Assessment of Radial basis function based meshfree method for the buckling analysis of rectangular FGM plate using HSDT and Strong form formulation

Document Type : Research Paper

Authors

1 Department of Mechanical Engineering, Sardar Vallabhbhai National Institute of Technology, Surat-395007, India

2 Department of Mechanical Engineering, BIT Mesra, off campus, Patna-80014, India

3 Institute of Business Management, GLA University, Mathura- 281406, India

4 Department of Mechanical Engineering, MMMUT, Gorakhpur-273010, India

Abstract

An effort has been made in modifying the radial distances of existing radial basis functions (RBFs) for the buckling analysis of functionally graded material (FGM) rectangular plates. The governing differential equations (GDE’s) and boundary conditions are developed by employing the energy principle. The novelty of the present modified RBFs is that they are suitable for analyzing plates with varying aspect ratios. In the present analysis, thirteen different RBFs available in the literature are analyzed. It is found that all RBFs are well suited for buckling analysis of FGM plates with a different aspect ratio which was not possible with existing RBFs. Existing RBFs were suitable for analyzing square plates. To demonstrate the accuracy and efficiency of the present method, results are obtained for the buckling load parameters with modified radial distances for different aspect ratios. The results of several numerical examples have shown that the present modified RBF-based meshfree methods are well suited and accurate for analyzing rectangular plates. The effect of aspect ratio with grading index , span to thickness ratio on the normalized critical buckling load is discussed.

Keywords

[1]          R. L. Hardy, "Multiquadric equations of topography and other irregular surfaces," Journal of geophysical research, vol. 76, no. 8, pp. 1905-1915, 1971.
[2]          E. J. Kansa, "Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates," Computers & Mathematics with applications, vol. 19, no. 8-9, pp. 127-145, 1990.
[3]          E. J. Kansa, "Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—II solutions to parabolic, hyperbolic and elliptic partial differential equations," Computers & mathematics with applications, vol. 19, no. 8-9, pp. 147-161, 1990.
[4]          V. Bayona, M. Moscoso, and M. Kindelan, "Optimal variable shape parameter for multiquadric based RBF-FD method," Journal of Computational Physics, vol. 231, no. 6, pp. 2466-2481, 2012.
[5]          C.-S. Huang, C.-F. Lee, and A.-D. Cheng, "Error estimate, optimal shape factor, and high precision computation of multiquadric collocation method," Engineering Analysis with Boundary Elements, vol. 31, no. 7, pp. 614-623, 2007.
[6]          C.-S. Huang, H.-D. Yen, and A.-D. Cheng, "On the increasingly flat radial basis function and optimal shape parameter for the solution of elliptic PDEs," Engineering analysis with boundary elements, vol. 34, no. 9, pp. 802-809, 2010.
[7]          L. Iurlaro, M. Gherlone, and M. Di Sciuva, "Energy based approach for shape parameter selection in radial basis functions collocation method," Composite Structures, vol. 107, pp. 70-78, 2014.
[8]          S. Rippa, "An algorithm for selecting a good value for the parameter c in radial basis function interpolation," Advances in Computational Mathematics, vol. 11, no. 2, pp. 193-210, 1999.
[9]          G. E. Fasshauer, "Solving partial differential equations by collocation with radial basis functions," in Proceedings of Chamonix, 1996, vol. 1997: Citeseer, pp. 1-8.
[10]        H. Wendland, "Meshless Galerkin methods using radial basis functions," Mathematics of computation, vol. 68, no. 228, pp. 1521-1531, 1999.
[11]        B. V. Farahani, J. Berardo, J. Belinha, A. Ferreira, P. J. Tavares, and P. Moreira, "On the optimal shape parameters of distinct versions of RBF meshless methods for the bending analysis of plates," Engineering Analysis with Boundary Elements, vol. 84, pp. 77-86, 2017.
[12]        A. Ferreira, "A formulation of the multiquadric radial basis function method for the analysis of laminated composite plates," Composite structures, vol. 59, no. 3, pp. 385-392, 2003.
[13]        A. Ferreira, E. Carrera, M. Cinefra, C. Roque, and O. Polit, "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, vol. 42, no. 5, pp. 1276-1284, 2011.
