Influence of porosity on thermal environment of functionally graded square (FGSP) and rectangular plates (FGRP)
Document Type : Research Paper
Authors
1 Civil Engineering and Public Works Department, Faculty of Technology, Djillali Liabes University, Sidi Bel Abbes 22000, Algeria
2 Department of Civil Engineering, Faculty of Science and Technology, University of Rélizane, Algeria
3 Department of Mechanical Engineering, Faculty of Engineering and Technology, Nile Valley University, Atbara, Sudan
Abstract
This study examines the influence of porosity on the thermo-mechanical environment of functionally graded square (FGSP) and rectangular plates (FGRP). The current theory suggests that only four unknown functions are involved, compared to five in other shear deformation theories, and the boundary conditions on the upper and lower surfaces of the plate do not require shear correction factors. It is assumed that the material properties of this plate (FGSP) and (FGRP) vary continuously over the thickness of the plate according to a power law function in terms of the volume fractions of the constituents. The porosity distribution of the plates (FGSP) and (FGRP) is uniform over their cross-sections. Using the concept of virtual work, the equilibrium equations of a plate (FGSP) and (FGRP) are derived. Numerical results for the rectangular plates have been provided and compared to those found in the literature. The impact of aspect ratios and porosity volume on the bending and thermo-mechanical environment properties of the square (FGSP) and rectangular plates (FGRP) is examined.
Keywords
- Functionally Graded(FG)
- thermo-mechanical environment
- square plates (SP)
- rectangular plates (RP)
- porosity
Main Subjects
[1] M. Koizumi, FGM activities in Japan, Composites Part B: Engineering, Vol. 28, No. 1-2, pp. 1-4, 1997/01//, 1997. en
[2] Ş. D. Akbaş, Wave propagation of a functionally graded beam in thermal environments, Steel and Composite Structures, Vol. 19, No. 6, pp. 1421-1447, 2015.
[3] S. S. Vel, R. Batra, Three-dimensional exact solution for the vibration of functionally graded rectangular plates, Journal of Sound and Vibration, Vol. 272, No. 3-5, pp. 703-730, 2004.
[4] R. Bennai, H. A. Atmane, A. Tounsi, A new higher-order shear and normal deformation theory for functionally graded sandwich beams, Steel and Composite Structures, Vol. 19, No. 3, pp. 521-546, 2015.
[5] M. Arefi, Elastic solution of a curved beam made of functionally graded materials with different cross sections, Steel and Composite Structures, Vol. 18, No. 3, pp. 659-672, 2015.
[6] F. Ebrahimi, S. Dashti, Free vibration analysis of a rotating non-uniform functionally graded beam, Steel and Composite Structures, Vol. 19, No. 5, pp. 1279-1298, 2015.
[7] K. Darılmaz, Vibration analysis of functionally graded material (FGM) grid systems, Steel and Composite Structures, Vol. 18, No. 2, pp. 395-408, 2015.
[8] V. R. Kar, S. K. Panda, Nonlinear thermomechanical deformation behaviour of P-FGM shallow spherical shell panel, Chinese Journal of Aeronautics, Vol. 29, No. 1, pp. 173-183, 2016, 2016.
[9] A. Hadj Mostefa, S. Merdaci, N. Mahmoudi, An overview of functionally graded materials «FGM», in Proceeding of, Springer, pp. 267-278.
[10] T.-H. Trinh1a, D.-K. Nguyen2b, B. S. Gan, S. Alexandrov3c, Post-buckling responses of elastoplastic FGM beams on nonlinear elastic foundation, Structural Engineering and Mechanics, Vol. 58, No. 3, pp. 515-532, 2016.
[11] H. A. Atmane, A. Tounsi, F. Bernard, S. Mahmoud, A computational shear displacement model for vibrational analysis of functionally graded beams with porosities, Steel Compos. Struct, Vol. 19, No. 2, pp. 369-384, 2015.
[12] A. Ferreira, R. Batra, C. Roque, L. Qian, R. Jorge, Natural frequencies of functionally graded plates by a meshless method, Composite Structures, Vol. 75, No. 1-4, pp. 593-600, 2006.
[13] L. Qian, R. Batra, L. Chen, Static and dynamic deformations of thick functionally graded elastic plates by using higher-order shear and normal deformable plate theory and meshless local Petrov–Galerkin method, Composites Part B: Engineering, Vol. 35, No. 6-8, pp. 685-697, 2004.
