Engineering Transactions, 68, 1, pp. 21–45, 2020

Three-Dimensional Analysis of Laminated Plates with Functionally Graded Layers by Two-Dimensional Numerical Model

Cracow University of Technology

Cracow University of Technology

This work presents a three-dimensional (3D) numerical analysis of multi-layered laminated plates in which selected layers may be made of functionally graded material (FGM), in which the Young’s modulus may change along the thickness as a consequence of a continuous and graded mixture of two materials. For the analysis, the method, known as FEM23, is applied, which uses a two-dimensional (2D) mesh, yet enables obtaining full 3D results for the layered structure. In FEM23, the layered structure may be a combination of thin and thick layers made of materials with significantly different properties. This paper presents two examples comparing the results to other numerical or analytical solutions. The examples confirm the correctness and flexibility of FEM23 for laminated plates with functionally graded layers.
Keywords: functionally graded material; laminated plates; postprocessing; Finite Element Method; bending plate
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Byfut A., Schröder A., Unsymmetric multi-level hanging nodes and anisotropic polynomial degrees in h1-conforming higher-order finite element methods, Computers & Mathematics with Applications, 73(9): 2092–2150, 2017, doi: /10.1016/j.camwa.2017.02.029.

Carrera E., c0 Reissner-Mindlin multilayered plate elements including zig-zag and interlaminar stress continuity, International Journal for Numerical Methods in Engineering, 39(11): 1797–1820, 1996, doi: 10.1002/(SICI)1097-0207(19960615)39:11<1797::AIDNME928>3.0.CO;2-W.

Carrera E., Historical review of Zig-Zag theories for multilayered plates and shells, ASME Applied Mechanics Reviews, 56(3): 287–308, 2003, doi: 10.1115/1.1557614.

Carrera E., Cinefra M., Li G., Refined finite element solutions for anisotropic laminated plates, Composite Structures, 183: 63–76, 2018, doi: 10.1016/j.compstruct.2017.01.014.

Carrera E., Kröplin B., Zigzag and interlaminar equilibria effects in large-deflection and postbuckling analysis of multilayered plates, Mechanics of Composite Materials and Structures, 4(1): 69–94, 1997, doi: 10.1080/10759419708945875.

Di Sciuva M., An improved shear-deformation theory for moderately thick multilayered anisotropic shells and plates, ASME Journal of Applied Mechanics, 54(3): 589–596, 1987, doi: 10.1115/1.3173074.

Fallah N., Delzendeh M., Free vibration analysis of laminated composite plates using meshless finite volume method, Engineering Analysis with Boundary Elements, 88: 132–144, 2018, doi: 10.1016/j.enganabound.2017.12.011.

Ferreira A.J.M., Analysis of composite plates using a layerwise theory and multiquadrics discretization, Mechanics of Advanced Materials and Structures, 12(2): 99–112, 2005, doi: 10.1080/15376490490493952.

Ferreira A., Carrera E., Cinefra M., Roque C., Polit O., Analysis of laminated shells by a sinusoidal shear deformation theory and radial basis functions collocation, accounting for through-the-thickness deformations, Composites Part B: Engineering, 42(5): 1276–1284, 2011, doi: 10.1016/j.compositesb.2011.01.031.

Filippi M., Carrera E., Zenkour A., Static analyses of fgm beams by various theories and finite elements, Composites Part B: Engineering, 72: 1–9, 2015, doi: 10.1016/j.compositesb.2014.12.004.

Golubovic A., Demirdžic I., Muzaferija S., Finite volume analysis of laminated composite plates, International Journal for Numerical Methods in Engineering, 109(11): 1607–1620, 2017, doi: 10.1002/nme.5347.

Jaskowiec J., Plucinski P., Three-dimensional modelling of heat conduction in laminated plate by two-dimensional numerical model, Composite Structures, 171: 562–575, 2017, doi: 10.1016/j.compstruct.2017.03.046.

Jaskowiec J., Plucinski P., Stankiewicz A., Cichon C., Three-dimensional modelling of laminated glass bending on two-dimensional in-plane mesh, Composites Part B: Engineering, 120: 63–82, 2017, doi: 10.1016/j.compositesb.2017.03.008.

Jha D., Kant T., Singh R., A critical review of recent research on functionally graded plates, Composite Structures, 96: 833–849, 2013, doi: 10.1016/j.compstruct.2012.09.001.

Kulikov G.M., Plotnikova S.V., Mamontov A.A., Sampling surfaces formulation for thermoelastic analysis of laminated functionally graded shells, Meccanica, 51(8): 1913–1929, 2016, doi: /10.1007/s11012-015-0347-1.

Kulikov G.M., Plotnikova S.V., Three-dimensional exact analysis of functionally graded laminated composite plates, [in:] Altenbach H., Mikhasev G.I. [Eds], Shell and Membrane Theories in Mechanics and Biology: From Macro- to Nanoscale Structures, pp. 223–241, Springer International Publishing, Cham, 2015, doi: 10.1007/978-3-319- 02535-3_13.

Kulikov G.M., Plotnikova S.V., On the use of sampling surfaces method for solution of 3D elasticity problems for thick shells, ZAMM – Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 92(11–12): 910–920, 2012, doi: 10.1002/zamm.201200028.

Kulikov G.M., Plotnikova S.V., Heat conduction analysis of laminated shells by a sampling surfaces method, Mechanics Research Communications, 55: 59–65, 2014, doi: 10.1016/j.mechrescom.2013.10.018.

