Free vibration behavior of bi-directional functionally graded plates with porosities using a refined first order shear deformation theory
Document Type : Research Paper
Authors
1 Department of Mechanical Engineering, Rajeev Gandhi Memorial College of Engineering and Technology, Nandyal
2 Department of Mechanical Engineering, Jawaharlal Nehru Technological University, Hyderabad
3 Mechanical Engineering Rajeev Gandhi Memorial College of Engineering & Technology, Nandyal
Abstract
This paper proposes the refined first order shear deformation theory to investigate the free vibration behavior of bidirectional functionally graded porous plates. This theory satisfies the transverse shear stress free conditions at the top and bottom of the plate, thus avoids the need of a shear correction factor. The rule of mixtures is employed to compute the effective material properties and assumed to be graded in both x and z-directions. The equations of motion are derived by means of Lagrange equations to investigate the free vibration response. The displacement functions in axial and transverse directions are expressed in simple algebraic polynomial series form, including admissible functions which are used to fulfill the simply supported boundary conditions. The admissible functions are generated using Pascal’s triangle. The accuracy of the present theory is assessed with the numerical results and is confirmed by comparing with 3-D exact solutions and with other higher order theories. The influence of thickness ratios, aspect ratios, gradation indexes, type of porosity distribution and the volume fraction of porosity on the free vibration behavior of bi-directional FGPs are discussed in detail. The presented numerical results can be used as benchmark solutions to assess the various plate theories and compare with solutions obtained by other analytical and finite element methods. From the present work, it can be concluded that the present theory allows examining the vibration behavior of bidirectional porous FG plates manufactured in the sintering process.
Keywords
[1] Shaw LL., 1998, The crack driving force of functionally graded materials, Journal of Materials Science Letters, doi: https://doi.org/10.1023/A:1006502026364.
[2] Valizadeh N., Natarajan S., GonZalez-Estrada O.A., Rabczuk T., Bui Q.T., Bardas P.A.S., 2013, NURBS-based finite element analysis of functionally graded plates: static bending, vibration, buckling and flutter, Composite Structures, doi: 10.1016/j.compstruct.2012.11.008.
[3] Liu P., Bui T.Q., Zhu D., Yu T.T., Wang J.W., Yin S.H., Hirose S., 2015, Buckling failure analysis of cracked functionally graded plates by a stabilized discrete shear gap extended 3-node triangular plate element, Composites Part B: Engineering, doi: https://doi.org/10.1016/j.compositesb.2015.03.036
[4] Sidda Reddy B.., Vijaya Kumar Reddy K., 2020, Flexural behavior of porous functionally graded plates using a novel higher order theory, Journal of computational applied mechanics, doi:10.22059/JCAMECH.2020.298540.488.
[5] Semsi Coskun, Jinseok Kim, Houssam Toutanji, 2019, Bending, Free Vibration, and Buckling Analysis of Functionally Graded Porous Micro-Plates Using a General Third-Order Plate Theory, Journal of composites science, doi: https://doi.org/10.3390/jcs3010015
[6] Sidda Reddy B.., Vijaya Kumar Reddy K., 2019, Thermomechanical behaviour of Functionally Graded Plates with HSDT,Journal of Computational and Applied Research in Mechanical Engineering (JCARME),doi:10.22061/jcarme.2019.5416.1674
[7] Mohammadi M., Ghayour M., Farajpour A., 2011, Analysis Of Free Vibration Sector Plate Based On Elastic Medium By Using New Version Of Differential Quadrature Method, Journal Of Simulation And Analysis Of Novel Technologies In Mechanical Engineering (Journal Of Solid Mechanics In Engineering) 3(2): 47-56.
[8] Sidda Reddy B., Suresh Kumar JEswara Reddy., C., 2011, Nonlinear bending analysis of functionally graded plates Using higher order theory, International Journal of Engineering Science and Technology (IJEST), 3 (4): 3010-3022.
[9] Sidda Reddy B., Suresh Kumar J., Eswara Reddy C., Vijaya Kumar Reddy K., 2011, Geometrically Nonlinear analysis of functionally graded material plates using higher order theory, International Journal of Engineering, Science and Technology(IJEST), 3 (1): 279-288, 2011.
