3390/sym11070863
  53. 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.ompscitech.2011.04.002.
  54. 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
  55. 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
  56. 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.
  57. Ng, C.H.W., Zhao, Y.B., Xiang, Y. and Wei, G.W. (2009), "On the accuracy and stability of a variety of differential quadrature formulation for the vibration analysis of beams", Int. J. Eng. Appl. Sci., 1.
  58. 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
  59. 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.
  60. 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.
  61. 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.
  62. 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.
  63. 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
  64. 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.
  65. Reddy, J.N. (2013), "An Introduction to Continuum Mechanics", Cambridge University Press.
  66. 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.
  67. 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.
  68. 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.
  69. 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.
  70. 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.
  71. 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.
  72. Sharma, K. and Marin, M. (2013), "Effect of distinct conductive an thermodynamic temperatures on the reflection of plane waves in micropolar elastic half-space", Sci. Bull., Series A Appl. Mathem. Phys., 75(2), 121-132.
  73. 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.
  74. 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.
  75. 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.
  76. 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.
  77. 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. https://doi.org/10.24200/sci.2016.3830
  78. Shu C., 2012, Differential Quadrature and its Application in Engineering, Springer Science & Business Media.
  79. 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.
  80. Strek, W., Tomala, R., Lukaszewicz, M., Cichy, B., Gerasymchuk, Y. and Gluchowski, P. (2017), "Laser induced white lighting of graphene foam", Sci. Rep., 7.
  81. 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.
  82. 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.
  83. 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.
  84. 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.
  85. 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.
  86. 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.
  87. 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.
  88. 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.
  89. Tucker C.L. and Liang E. (1999), "Stiffness predictions for unidirectional short-fiber composites: Review and evaluation", Compos. Sc.i Technol., 59, 655-671. https://doi.org/10.1016/S0266-3538(98)00120-1
  90. 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.
  91. 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.
  92. 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.
  93. 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.
  94. 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
  95. 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.
  96. 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. https://doi.org/10.1021/nn101187z
  97. 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.
  98. 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.
  99. 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.
  100. 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.
  101. 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.
  102. 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.
  103. 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.