Kulikov G.M., Plotnikova S.V., Strong sampling surfaces formulation for layered shells, International Journal of Solids and Structures, 121: 75–85, 2017, doi: 10.1016/j.ijsolstr.2017.05.017.

Liu B., Ferreira A.J.M., Xing Y.F., Neves A.M.A., Analysis of functionally graded sandwich and laminated shells using a layerwise theory and a differential quadrature finite element method, Composite Structures, 136: 546–553, 2016, doi: 10.1016/j.compstruct.2015.10.044.

Naebe M., Shirvanimoghaddam K., Functionally graded materials: A review of fabrication and properties, Applied Materials Today, 5: 223–245, 2016, doi: 10.1016/j.apmt.2016.10.001.

Nguyen T.-K., Nguyen V.-H., Chau-Dinh T., Vo T.P., Nguyen-Xuan H., Static and vibration analysis of isotropic and functionally graded sandwich plates using an edge-based MITC3 finite elements, Composites Part B, 107: 162–173, 2016, doi: 10.1016/j.compositesb.2016.09.058.

Pandey S., Pradyumna S., Free vibration of functionally graded sandwich plates in thermal environment using a layerwise theory, European Journal of Mechanics – A/Solids, 51: 55–66, 2015, doi: 10.1016/j.euromechsol.2014.12.001.

Pandey S., Pradyumna S., A layerwise finite element formulation for free vibration analysis of functionally graded sandwich shells, Composite Structures, 133: 438–450, 2015, doi: 10.1016/j.compstruct.2015.07.087.

Pandey S., Pradyumna S., Transient stress analysis of sandwich plate and shell panels with functionally graded material core under thermal shock, Journal of Thermal Stresses, 41(5): 543–567, 2018, doi: 10.1080/01495739.2017.1422999.

Rachowicz W., Pardo D., Demkowicz L., Fully automatic hp-adaptivity in three dimensions, Computer Methods in Applied Mechanics and Engineering, 195(37): 4816–4842, 2006, doi: 10.1016/j.cma.2005.08.022.

Rachowicz W., Zdunek A., An h-adaptive mortar finite element method for finite deformation contact with higher order p extension, Computers & Mathematics with Applications, 73(8): 1834–1854, 2017, doi: 10.1016/j.camwa.2017.02.022.

Reddy J.N., Mechanics of laminated composite plates and shells: theory and analysis, 2 nd Ed., Taylor & Francis, 2004.

Ren J., A new theory of laminated plate, Composites Science and Technology, 26(3): 225–239, 1986, doi: 10.1016/0266-3538(86)90087-4.

Swaminathan K., Naveenkumar D.T., Zenkour A.M., Carrera E., Stress, vibration and buckling analyses of FGM plates – a state-of-the-art review, Composite Structure, 120: 10–31, 2015, doi: 10.1016/j.compstruct.2014.09.070.

Swaminathan K., Sangeetha D.M., Thermal analysis of FGM plates – A critical review of various modeling techniques and solution methods, Composite Structures, 160: 43–60, 2017, doi: 10.1016/j.compstruct.2016.10.047.

Szabó B.A., Sahrmann G.J., Hierarchic plate and shell models based on p-extension, International Journal for Numerical Methods in Engineering, 26(8): 1855–1881, 1988, doi: 10.1002/nme.1620260812.

Taylor R.L., Auricchio F., Linked interpolation for Reissner-Mindlin plate elements: Part II – A simple triangle, International Journal for Numerical Methods in Engineering, 36(18): 3057–3066, 1993, doi: 10.1002/nme.1620361803.

Zappino E., Carrera E., Refined One-Dimensional Models for the Multi-Field Analysis of Layered Smart Structures, pp. 343–366, Springer Singapore, Singapore, 2018.

Zboinski G., Application of the three-dimensional triangular-prism hpq adaptive finite element to plate and shell analysis, Computers & Structures, 65(4): 497–514, 1997, doi: 10.1016/S0045-7949(96)00415-4.

Zboinski G., Hierarchical modeling and finite element approximation for adaptive analysis of complex structures, DSc Thesis, Gdansk (Poland), 2001.

Zboinski G., 3D-based hp-adaptive first order shell finite element for modelling and analysis of complex structures – Part 2: Application to structural analysis, International Journal for Numerical Methods in Engineering, 70(13): 1546–1580, 2007, doi: 10.1002/nme.1919.

Zboinski G., Adaptive hpq finite element methods for the analysis of 3D-based models of complex structures. Part 1. Hierarchical modeling and approximations estimation, Computer Methods in Applied Mechanics and Engineering, 1999(45–48): 2913–2940, 2010, doi: 10.1016/j.cma.2010.06.003.

Zboinski G., Adaptive hpq finite element methods for the analysis of 3D-based models of complex structures. Part 2. A posteriori error estimation, Computer Methods in Applied Mechanics and Engineering, 267: 531–565, 2013, doi: 10.1016/j.cma.2013.08.018.

Zboinski G., Jasinski M., 3D-based hp-adaptive first-order shell finite element for modelling and analysis of complex structures – Part 1: The model and the approximation, International Journal for Numerical Methods in Engineering, 70(13): 1513–1545, 2007, doi: 10.1002/nme.1920.

Zienkiewicz O.C., Zhu J.Z., The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique, International Journal for Numerical Methods in Engineering, 33(7): 1331–1364, 1992, doi: 10.1002/nme.1620330702.

Zienkiewicz O.C., Zhu J.Z., The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity, International Journal for Numerical Methods in Engineering, 33(7): 1365–1382, 1992, doi: 10.1002/nme.1620330703.

DOI: 10.24423/EngTrans.1063.20200102