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An extended cell-based smoothed three-node Mindlin plate element (XCS-MIN3) for free vibration analysis of cracked FGM plates. (English) Zbl 1404.74167

Summary: A cell-based smoothed three-node Mindlin plate element (CS-MIN3) was recently proposed and proven to be robust for static and free vibration analyses of Mindlin plates. The method significantly improves the accuracy of the solution due to a softening effect of the cell-based strain smoothing technique. In addition, it is very flexible to apply for arbitrary complicated geometric domains due to using only three-node triangular elements which can be easily generated automatically. However so far, the CS-MIN3 has been only developed for isotropic material and for analyzing intact structures without possessing internal cracks. Hence, the paper tries to extend the CS-MIN3 by integrating itself with functionally graded material (FGM) and enriched functions of the extended finite element method (XFEM) to give a so-called extended cell-based smoothed three-node Mindlin plate (XCS-MIN3) for free vibration analysis of cracked FGM plates. Three numerical examples with different conditions are solved and compared with previously published results to illustrate the accuracy and reliability of the XCS-MIN3 for free vibration analysis of cracked FGM plates.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74H45 Vibrations in dynamical problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74K20 Plates

Software:

XFEM
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Full Text: DOI

References:

[1] Aggarwala, B. D. and Ariel, P. D. [1981] “ Vibration and bending of a cracked plate,” Rozpr. Inz. — Eng. Trans.29, 295-310. · Zbl 0498.73064
[2] Areias, P. and Rabczuk, T. [2013] “ Finite strain fracture of plates and shells with configurational forces and edge rotations,” Int. J. Numer. Methods Eng.94, 1099-1122. · Zbl 1352.74316
[3] Areias, P., Rabczuk, T. and Camanho, P. P. [2013a] “ Initially rigid cohesive laws and fracture based on edge rotations,” Comput. Mech.52, 931-947. · Zbl 1311.74103
[4] Areias, P., Rabczuk, T. and Dias-da-Costa, D. [2013b], “ Element-wise fracture algorithm based on rotation of edges,” Eng. Fract. Mech.110, 113-137.
[5] Areias, P., Rabczuk, T. and Camanho, P. P. [2014] “ Finite strain fracture of 2D problems with injected anisotropic softening elements,” Theor. Appl. Fract. Mech.72, 50-63.
[6] Babuška, I., Caloz, G. and Osborn, J. [1994] “ Special sinite element methods for a class of second order elliptic problems with rough coefficients,” SIAM J. Numer. Anal.31, 945-981. · Zbl 0807.65114
[7] Babuška, I. and Melenk, J. [1997] “ The partition of unity finite element method,” Int. J. Numer. Methods Eng.40, 727-758. · Zbl 0949.65117
[8] Babuška, I., Nistor, V. and Tarfulea, N. [2008] “ Generalized finite element method for second-order elliptic operators with Dirichlet boundary conditions,” J. Comput. Appl. Math.218, 175-183. · Zbl 1153.65106
[9] Belytschko, T.et al., [2001] “ Arbitrary discontinuities in finite elements,” Int. J. Numer. Methods Eng.50, 993-1013. · Zbl 0981.74062
[10] Belytschko, T. and Black, T. [1999] “ Elastic crack growth in finite elements with minimal remeshing,” Int. J. Numer. Methods Eng.45, 601-620. · Zbl 0943.74061
[11] Bordas, S. P. A.et al., [2010] “ Strain smoothing in FEM and XFEM,” Comput. Struct.88, 1419-1443.
[12] Bordas, S. P. A.et al., [2011] “ On the performance of strain smoothing for quadratic and enriched finite element approximations (XFEM/GFEM/PUFEM),” Int. J. Numer. Methods Eng.86, 637-666. · Zbl 1216.74019
[13] Chau-Dinh, T.et al., [2012] “ Phantom-node method for shell models with arbitrary cracks,” Comput. Struct.92-93, 242-256.