[14]        A. Ferreira and G. Fasshauer, "Computation of natural frequencies of shear deformable beams and plates by an RBF-pseudospectral method," Computer Methods in Applied Mechanics and Engineering, vol. 196, no. 1-3, pp. 134-146, 2006.
[15]        A. Ferreira, G. Fasshauer, R. Batra, and J. Rodrigues, "Static deformations and vibration analysis of composite and sandwich plates using a layerwise theory and RBF-PS discretizations with optimal shape parameter," Composite Structures, vol. 86, no. 4, pp. 328-343, 2008.
[16]        A. Ferreira, C. Roque, and R. Jorge, "Free vibration analysis of symmetric laminated composite plates by FSDT and radial basis functions," Computer Methods in Applied Mechanics and Engineering, vol. 194, no. 39-41, pp. 4265-4278, 2005.
[17]        A. Ferreira, C. Roque, R. Jorge, and E. Kansa, "Static deformations and vibration analysis of composite and sandwich plates using a layerwise theory and multiquadrics discretizations," Engineering Analysis with Boundary Elements, vol. 29, no. 12, pp. 1104-1114, 2005.
[18]        A. Ferreira, C. Roque, and P. Martins, "Analysis of composite plates using higher-order shear deformation theory and a finite point formulation based on the multiquadric radial basis function method," Composites Part B: Engineering, vol. 34, no. 7, pp. 627-636, 2003.
[19]        A. Ferreira, C. Roque, A. Neves, R. Jorge, C. Soares, and K. M. Liew, "Buckling and vibration analysis of isotropic and laminated plates by radial basis functions," Composites Part B: Engineering, vol. 42, no. 3, pp. 592-606, 2011.
[20]        A. Ferreira, C. Roque, A. Neves, R. Jorge, C. M. Soares, and J. Reddy, "Buckling analysis of isotropic and laminated plates by radial basis functions according to a higher-order shear deformation theory," Thin-Walled Structures, vol. 49, no. 7, pp. 804-811, 2011.
[21]        M. Jakomin, F. Kosel, and T. Kosel, "Thin double curved shallow bimetallic shell of translation in a homogenous temperature field by non-linear theory," Thin-walled structures, vol. 48, no. 3, pp. 243-259, 2010.
[22]        S. Xiang, Z.-y. Bi, S.-x. Jiang, Y.-x. Jin, and M.-s. Yang, "Thin plate spline radial basis function for the free vibration analysis of laminated composite shells," Composite Structures, vol. 93, no. 2, pp. 611-615, 2011.
[23]        S. Xiang, S.-x. Jiang, Z.-y. Bi, Y.-x. Jin, and M.-s. Yang, "A nth-order meshless generalization of Reddy’s third-order shear deformation theory for the free vibration on laminated composite plates," Composite Structures, vol. 93, no. 2, pp. 299-307, 2011.
[24]        S. Xiang, H. Shi, K.-m. Wang, Y.-t. Ai, and Y.-d. Sha, "Thin plate spline radial basis functions for vibration analysis of clamped laminated composite plates," European Journal of Mechanics-A/Solids, vol. 29, no. 5, pp. 844-850, 2010.
[25]        S. Xiang and K.-m. Wang, "Free vibration analysis of symmetric laminated composite plates by trigonometric shear deformation theory and inverse multiquadric RBF," Thin-Walled Structures, vol. 47, no. 3, pp. 304-310, 2009.
[26]        J. Rodrigues, C. Roque, A. Ferreira, E. Carrera, and M. Cinefra, "Radial basis functions–finite differences collocation and a Unified Formulation for bending, vibration and buckling analysis of laminated plates, according to Murakami’s zig-zag theory," Composite Structures, vol. 93, no. 7, pp. 1613-1620, 2011.
[27]        C. Roque, D. Cunha, C. Shu, and A. Ferreira, "A local radial basis functions—Finite differences technique for the analysis of composite plates," Engineering Analysis with Boundary Elements, vol. 35, no. 3, pp. 363-374, 2011.
[28]        R. B. Prasad, J. Singh, and K. K. Shukla, "Application of radial basis function-based meshless method for torsional analysis of elliptical and bone shaped-irregular sections," Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications, vol. 236, no. 3, pp. 611-632, 2022.