[14] D. Jha, T. Kant, R. K. Singh, Higher order shear and normal deformation theory for natural frequency of functionally graded rectangular plates, Nuclear Engineering and Design, Vol. 250, pp. 8-13, 2012.
[15] B. Uymaz, M. Aydogdu, Three-dimensional vibration analyses of functionally graded plates under various boundary conditions, Journal of Reinforced Plastics and Composites, Vol. 26, No. 18, pp. 1847-1863, 2007.
[16] M. Mohamed, T. Abdelouahed, M. Slimane, A refined of trigonometric shear deformation plate theory based on neutral surface position is proposed for static analysis of FGM plate, Procedia Structural Integrity, Vol. 26, pp. 129-138, 2020.
[17] S. Hosseini-Hashemi, M. Fadaee, S. R. Atashipour, A new exact analytical approach for free vibration of Reissner–Mindlin functionally graded rectangular plates, International Journal of Mechanical Sciences, Vol. 53, No. 1, pp. 11-22, 2011, 2011.
[18] H. Matsunaga, Free vibration and stability of functionally graded plates according to a 2-D higher-order deformation theory, Composite structures, Vol. 82, No. 4, pp. 499-512, 2008.
[19] A. Rezaei, A. Saidi, Application of Carrera Unified Formulation to study the effect of porosity on natural frequencies of thick porous–cellular plates, Composites Part B: Engineering, Vol. 91, pp. 361-370, 2016.
[20] M. Askari, A. R. Saidi, A. S. Rezaei, M. A. Badizi, Navier-type free vibration analysis of porous smart plates according to reddy’s plate theory, in Proceeding of, 13.
[21] S. Merdaci, A. H. Mostefa, Influence of porosity on the analysis of sandwich plates FGM using of high order shear-deformation theory, Frattura ed Integrità Strutturale, Vol. 14, No. 51, pp. 199-214, 2020, 2020. en
[22] A. Benachour, H. D. Tahar, H. A. Atmane, A. Tounsi, M. S. Ahmed, A four variable refined plate theory for free vibrations of functionally graded plates with arbitrary gradient, Composites Part B: Engineering, Vol. 42, No. 6, pp. 1386-1394, 2011.
[23] F. Ebrahimi, S. Habibi, Deflection and vibration analysis of higher-order shear deformable compositionally graded porous plate, Steel and Composite Structures, Vol. 20, No. 1, pp. 205-225, 2016.
[24] J. Zhao, Q. Wang, X. Deng, K. Choe, R. Zhong, C. Shuai, Free vibrations of functionally graded porous rectangular plate with uniform elastic boundary conditions, Composites Part B: Engineering, Vol. 168, pp. 106-120, 2019.
[25] M. Slimane, Free vibration analysis of composite material plates" Case of a typical functionally graded FG plates ceramic/metal" with porosities, Nano Hybrids and Composites, Vol. 25, pp. 69-83, 2019.
[26] F. Mouaici, S. Benyoucef, H. A. Atmane, A. Tounsi, Effect of porosity on vibrational characteristics of non-homogeneous plates using hyperbolic shear deformation theory, Wind & structures, Vol. 22, No. 4, pp. 429-454, 2016.
[27] M. Slimane, H. M. Adda, M. Mohamed, B. Hakima, H. Hadjira, B. Sabrina, Effects of even pores distribution of functionally graded plate porous rectangular and square, Procedia Structural Integrity, Vol. 26, pp. 35-45, 2020, 2020. en
[28] S. Merdaci, A. Hadj Mostefa, Free vibration analysis of composite material plates with porosities based on the first-order shear deformation theory, Journal of Mineral and Material Science (JMMS), Vol. 1, No. 3, pp. 1-2, 2020.
[29] J. Zhu, Z. Lai, Z. Yin, J. Jeon, S. Lee, Fabrication of ZrO2–NiCr functionally graded material by powder metallurgy, Materials chemistry and physics, Vol. 68, No. 1-3, pp. 130-135, 2001.
[30] N. Wattanasakulpong, B. G. Prusty, D. W. Kelly, M. Hoffman, Free vibration analysis of layered functionally graded beams with experimental validation, Materials & Design (1980-2015), Vol. 36, pp. 182-190, 2012.
[31] M. Slimane, Analysis of Bending of Ceramic-Metal Functionally Graded Plates with Porosities Using of High Order Shear Theory, Advanced Engineering Forum, Vol. 30, pp. 54-70, 2018, 2018. en
[32] A. M. Zenkour, A quasi-3D refined theory for functionally graded single-layered and sandwich plates with porosities, Composite Structures, Vol. 201, pp. 38-48, 2018/10//, 2018. en
[33] S. Merdaci, H. Belghoul, High-order shear theory for static analysis of functionally graded plates with porosities, Comptes Rendus Mécanique, Vol. 347, No. 3, pp. 207-217, 2019/03/01/, 2019.
[34] N. Wattanasakulpong, V. Ungbhakorn, Linear and nonlinear vibration analysis of elastically restrained ends FGM beams with porosities, Aerospace Science and Technology, Vol. 32, No. 1, pp. 111-120, 2014/01/01/, 2014. en
[35] A. Rezaei, A. Saidi, M. Abrishamdari, M. P. Mohammadi, Natural frequencies of functionally graded plates with porosities via a simple four variable plate theory: an analytical approach, Thin-Walled Structures, Vol. 120, pp. 366-377, 2017.
[36] J. M. Whitney, N. Pagano, Shear deformation in heterogeneous anisotropic plates, 1970.
[37] T.-K. Nguyen, K. Sab, G. Bonnet, First-order shear deformation plate models for functionally graded materials, Composite Structures, Vol. 83, No. 1, pp. 25-36, 2008.
[38] J. Reddy, Analysis of functionally graded plates, International Journal for numerical methods in engineering, Vol. 47, No. 1‐3, pp. 663-684, 2000.
[39] A. M. Zenkour, Generalized shear deformation theory for bending analysis of functionally graded plates, Applied Mathematical Modelling, Vol. 30, No. 1, pp. 67-84, 2006/01//, 2006.
[40] A. Zenkour, A. Radwan, Bending response of FG plates resting on elastic foundations in hygrothermal environment with porosities, Composite Structures, Vol. 213, pp. 133-143, 2019.
[41] I. Belkorissat, M. Ameur, Influence of an initial imperfection on hygro-thermo-mechanical behaviors of FG plates laid on elastic foundation, Journal of Mechanical Science and Technology, Vol. 37, No. 5, pp. 2471-2477, 2023.
[42] B. Bouderba, M. S. A. Houari, A. Tounsi, Thermomechanical bending response of FGM thick plates resting on Winkler-Pasternak elastic foundations, Steel & Composite structures, Vol. 14, No. 1, pp. 85-104, 2013/01//, 2013.
[43] Z. Otmane, M. Slimane, H. M. Adda, Thermo-Mechanical Bending for Hybrid Material Plates Perfect-Imperfect Rectangular Using High Order Theory, Applied Mechanics and Materials, Vol. 909, pp. 29-44, 2022.
[2] Ş. D. Akbaş, Wave propagation of a functionally graded beam in thermal environments, Steel and Composite Structures, Vol. 19, No. 6, pp. 1421-1447, 2015.
[3] S. S. Vel, R. Batra, Three-dimensional exact solution for the vibration of functionally graded rectangular plates, Journal of Sound and Vibration, Vol. 272, No. 3-5, pp. 703-730, 2004.
[4] R. Bennai, H. A. Atmane, A. Tounsi, A new higher-order shear and normal deformation theory for functionally graded sandwich beams, Steel and Composite Structures, Vol. 19, No. 3, pp. 521-546, 2015.
[5] M. Arefi, Elastic solution of a curved beam made of functionally graded materials with different cross sections, Steel and Composite Structures, Vol. 18, No. 3, pp. 659-672, 2015.
[6] F. Ebrahimi, S. Dashti, Free vibration analysis of a rotating non-uniform functionally graded beam, Steel and Composite Structures, Vol. 19, No. 5, pp. 1279-1298, 2015.
[7] K. Darılmaz, Vibration analysis of functionally graded material (FGM) grid systems, Steel and Composite Structures, Vol. 18, No. 2, pp. 395-408, 2015.
[8] V. R. Kar, S. K. Panda, Nonlinear thermomechanical deformation behaviour of P-FGM shallow spherical shell panel, Chinese Journal of Aeronautics, Vol. 29, No. 1, pp. 173-183, 2016, 2016.
[9] A. Hadj Mostefa, S. Merdaci, N. Mahmoudi, An overview of functionally graded materials «FGM», in Proceeding of, Springer, pp. 267-278.
[10] T.-H. Trinh1a, D.-K. Nguyen2b, B. S. Gan, S. Alexandrov3c, Post-buckling responses of elastoplastic FGM beams on nonlinear elastic foundation, Structural Engineering and Mechanics, Vol. 58, No. 3, pp. 515-532, 2016.
[11] H. A. Atmane, A. Tounsi, F. Bernard, S. Mahmoud, A computational shear displacement model for vibrational analysis of functionally graded beams with porosities, Steel Compos. Struct, Vol. 19, No. 2, pp. 369-384, 2015.
[12] A. Ferreira, R. Batra, C. Roque, L. Qian, R. Jorge, Natural frequencies of functionally graded plates by a meshless method, Composite Structures, Vol. 75, No. 1-4, pp. 593-600, 2006.
[13] L. Qian, R. Batra, L. Chen, Static and dynamic deformations of thick functionally graded elastic plates by using higher-order shear and normal deformable plate theory and meshless local Petrov–Galerkin method, Composites Part B: Engineering, Vol. 35, No. 6-8, pp. 685-697, 2004.
[14] D. Jha, T. Kant, R. K. Singh, Higher order shear and normal deformation theory for natural frequency of functionally graded rectangular plates, Nuclear Engineering and Design, Vol. 250, pp. 8-13, 2012.
[15] B. Uymaz, M. Aydogdu, Three-dimensional vibration analyses of functionally graded plates under various boundary conditions, Journal of Reinforced Plastics and Composites, Vol. 26, No. 18, pp. 1847-1863, 2007.
[16] M. Mohamed, T. Abdelouahed, M. Slimane, A refined of trigonometric shear deformation plate theory based on neutral surface position is proposed for static analysis of FGM plate, Procedia Structural Integrity, Vol. 26, pp. 129-138, 2020.
[17] S. Hosseini-Hashemi, M. Fadaee, S. R. Atashipour, A new exact analytical approach for free vibration of Reissner–Mindlin functionally graded rectangular plates, International Journal of Mechanical Sciences, Vol. 53, No. 1, pp. 11-22, 2011, 2011.
[18] H. Matsunaga, Free vibration and stability of functionally graded plates according to a 2-D higher-order deformation theory, Composite structures, Vol. 82, No. 4, pp. 499-512, 2008.
[19] A. Rezaei, A. Saidi, Application of Carrera Unified Formulation to study the effect of porosity on natural frequencies of thick porous–cellular plates, Composites Part B: Engineering, Vol. 91, pp. 361-370, 2016.
[20] M. Askari, A. R. Saidi, A. S. Rezaei, M. A. Badizi, Navier-type free vibration analysis of porous smart plates according to reddy’s plate theory, in Proceeding of, 13.
[21] S. Merdaci, A. H. Mostefa, Influence of porosity on the analysis of sandwich plates FGM using of high order shear-deformation theory, Frattura ed Integrità Strutturale, Vol. 14, No. 51, pp. 199-214, 2020, 2020. en
[22] A. Benachour, H. D. Tahar, H. A. Atmane, A. Tounsi, M. S. Ahmed, A four variable refined plate theory for free vibrations of functionally graded plates with arbitrary gradient, Composites Part B: Engineering, Vol. 42, No. 6, pp. 1386-1394, 2011.
[23] F. Ebrahimi, S. Habibi, Deflection and vibration analysis of higher-order shear deformable compositionally graded porous plate, Steel and Composite Structures, Vol. 20, No. 1, pp. 205-225, 2016.
[24] J. Zhao, Q. Wang, X. Deng, K. Choe, R. Zhong, C. Shuai, Free vibrations of functionally graded porous rectangular plate with uniform elastic boundary conditions, Composites Part B: Engineering, Vol. 168, pp. 106-120, 2019.
[25] M. Slimane, Free vibration analysis of composite material plates" Case of a typical functionally graded FG plates ceramic/metal" with porosities, Nano Hybrids and Composites, Vol. 25, pp. 69-83, 2019.