[14] Chen, L.et al., [2012] “ Extended finite element method with edge-based strain smoothing (ESm-XFEM) for linear elastic crack growth,” Comput. Methods Appl. Mech. Eng.209-212, 250-265. · Zbl 1243.74170
[15] Cui, X. Y. et al., [2008a] “ A rotation free formulation for static and free vibration analysis of thin beams using gradient smoothing technique,” C. Model. Eng. Sci38, 217-229. · Zbl 1357.74059
[16] Cui, X. Y.et al., [2008b] “ A smoothed finite element method (SFEM) for linear and geometrically nonlinear analysis of plates and shells,” Comput. Model. Eng. Sci.28, 109-125. · Zbl 1232.74099
[17] Cui, X. Y.et al., [2009a] “ Analysis of plates and shells using an edge-based smoothed finite element method,” Comput. Mech.45, 141-156. · Zbl 1202.74165
[18] Cui, X. Y., Liu, G.-R. and Li, G. Y. [2009b] “ A smoothed Hermite radial point interpolation method for thin plate analysis,” Arch. Appl. Mech.81, 1-18. · Zbl 1271.74425
[19] Cui, X. Y., Liu, G. R. and Li, G. Y. [2010] “ Analysis of mindlin-reissner plates using cell-based smoothed radial point interpolation method,” Int. J. Appl. Mech.2, 653-680.
[20] Cui, X. Y.et al., [2011a] “ A thin plate formulation without rotation DOFs based on the radial point interpolation method and triangular cells,” Int. J. Numer. Methods Eng.85, 958-986. · Zbl 1217.74143
[21] Cui, X. Y., Lin, S. and Li, G. Y. [2011b] “ Nodal integration thin plate formulation using linear interpolation and triangular cells,” Int. J. Comput. Methods.8, 813-824. · Zbl 1245.74040
[22] Cui, X. Y., Liu, G. R. and Li, G. Y. [2011c] “ Bending and vibration responses of laminated composite plates using an edge-based smoothing technique,” Eng. Anal. Bound. Elem.35, 818-826. · Zbl 1259.74029
[23] Dolbow, J., Moës, N. and Belytschko, T. [2000] “ Modeling fracture in Mindlin-Reissner plates with the extended finite element method,” Int. J. Solids Struct.37, 7161-7183. · Zbl 0993.74061
[24] Feng, S. Z.et al., [2013] “ Thermo-mechanical analysis of functionally graded cylindrical vessels using edge-based smoothed finite element method,” Int. J. Press. Vessel. Pip.111-112, 302-309.
[25] Ferreira, A. J. M.et al., [2005] “ Static analysis of functionally graded plates using third-order shear deformation theory and a meshless method,” Compos. Struct.69, 449-457.
[26] Ferreira, A. J. M.et al., [2006] “ Natural frequencies of functionally graded plates by a meshless method,” Compos. Struct.75, 593-600.
[27] Ghorashi, S. S.et al., [2015] “ T-spline based XIGA for fracture analysis of orthotropic media,” Comput. Struct.147, 138-146.
[28] He, X. Q.et al., [2001] “ Active control of FGM plates with integrated piezoelectric sensors and actuators,” Int. J. Solids Struct.38, 1641-1655. · Zbl 1012.74047
[29] Huang, C. S., Leissa, A. W. and Chan, C. W. [2011a] “ Vibrations of rectangular plates with internal cracks or slits,” Int. J. Mech. Sci.53, 436-445.
[30] Huang, C. S., Leissa, A. W. and Li, R. S. [2011b] “ Accurate vibration analysis of thick, cracked rectangular plates,” J. Sound Vib.330, 2079-2093.
[31] Huang, C. S., McGee, O. G. and Chang, M. J. [2011c] “ Vibrations of cracked rectangular FGM thick plates,” Compos. Struct.93, 1747-1764.
[32] Huang, C. S., McGee, O. G. and Wang, K. P. [2013] “ Three-dimensional vibrations of cracked rectangular parallelepipeds of functionally graded material,” Int. J. Mech. Sci.70, 1-25.
[33] Khadem, S. E. and Rezaee, M. [2000] “ Introduction of modified comparison functions for vibration analysis of a rectangular cracked plate,” J. Sound Vib.236, 245-258.
[34] Kitipornchai, S.et al., [2009] “ Nonlinear vibration of edge cracked functionally graded Timoshenko beams,” J. Sound Vib.324, 962-982.
