# zbMATH — the first resource for mathematics

A moving Kriging meshfree method with naturally stabilized nodal integration for analysis of functionally graded material sandwich plates. (English) Zbl 1397.74202
Summary: This paper presents a moving Kriging meshfree method based on a naturally stabilized nodal integration (NSNI) for bending, free vibration and buckling analyses of isotropic and sandwiched functionally graded plates within the framework of higher-order shear deformation theories. A key feature of the present formulation is to develop a NSNI technique for the moving Kriging meshfree method. Using this scheme, the strains are directly evaluated at the same nodes as the direct nodal integration (DNI). Importantly, the computational approach alleviates instability solutions in the DNI and significantly decreases the computational cost from using the traditional high-order Gauss quadrature. Being different from the stabilized conforming nodal integration scheme which uses the divergence theorem to evaluate the strains by boundary integrations, the NSNI adopts a naturally implicit gradient expansion. The NSNI is then integrated into the Galerkin weak form for deriving the discrete system equations. Due to satisfying the Kronecker delta function property of the moving Kriging integration shape function, the enforcement of essential boundary conditions in the present method is similar to the finite element method. Through numerical examples, the effects of geometries, stiffness ratios, volume fraction and boundary conditions are studied to prove the efficiency of the present approach.

##### MSC:
 74S30 Other numerical methods in solid mechanics (MSC2010) 74K20 Plates 74A40 Random materials and composite materials
##### Software:
KOKO Mesh Generator; MUL2
Full Text:
##### References:
 [1] Brischetto, S; Tornabene, F; Fantuzzi, N; Viola, E, 3D exact and 2D generalized differential quadrature models for free vibration analysis of functionally graded plates and cylinders, Meccanica, 51, 2059-2098, (2016) · Zbl 1386.74083 [2] Swaminathan, K; Naveenkumar, DT; Zenkour, AM; Carrera, E, Stress, vibration and buckling analyses of FGM plates—a state-of-the-art review, Compos. Struct., 120, 10-31, (2015) [3] Pan, E, Exact solution for functionally graded anisotropic elastic composite laminates, J. Compos. Mater., 37, 1903-1920, (2003) [4] Pagano, NJ, Exact solutions for rectangular bidirectional composites and sandwich plates, J. Compos. Mater., 4, 20-34, (1970) [5] Vel, SS; Batra, RC, Three-dimensional exact solution for the vibration of functionally graded rectangular plates, J. Sound Vib., 272, 703-730, (2004) [6] Kashtalyan, M, Three-dimensional elasticity solution for bending of functionally graded rectangular plates, Eur. J. Mech. A. Solids, 23, 853-864, (2004) · Zbl 1058.74569 [7] Zenkour, AM, Benchmark trigonometric and 3-D elasticity solutions for an exponentially graded thick rectangular plate, Arch. Appl. Mech., 77, 197-214, (2006) · Zbl 1161.74436 [8] Reddy, JN; Cheng, ZQ, Three-dimensional thermomechanical deformations of functionally graded rectangular plates, Eur. J. Mech. A. Solids, 20, 841-855, (2001) · Zbl 1002.74061 [9] Cheng, ZQ; Batra, RC, Deflection relationships between the homogeneous Kirchhoff plate theory and different functionally graded plate theories, Arch. Mech., 52, 143-158, (2000) · Zbl 0972.74042 [10] Natarajan, S; Baiz, PM; Bordas, SPA; Rabczuk, T; Kerfriden, P, Natural frequencies of cracked functionally graded material plates by the extended finite element method, Compos. Struct., 93, 3082-3092, (2011) [11] Natarajan, S; Ferreira, AJM; Bordas, SPA; Carrera, E; Cinefra, M, Analysis of composite plates by a unified formulation-cell based smoothed finite element method and field