[26] F. Mouaici, S. Benyoucef, H. A. Atmane, A. Tounsi, Effect of porosity on vibrational characteristics of non-homogeneous plates using hyperbolic shear deformation theory, Wind & structures, Vol. 22, No. 4, pp. 429-454, 2016.
[27] M. Slimane, H. M. Adda, M. Mohamed, B. Hakima, H. Hadjira, B. Sabrina, Effects of even pores distribution of functionally graded plate porous rectangular and square, Procedia Structural Integrity, Vol. 26, pp. 35-45, 2020, 2020. en
[28] S. Merdaci, A. Hadj Mostefa, Free vibration analysis of composite material plates with porosities based on the first-order shear deformation theory, Journal of Mineral and Material Science (JMMS), Vol. 1, No. 3, pp. 1-2, 2020.
[29] J. Zhu, Z. Lai, Z. Yin, J. Jeon, S. Lee, Fabrication of ZrO2–NiCr functionally graded material by powder metallurgy, Materials chemistry and physics, Vol. 68, No. 1-3, pp. 130-135, 2001.
[30] N. Wattanasakulpong, B. G. Prusty, D. W. Kelly, M. Hoffman, Free vibration analysis of layered functionally graded beams with experimental validation, Materials & Design (1980-2015), Vol. 36, pp. 182-190, 2012.
[31] M. Slimane, Analysis of Bending of Ceramic-Metal Functionally Graded Plates with Porosities Using of High Order Shear Theory, Advanced Engineering Forum, Vol. 30, pp. 54-70, 2018, 2018. en
[32] A. M. Zenkour, A quasi-3D refined theory for functionally graded single-layered and sandwich plates with porosities, Composite Structures, Vol. 201, pp. 38-48, 2018/10//, 2018. en
[33] S. Merdaci, H. Belghoul, High-order shear theory for static analysis of functionally graded plates with porosities, Comptes Rendus Mécanique, Vol. 347, No. 3, pp. 207-217, 2019/03/01/, 2019.
[34] N. Wattanasakulpong, V. Ungbhakorn, Linear and nonlinear vibration analysis of elastically restrained ends FGM beams with porosities, Aerospace Science and Technology, Vol. 32, No. 1, pp. 111-120, 2014/01/01/, 2014. en
[35] A. Rezaei, A. Saidi, M. Abrishamdari, M. P. Mohammadi, Natural frequencies of functionally graded plates with porosities via a simple four variable plate theory: an analytical approach, Thin-Walled Structures, Vol. 120, pp. 366-377, 2017.
[36] J. M. Whitney, N. Pagano, Shear deformation in heterogeneous anisotropic plates, 1970.
[37] T.-K. Nguyen, K. Sab, G. Bonnet, First-order shear deformation plate models for functionally graded materials, Composite Structures, Vol. 83, No. 1, pp. 25-36, 2008.
[38] J. Reddy, Analysis of functionally graded plates, International Journal for numerical methods in engineering, Vol. 47, No. 1‐3, pp. 663-684, 2000.
[39] A. M. Zenkour, Generalized shear deformation theory for bending analysis of functionally graded plates, Applied Mathematical Modelling, Vol. 30, No. 1, pp. 67-84, 2006/01//, 2006.
[40] A. Zenkour, A. Radwan, Bending response of FG plates resting on elastic foundations in hygrothermal environment with porosities, Composite Structures, Vol. 213, pp. 133-143, 2019.
[41] I. Belkorissat, M. Ameur, Influence of an initial imperfection on hygro-thermo-mechanical behaviors of FG plates laid on elastic foundation, Journal of Mechanical Science and Technology, Vol. 37, No. 5, pp. 2471-2477, 2023.
[42] B. Bouderba, M. S. A. Houari, A. Tounsi, Thermomechanical bending response of FGM thick plates resting on Winkler-Pasternak elastic foundations, Steel & Composite structures, Vol. 14, No. 1, pp. 85-104, 2013/01//, 2013.
[43] Z. Otmane, M. Slimane, H. M. Adda, Thermo-Mechanical Bending for Hybrid Material Plates Perfect-Imperfect Rectangular Using High Order Theory, Applied Mechanics and Materials, Vol. 909, pp. 29-44, 2022.
)
Volume 56, Issue 1
January 2025Pages 76-86
- Receive Date: 25 July 2024
- Revise Date: 13 August 2024
- Accept Date: 25 August 2024