[29]        J. Singh and K. Shukla, "Nonlinear flexural analysis of functionally graded plates under different loadings using RBF based meshless method," Engineering Analysis with Boundary Elements, vol. 36, no. 12, pp. 1819-1827, 2012.
[30]        J. Singh and K. Shukla, "Nonlinear flexural analysis of laminated composite plates using RBF based meshless method," Composite Structures, vol. 94, no. 5, pp. 1714-1720, 2012.
[31]        J. Singh, S. Singh, and K. Shukla, "RBF-based meshless method for free vibration analysis of laminated composite plates," International Journal of Mechanical and Mechatronics Engineering, vol. 5, no. 7, pp. 1290-1295, 2011.
[32]        J. Singh, S. Singh, and K. Shukla, "Meshless analysis of laminated composite and sandwich plates subjected to various types of loads," International Journal for Computational Methods in Engineering Science and Mechanics, vol. 15, no. 2, pp. 158-171, 2014.
[33]        S. Singh, J. Singh, and K. Shukla, "Buckling of laminated composite plates subjected to mechanical and thermal loads using meshless collocations," Journal of Mechanical Science and Technology, vol. 27, no. 2, pp. 327-336, 2013.
[34]        S. Singh, J. Singh, and K. K. Shula, "Buckling of laminated composite and sandwich plates using radial basis function collocations," International Journal of Structural Stability and Dynamics, vol. 15, no. 01, p. 1540002, 2015.
[35]        M. K. Solanki, R. Kumar, and J. Singh, "Flexure analysis of laminated plates using multiquadratic RBF based meshfree method," International Journal of Computational Methods, vol. 15, no. 06, p. 1850049, 2018.
[36]        M. K. Solanki, S. K. Mishra, K. Shukla, and J. Singh, "Nonlinear free vibration of laminated composite and sandwich plates using multiquadric collocations," Materials Today: Proceedings, vol. 2, no. 4-5, pp. 3049-3055, 2015.
[37]        M. K. Solanki, S. K. Mishra, and J. Singh, "Meshfree approach for linear and nonlinear analysis of sandwich plates: A critical review of twenty plate theories," Engineering Analysis with Boundary Elements, vol. 69, pp. 93-103, 2016.
[38]        P. C. Vishwakarma, J. Damania, and J. Singh, "Improving Vibration analysis of laminated composite plate by using WU-C2 RBF based Meshfree Method," in IOP Conference Series: Materials Science and Engineering, 2018, vol. 404, no. 1: IOP Publishing, p. 012036.
[39]        A. Kumar, R. Kumar, J. Damania, and J. Singh, "Buckling Analysis of FGM Plates by thin plate spline RBF based Meshfree Approach," in IOP Conference Series: Materials Science and Engineering, 2018, vol. 404, no. 1: IOP Publishing, p. 012037.
[40]        R. Kumar, M. Bajaj, J. Singh, and K. K. Shukla, "New HSDT for free vibration analysis of elastically supported porous bidirectional functionally graded sandwich plate using collocation method," Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, p. 09544062221090075, 2022.
[41]        R. Kumar, A. Lal, B. Singh, and J. Singh, "New transverse shear deformation theory for bending analysis of FGM plate under patch load," Composite Structures, vol. 208, pp. 91-100, 2019.
[42]        R. Kumar, A. Lal, B. Singh, and J. Singh, "Meshfree approach on buckling and free vibration analysis of porous FGM plate with proposed IHHSDT resting on the foundation," Curved and Layered Structures, vol. 6, no. 1, pp. 192-211, 2019.
[43]        R. Kumar, A. Lal, B. Singh, and J. Singh, "Non-linear analysis of porous elastically supported FGM plate under various loading," Composite Structures, vol. 233, p. 111721, 2020.
[44]        R. Kumar, A. Lal, B. N. Singh, and J. Singh, "Numerical simulation of the thermomechanical buckling analysis of bidirectional porous functionally graded plate using collocation meshfree method," Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications, vol. 236, no. 4, pp. 787-807, 2022.
[45]        R. Kumar, A. Lal, and J. Singh, "Meshfree approach for the vibration analysis of FGM plates using two shear displacement model," in Proceedings of the third Indian conference on applied mechanics, 2017.