consistent elements, Compos. Struct., 105, 75-81, (2013) [12] Rodrigues, JD; Natarajan, S; Ferreira, AJM; Carrera, E; Cinefra, M; Bordas, SPA, Analysis of composite plates through cell-based smoothed finite element and 4-noded mixed interpolation of tensorial components techniques, Comput. Struct., 135, 83-87, (2014) [13] Nguyen-Xuan, H; Tran, VL; Nguyen-Thoi, T; Vu-Do, HC, Analysis of functionally graded plates using an edge-based smoothed finite element method, Compos. Struct., 93, 3019-3039, (2011) [14] Do, VVN; Thai, CH, A modified Kirchhoff plate theory for analyzing thermo-mechanical static and buckling responses of functionally graded material plates, Thin Walled Struct., 117, 113-126, (2017) [15] Nguyen, NT; Hui, D; Lee, J; Nguyen-Xuan, H, An efficient computational approach for size-dependent analysis of functionally graded nanoplates, Comput. Methods Appl. Mech. Eng., 297, 191-218, (2015) · Zbl 1423.74550 [16] Ambartsumian, SA, On the theory of bending plates, Izv Otd Tech Nauk ANSSSR, 5, 269-277, (1958) [17] Reddy, JN, Analysis of functionally graded plates, Int. J. Numer. Methods Eng., 684, 663-684, (2000) · Zbl 0970.74041 [18] Nguyen-Xuan, H; Thai, HC; Nguyen-Thoi, T, Isogeometric finite element analysis of composite sandwich plates using a higher order shear deformation theory, Compos. Part B: Eng., 55, 558-574, (2013) [19] Nguyen, NT; Thai, HC; Nguyen-Xuan, H, On the general framework of high order shear deformation theories for laminated composite plate structures: a novel unified approach, Int. J. Mech. Sci., 110, 242-255, (2016) [20] Soldatos, KP, A transverse shear deformation theory for homogenous monoclinic plates, Acta Mech., 94, 195-220, (1992) · Zbl 0850.73130 [21] Touratier, M, An efficient standard plate theory, Int. J. Eng. Sci., 29, 745-752, (1991) · Zbl 0825.73299 [22] Thai, HC; Ferreira, AJM; Rabczuk, T; Bordas, SPA; Nguyen-Xuan, H, Isogeometric analysis of laminated composite and sandwich plates using a new inverse trigonometric shear deformation theory, Eur. J. Mech. A. Solids, 43, 89-108, (2014) · Zbl 1406.74453 [23] Thai, HC; Kulasegaram, S; Tran, VL; Nguyen-Xuan, H, Generalized shear deformation theory for functionally graded isotropic and sandwich plates based on isogeometric approach, Comput. Struct., 141, 94-112, (2014) [24] Karama, M; Afaq, KS; Mistou, S, Mechanical behavior of laminated composite beam by new multi-layered laminated composite structures model with transverse shear stress continuity, Int. J. Solids Struct., 40, 1525-1546, (2003) · Zbl 1087.74579 [25] Aydogdu, M, A new shear deformation theory for laminated composite plates, Compos. Struct., 89, 94-101, (2009) [26] Zenkour, AM, A comprehensive analysis of functionally graded sandwich plates: part 1 deflection and stresses, Int. J. Solids Struct., 42, 5224-5242, (2005) · Zbl 1119.74471 [27] Zenkour, AM, A comprehensive analysis of functionally graded sandwich plates: part 2 buckling and free vibration, Int. J. Solids Struct., 42, 5243-5258, (2005) · Zbl 1119.74472 [28] Nguyen, TN; Thai, CH; Nguyen-Xuan, H, A novel computational approach for functionally graded isotropic and sandwich plate structures based on a rotation-free meshfree method, Thin Walled Struct., 107, 473-488, (2016) [29] Tran, VL; Thai, HC; Nguyen-Xuan, H, An isogeometric finite element formulation for thermal buckling analysis of functionally graded plates, Finite Elem. Anal. Des., 73, 65-76, (2013) [30] Neves, AMA; Ferreira, AJM; Carrera, E; Roque, CMC; Cinefra, M; Jorge, RMN; Soares, CMM, A quasi-3D sinusoidal shear deformation theory for the static and free vibration analysis of functionally graded plates, Compos. Part B: Eng., 43, 711-725, (2012) · Zbl 1347.74038 [31] Neves, AMA; Ferreira, AJM; Carrera, E; Cinefra, M; Roque, CMC; Jorge, RMN; Soares, CMM, 