[10] Sidda Reddy B., Suresh Kumar J., Eswara Reddy C., Vijaya Kumar Reddy K., 2011, Nonlinear thermal analysis of functionally graded plates using higher order theory, Innovative Systems Design and Engineering, 2 (5):1-13.
[11] Sidda Reddy B., Suresh Kumar J., Eswara Reddy C., Vijaya Kumar Reddy K., 2011, Higher order theory for free vibration analysis of functionally graded material plates, ARPN Journal of Engineering and Applied Sciences, 6 (10): 105-111.
[12] Mohammadi M., Farajpour A., Goodarzi M., Mohammadi H., 2013, Temperature Effect on Vibration Analysis of Annular Graphene Sheet Embedded on Visco-Pasternak Foundation, Journal of Solid Mechanics 5(3): 305-323.
[13] Safarabadi M., Mohammadi M., Farajpour A., Goodarzi M., 2015, Effect of Surface Energy on the Vibration Analysis of Rotating Nanobeam, Journal of Solid Mechanics 7(3): 299-311.
[14] Sidda Reddy B., Suresh Kumar J., Vijaya Kumar Reddy K., Buckling analysis of functionally graded material plates using higher order shear deformation theory, Journal of composites, doi: http://dx.doi.org/10.1155/2013/808764
[15] Sidda Reddy B., Suresh Kumar J., Eswara Reddy C., Vijaya Kumar Reddy K., 2014, Static Analysis of Functionally Graded Plates Using Higher-Order Shear Deformation Theory, International Journal of Applied Science and Engineering, 12 (1): 23-41.
[16] Sidda Reddy B., Suresh Kumar J., Eswara Reddy C., Vijaya Kumar Reddy K., 2014, Free Vibration Behaviour of Functionally Graded Plates Using Higher-Order Shear Deformation Theory, Journal of Applied Science and Engineering, doi: 10.6180/jase.2014.17.3.03.
[17] Sidda Reddy B., Suresh Kumar J., Eswara Reddy C., Vijaya Kumar Reddy K., 2014, Static Bending Behavior of Functionally Graded Plates Subjected to Mechanical Loading, Jordan Journal of Mechanical and Industrial Engineering, 8 (4): 169 -176.
[18] Sidda Reddy B., Suresh Kumar J., Eswara Reddy C., Vijaya Kumar Reddy K., 2015, Buckling analysis of functionally graded plates using higher order theory with thickness stretching effect, International journal of applied science and Engineering, 13 (1):19-35.
[19] Baghani M., Mohammadi M., Farajpour A., 2016, Dynamic and Stability Analysis of the Rotating Nanobeam in a Nonuniform Magnetic Field Considering the Surface Energy, International Journal of Applied Mechanics, doi:10.1142/S1758825116500484.
[20] Goodarzi M., Mohammadi M. , Khooran M., Saadi F., 2016, Thermo-Mechanical Vibration Analysis of FG Circular and Annular Nanoplate Based on the Visco-Pasternak Foundation, Journal of Solid Mechanics 8(4): 788-805.
[21] Rezaei A.S., Saidi A.R., 2015, Exact solution for free vibration of thick rectangular plates made of porous materials, Composite Structures, doi: https://doi.org/10.1016/j.compstruct.2015.08.125.
[22] Shafiei N., Mirjavadi S.S., MohaselAfshari B., Rabby S., Kazemi M., Vibration of two-dimensional imperfect functionally graded (2D-FG) porous nano-/micro-beams, Computer Methods in Applied Mechanics and Engineering, doi: https://doi.org/10.1016/j.cma.2017.05.007.
[23] Zenkour A. M., 2018, A quasi-3D refined theory for functionally graded single-layered and sandwich plates with porosities, Composite structures, doi: https://doi.org/10.1016/j.compstruct.2018.05.147
[24] Barati M.R., 2018, Vibration analysis of porous FG nanoshells with even and uneven porosity distributions using nonlocal strain gradient elasticity, Acta Mechanica, doi:https://doi.org/10.1007/s00707-017-2032-z.
[25] Jing Zhao., Kwangnam Choe., Fei Xie, Ailun Wang, Cijun Shuai and Qingshan Wang, 2018, Three-dimensional exact solution for vibration analysis of thick functionally graded porous (FGP) rectangular plates with arbitrary boundary conditions, Composites Part B: Engineering, doi: https://doi.org/10.1016/j.compositesb.2018.09.001.