[46]        R. Kumar, B. Singh, and J. Singh, "Geometrically nonlinear analysis for flexure response of FGM plate under patch load," Mechanics Based Design of Structures and Machines, pp. 1-25, 2022.
[47]        R. Kumar and J. Singh, "Assessment of higher order transverse shear deformation theories for modeling and buckling analysis of FGM plates using RBF based meshless approach," Multidiscipline Modeling in Materials and Structures, 2018.
[48]        A. Neves, A. Ferreira, E. Carrera, M. Cinefra, R. Jorge, and C. Soares, "Buckling analysis of sandwich plates with functionally graded skins using a new quasi‐3D hyperbolic sine shear deformation theory and collocation with radial basis functions," ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, vol. 92, no. 9, pp. 749-766, 2012.
[49]        A. Neves et al., "A quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates," Composite Structures, vol. 94, no. 5, pp. 1814-1825, 2012.
[50]        A. Neves et al., "Static, free vibration and buckling analysis of isotropic and sandwich functionally graded plates using a quasi-3D higher-order shear deformation theory and a meshless technique," Composites Part B: Engineering, vol. 44, no. 1, pp. 657-674, 2013.
[51]        D. Maturi, A. Ferreira, A. Zenkour, and D. Mashat, "Analysis of sandwich plates with a new layerwise formulation," Composites Part B: Engineering, vol. 56, pp. 484-489, 2014.
[52]        C. H. Thai, V. N. Do, and H. Nguyen-Xuan, "An improved Moving Kriging-based meshfree method for static, dynamic and buckling analyses of functionally graded isotropic and sandwich plates," Engineering Analysis with Boundary Elements, vol. 64, pp. 122-136, 2016.
[53]        T. Bui, M. Nguyen, and C. Zhang, "Buckling analysis of Reissner–Mindlin plates subjected to in-plane edge loads using a shear-locking-free and meshfree method," Engineering analysis with boundary elements, vol. 35, no. 9, pp. 1038-1053, 2011.
[54]        J. Rodrigues, C. Roque, A. Ferreira, M. Cinefra, and E. Carrera, "Radial basis functions-differential quadrature collocation and a unified formulation for bending, vibration and buckling analysis of laminated plates, according to Murakami’s Zig-Zag theory," Computers & structures, vol. 90, pp. 107-115, 2012.
[55]        F. Chu, J. He, L. Wang, and Z. Zhong, "Buckling analysis of functionally graded thin plate with in-plane material inhomogeneity," Engineering Analysis with Boundary Elements, vol. 65, pp. 112-125, 2016.
[56]        A. Farajpour, M. Danesh, and M. Mohammadi, "Buckling analysis of variable thickness nanoplates using nonlocal continuum mechanics," Physica E: Low-dimensional Systems and Nanostructures, vol. 44, no. 3, pp. 719-727, 2011.
[57]        A. Farajpour, M. Mohammadi, A. Shahidi, and M. Mahzoon, "Axisymmetric buckling of the circular graphene sheets with the nonlocal continuum plate model," Physica E: Low-dimensional Systems and Nanostructures, vol. 43, no. 10, pp. 1820-1825, 2011.
[58]        A. Farajpour, A. Rastgoo, and M. Mohammadi, "Vibration, buckling and smart control of microtubules using piezoelectric nanoshells under electric voltage in thermal environment," Physica B: Condensed Matter, vol. 509, pp. 100-114, 2017.
[59]        A. Farajpour, A. Shahidi, M. Mohammadi, and M. Mahzoon, "Buckling of orthotropic micro/nanoscale plates under linearly varying in-plane load via nonlocal continuum mechanics," Composite Structures, vol. 94, no. 5, pp. 1605-1615, 2012.
[60]        A. Hadi, M. Z. Nejad, A. Rastgoo, and M. Hosseini, "Buckling analysis of FGM Euler-Bernoulli nano-beams with 3D-varying properties based on consistent couple-stress theory," Steel and Composite Structures, An International Journal, vol. 26, no. 6, pp. 663-672, 2018.
[61]        M. Mohammadi, A. Farajpour, A. Moradi, and M. Ghayour, "Shear buckling of orthotropic rectangular graphene sheet embedded in an elastic medium in thermal environment," Composites Part B: Engineering, vol. 56, pp. 629-637, 2014.