[35] Lee, H. P. and Lim, S. P. [1993] “ Vibration of cracked rectangular plates including transverse shear deformation and rotary inertia,” Comput. Struct.49, 715-718. · Zbl 0800.73539
[36] Liew, K. M., Hung, K. C. and Lim, M. K. [1994] “ A solution method for analysis of cracked plates under vibration,” Eng. Fract. Mech.48, 393-404.
[37] Liu, G. R.et al., [2009] “ A node-based smoothed finite element method (NS-FEM) for upper bound solutions to solid mechanics problems,” Comput. Struct.87, 14-26.
[38] Liu, G. R., Dai, K. Y. and Nguyen, T. T. [2007] “ A smoothed finite element method for mechanics problems,” Comput. Mech.39, 859-877. · Zbl 1169.74047
[39] Liu, G. R. and Nguyen-Thoi, T. [2010] Smoothed Finite Element Methods (Taylor and Francis Group, NewYork).
[40] Luong-Van, H.et al., [2014] “ A cell-based smoothed finite element method using three-node shear-locking free Mindlin plate element (CS-FEM-MIN3) for dynamic response of laminated composite plates on viscoelastic foundation,” Eng. Anal. Bound. Elem.42, 8-19. · Zbl 1297.74122
[41] Lynn, P, K. N. [1967] “ Free vibrations of thin rectangular plates having narrow cracks with simply supported edges,” Dev. Mech.4, 928-991.
[42] Melenk, J. M. [1995] On generalized finite element methods, Ph.D. dissertation, University of Maryland, College Park.
[43] Natarajan, S.et al., [2011] “ Natural frequencies of cracked functionally graded material plates by the extended finite element method,” Compos. Struct.93, 3082-3092.
[44] Nguyen-Thoi, T.et al., [2009a] “ A face-based smoothed finite element method (FS-FEM) for 3D linear and geometrically nonlinear solid mechanics problems using 4-node tetrahedral elements,” Int. J. Numer. Methods Eng.78, 324-353. · Zbl 1183.74299
[45] Nguyen-Thoi, T.et al., [2009b] “ A face-based smoothed finite element method (FS-FEM) for visco-elastoplastic analyses of 3D solids using tetrahedral mesh,” Comput. Methods Appl. Mech. Eng.198, 3479-3498. · Zbl 1230.74193
[46] Nguyen-Thoi, T.et al., [2009c] “ An edge-based smoothed finite element method for visco-elastoplastic analyses of 2D solids using triangular mesh,” Comput. Mech.45, 23-44. · Zbl 1398.74382
[47] Nguyen-Thoi, T.et al., [2012] “ A cell-based smoothed three-node Mindlin plate element (CS-MIN3) for static and free vibration analyses of plates,” Comput. Mech.35, 466-485.
[48] Nguyen-Thoi, T.et al., [2013a] “ A cell-based smoothed discrete shear gap method (CS-DSG3) using triangular elements for static and free vibration analyses of shell structures,” Int. J. Mech. Sci.74, 32-45.
[49] Nguyen-Thoi, T.et al., [2013b] “ A cell-based smoothed three-node Mindlin plate element (CS-MIN3) for static and free vibration analyses of plates,” Comput. Mech.51, 65-81. · Zbl 1294.74064
[50] Nguyen-Thanh, N.et al., [2015] “ An extended isogeometric thin shell analysis based on Kirchhoff-Love theory,” Comput. Methods Appl. Mech. Eng.284, 265-291. · Zbl 1423.74811
[51] Nguyen-Xuan, H.et al., [2009] “ An edge-based smoothed finite element method for analysis of two-dimensional piezoelectric structures,” Smart Mater. Struct.18, 065015, 12 pp.
[52] Nguyen-Xuan, H.et al., [2013] “ An adaptive singular ES-FEM for mechanics problems with singular field of arbitrary order,” Comput. Methods Appl. Mech. Eng.253, 252-273. · Zbl 1297.74126
[53] Phung-Van, P.et al., [2013] “ A cell-based smoothed discrete shear gap method (CS-DSG3) based on the C0-type higher-order shear deformation theory for static and free vibration analyses of functionally graded plates,” Comput. Mater. Sci.79, 857-872.