[26] Barati M.R., Shahverdi H., Nonlinear vibration of nonlocal four-variable graded plates with porosities implementing homotopy perturbation and Hamil-tonian methods, Acta Mechanica. 229 (2018) 343–362. doi:https://doi.org/10.1007/s00707-017-1952-y
[27] Pinar Aydan DEMIRHAN, Vedat TASKIN, 2019,Bending and free vibration analysis of Levy-type porous functionally graded plate using state space approach, Composite Part B: Engineering160, 661-676. doi: https://doi.org/10.1016/j.compositesb.2018.12.020
[28] Jinseok Kim, Krzysztof KamilŻur, Reddy J.N, 2019, Bending, free vibration, and buckling of modified couples stress-based functionally graded porous micro-plates, Composite structures, 209, 879-888. doi:https://doi.org/10.1016/j.compstruct.2018.11.023
[29] Chen D., Yang J., Kitipornchai S, 2019, Buckling and bending analyses of a novel functionally graded porous plate using Chebyshev-Ritz method, Archieves of Civil and Mechanical Engineering, doi: https://doi.org/10.1016/j.acme.2018.09.004
[30] Ahmed Amine Daikh, Zenkour A. M, 2019, Effect of porosity on the bending analysis of various functionally graded sandwich plates, Materials research Express, doi:https://doi.org/10.1088/2053-1591/ab0971.
[31] Slimane Merdaci, Hakima Belghoul, 2019, High-order shear theory for static analysis of functionally graded plates with porosities, C. R. Mécanique, doi: https://doi.org/10.1016/j.crme.2019.01.001
[32] Amir Farzam, Behrooz Hassani, 2019, Isogeometric analysis of in-plane functionally graded porous microplates using modified couple stress theory, Aerospace Science and Technology, doi: https://doi.org/10.1016/j.ast.2019.05.012
[33] Zamani Nejad M., Rastgoo A., Hadi A., 2014, Effect of Exponentially-Varying Properties on Displacements and Stresses in Pressurized Functionally Graded Thick Spherical Shells with Using Iterative Technique, Journal of Solid Mechanics, 6 (4): 366-377.
[34] Mohammad Zamani Nejad, Abbas Rastgoo, Amin Hadi, 2014, Exact elasto-plastic analysis of rotating disks made of functionally graded materials, International Journal of Engineering Science, doi: http://dx.doi.org/10.1016/j.ijengsci.2014.07.009
[35] Mohammad Hosseini, Mohammad Shishesaz, Khosro Naderan Tahan, Amin Hadi, 2016, Stress analysis of rotating nano-disks of variable thickness made of functionally graded materials, International Journal of Engineering Science, doi: http://dx.doi.org/10.1016/j.ijengsci.2016.09.002
[36] Mohammad Shishesaz, Mohammad Hosseini, Khosro Naderan Tahan, Amin Hadi, 2017, Analysis of functionally graded nanodisks under thermoelastic loading based on the strain gradient theory, Acta Mech, doi: 10.1007/s00707-017-1939-8
[37] Mohammad Hosseini, Mohammad Shishesaz, Amin Hadi, 2019, Thermoelastic analysis of rotating functionally graded micro/nanodisks of variable thickness, Thin-Walled Structures, doi: https://doi.org/10.1016/j.tws.2018.10.030
[38] Zeinab Mazarei and Mohammad Zamani Nejad, 2016, Thermo-Elasto-Plastic Analysis of Thick-Walled Spherical Pressure Vessels Made of Functionally Graded Materials, International Journal of Applied Mechanics, doi: 10.1142/S175882511650054X
[39] Mahboobeh Gharibi, Mohammad Zamani Nejad , Amin Hadi, 2017, Elastic analysis of functionally graded rotating thick cylindrical pressure vessels with exponentially-varying properties using power series method of Frobenius, JCAMECH, doi: 10.22059/jcamech.2017.233633.143
[40] Mohammad Zamani Nejad, Mehdi Jabbari and Amin Hadi, 2017, A review of functionally graded thick cylindrical and conical shells, JCAMECH, doi: 10.22059/JCAMECH.2017.247963.220
[41] Amin Hadi, Abbas Rastgoo, Nooshin Haghighipour and Azam Bolhassani, 2018, Numerical modelling of a spheroid living cell membrane under hydrostatic pressure, Journal of Statistical Mechanics: Theory and Experiment, doi: https://doi.org/10.1088/1742-5468/aad369
[42] Nejad M Z., Alamzadeh N., Hadi A., 2018, Thermoelastoplastic analysis of FGM rotating thick cylindrical pressure vessels in linear elastic-fully plastic condition, Composites Part B, doi: 10.1016/j.compositesb.2018.09.022.