[62]        M. Z. Nejad, A. Hadi, and A. Rastgoo, "Buckling analysis of arbitrary two-directional functionally graded Euler–Bernoulli nano-beams based on nonlocal elasticity theory," International Journal of Engineering Science, vol. 103, pp. 1-10, 2016.
[63]        K. Liew, J. Wang, T. Ng, and M. Tan, "Free vibration and buckling analyses of shear-deformable plates based on FSDT meshfree method," Journal of Sound and Vibration, vol. 276, no. 3-5, pp. 997-1017, 2004.
[64]        L. Zhang, P. Zhu, and K. Liew, "Thermal buckling of functionally graded plates using a local Kriging meshless method," Composite Structures, vol. 108, pp. 472-492, 2014.
[65]        P. Tan, N. Nguyen-Thanh, T. Rabczuk, and K. Zhou, "Static, dynamic and buckling analyses of 3D FGM plates and shells via an isogeometric-meshfree coupling approach," Composite Structures, vol. 198, pp. 35-50, 2018.
[66]        A. Zarei and A. Khosravifard, "A meshfree method for static and buckling analysis of shear deformable composite laminates considering continuity of interlaminar transverse shearing stresses," Composite Structures, vol. 209, pp. 206-218, 2019.
[67]        H. Asemi, S. Asemi, A. Farajpour, and M. Mohammadi, "Nanoscale mass detection based on vibrating piezoelectric ultrathin films under thermo-electro-mechanical loads," Physica E: Low-dimensional Systems and Nanostructures, vol. 68, pp. 112-122, 2015.
[68]        S. Asemi, A. Farajpour, H. Asemi, and M. Mohammadi, "Influence of initial stress on the vibration of double-piezoelectric-nanoplate systems with various boundary conditions using DQM," Physica E: Low-dimensional Systems and Nanostructures, vol. 63, pp. 169-179, 2014.
[69]        S. Asemi, A. Farajpour, and M. Mohammadi, "Nonlinear vibration analysis of piezoelectric nanoelectromechanical resonators based on nonlocal elasticity theory," Composite Structures, vol. 116, pp. 703-712, 2014.
[70]        A. Barati, A. Hadi, M. Z. Nejad, and R. Noroozi, "On vibration of bi-directional functionally graded nanobeams under magnetic field," Mechanics Based Design of Structures and Machines, vol. 50, no. 2, pp. 468-485, 2022.
[71]        M. Danesh, A. Farajpour, and M. Mohammadi, "Axial vibration analysis of a tapered nanorod based on nonlocal elasticity theory and differential quadrature method," Mechanics Research Communications, vol. 39, no. 1, pp. 23-27, 2012.
[72]        A. Farajpour, M. H. Yazdi, A. Rastgoo, M. Loghmani, and M. Mohammadi, "Nonlocal nonlinear plate model for large amplitude vibration of magneto-electro-elastic nanoplates," Composite Structures, vol. 140, pp. 323-336, 2016.
[73]        M. R. Farajpour, A. Rastgoo, A. Farajpour, and M. Mohammadi, "Vibration of piezoelectric nanofilm‐based electromechanical sensors via higher‐order non‐local strain gradient theory," Micro & Nano Letters, vol. 11, no. 6, pp. 302-307, 2016.
[74]        A. Hadi, M. Z. Nejad, and M. Hosseini, "Vibrations of three-dimensionally graded nanobeams," International Journal of Engineering Science, vol. 128, pp. 12-23, 2018.
[75]        A. Hadi, A. Rastgoo, A. Daneshmehr, and F. Ehsani, "Stress and strain analysis of functionally graded rectangular plate with exponentially varying properties," Indian Journal of Materials Science, vol. 2013, 2013.
[76]        M. Mohammadi, A. Farajpour, M. Goodarzi, and F. Dinari, "Thermo-mechanical vibration analysis of annular and circular graphene sheet embedded in an elastic medium," Latin American Journal of Solids and Structures, vol. 11, pp. 659-682, 2014.
[77]        M. Mohammadi, A. Farajpour, M. Goodarzi, and H. Mohammadi, "Temperature Effect on Vibration Analysis of Annular Graphene Sheet Embedded on Visco-Pasternak Foundati," Journal of Solid Mechanics, vol. 5, no. 3, pp. 305-323, 2013. [Online]. Available: http://jsm.iau-arak.ac.ir/article_514574_947d5d6914c8c336b9a9ba095eeac2f6.pdf.
[78]        M. Mohammadi, M. Ghayour, and A. Farajpour, "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, vol. 3, no. 2, pp. 47-56, 2010.
[79]        M. Mohammadi, M. Ghayour, and A. Farajpour, "Free transverse vibration analysis of circular and annular graphene sheets with various boundary conditions using the nonlocal continuum plate model," Composites Part B: Engineering, vol. 45, no. 1, pp. 32-42, 2013.
[80]        M. Mohammadi, M. Goodarzi, M. Ghayour, and A. Farajpour, "Influence of in-plane pre-load on the vibration frequency of circular graphene sheet via nonlocal continuum theory," Composites Part B: Engineering, vol. 51, pp. 121-129, 2013.
[81]        M. Mohammadi, M. Hosseini, M. Shishesaz, A. Hadi, and A. Rastgoo, "Primary and secondary resonance analysis of porous functionally graded nanobeam resting on a nonlinear foundation subjected to mechanical and electrical loads," European Journal of Mechanics-A/Solids, vol. 77, p. 103793, 2019.
[82]        M. Mohammadi, A. Moradi, M. Ghayour, and A. Farajpour, "Exact solution for thermo-mechanical vibration of orthotropic mono-layer graphene sheet embedded in an elastic medium," Latin American Journal of Solids and Structures, vol. 11, pp. 437-458, 2014.
[83]        M. Mohammadi and A. Rastgoo, "Nonlinear vibration analysis of the viscoelastic composite nanoplate with three directionally imperfect porous FG core," Structural Engineering and Mechanics, An Int'l Journal, vol. 69, no. 2, pp. 131-143, 2019.
[84]        M. Mohammadi and A. Rastgoo, "Primary and secondary resonance analysis of FG/lipid nanoplate with considering porosity distribution based on a nonlinear elastic medium," Mechanics of Advanced Materials and Structures, vol. 27, no. 20, pp. 1709-1730, 2020/10/15 2020, doi: 10.1080/15376494.2018.1525453.
[85]        M. Mohammadi, M. Safarabadi, A. Rastgoo, and A. Farajpour, "Hygro-mechanical vibration analysis of a rotating viscoelastic nanobeam embedded in a visco-Pasternak elastic medium and in a nonlinear thermal environment," Acta Mechanica, vol. 227, no. 8, pp. 2207-2232, 2016.
[86]        H. Moosavi, M. Mohammadi, A. Farajpour, and S. Shahidi, "Vibration analysis of nanorings using nonlocal continuum mechanics and shear deformable ring theory," Physica E: Low-dimensional Systems and Nanostructures, vol. 44, no. 1, pp. 135-140, 2011.
[87]        R. Noroozi, A. Barati, A. Kazemi, S. Norouzi, and A. Hadi, "Torsional vibration analysis of bi-directional FG nano-cone with arbitrary cross-section based on nonlocal strain gradient elasticity," Advances in nano research, vol. 8, no. 1, pp. 13-24, 2020.
[88]        N. GHAYOUR, A. SEDAGHAT, and M. MOHAMMADI, "WAVE PROPAGATION APPROACH TO FLUID FILLED SUBMERGED VISCO-ELASTIC FINITE CYLINDRICAL SHELLS," JOURNAL OF AEROSPACE SCIENCE AND TECHNOLOGY (JAST), vol. 8, no. 1, pp. -, 2011. [Online]. Available: https://www.sid.ir/en/Journal/ViewPaper.aspx?ID=254379.
[89]        H. Moosavi, M. Mohammadi, A. Farajpour, and S. H. Shahidi, "Vibration analysis of nanorings using nonlocal continuum mechanics and shear deformable ring theory," Physica E: Low-dimensional Systems and Nanostructures, vol. 44, no. 1, pp. 135-140, 2011/10/01/ 2011, doi: https://doi.org/10.1016/j.physe.2011.08.002.
