Size-dependent flexoelectricity-based vibration characteristics of honeycomb sandwich plates with various boundary conditions |
Soleimani-Javid, Zeinab
(Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan)
Arshid, Ehsan (Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan) Khorasani, Mohammad (Department of Basic and Applied Sciences for Engineering, Sapienza University) Amir, Saeed (Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan) Tounsi, Abdelouahed (Yonsei Frontier Laboratory, Yonsei University) |
1 | Bendaho, B., Belabed, Z., Bourada, M., Benatta, M.A., Bourada, F. and Tounsi, A. (2019), "Assessment of new 2D and quasi-3D nonlocal theories for free vibration analysis of size-dependent functionally graded (FG) nanoplates", Adv. Nano. Res., Int. J., 7(4), 277-292. https://doi.org/10.12989/anr.2019.7.4.277. DOI |
2 | Bisen, H.B., Hirwani, C.K., Satankar, R.K., Panda, S.K., Mehar, K. and Patel, B. (2020), "Numerical study of frequency and deflection responses of natural fiber (Luffa) reinforced polymer composite and experimental validation", J. Natural Fibers, 17(4), 505-519. https://doi.org/10.1080/15440478.2018.1503129. DOI |
3 | Dewangan, H.C., Sharma, N., Hirwani, C.K. and Panda, S.K. (2020), "Numerical eigenfrequency and experimental verification of variable cutout (square/rectangular) borne layered glass/epoxy flat/curved panel structure", Mech. Based Des. Struct., 1-18. https://doi.org/10.1080/15397734.2020.1759432. DOI |
4 | Ebrahimi, F., Karimiasl, M., Civalek, O . and Vinyas, M. (2019b), "Surface effects on scale-dependent vibration behavior of flexoelectric sandwich nanobeams", Adv. Nano. Res., Int. J., 7(2), 77-88. https://doi.org/10.12989/anr.2019.7.2.077. DOI |
5 | Arshid, E., Kiani, A. and Amir, S. (2019b), "Magneto-electroelastic vibration of moderately thick FG annular plates subjected to multi physical loads in thermal environment using GDQ method by considering neutral surface", Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications, 233(10), 2140-2159. https://doi.org/10.1177/1464420719832626. DOI |
6 | Amir, S., Arshid, E. and Khoddami Maraghi, Z. (2020d), "Free vibration analysis of magneto-rheological smart annular three-layered plates subjected to magnetic field in viscoelastic medium", Smart Struct. Syst., Int. J., 25(5), 581-592. https://doi.org/https://doi.org/10.12989/sss.2020.25.5.581. DOI |
7 | Arshid, E. and Khorshidvand, A.R. (2018), "Thin-Walled Structures Free vibration analysis of saturated porous FG circular plates integrated with piezoelectric actuators via differential quadrature method", Thin Wall. Struct., 125(January), 220-233. https://doi.org/10.1016/j.tws.2018.01.007. DOI |
8 | Arshid, E., Amir, S. and Loghman, A. (2020a), "Static and dynamic analyses of FG-GNPs reinforced porous nanocomposite annular micro-plates based on MSGT", Int. J. Mech. Sci., 180, 105656. https://doi.org/10.1016/j.ijmecsci.2020.105656. DOI |
9 | Ma, W. and Cross, L.E. (2002), "Flexoelectric polarization of barium strontium titanate in the paraelectric state", Appl. Phys. Lett., 81(18), 3440-3442. https://doi.org/10.1063/1.1518559. DOI |
10 | Ma, W. and Cross, L.E. (2006), "Flexoelectricity of barium titanate", Appl. Phys. Lett., 88(23), 2004-2007. https://doi.org/10.1063/1.2211309. DOI |
11 | Maheri, M.R. and Adams, R.D. (1994), "Steady-state flexural vibration damping of honeycomb sandwich beams", Compos. Sci. Technol., 52(3), 333-347. https://doi.org/10.1016/0266-3538(94)90168-6. DOI |
12 | Maranganti, R., Sharma, N.D. and Sharma, P. (2006), "Electromechanical coupling in nonpiezoelectric materials due to nanoscale nonlocal size effects: Green's function solutions and embedded inclusions", Phys. Rev. B., 74(1), 1-14. https://doi.org/10.1103/PhysRevB.74.014110. DOI |
13 | Mashkevich, V.S. (1957), "Electrical, Optical, and Elastic Properties of Diamond-Type Crystals", J. Exptl. Theoret. Phys., 5(4), 707-713. |
14 | Mehar, K., Mahapatra, T.R., Panda, S.K., Katariya, P.V. and Tompe, U.K. (2018), "Finite-element solution to nonlocal elasticity and scale effect on frequency behavior of shear deformable nanoplate structure", J. Eng. Mech., 144(9), 04018094. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001519. DOI |
15 | Mehar, K., Panda, S.K. and Sharma, N. (2020b), "Numerical investigation and experimental verification of thermal frequency of carbon nanotube-reinforced sandwich structure", Eng. Struct., 211, 110444. https://doi.org/10.1016/j.engstruct.2020.110444. DOI |
16 | Sharma, N.D.D., Maranganti, R. and Sharma, P. (2007), "On the possibility of piezoelectric nanocomposites without using piezoelectric materials", J. Mech. Phys. Solids, 55(11), 2328-2350. https://doi.org/10.1016/j.jmps.2007.03.016. DOI |
17 | Shen, S. and Hu, S. (2010), "A theory of flexoelectricity with surface effect for elastic dielectrics", J. Mech. Phys. Solids, 58(5), 665-677. https://doi.org/10.1016/j.jmps.2010.03.001. DOI |
18 | Mehrabadi, S.J., Aragh, B.S., Khoshkhahesh, V. and Taherpour, A. (2012), "Mechanical buckling of nanocomposite rectangular plate reinforced by aligned and straight single-walled car", Compos. Part B-Eng., 43(4), 2031-2040. https://doi.org/10.1016/j.compositesb.2012.01.067 DOI |
19 | Ansari, R. and Sahmani, S. (2011), "Surface stress effects on the free vibration behavior of nanoplates", Int. J. Eng. Sci., 49(11), 1204-1215. https://doi.org/10.1016/j.ijengsci.2011.06.005. DOI |
20 | Khorasani, M., Soleimani-Javid, Z., Arshid, E., Lampani, L. and Civalek, O . (2021), "Thermo-elastic buckling of honeycomb micro plates integrated with FG-GNPs reinforced Epoxy skins with stretching effect", Compos. Struct., 258, 113430. https://doi.org/10.1016/j.compstruct.2020.113430. DOI |
21 | Arshid, H., Khorasani, M., Soleimani-Javid, Z., Dimitri, R. and Tornabene, F. (2020c), "Quasi-3D Hyperbolic Shear Deformation Theory for the Free Vibration Study of Honeycomb Microplates with Graphene Nanoplatelets-Reinforced Epoxy Skins", Molecules, 25(21), 5085. https://doi.org/10.3390/molecules25215085. DOI |
22 | Nakamoto, H., Adachi, T. and Araki, W. (2009), "In-plane impact behavior of honeycomb structures filled with linearly arranged inclusions", Int. J. Impact Eng., 36(8), 1019-1026. https://doi.org/10.1016/j.ijimpeng.2009.01.004. DOI |
23 | Mukhopadhyay, T. and Adhikari, S. (2016), "Free-Vibration analysis of sandwich panels with randomly irregular Honeycomb Core", J. Eng. Mech., 142, 06016008. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001153. DOI |
24 | Patle, B.K., Hirwani, C.K., Singh, R.P. and Panda, S.K. (2018), "Eigenfrequency and deflection analysis of layered structure using uncertain elastic properties - a fuzzy finite element approach", Int. J. Approx. Reason., 98, 163-176. https://doi.org/10.1016/j.ijar.2018.04.013. DOI |
25 | Hirwani, C.K. and Panda, S.K. (2019b), "Nonlinear finite element solutions of thermoelastic deflection and stress responses of internally damaged curved panel structure", Appl. Math. Model., 65, 303-317. https://doi.org/10.1016/j.apm.2018.08.014. DOI |
26 | Fu, J.Y., Zhu, W., Li, N. and Cross, L.E. (2006), "Experimental studies of the converse flexoelectric effect induced by inhomogeneous electric field in a barium strontium titanate composition", J. Appl. Phys., 100(2). https://doi.org/10.1063/1.2219990. DOI |
27 | Guo, Y. and Jinghui, Z. (2004), "Shock Absorbing Characteristics and Vibration Transmissibility of Honeycomb Paperboard", Shock Vib., 11(5,6), 521-531. DOI |
28 | He, L., Lou, J., Zhang, E., Wang, Y. and Bai, Y. (2015), "A size-dependent four variable refined plate model for functionally graded microplates based on modified couple stress theory", Compos. Struct., 130, 107-115. https://doi.org/10.1016/j.compstruct.2015.04.033. DOI |
29 | Hebali, H., Tounsi, A., Sid, M. and Bessaim, A. (2014), "New Quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates", J. Eng. Mech., 140, 374-383. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000665. DOI |
30 | Hirwani, C.K. and Panda, S.K. (2019a), "Nonlinear thermal free vibration frequency analysis of delaminated shell panel using FEM", Compos. Struct., 224, 111011. https://doi.org/10.1016/j.compstruct.2019.111011. DOI |
31 | Ghobadi, A., Tadi, B.Y. and Golestanian, H. (2019), "Size Dependent Nonlinear Bending Analysis of a Flexoelectric Functionally Graded Nano-Plate Under Thermo-Electro-Mechanical Loads", J. Solid. Mech., 12(1), 33-56. https://doi.org/10.22034/JSM.2019.569280.1296. DOI |
32 | Nguyen, T.D., Mao, S., Yeh, Y.-W., Purohit, P.K. and McAlpine, M.C. (2013), "Nanoscale Flexoelectricity", Adv. Mater., 25(7), 946-974. https://doi.org/10.1002/adma.201203852. DOI |
33 | Sahoo, S.S., Panda, S.K., Mahapatra, T.R. and Hirwani, C.K. (2019), "Numerical Analysis of Transient Responses of Delaminated Layered Structure Using Different Mid-plane Theories and Experimental Validation", Irann. J. Sci. Technol., 43(1), 41-56. https://doi.org/10.1007/s40997-017-0111-3. DOI |
34 | Salari, E., Ashoori, A. and Vanini, S.A.S. (2019), "Porosity-dependent asymmetric thermal buckling of inhomogeneous annular nanoplates resting on elastic substrate", Adv. Nano. Res., Int. J., 7(1), 25-38. https://doi.org/10.12989/anr.2019.7.1.025. DOI |
35 | Sharma, N.D., Landis, C.M. and Sharma, P. (2010), "Piezoelectric thin-film super-lattices without using piezoelectric materials", J. Appl. Phys., 108(2), 024304. https://doi.org/10.1063/1.3443404. DOI |
36 | Singh, V.K., Mahapatra, T.R. and Panda, S.K. (2016b), "Nonlinear transient analysis of smart laminated composite plate integrated with PVDF sensor and AFC actuator", Compos. Struct., 157, 121-130. https://doi.org/10.1016/j.compstruct.2016.08.020. DOI |
37 | Shingare, K.B. and Kundalwal, S.I. (2020), "Flexoelectric and surface effects on the electromechanical behavior of graphene-based nanobeams", Appl. Math. Model., 81, 70-91. https://doi.org/10.1016/j.apm.2019.12.021. DOI |
38 | Shu, L., Wei, X., Pang, T., Yao, X. and Wang, C. (2011), "Symmetry of flexoelectric coefficients in crystalline medium", J. Appl. Phys., 110(10), 104106. https://doi.org/10.1063/1.3662196. DOI |
39 | Singh, V.K., Hirwani, C.K., Panda, S.K., Mahapatra, T.R. and Mehar, K. (2019), "Numerical and experimental nonlinear dynamic response reduction of smart composite curved structure using collocation and non-collocation configuration", Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 233(5), 1601-1619. https://doi.org/10.1177/0954406218774362. DOI |
40 | Singh, V.K., