Steel and Composite Structures

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Vibrational characteristics of sandwich annular plates with damaged core and FG face sheets

  • Xi, Fei (School of Art and Design, Nanjing University of Finance & Economics)
  • Received : 2021.11.26
  • Accepted : 2022.06.21
  • Published : 2022.07.10
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Abstract

The main goal of this paper is to study the vibration of damaged core laminated annular plates with FG face sheets based on a three-dimensional theory of elasticity. The structures are made of a damaged isotropic core and two external face sheets. These skins are strengthened at the nanoscale level by randomly oriented Carbon nanotubes (CNTs) and are reinforced at the microscale stage by oriented straight fibers. These reinforcing phases are included in a polymer matrix and a three-phase approach based on the Eshelby-Mori-Tanaka scheme and on the Halpin-Tsai approach, which is developed to compute the overall mechanical properties of the composite material. In this study the effect of microcracks on the vibrational characteristic of the sandwich plate is considered. In particular, the structures are made by an isotropic core that undergoes a progressive uniform damage, which is modeled as a decay of the mechanical properties expressed in terms of engineering constants. These defects are uniformly distributed and affect the central layer of the plates independently from the direction, this phenomenon is known as "isotropic damage" and it is fully described by a scalar parameter. Three complicated equations of motion for the sectorial plates under consideration are semi-analytically solved by using 2-D differential quadrature method. Using the 2-D differential quadrature method in the r- and z-directions, allows one to deal with sandwich annular plate with arbitrary thickness distribution of material properties and also to implement the effects of different boundary conditions of the structure efficiently and in an exact manner. The fast rate of convergence and accuracy of the method are investigated through the different solved examples. The sandwich annular plate is assumed to have any arbitrary boundary conditions at the circular edges including simply supported, clamped and, free. Several parametric analyses are carried out to investigate the mechanical behavior of these multi-layered structures depending on the damage features, through-the-thickness distribution, and boundary conditions.

Keywords

Acknowledgement

Natural Science Foundation of Jiangsu, General project of philosophy and Social Sciences in Jiangsu, Social Science Applied Research Project of Jiangsu, Project of Anhui Sentai wood plastic Company and Educational Reform Project of Nanjing University of Finance and Economics