[90]        M. MOHAMMADI, M. GOODARZI, M. GHAYOUR, and S. ALIVAND, "SMALL SCALE EFFECT ON THE VIBRATION OF ORTHOTROPIC PLATES EMBEDDED IN AN ELASTIC MEDIUM AND UNDER BIAXIAL IN-PLANE PRE-LOAD VIA NONLOCAL ELASTICITY THEORY," JOURNAL OF SOLID MECHANICS, vol. 4, no. 2, pp. -, 2012. [Online]. Available: https://www.sid.ir/en/Journal/ViewPaper.aspx?ID=303684.
[91]        M. Mohammadi, M. Goodarzi, M. Ghayour, and S. Alivand, "Small scale effect on the vibration of orthotropic plates embedded in an elastic medium and under biaxial in-plane pre-load via nonlocal elasticity theory," 2012.
[92]        M. Mohammadi, A. Farajpour, M. Goodarzi, and R. Heydarshenas, "Levy Type Solution for Nonlocal Thermo-Mechanical Vibration of Orthotropic Mono-Layer Graphene Sheet Embedded in an Elastic Medium," Journal of Solid Mechanics, vol. 5, no. 2, pp. 116-132, 2013. [Online]. Available: http://jsm.iau-arak.ac.ir/article_514544_042ba549d411678ff6d4a1c27993d17c.pdf.
[93]        M. Mohammadi, A. Farajpour, M. Goodarzi, and H. Shehni nezhad pour, "Numerical study of the effect of shear in-plane load on the vibration analysis of graphene sheet embedded in an elastic medium," Computational Materials Science, vol. 82, pp. 510-520, 2014/02/01/ 2014, doi: https://doi.org/10.1016/j.commatsci.2013.10.022.
[94]        A. Farajpour, A. Rastgoo, and M. Mohammadi, "Surface effects on the mechanical characteristics of microtubule networks in living cells," Mechanics Research Communications, vol. 57, pp. 18-26, 2014/04/01/ 2014, doi: https://doi.org/10.1016/j.mechrescom.2014.01.005.
[95]        M. GOODARZI, M. MOHAMMADI, A. FARAJPOUR, and M. KHOORAN, "INVESTIGATION OF THE EFFECT OF PRE-STRESSED ON VIBRATION FREQUENCY OF RECTANGULAR NANOPLATE BASED ON A VISCO-PASTERNAK FOUNDATION," JOURNAL OF SOLID MECHANICS, vol. 6, no. 1, pp. -, 2014. [Online]. Available: https://www.sid.ir/en/Journal/ViewPaper.aspx?ID=376801.
[96]        S. R. Asemi, M. Mohammadi, and A. Farajpour, "A study on the nonlinear stability of orthotropic single-layered graphene sheet based on nonlocal elasticity theory," Latin American Journal of Solids and Structures, vol. 11, no. 9, pp. 1515-1540, 2014.
[97]        M. Mohammadi, A. Moradi, M. Ghayour, and A. Farajpour, "Exact solution for thermo-mechanical vibration of orthotropic mono-layer graphene sheet embedded in an elastic medium," Latin American Journal of Solids and Structures, vol. 11, no. 3, pp. 437-458, 2014.
[98]        M. Mohammadi, A. Farajpour, and M. Goodarzi, "Numerical study of the effect of shear in-plane load on the vibration analysis of graphene sheet embedded in an elastic medium," Computational Materials Science, vol. 82, pp. 510-520, 2014.
[99]        M. Safarabadi, M. Mohammadi, A. Farajpour, and M. Goodarzi, "Effect of surface energy on the vibration analysis of rotating nanobeam," 2015.
[100]     M. Goodarzi, M. Mohammadi, M. Khooran, and F. Saadi, "Thermo-Mechanical Vibration Analysis of FG Circular and Annular Nanoplate Based on the Visco-Pasternak Foundation," Journal of Solid Mechanics, vol. 8, no. 4, pp. 788-805, 2016. [Online]. Available: http://jsm.iau-arak.ac.ir/article_527024_c73ec8e3fa8e3b28dd54dbd53891d503.pdf.
[101]     M. Baghani, M. Mohammadi, and A. Farajpour, "Dynamic and Stability Analysis of the Rotating Nanobeam in a Nonuniform Magnetic Field Considering the Surface Energy," International Journal of Applied Mechanics, vol. 08, no. 04, p. 1650048, 2016, doi: 10.1142/s1758825116500484.
[102]     M. R. Farajpour, A. Rastgoo, A. Farajpour, and M. Mohammadi, "Vibration of piezoelectric nanofilm-based electromechanical sensors via higher-order non-local strain gradient theory," Micro & Nano Letters, vol. 11, no. 6, pp. 302-307, 2016.
[103]     A. Farajpour, M. Yazdi, A. Rastgoo, and M. Mohammadi, "A higher-order nonlocal strain gradient plate model for buckling of orthotropic nanoplates in thermal environment," Acta Mechanica, vol. 227, no. 7, pp. 1849-1867, 2016.
[104]     M. Mohammadi, M. Hosseini, M. Shishesaz, A. Hadi, and A. Rastgoo, "Primary and secondary resonance analysis of porous functionally graded nanobeam resting on a nonlinear foundation subjected to mechanical and electrical loads," European Journal of Mechanics - A/Solids, vol. 77, p. 103793, 2019/09/01/ 2019, doi: https://doi.org/10.1016/j.euromechsol.2019.05.008.
[105]     M. Mohammadi, A. Farajpour, A. Moradi, and M. Hosseini, "Vibration analysis of the rotating multilayer piezoelectric Timoshenko nanobeam," Engineering Analysis with Boundary Elements, vol. 145, pp. 117-131, 2022/12/01/ 2022, doi: https://doi.org/10.1016/j.enganabound.2022.09.008.
[106]     M. Hosseini, M. Shishesaz, K. N. Tahan, and A. Hadi, "Stress analysis of rotating nano-disks of variable thickness made of functionally graded materials," International Journal of Engineering Science, vol. 109, pp. 29-53, 2016.
[107]     M. Z. Nejad, A. Hadi, A. Omidvari, and A. Rastgoo, "Bending analysis of bi-directional functionally graded Euler-Bernoulli nano-beams using integral form of Eringen's non-local elasticity theory," Structural engineering and mechanics: An international journal, vol. 67, no. 4, pp. 417-425, 2018.
[108]     M. M. Khoram, M. Hosseini, A. Hadi, and M. Shishehsaz, "Bending analysis of bidirectional FGM Timoshenko nanobeam subjected to mechanical and magnetic forces and resting on Winkler–Pasternak foundation," International Journal of Applied Mechanics, vol. 12, no. 08, p. 2050093, 2020.
[109]     V. N. Van Do, M. T. Tran, and C.-H. Lee, "Nonlinear thermal buckling analyses of functionally graded plates by a mesh-free radial point interpolation method," Engineering Analysis with Boundary Elements, vol. 87, pp. 153-164, 2018.
[110]     S. R. Asemi, M. Mohammadi, and A. Farajpour, "A study on the nonlinear stability of orthotropic single-layered graphene sheet based on nonlocal elasticity theory," Latin American Journal of Solids and Structures, vol. 11, pp. 1541-1546, 2014.
[111]     A. Hadi, A. Daneshmehr, S. N. Mehrian, M. Hosseini, and F. Ehsani, "Elastic analysis of functionally graded Timoshenko beam subjected to transverse loading," Technical Journal of Engineering and Applied Sciences, vol. 3, no. 13, pp. 1246-1254, 2013.
[112]     T.-K. Nguyen, "A higher-order hyperbolic shear deformation plate model for analysis of functionally graded materials," International Journal of Mechanics and Materials in Design, vol. 11, no. 2, pp. 203-219, 2015.
[113]     M. Sekkal, B. Fahsi, A. Tounsi, and S. Mahmoud, "A new quasi-3D HSDT for buckling and vibration of FG plate," Structural engineering and mechanics: An international journal, vol. 64, no. 6, pp. 737-749, 2017.
[114]     T. Teo and K. Liew, "A differential quadrature procedure for three-dimensional buckling analysis of rectangular plates," International journal of solids and structures, vol. 36, no. 8, pp. 1149-1168, 1999.
Volume 53, Issue 3
September 2022
Pages 332-347
  • Receive Date: 27 April 2022
  • Revise Date: 19 June 2022
  • Accept Date: 26 June 2022
  • First Publish Date: 26 June 2022