[43] Behrouz Karami, Davood Shahsavari , Maziar Janghorban , Li Li, 2019, On the resonance of functionally graded nanoplates using bi-Helmholtz nonlocal strain gradient theory, International Journal of Engineering Science, doi: https://doi.org/10.1016/j.ijengsci.2019.103143
[44] Esmail Zarezadeh, Vahid Hosseini & Amin Hadi, 2019, Torsional vibration of functionally graded nano-rod under magnetic field supported by a generalized torsional foundation based on nonlocal elasticity theory, Mechanics Based Design of Structures and Machines, doi: 10.1080/15397734.2019.1642766
[45] Nemat-Alla M., 2003, Reduction of thermal stresses by developing two-dimensional functionally graded materials, Internal journal of solids and structures, doi: https://doi.org/10.1016/j.ijsolstr.2003.08.017
[46] Nemat-Alla M., 2009, Reduction of thermal stresses by composition optimization of twodimensional functionally graded materials, Acta Mechanica, doi: https://doi.org/10.1007/s00707-008-0136-1
[47] Asgari M., Akhlaghi M., 2011, Natural frequency analysis of 2D-FGM thick hollow cylinder based on three-dimensional elasticity equations, European journal of Mechanics- A/Solids, doi: https://doi.org/10.1016/j.euromechsol.2010.10.002.
[48] Ebrahimi M.J., Najafizadeh M.M., 2014, Free vibration analysis of two-dimensional functionally graded cylindrical shells, Applied mathematical modelling, doi: https://doi.org/10.1016/j.apm.2013.06.015
[49] Li L., Li X., Hu Y., 2017, Nonlinear bending of a two-dimensionally functionally graded beam, Composite Structures, doi: https://doi.org/10.1016/j.compstruct.2017.10.087
[50] Simsek M., 2015, Bi-directional functionally graded materials (BDFGMs) for free and forced vibration of Timoshenko beams with various boundary conditions, Composite Structures, doi: https://doi.org/10.1016/j.compstruct.2015.08.021.
[51] Nejad M.Z., Hadi A., 2016, Non-local analysis of free vibration of bi-directional functionally graded Euler-Bernoulli nano-beams, International journal of Engineering Science, doi:
https://doi.org/10.1016/j.ijengsci.2016.04.011.
[52] Nguyen D.K., Nguyen Q.H., Tran T.T., Bui V.T., 2017, Vibration of bi-dimensional functionally graded Timoshenko beams excited by a moving load, Acta Mechanica, doi: https://doi.org/10.1007/s00707-016-1705-3
[53] Armağan Karamanlı, 2018, Free vibration analysis of two directional functionally graded beams using a third order shear deformation theory, Composite Structures doi: https://doi.org/10.1016/j.compstruct.2018.01.060.
[54] Ye Tang, Qian Ding, 2019, Nonlinear vibration analysis of a bi-directional functionally graded beam under hygro-thermal loads, Composite Structures, doi: https://doi.org/10.1016/j.compstruct.2019.111076
[55] Simsek M., 2016, Buckling of Timoshenko beams composed of two-dimensional functionally graded materials (2D-FGM) having different boundary conditions, Composite Structures, doi: https://doi.org/10.1016/j.compstruct.2016.04.034
[56] Nejad M.Z., Hadi A., Rastgoo A., 2016, Buckling analysis of arbitrary two-directional functionally graded Euler-Bernoulli nano-beams based on nonlocal elasticity theory, International Journal of Engineering science, doi: https://doi.org/10.1016/j.ijengsci.2016.03.001
[57] Karamanlı, A., 2017, Bending behaviour of two directional functionally graded sandwich beams by using a quasi-3d shear deformation theory, Composite Structures, doi: http://dx.doi.org/10.1016/j.compstruct.2017.04.046
[58] Lezgy-Nazargah M., 2015, Fully coupled thermo-mechanical analysis of bi-directional FGM beams using NURBS isogeometric finite element approach, Aerospace Science and Technology, doi: https://doi.org/10.1016/j.ast.2015.05.006
[59] Apalak MK., Demirbas MD., 2016, Thermal stress analysis of in-plane two-directional functionally graded plates subjected to in-plane edge heat fluxes, Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications, doi: 10.1177/1464420716643857.
[60] Thom Van Doa, Dinh Kien Nguyenb, Nguyen Dinh Ducc,d, Duc Hong Doanc,, 2017, Tinh Quoc Bui, Analysis of bi-directional functionally graded plates by FEM and a new third-order shear deformation plate theory, Thin-Walled Structures, doi: https://doi.org/10.1016/j.tws.2017.07.022
[61] Lieu, Q.X., Lee, S., Kang, J., Lee, J., 2018, Bending and free vibration analyses of in-plane bidirectional functionally graded plates with variable thickness using isogeometric analysis, Composite Structures, doi: https://doi.org/10.1016/j.compstruct.2018.03.021
[62] Qui X. Lieu, Dongkyu Lee, Joowon Kang & Jaehong Lee 2018, NURBSbased modeling and analysis for free vibration and buckling problems of in-plane bi-directional functionally graded plates, Mechanics of Advanced Materials and Structures, doi: https://doi.org/10.1080/15376494.2018.1430273.
[63] Chen M., Jin G., Ma X., Zhang Y., Ye T., Liu Z., 2018, Vibration analysis for sector cylindrical shells with bi-directional functionally graded materials and elastically restrained edges, Composites Part B:Engineering, doi: 10.1016/j.compositesb.2018.08.129.
[64] Esmaeilzadeh M., Kadkhodayan M., 2019, Dynamic analysis of stiffened bi-directional functionally graded plates with porositites under a moving load by dynamic relaxation method with kinematic damping, Aerospace Science and Technology, doi: https://doi.org/10.1016/j.ast.2019.105333.
[65] Abbas Barati, Amin Hadi, Mohammad Zamani Nejad & Reza Noroozi, 2020, On vibration of bi-directional functionally graded nanobeams under magnetic field, Mechanics Based Design of Structures and Machines, doi: 10.1080/15397734.2020.1719507
[66] Abbas Barati, Mohsen Mahdavi Adeli, Amin Hadi, 2020, Static Torsion of Bi-Directional Functionally Graded Microtube Based on the Couple Stress Theory Under Magnetic Field, International Journal of Applied Mechanics, doi: 10.1142/S1758825120500210
[67] Reza Noroozi, Abbas Barati, Amin Kazemi, Saeed Norouzi and Amin Hadi, 2020, Torsional vibration analysis of bi-directional FG nano-cone with arbitrary cross-section based on nonlocal strain gradient elasticity, Advances in Nano Research, doi: http://dx.doi.org/10.12989/anr.2020.8.1.013
[68] Zhu J., Lai Z., Yin Z., Jeon J., Lee S., 2001, Fabrication of ZrO2–NiCr functionally graded material by powder metallurgy, Materials Chemistry and Physics , doi: https://doi.org/10.1016/S0254-0584(00)00355-2
[69] Timoshenko S., Woinowsky-Krieger S., 1959, Theory of platesandshells. 2nd edition. Singapore: McGraw-Hill..
[70] Vel SS., Batra RC., 2004, Three-dimensional exact solution for the vibration of functionally graded rectangular plates, Journal of sound and vibration, doi:https://doi.org/10.1016/S0022-460X(03)00412-7
[71] Hosseini-Hashemi Sh., Fadaee M., Atashipour S.R., 2011, Study on the free vibration of thick functionally graded rectangular plates according to a new exact closed-form procedure, Composite Structures, doi:10.1016/j.compstruct.2010.08.007
[72] Nguyen Van Long., Tran Huu Quoc., Tran Minh Tu., 2016, Bending and free vibration analysis of functionally graded plate using new eight-unknown shear deformation theory by finite-element method, International journal of Advanced Structural Engineering, doi: 10.1007/s40091-016-0140-y
[73] Wang WH., 2011, The elastic properties, elastic models and elastic perspectives of metallic glasses, Progress in Material Science, doi:10.1016/j.pmatsci.2011.07.001.
)
Volume 51, Issue 2
December 2020Pages 374-388
- Receive Date: 18 May 2020
- Revise Date: 30 June 2020
- Accept Date: 01 July 2020