References

  1. 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.
  2. Bacciocchi, M., Tarantino, A.M., (2019), "Time-dependent behavior of viscoelastic three-phase composite plates reinforced by carbon nanotubes", Compos. Struct., 216, 20-31. https://doi.org/10.1016/j.compstruct.2019.02.083.
  3. Bennai, R., Ait Atmane, H. and Tounsi, A. (2015), "A new higherorder 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.
  4. Benson, P.R. and Hinton, E. (1976), "A thick finite strip solution for static, free vibration and stability problems", Int. J. Numer. Methods Eng., 10(3), 665-678. https://doi.org/10.1002/nme.1620100314.
  5. 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.
  6. 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.
  7. Cheung, M.S. and Chan, M.Y.T. (1981), "Static and dynamic analysis of thin and thick sectorial plates by the finite strip method", Comput. Struct., 14(1-2), 79-88. https://doi.org/10.1016/0045-7949(81)90086-9.
  8. Dong, C.Y. (2008), "Three-dimensional free vibration analysis of functionally graded annular plates using the Chebyshev-Ritz method", Mater. Des., 29(8),1518-1525. https://doi.org/10.1016/j.matdes.2008.03.001.
  9. Eshelby, J.D. (1957), "The determination of the elastic field of an ellipsoidal inclusion, and related problems", P. Roy. Soc. Lond. A Mat., 241(1226), 376-396. https://www.jstor.org/stable/100095. 100095
  10. Fidelus, J.D., Wiesel, E., Gojny, F.H., Schulte, K. and Wagner, H.D. (2005), "Thermo-mechanical properties of randomly oriented carbon/epoxy nanocomposites", Compos. Part A, 36(11), 1555-1561. https://doi.org/10.1016/j.compositesa.2005.02.006.
  11. Ghavamian, A., Rahmandoust, M. and Ochsner, A. (2012), 62, "A numerical evaluation of the influence of defects on the elastic modulus of single and multi-walled carbon nanotubes", Comput. Mater. Sci., 62, 110-116. https://doi.org/10.1016/j.commatsci.2012.05.003.
  12. Gojny, F.H., Wichmann, M.H.G., Fiedler, B. and Schulte, K. (2005), "Influence of different carbon nanotubes on the mechanical properties of epoxy matrix composites-A comparative study", Compos. Sci. Technol., 65(15-16), 2300-2313.https://doi.org/10.1016/j.compscitech.2005.04.021.
  13. Guruswamy, P. and Yang, T.Y. (1979), "A sector finite element for dynamic analysis of thick plates", J. Sound Vib., 62(4), (1979) 505-516. https://doi.org/10.1016/0022-460X(79)90459-0.
  14. Halpin, J.C. and Tsai, S.W. (1969), "Effects of environmental factors on composite materials", AFML-TR-67-423; Air Force Research Laboratory.
  15. Hill, R. (1964a), "Theory ofmechanical properties of fibrestrengthened materials Elastic behavior", J. Mech. Phys. Solids, 12(4), 199-212. https://doi.org/10.1016/0022-5096(64)90019-5.
  16. Hill, R. (1964b), "Theory of mechanical properties of fibre"strengthened materials: II. Inelastic behavior", J. Mech. Phys. Solids, 12(4), 213-218. https://doi.org/10.1016/0022-5096(64)90020-1.
  17. 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.
  18. 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.
  19. Kitipornchai, S., Chen, D., Yang, J. (2017), "Free vibration and elastic buckling of functionally graded porous beams reinforced by graphene platelets", Mater. Design, 116, 656-665. https://doi.org/10.1016/j.matdes.2016.12.061.
  20. Koizumi, M. (1993), "The concept of FGM", Ceram. Trans. Funct. Grad. Mater., 34, 3-10.
  21. Lanzoni, L. and Tarantino, A.M. (2014), "Damaged hyperelastic membranes", Int. J. Nonlinear Mech., 60, 9-22. https://doi.org/10.1016/j.ijnonlinmec.2013.12.001.
  22. Lanzoni, L. and Tarantino, A.M. (2015), "Equilibrium configurations and stability of a damaged body under uniaxial tractions", Z. Angew. Math. Phys., 66(1), 171-190. https://doi.org/10.1007/s00033-014-0397-6.
  23. Leissa, A.W., McGee, O.G. and Huang, C.S. (1993), "Vibrations of sectorial plates having corner stress singularities", J. Appl. Mech. Transactions ASME, 60(1), 134-140. https://doi.org/10.1115/1.2900735.
  24. Lemaitre, J., Chaboche, J.L. (1990), Mechanics of Solid Materials, Cambridge University Press, New York, NY, USA.
  25. 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.
  26. 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.
  27. Liu, R., Wang, L. (2015), "Thermal vibration of a single-walled carbon nanotube predicted by semiquantum molecular dynamics", Phys. Chem. Chem. Phys., 17(7), 5194-5201. https://doi.org/10.1039/C4CP05495D.
  28. McGee, O.G., Huang, C.S. and Leissa, A.W. (1995), "Comprehensive exact solutions for free vibrations of thick annular sectorial plates with simply supported radial edges", Int. J. Mech. Sci., 37(5), 537-566. https://doi.org/10.1016/0020-7403(94)00050-T.
  29. Mizusawa, T. (1991), "Vibration of thick annular sector plates using semi-analytical methods", J. Sound Vib., 150(2), 245-259. https://doi.org/10.1016/0022-460X(91)90619-U.
  30. Mori, T., Tanaka, K. (1973), "Average stress in matrix and average elastic energy of materials with misfitting inclusions", Acta Metall., 21, 571-574. https://doi.org/10.1016/0001-6160(73)90064-3.
  31. 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.
  32. 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.
  33. Nie, G.J. and Zhong, Z. (2007), "Semi-analytical solution for three-dimensional vibration of functionally graded circular plates", Comput. Methods Appl. Mech. Eng., 196(49-52), 4901-4910. https://doi.org/10.1016/j.cma.2007.06.028.
  34. Odegard, G.M., Gates, T.S., Wise, K.E., Park, C., 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.
  35. 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.
  36. 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.
  37. Ramakris, R. and Kunukkas, V.X. (1973), "Free vibration of annular sector plates", J. Sound Vib., 30(1), 127-129. https://doi.org/10.1016/0022-460X(83)90546-1.
  38. Rao, M.N., Guruswamy, P. and Sampath Kumaran, K.S. (1977), "Finite element analysis of thick annular and sector plates", Nucl. Eng. Des., 41(2), 247-255. https://doi.org/10.1016/0029- 5493(77)90113-3.
  39. Reddy, J.N. and Miravete, A. (1995), Practical Analysis of Composite Laminates, CRC Press, Boca Raton, FL, USA.
  40. 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.
  41. 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.
  42. 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.
  43. Shi, D.L., Huang, Y.Y., Hwang, K.C., Gao, H., (2004), "The effect of nanotube waviness and agglomeration on the elastic property of carbon nanotube reinforced composites", J. Eng. Mater. T. ASME, 126(3), 250-257. https://doi.org/10.1115/1.1751182.
  44. Shu, C. (2000), Differential Quadrature and its Application in Engineering. Springer, Berlin, Germany.
  45. Sinha, L., Mishra, S.S., Nayak, A.N. and Sahu, S.K. (2020), "Free vibration characteristics of laminated composite stiffened plates: Experimental and numerical investigation", Compos. Struct., 233, 11157. https://doi.org/10.1016/j.compstruct.2019.111557.
  46. Sinha, L., Tripathy, A., Nayak, A.N. and Sahu, S.K. (2021), "Free vibration behavior of angle-ply laminated composite stiffened plates", J. Struct. Stability Dynam., 21(13), 2150187. https://doi.org/10.1142/S021945542150187X.
  47. Sinha, L., Das, D., Nayak, A.N. and Kumar Sahu, S. (2021), "Experimental and numerical study on free vibration characteristics of laminated composite plate with/without cutout", Compos. Struct., 256, 113051. https://doi.org/10.1016/j.compstruct.2020.113051.
  48. Srinivasan, R.S. and Thiruvenkatachari, V. (1983), "Free vibration of annular sector plates by an integral equation technique", J. Sound Vib., 89(3), 425-432. https://doi.org/10.1016/0022-460X(83)90546-1
  49. Srinivasan, R.S. and Thiruvenkatachari, V. (1986), "Free vibration analysis of laminated annular sector plates", J. Sound Vib., 109(1), 89-96. https://doi.org/10.1016/0022-460X(83)90546-1.
  50. Swaminadham, M., Danielski, J. and Mahrenholtz, O. (1984), "Free vibration analysis of annular sector plates by holographic experiments", J. Sound Vib., 95(3), 333-340. https://doi.org/10.1016/0022-460X(83)90546-1.
  51. 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.
  52. 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.
  53. Tahouneh, V., Naei, M.H., Mosavi Mashhadi, M. (2019), "Using IGA and trimming approaches for vibrational analysis of Lshape graphene sheets via nonlocal elasticity theory", Steel Compos. Struct., 33(5), 717-727. https://doi.org/10.12989/scs.2019.33.5.717.
  54. Tahouneh, V., Naei, M.H., 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.
  55. Tarantino, A.M. (2014), "Equilibrium paths of a hyperelastic body under progressive damage", J. Elast., 114, 225-250. https://doi.org/10.1007/s10659-013-9439-0 .
  56. Tornabene, F., Bacciocchi, M., Fantuzzi, N. and Reddy, J.N. (2019), "Multiscale Approach for three-phase cnt/polymer/fiber laminated nanocomposite structures", Polymer Compos., 40(S1), E102-E126. ttps://doi.org/10.1002/pc.24520.
  57. Tornabene, F., Fantuzzi, N., 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.
  58. 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.
  59. Tsai, S.W. (1965). Strength Characteristics of Composite Materials, Philco Corporation, Newport Beach, CA, USA.
  60. Tsai, S.W. (1964) Structural Behavior of Composite Materials, Philco Corporation, Newport Beach, CA, USA.
  61. Wagner, H.D., Lourie, O. and Feldman, Y. (1997), "Stress-induced fragmentation of multiwall carbon nanotubes in a polymer matrix", Appl. Phys. Lett., 72(2), 188-190. https://doi.org/10.1063/1.120680.
  62. 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.
  63. Xiang, Y., Liew, K.M. and Kitipornchai, S. (1993), "Transverse vibration of thick annular sector plates", J. Eng. Mech. ASCE, 119(8), 1579-1599. https://doi.org/10.1061/(ASCE)07339399(1993)119:8(1579).
  64. Xu, W., Wang, L. and Jiang, J. (2016), "Strain gradient finite element analysis on the vibration of double-layered graphene sheets", Int. J. Comput. Meth., 13(3), 1650011. https://doi.org/10.1142/S0219876216500110.
  65. Zhang, Y. and Wang, L. (2018), "Thermally stimulated nonlinear vibration of rectangular single-layered black phosphorus", J. Appl. Phy., 124(13), https://doi.org/10.1063/1.5047584. https://doi.org/10.1063/1.5047584.
  66. Zhou, D., Lo, S.H. and Cheung, Y.K. (2009), "3-D vibration analysis of annular sector plates using the Chebyshev-Ritz method", J. Sound Vib., 320(1-2), 421-437. https://doi.org/10.1016/j.jsv.2008.08.001.
  67. Zhu, X.H. and Meng, Z.Y. (1995), "Operational principle fabrication and displacement characteristics of a functionally gradient piezoelectric ceramic actuator", Sens. Actuators, 48(3), 169-176. https://doi.org/10.1016/0924-4247(95)00996-5.