×

Three-dimensional static and dynamic analysis of thick functionally graded plates by the meshless local Petrov-Galerkin (MLPG) method. (English) Zbl 1259.74093

Summary: In this paper, three dimensional (3D) static and dynamic analysis of thick functionally graded plates based on the meshless local Petrov-Galerkin (MLPG) is presented. Using the kinematics of a three-dimensional continuum, the local weak form of the equilibrium equations is derived. A weak formulation for the set of governing equations is transformed into local integral equations on local sub-domains using a Heaviside step function as test function. In this case, governing equations corresponding to the stiffness matrix do not involve any domain integration or singular integrals. Nodal points are distributed in the 3D analyzed domain and each node is surrounded by a cubic sub-domain to which a local integral equation is applied. The meshless approximation based on the three-dimensional moving least-square (MLS) is employed as shape function to approximate the field variable of scattered nodes in the problem domain. The Newmark time integration method is used to solve the system of coupled second-order ODEs. Effective material properties of the plate, made of two isotropic constituents with volume fractions varying only in the thickness direction, are computed using the Mori-Tanaka homogenization technique. Numerical examples for solving the static and dynamic response of elastic thick functionally graded plates are demonstrated. As a result, the distributions of the deflection and stresses through the plate thickness are presented for different material gradients and boundary conditions. The effects of the volume fractions of the constituents on the centroidal deflection are also investigated. The numerical efficiency of the proposed meshless method is illustrated by the comparison of results obtained from previous literatures.

MSC:

74S30 Other numerical methods in solid mechanics (MSC2010)
74K20 Plates
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Berger R, Kwon P, Dharan CKH. High speed centrifugal casting of metal matrix composites. In: the fifth international symposium on transport phenomena and dynamics of rotating machinery, Maui, Hawaii, May 8-11; 1994.
[2] Fukui, Y., Fundamental investigation of functionally gradient materials manufacturing system using centrifugal force, JSME int J ser III, 34, 144-148, (1991)
[3] Choy, K.L.; Felix, E., Functionally graded diamond-like carbon coatings on metallic substrates, Mater sci eng, 278, 162-169, (2000)
[4] Khor, K.A.; Gu, Y.W., Effects of residual stress on the performance of plasma sprayed functionally graded zro_{2}/nicocraly coatings, Mater sci eng, 277, 64-76, (2000)
[5] Lambros, A.; Narayanaswamy, A.; Santare, M.H.; Anlas, G., Manufacturing and testing of a functionally graded material, J eng mater technol, 121, 488-493, (1999)
[6] Breval, E.; Aghajanian, K.; Luszcz, S.J., Microstructure and composition of alumina/aluminum composites made by the directed oxidation of aluminum, J am ceram soc, 73, 2610-2614, (1990)
[7] Manor, E.; Ni, H.; Levi, C.G.; Mehrabian, R., Microstructure evaluation of sic/al_{2}O3/al alloy composite produced by melt oxidation, J am ceram soc, 26, 1777-1787, (1993)
[8] Mian, A.M.; Spencer, A.J.M., Exact solutions for functionally graded and laminated elastic materials, J mech phys solids, 46, 2283-2295, (1998) · Zbl 1043.74008
[9] Reddy, J.N., Analysis of functionally graded plates, Int J num meth eng, 47, 663-684, (2000) · Zbl 0970.74041
[10] Cheng, Z.Q.; Batra, R.C., Deflection relationships between the homogeneous Kirchhoff plate theory and different functionally graded plate theories, Arch mech, 52, 143-158, (2000) · Zbl 0972.74042
[11] Vel, S.S.; Batra, R.C., Exact solution for thermoelastic deformations of functionally graded thick rectangular plates, Aiaa j, 40, 1421-1433, (2002)
[12] Vel, S.S.; Batra, R.C., Three-dimensional analysis of transient thermal stresses in functionally graded rectangular plates, Int J solids struct, 40, 7181-7196, (2003) · Zbl 1076.74037
[13] Vel, S.S.; Batra, Three-dimensional exact solution for the vibration of functionally graded rectangular plates, J sound vib, 272, 703-730, (2004)
[14] Belytschko, T.; Lu, Y.Y.; Gu, L., Element-free Galerkin methods, Int J numer meth eng, 37, 229-256, (1994) · Zbl 0796.73077
[15] Liu, W.K.; Jun, S.; Zhang, Y., Reproducing kernel particle methods, Int J numer meth fluids, 20, 1081-1106, (1995) · Zbl 0881.76072
[16] Duarte, C.A.; Oden, J.T., An h – p adaptive method using clouds, Comput meth appl mech eng, 139, 237-262, (1996) · Zbl 0918.73328
[17] Babuska, I.; Melenk, J., The partition of unity method, Int J numer meth eng, 40, 727-758, (1997) · Zbl 0949.65117
[18] Wendland, H., Piecewise polynomial positive definite and compactly supported radial basis functions of minimal degree, Adv comput meth, 4, 389-396, (1995) · Zbl 0838.41014
[19] Nayroles, B.; Touzot, G.; Villon, P., Generalizing the finite element method: diffuse approximation and diffuse elements, Comput mech, 10, 307-318, (1992) · Zbl 0764.65068
[20] Sukumar, N.; Moran, B.; Belytschko, T., The natural element method in solid mechanics, Int J numer meth eng, 43, 839-887, (1998) · Zbl 0940.74078
[21] Lucy, L.B., A numerical approach to the testing of the fission hypothesis, Astronom J, 82, 1013-1024, (1977)
[22] Fasshauer GE. Solving partial differential equations by collocation with radial basis functions In: Proceedings of the 3rd international conference on curves and surfaces, surface fitting and multiresolution methods, vol 2; 1997. p. 131-38.