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, Compos. Part B: Eng., 44, 657-674, (2013) · Zbl 1347.74038 [32] Zenkour, AM, Bending analysis of functionally graded sandwich plates using a simple four-unknown shear and normal deformations theory, J. Sandw. Struct. Mater., 15, 629-656, (2013) [33] Zenkour, AM, A simple four-unknown refined theory for bending analysis of functionally graded plates, Appl. Math. Model., 37, 9041-9051, (2013) · Zbl 1426.74208 [34] Thai, HC; Zenkour, AM; Wahab, MA; Nguyen-Xuan, H, A simple four unknown shear and normal deformations theory for functionally graded isotropic and sandwich plates based on isogeometric analysis, Compos. Struct., 139, 77-95, (2016) [35] Thai, HT; Kim, SE, A simple quasi-3D sinusoidal shear deformation theory for functionally graded plates, Compos. Struct., 99, 172-180, (2013) [36] Mantari, JL; Soares, CG, A trigonometric plate theory with 5-unknowns and stretching effect for advanced composite plates, Compos. Struct., 107, 396-405, (2014) [37] Nguyen-Thanh, N; Rabczuk, T; Nguyen-Xuan, H; Bordas, SPA, A smoothed finite element method for shell analysis, Comput. Methods Appl. Mech. Eng., 198, 165-177, (2008) · Zbl 1194.74453 [38] Nguyen-Xuan, H; Rabczuk, T; Nguyen-Thanh, N; Nguyen-Thoi, T; Bordas, SPA, A node-based smoothed finite element method with stabilized discrete shear gap technique for analysis of Reissner-Mindlin plates, Comput. Mech., 46, 679-701, (2010) · Zbl 1260.74029 [39] Nguyen-Thanh, N; Rabczuk, T; Nguyen-Xuan, H; Bordas, SPA, An alternative alpha finite element method with discrete shear gap technique for analysis of isotropic Mindlin-Reissner plates, Finite Elem. Anal. Des., 47, 519-535, (2011) [40] Thai, CH; Tran, LV; Tran, DT; Nguyen-Thoi, T; Nguyen-Xuan, H, Analysis of laminated composite plates using higher-order shear deformation plate theory and node-based smoothed discrete shear gap method, Appl. Math. Model., 36, 5657-5677, (2012) · Zbl 1254.74079 [41] Natarajan, S; Ferreira, AJM; Bordas, SPA; Carrera, E; Cinefra, M; Zenkour, AM, Analysis of functionally graded material plates using triangular elements with cell-based smoothed discrete shear gap method, Math. Probl. Eng., 2014, 247932, (2014) · Zbl 1407.74060 [42] Nguyen-Thanh, N; Kiendl, J; Nguyen-Xuan, H; Wüchner, R; Bletzinger, KU; Bazilevs, Y; Rabczuk, T, Rotation free isogeometric thin shell analysis using PHT-splines, Comput. Methods Appl. Mech. Eng., 200, 3410-3424, (2011) · Zbl 1230.74230 [43] Gu, L, Moving Kriging interpolation and element-free Galerkin method, Int. J. Numer. Methods Eng., 56, 1-11, (2003) · Zbl 1062.74652 [44] Chen, JS; Wu, CT; Yoon, S; You, Y, A stabilized conforming nodal integration for Galerkin mesh-free methods, Int. J. Numer. Methods Eng., 50, 435-466, (2001) · Zbl 1011.74081 [45] Puso, M; Chen, JS; Zywicz, E; Elmer, W, Meshfree and finite element nodal integration methods, Int. J. Numer. Methods Eng., 74, 416-446, (2008) · Zbl 1159.74456 [46] Hillman, M; Chen, JS; Chi, SW, Stabilized and variationally consistent nodal integration for meshfree modeling of impact problems, Comput. Part. Mech., 1, 245-256, (2014) [47] Beissel, S; Belytschko, T, Nodal integration of the element-free Galerkin method, Comput. Methods Appl. Mech. Eng., 139, 49-74, (1996) · Zbl 0918.73329 [48] Nagashima, T, Node-by-node meshless approach and its applications to structural analyses, Int. J. Numer. Methods Eng., 46, 341-385, (1999) · Zbl 0965.74079 [49] Liu, GR; Zhang, GY; Wang, YY; Zhong, ZH; Li, GY; Han, X, A nodal integration technique for meshfree radial point interpolation method (NI-RPIM), Int. J. Solids Struct., 44, 3840-3890, (2007) · Zbl 1135.74050 [50] Wu, CT; Koishi, M; Hu, W, A displacement smoothing induced strain gradient stabilization for the meshfree Galerkin nodal integration method, Comput. Mech., 56, 19-37, (2015) · Zbl 1329.74292 [51] Hillman, M; Chen, JS, An accelerated, convergent, and stable nodal integration in Galerkin meshfree methods for linear and nonlinear mechanics, Int. J. Numer. Methods Eng., 107, 603-630, (2016) · Zbl 1352.74119 [52] Li, Q; Lu, V; Kou, K, Three-dimensional vibration analysis of functionally graded material sandwich plates, J. Sound Vib., 311, 498-515, (2008) [53] Thai, HC; Do, NVV; Nguyen-Xuan, H, An improved moving Kriging meshfree method for analysis of isotropic and sandwich functionally graded material plates using higher-order shear deformation theory, Eng. Anal. Bound. Elem., 64, 122-136, (2016) · Zbl 1403.74034 [54] Thai, HC; Nguyen, NT; Rabczuk, T; Nguyen-Xuan, H, An improved moving Kriging meshfree method for plate analysis using a refined plate theory, Comput. Struct., 176, 34-49, (2016) [55] Liu, G.R.: Meshfree Methods: Moving Beyond the Finite Element Method. CRC Press, Boca Raton (2003) [56] Thai, CH; Ferreira, AJM; Nguyen-Xuan, H, Naturally stabilized nodal integration meshfree formulations for analysis of laminated composite and sandwich plates, Compos. Struct., 178, 260-276, (2017) [57] Thai, CH; Ferreira, AJM; Rabczuk, T; Nguyen-Xuan, H, A naturally stabilized nodal integration meshfree formulation for carbon nanotube-reinforced composite plate analysis, Eng. Anal. Bound. Elem., (2017) · Zbl 1403.74322 [58] Liu, WK; Ong, JS; Uras, RA, Finite element stabilization matrices—a unification approach, Comput. Methods Appl. Mech. Eng., 53, 13-46, (1985) · Zbl 0553.73065 [59] Koko, J, A Matlab mesh generator for the two-dimensional finite element method, Appl. Math. Comput., 250, 650-664, (2015) · Zbl 1328.65245 [60] Carrera, E; Brischetto, S; Cinefra, M; Soave, M, Effects of thickness stretching in functionally graded plates and shells, Compos. Part B: Eng., 42, 123-133, (2011) [61] Li, XY; Ding, HJ; Chen, WQ, Elasticity solutions for a transversely isotropic functionally graded circular plate subject to an axisymmetric transverse load qr$$^{k}$$, Int. J. Solids Struct., 45, 191-210, (2008) · Zbl 1167.74486 [62] Reddy, JN; Wang, CM; Kitipornchai, S, Axisymmetric bending of functionally graded circular and annular plates, Eur. J. Mech. A. Solids, 18, 185-199, (1999) · Zbl 0942.74044 [63] Yin, S; Hale, JS; Yu, T; Bui, TQ; Bordas, SPA, Isogeometric locking-free plate element: a simple first order shear deformation theory for functionally graded plates, Compos. Struct., 118, 121-138, (2014) [64] Reddy, J.N.: Theory and Analysis of Elastic Plates and Shells. CRC Press, Boca Raton (2007) [65] Nguyen, KD; Nguyen-Xuan, H, An isogeometric finite element approach for three-dimensional static and dynamic analysis of functionally graded material plate structures, Compos. Struct., 132, 423-439, (2015) [66] Natarajan, S; Manickam, G, Bending and vibration of functionally graded material sandwich plates using an accurate theory, Finite Elem. Anal. Des., 57, 32-42, (2012) [67] Tornabene, F, Free vibration analysis of functionally graded conical, cylindrical shell and annular plate structures with a four-parameter power-law distribution, Comput. Methods Appl. Mech. Eng., 198, 2911-2935, (2009) · Zbl 1229.74062 [68] Ma, LS; Wang, TJ, Relationship between axisymmetric bending and buckling solutions of FGM circular plates based on third-order plate theory and classical plate theory, Int. J. Solids Struct., 41, 85-101, (2004) · Zbl 1076.74036 [69] Saidi, AR; Rasouli, A; Sahraee, S, Axisymmetric bending and buckling analysis of thick functionally graded circular plates using unconstrained third-order shear deformation plate theory, Compos. Struct., 89, 110-119, (2009)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.