[54] Qian, G. L., Gu, S. N. and Jiang, J. S. [1991] “ A finite element model of cracked plates and application to vibration problems,” Comput. Struct.39, 483-487.
[55] Rabczuk, T. and Areias, P. M. A. [2006] “ A meshfree thin shell for arbitrary evolving cracks based on an external enrichment,” Comput. Model. Eng. Sci.16, 115-130.
[56] Rabczuk, T., Areias, P. M. A. and Belytschko, T. [2007] “ A meshfree thin shell method for nonlinear dynamic fracture,” Int. J. Numer. Methods Eng.72, 524-548. · Zbl 1194.74537
[57] Reddy, J. N. [2000] “ Analysis of functionally graded plates,” Int. J. Numer. Methods Eng.47, 663-684. · Zbl 0970.74041
[58] Reddy, J. N. [2003] Mechanics of Laminated Composite Plates and Shell: Theory and Analysis, 2nd Edition (CRC Press, Boca Raton, FL).
[59] Simone, A., Duarte, C. A. and Van der Giessen, E. [2006] “ A Generalized Finite Element Method for polycrystals with discontinuous grain boundaries,” Int. J. Numer. Methods Eng.67, 1122-1145. · Zbl 1113.74076
[60] Solecki, R. [1985] “ Bending vibration of a rectangular plate with arbitrarily located rectilinear crack,” Eng. Fract. Mech.22, 687-695.
[61] Stahl, B. and Keer, L. M. [1972] “ Vibration and stability of cracked rectangular plates,” Int. J. Solids Struct.8, 69-91. · Zbl 0227.73107
[62] Tessler, A. and Hughes, T. J. R. [1985] “ A three-node mindlin plate element with improved transverse shear,” Comput. Methods Appl. Mech. Eng.50, 71-101. · Zbl 0562.73069
[63] Tran, L. V, Thai, C. H. and Nguyen-Xuan, H. [2013] “ An isogeometric finite element formulation for thermal buckling analysis of functionally graded plates,” Finite Elem. Anal. Des.73, 65-76.
[64] Vel, S. S. and Batra, R. C. [2004] “ Three-dimensional exact solution for the vibration of functionally graded rectangular plates,” J. Sound Vib.272, 703-730.
[65] Ventura, G. [2006] “ On the elimination of quadrature subcells for discontinuous functions in the eXtended Finite-Element Method,” Int. J. Numer. Methods Eng.66, 761-795. · Zbl 1110.74858
[66] Vu-Bac, N.et al., [2011] “ A node-based smoothed extended finite element method (NS-XFEM) for fracture analysis,” Comput. Model. Eng. Sci.73, 331. · Zbl 1231.74444
[67] Vu-Bac, N.et al., [2013] “ A phantom-node method with edge-based strain smoothing for linear elastic fracture mechanics,” J. Appl. Math.2013. · Zbl 1271.74418
[68] Wang, G.et al., [2015] “ A coupled smoothed finite element method (S-FEM) for structural-acoustic analysis of shells,” Eng. Anal. Bound. Elem.61, 207-217. · Zbl 1403.74325
[69] Wu, G.-Y. and Shih, Y.-S. [2005] “ Dynamic instability of rectangular plate with an edge crack,” Comput. Struct.84, 1-10.
[70] Xiangyang, C., Gang, Z. and Guangyao, L. [2010] “ An efficient triangular shell element based on edge-based smoothing technique for sheet metal forming simulation,” AIP Conf. Proc. Vol. 1252, pp. 1118-1125.
[71] Yang, J. and Shen, H.-S. [2001] “ Dynamic response of initially stressed functionally graded rectangular thin plates,” Compos. Struct.54, 497-508.
[72] Zheng, G.et al., [2011] “ An edge-based smoothed triangle element for nonlinear explicit dynamic analysis of shells,” Comput. Mech.48, 65-80. · Zbl 1398.74434
[73] Zhuang, X., Augarde, C. E. and Mathisen, K. M. [2012] “ Fracture modeling using meshless methods and level sets in 3D: Framework and modeling,” Int. J. Numer. Methods Eng.92, 969-998. · Zbl 1352.74312
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