[23] Zhang, G.M.; Batra, R.C., Modified smoothed particle hydrodynamics method and its application to transient problems, Comput mech, 34, 137-146, (2004) · Zbl 1138.74422
[24] Atluri, S.N.; Zhu, T., A new meshless local petrov – galerkin (MLPG) approach in computational mechanics, Comput mech, 22, 117-127, (1998) · Zbl 0932.76067
[25] Atluri, S.N.; Kim, H.G.; Cho, J.Y., A critical assessment of the truly meshless local petrov – galerkin (MLPG), and local boundary integral equation (LBIE) methods, Comput mech, 24, 348-372, (1999) · Zbl 0977.74593
[26] Atluri, S.N.; Zhu, T., The meshless local petrov – galerkin (MLPG) approach for solving problems in elasto-statics, Comput mech, 25, 169-179, (2000) · Zbl 0976.74078
[27] Atluri, S.N.; Shen, S., The meshless local petrov – galerkin (MLPG) method, (2002), Tech Science Press · Zbl 1012.65116
[28] Batra, R.C.; Ching, H.K., Analysis of elastodynamic deformation near a crack/notch tip by the meshless local petrov – galerkin (MLPG) method, CMES: comput modeling eng sci, 3, 717-730, (2002) · Zbl 1152.74343
[29] Lin, H.; Atluri, S.N., The meshless local petrov – galerkin (MLPG) method for solving incompressible navier – stokes equations, CMES: comput modeling eng sci, 2, 117-142, (2001)
[30] Lin, H.; Atluri, S.N., Meshless local petrov – galerkin (MLPG) method for convection-diffusion problems, CMES: comput modeling eng sci, 1, 45-60, (2000)
[31] Sladek, J.; Sladek, V.; Atluri, S.N., A pure contour formulation for the meshless local boundary intergral equation method in thermoelasticity, CMES: comput modeling eng sci, 2, 423-433, (2001) · Zbl 1060.74069
[32] Atluri, S.N.; Cho, J.Y.; Kim, H.G., Analysis of thin beams, using the meshless local petrov – galerkin method, with generalized moving least squares interpolations, Comput mech, 24, 334-347, (1999) · Zbl 0968.74079
[33] Gu, Y.T.; Liu, G.R., A meshless local petrov – galerkin (MLPG) formulation for static and free vibration analyses of thin plates, CMES: comput modeling eng sci, 2, 463-476, (2001) · Zbl 1102.74310
[34] Long, S.; Atluri, S.N., A meshless local petrov – galerkin (MLPG) method for solving the bending problem of a thin plate, CMES: comput modeling eng sci, 3, 53-64, (2002) · Zbl 1147.74414
[35] Qian, L.F.; Batra, R.C.; Chen, L.M., Elastostatic deformations of a thick plate by using a higher-order shear and normal deformable plate theory and two meshless local Petrov-Galerkin (MLPG) methods, CMES: comput modeling eng sci, 4, 161-175, (2003) · Zbl 1148.74344
[36] Soric, J.; Li, Q.; Jarak, T.; Atluri, S.N., Meshless local Petrov-Galerkin (MLPG) formulation for analysis of thick plates, CMES: comput modeling eng sci, 6, 349-357, (2004) · Zbl 1075.74083
[37] Li, Q.; Soric, J.; Jarak, T.; Atluri, S.N., A locking-free meshless local petrov – galerkin formulation for thick and thin plates, J comput phys, 208, 116-133, (2005) · Zbl 1115.74369
[38] Xiao, J.R.; Batra, R.C.; Gilhooley, D.F.; Gillespie, J.W.; McCarthy, M.A., Analysis of thick plates by using a higher-order shear and normal deformable plate theory and MLPG method with radial basis functions, Comput meth appl mech eng, 196, 979-987, (2007) · Zbl 1120.74865
[39] Sladek, J.; Sladek, V.; Krivacek, J.; Wen, P.H.; Zhang, Ch., Meshless local petrov – galerkin (MLPG) method for reissner – mindlin plates under dynamic load, Comput meth appl mech eng, 196, 2681-2691, (2007) · Zbl 1173.74482
[40] Kim, H.G.; Atluri, S.N., Arbitrary placement of secondary nodes, and error control, in the meshless local petrov – galerkin (MLPG) method, CMES: comput modeling eng sci, 1, 11-32, (2000) · Zbl 1147.65326
[41] Ching, H.K.; Batra, R.C., Determination of crack tip fields in linear elastostatics by the meshless local petrov – galerkin (MLPG) method, CMES: comput modeling eng sci, 2, 273-290, (2001)
[42] Tang, Z.; Shen, S.; Atluri, S.N., Analysis of materials with strain-gradient effects: a meshless local petrov – galerkin (MLPG) approach, with nodal displacements only, CMES: comput modeling eng sci, 4, 177-196, (2003) · Zbl 1148.74346
[43] Ching, H.K.; Yen, S.C., Meshless local Petrov-Galerkin analysis for 2D functionally graded elastic solids under mechanical and thermal loads, Compos: part B, 36, 223-240, (2005)
[44] Ching, H.K.; Yen, S.C., Transient thermoelastic deformations of 2-D functionally graded beams under nonuniformly convective heat supply, Compos struct, 73, 381-393, (2006)
[45] Sladek, J.; Sladek, V.; Zhang, Ch., Stress analysis in anisotropic functionally graded materials by the MLPG method, Eng anal bound elem, 29, 597-609, (2005) · Zbl 1182.74258
[46] Qiana, L.F.; Batra, R.C.; Chena, L.M., 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, Compos: part B, 35, 685-697, (2004)
[47] Gilhooley, D.F.; Batra, R.C.; Xiao, J.R.; McCarthy, M.A.; Gillespie, J.W., Analysis of thick functionally graded plates by using higher-order shear and normal deformable plate theory and MLPG method with radial basis functions, Compos struct, 80, 539-552, (2007) · Zbl 1120.74865
[48] Sladek, J.; Sladek, V.; Solek, P., Elastic analysis in 3D anisotropic functionally graded solids by the MLPG, CMES: comput modeling eng sci, 43, 223-251, (2009) · Zbl 1232.74023
[49] Li, Q.; Shen, S.; Han, Z.D.; Atluri, S.N., Application of meshless local Petrov-Galerkin (MLPG) to problems with singularities, and material discontinuities, in 3-D elasticity, CMES: comput modeling eng sci, 4, 567-581, (2003) · Zbl 1108.74387
[50] Han, Z.D.; Atluri, S.N., Meshless local Petrov-Galerkin (MLPG) approaches for solving 3D problems in elasto-statics, CMES: comput modeling eng sci, 6, 168-188, (2004) · Zbl 1087.74654
[51] Han, Z.D.; Atluri, S.N., A meshless local Petrov-Galerkin (MLPG) approach for 3-dimensional elasto-dynamics, CMC: comput mater continua, 1, 129-140, (2004) · Zbl 1181.74152
[52] Han, Z.D.; Atluri, S.N., Truly meshless local Petrov-Galerkin (MLPG) solutions of traction & displacement bies, CMES: comput modeling eng sci, 4, 665-678, (2003) · Zbl 1064.74175
[53] Atluri, S.N.; Shen, S., The meshless local Petrov-Galerkin (MLPG) method: a simple & less costly alternative to the finite element and boundary element methods, CMES: comput modeling eng sci, 3, 11-52, (2002) · Zbl 0996.65116
[54] Newmark, N.M., A method of computation for structural dynamics, J eng mech div, ASCE, 85, 67-94, (1959)
[55] Belytschko, T.; Guo, Y.; Liu, W.K.; Xiao, S.P., A unified stability of meshless particle methods, Int J num meth eng, 48, 1359-1400, (2000) · Zbl 0972.74078
[56] LIU, G.R.; GU, Y.T., An introduction to meshfree methods and their programming, (2005), Springer Dordrecht, Netherlands
[57] Frohlich, H.; Sack, R., Theory of the rheological properties of dispersions, Proc R soc, A185, 415-430, (1946)
[58] Hill, R., A self-consistent mechanics of composite materials, J mech phys solids, 13, 213-222, (1965)
[59] Mori, T.; Tanaka, K., Average stress in matrix and average elastic energy of materials with misfitting inclusions, Acta metall, 21, 571-574, (1973)
[60] Jiang, B.; Batra, R.C., Effective properties of a piezocomposite containing shape memory alloy and inert inclusions, Continuum mech thermodyn, 14, 87-111, (2002) · Zbl 1100.74614
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.