×

Free vibration analysis of functionally graded anisotropic microplates using modified strain gradient theory. (English) Zbl 1464.74074

Summary: This study presents a size dependent model using the higher-order shear deformation theory (HSDT) in conjunction with modified strain gradient theory (MSGT) for free vibration analysis of functionally graded (FG) anisotropic microplates. The FG anisotropic material is made of hexagonal beryllium crystals which can be considered as a hexagonal one. To consider size effects, three material length scale parameters (MLSPs) are added into the elastic constants of the anisotropic material. Based on the principle of virtual work, discretized governing equations of the FG hexagonal microplates are obtained. Subsequently, the natural frequency of the FG anisotropic microplates is determined by using isogeometric analysis (IGA). Numerical results show that the natural frequency of the FG anisotropic microplates is influenced by the geometry, boundary condition, length-to-thickness ratio, exponential factor and material length scale parameter. The results of classical HSDT model can be restored from the present model when three MLSPs equal to zero. Moreover, the differences of the natural frequency predicted by MSGT and classical HSDT can grow up more than 4.5 times.

MSC:

74H45 Vibrations in dynamical problems in solid mechanics
65D07 Numerical computation using splines
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74E10 Anisotropy in solid mechanics
74K20 Plates
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Eringen, A. C., Nonlocal polar elastic continua, Int J Eng Sci, 10, 1, 1-16 (1972) · Zbl 0229.73006
[2] Salehipour, H.; Shahidi, A. R.; Nahvi, H., Modified nonlocal elasticity theory for functionally graded materials, Int J Eng Sci, 90, 44-57 (2015) · Zbl 1423.74138
[3] Mindlin, R. D.; Eshel, N. N., On first strain-gradient theories in linear elasticity, Int J Solids Struct, 4, 1, 109-124 (1968) · Zbl 0166.20601
[4] Yang, F.; Chong, A. C.M.; Lam, D. C.C.; Tong, P., Couple stress based strain gradient theory for elasticity, Int J Solids Struct, 39, 10, 2731-2743 (2002) · Zbl 1037.74006
[5] Gurtin, M. E.; Weissmüller, J.; Larché, F., A general theory of curved deformable interfaces in solids at equilibrium, Philos Mag A, 78, 5, 1093-1109 (1998)
[6] Aifantis, E. C., Strain gradient interpretation of size effects, Int J Fract, 95, 1, 299 (1999)
[7] Mindlin, R. D., Micro-structure in linear elasticity, Arch Ration Mech Anal, 16, 1, 51-78 (1964) · Zbl 0119.40302
[8] Lam, D. C.C.; Yang, F.; Chong, A. C.M.; Wang, J.; Tong, P., Experiments and theory in strain gradient elasticity, J Mech Phys Solids, 51, 8, 1477-1508 (2003) · Zbl 1077.74517
[9] Wang, B.; Zhou, S.; Zhao, J.; Chen, X., A size-dependent Kirchhoff micro-plate model based on strain gradient elasticity theory, Eur J Mech A Solids, 30, 4, 517-524 (2011) · Zbl 1278.74103
[10] Ashoori Movassagh, A.; Mahmoodi, M. J., A micro-scale modeling of Kirchhoff plate based on modified strain-gradient elasticity theory, Eur J Mech A Solids, 40, 50-59 (2013) · Zbl 1406.74067
[11] Mirsalehi, M.; Azhari, M.; Amoushahi, H., Buckling and free vibration of the FGM thin micro-plate based on the modified strain gradient theory and the spline finite strip method, Eur J Mech A Solids, 61, 1-13 (2017) · Zbl 1406.74313
[12] Li, A.; Zhou, S.; Zhou, S.; Wang, B., A size-dependent model for bi-layered Kirchhoff micro-plate based on strain gradient elasticity theory, Compos Struct, 113, 272-280 (2014)
[13] Ansari, R.; Gholami, R.; Faghih Shojaei, M.; Mohammadi, V.; Sahmani, S., Bending, buckling and free vibration analysis of size-dependent functionally graded circular/annular microplates based on the modified strain gradient elasticity theory, Eur J Mech A Solids, 49, 251-267 (2015) · Zbl 1406.74526
[14] Zhang, B.; He, Y.; Liu, D.; Lei, J.; Shen, L.; Wang, L., A size-dependent third-order shear deformable plate model incorporating strain gradient effects for mechanical analysis of functionally graded circular/annular microplates, Compos Part B, 79, 553-580 (2015)
[15] Zhang, B.; He, Y.; Liu, D.; Shen, L.; Lei, J., An efficient size-dependent plate theory for bending, buckling and free vibration analyses of functionally graded microplates resting on elastic foundation, Appl Math Modell, 39, 13, 3814-3845 (2015) · Zbl 1443.74228
[16] Thai, C. H.; Ferreira, A. J.M.; Nguyen-Xuan, H., Isogeometric analysis of size-dependent isotropic and sandwich functionally graded microplates based on modified strain gradient elasticity theory, Compos Struct, 192, 274-288 (2018)
[17] Farzam, A.; Hassani, B., Size-dependent analysis of FG microplates with temperature-dependent material properties using modified strain gradient theory and isogeometric approach, Compos B Eng, 161, 150-168 (2019)
[18] Thai, S.; Thai, H.-T.; Vo, T. P.; Patel, V. I., Size-dependant behaviour of functionally graded microplates based on the modified strain gradient elasticity theory and isogeometric analysis, Computers & Structures, 190, 219-241 (2017)
[19] Thai, C. H.; Ferreira, A. J.M.; Rabczuk, T.; Nguyen-Xuan, H., Size-dependent analysis of FG-CNTRC microplates based on modified strain gradient elasticity theory, Eur J Mech A Solids, 72, 521-538 (2018) · Zbl 1406.74454
[20] Thai, C. H.; Ferreira, A. J.M.; Phung-Van, P., Size dependent free vibration analysis of multilayer functionally graded GPLRC microplates based on modified strain gradient theory, Compos B Eng, 169, 174-188 (2019)
[21] Salehipour, H.; Shahsavar, A., A three dimensional elasticity model for free vibration analysis of functionally graded micro/nano plates: Modified strain gradient theory, Compos Struct, 206, 415-424 (2018)
[22] Sankar, B. V., An elasticity solution for functionally graded beams, Compos Sci Technol, 61, 5, 689-696 (2001)
[23] Pan, E., Exact Solution for Functionally Graded Anisotropic Elastic Composite Laminates, J Compos Mater, 37, 21, 1903-1920 (2003)
[24] Batra, R. C.; Qian, L. F.; Chen, L. M., Natural frequencies of thick square plates made of orthotropic, trigonal, monoclinic, hexagonal and triclinic materials, J Sound Vib, 270, 4, 1074-1086 (2004)
[25] Batra, R. C.; Jin, J., Natural frequencies of a functionally graded anisotropic rectangular plate, J Sound Vib, 282, 1-2, 509-516 (2005)
[26] Ferreira, A. J.M.; Batra, R. C., Natural frequencies of orthotropic, monoclinic and hexagonal plates by a meshless method, J Sound Vib, 285, 3, 734-742 (2005)
[27] Ramirez, F.; Heyliger, P. R.; Pan, E., Static analysis of functionally graded elastic anisotropic plates using a discrete layer approach, Compos Part B, 37, 1, 10-20 (2006)
[28] Guo, J.; Chen, J.; Pan, E., Size-dependent behavior of functionally graded anisotropic composite plates, Int J Eng Sci, 106, 110-124 (2016) · Zbl 1423.74544
[29] Guo, J.; Chen, J.; Pan, E., Analytical three-dimensional solutions of anisotropic multilayered composite plates with modified couple-stress effect, Compos Struct, 153, 321-331 (2016)
[30] Karami, B.; Janghorban, M., A new size-dependent shear deformation theory for free vibration analysis of functionally graded/anisotropic nanobeams, Thin-Walled Struct, 143, Article 106227 pp. (2019)
[31] Karami, B.; Janghorban, M.; Rabczuk, T., Static analysis of functionally graded anisotropic nanoplates using nonlocal strain gradient theory, Compos Struct, 227, Article 111249 pp. (2019)
[32] Thai, H.-T.; Vo, T. P.; Nguyen, T.-K.; Kim, S.-E., A review of continuum mechanics models for size-dependent analysis of beams and plates, Compos Struct, 177, 196-219 (2017)
[33] Hughes, T. J.R.; Cottrell, J. A.; Bazilevs, Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput Meth Appl Mech Eng, 194, 39, 4135-4195 (2005) · Zbl 1151.74419
[34] Cottrell, J. A.; Reali, A.; Bazilevs, Y.; Hughes, T. J.R., Isogeometric analysis of structural vibrations, Comput Meth Appl Mech Eng, 195, 41, 5257-5296 (2006) · Zbl 1119.74024
[35] Shojaee, S.; Izadpanah, E.; Valizadeh, N.; Kiendl, J., Free vibration analysis of thin plates by using a NURBS-based isogeometric approach, Finite Elem Anal Des, 61, 23-34 (2012)
[36] Thai, C. H.; Nguyen-Xuan, H.; Nguyen-Thanh, N.; Le, T. H.; Nguyen-Thoi, T.; Rabczuk, T., Static, free vibration, and buckling analysis of laminated composite Reissner-Mindlin plates using NURBS-based isogeometric approach, Int J Numer Methods Eng, 91, 6, 571-603 (2012) · Zbl 1253.74007
[37] Thai, C. H.; Ferreira, A. J.M.; Carrera, E.; Nguyen-Xuan, H., Isogeometric analysis of laminated composite and sandwich plates using a layerwise deformation theory, Compos Struct, 104, 196-214 (2013)
[38] Thai, C. H.; Ferreira, A. J.M.; Bordas, S. P.A.; Rabczuk, T.; 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
[39] Thai, C. H.; Kulasegaram, S.; Tran, L. V.; Nguyen-Xuan, H., Generalized shear deformation theory for functionally graded isotropic and sandwich plates based on isogeometric approach, Comput Struct, 141, 94-112 (2014)
[40] Phung-Van, P.; De Lorenzis, L.; Thai, C. H.; Abdel-Wahab, M.; Nguyen-Xuan, H., Analysis of laminated composite plates integrated with piezoelectric sensors and actuators using higher-order shear deformation theory and isogeometric finite elements, Comput Mater Sci, 96, 495-505 (2015)
[41] Phung-Van, P.; Tran, L. V.; Ferreira, A.; Nguyen-Xuan, H.; Abdel-Wahab, M., Nonlinear transient isogeometric analysis of smart piezoelectric functionally graded material plates based on generalized shear deformation theory under thermo-electro-mechanical loads, Nonlinear Dyn, 87, 2, 879-894 (2017) · Zbl 1372.74058
[42] Phung-Van, P.; Abdel-Wahab, M.; Liew, K.; Bordas, S.; Nguyen-Xuan, H., Isogeometric analysis of functionally graded carbon nanotube-reinforced composite plates using higher-order shear deformation theory, Compos Struct, 123, 137-149 (2015)
[43] Phung-Van, P.; Nguyen, L. B.; Tran, L. V.; Dinh, T. D.; Thai, C. H.; Bordas, S., An efficient computational approach for control of nonlinear transient responses of smart piezoelectric composite plates, Int J Non Linear Mech, 76, 190-202 (2015)
[44] Phung-Van, P.; Thai, C. H.; Ferreira, A.; Rabczuk, T., Isogeometric nonlinear transient analysis of porous FGM plates subjected to hygro-thermo-mechanical loads, Thin-Walled Struct, 148, Article 106497 pp. (2019)
[45] Kiendl, J.; Bletzinger, K. U.; Linhard, J.; Wüchner, R., Isogeometric shell analysis with Kirchhoff-Love elements, Comput Meth Appl Mech Eng, 198, 49, 3902-3914 (2009) · Zbl 1231.74422
[46] Fischer, P.; Klassen, M.; Mergheim, J.; Steinmann, P.; Müller, R., Isogeometric analysis of 2D gradient elasticity, Comput Mech, 47, 3, 325-334 (2011) · Zbl 1398.74329
[47] Niiranen, J.; Kiendl, J.; Niemi, A. H.; Reali, A., Isogeometric analysis for sixth-order boundary value problems of gradient-elastic Kirchhoff plates, Comput Meth Appl Mech Eng, 316, 328-348 (2017) · Zbl 1439.74037
[48] Makvandi, R.; Reiher, J. C.; Bertram, A.; Juhre, D., Isogeometric analysis of first and second strain gradient elasticity, Comput Mech, 61, 3, 351-363 (2018) · Zbl 1451.74218
[49] Phung-Van, P.; Ferreira, A. J.M.; Nguyen-Xuan, H.; Abdel Wahab, M., An isogeometric approach for size-dependent geometrically nonlinear transient analysis of functionally graded nanoplates, Compos Part B, 118, 125-134 (2017)
[50] Phung-Van, P.; Lieu, Q. X.; Nguyen-Xuan, H.; Abdel Wahab, M., Size-dependent isogeometric analysis of functionally graded carbon nanotube-reinforced composite nanoplates, Compos Struct, 166, 120-135 (2017)
[51] Phung-Van, P.; Thai, C. H.; Nguyen-Xuan, H.; Abdel Wahab, M., Porosity-dependent nonlinear transient responses of functionally graded nanoplates using isogeometric analysis, Compos B Eng, 164, 215-225 (2019)
[52] Phung-Van, P.; Thanh, C.-L.; Nguyen-Xuan, H.; Abdel-Wahab, M., Nonlinear transient isogeometric analysis of FG-CNTRC nanoplates in thermal environments, Compos Struct, 201, 882-892 (2018)
[53] Phung-Van, P.; Thai, C. H.; Nguyen-Xuan, H.; Abdel-Wahab, M., An isogeometric approach of static and free vibration analyses for porous FG nanoplates, Eur J Mech-A/Solids, 78, Article 103851 pp. (2019) · Zbl 1472.74206
[54] Phung-Van, P.; Thai, C. H.; Abdel-Wahab, M.; Nguyen-Xuan, H., Optimal design of FG sandwich nanoplates using size-dependent isogeometric analysis, Mech Mater, 142, Article 103277 pp. (2020)
[55] Reddy, J. N., A Simple Higher-Order Theory for Laminated Composite Plates, J Appl Mech, 51, 4, 745-752 (1984) · Zbl 0549.73062
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.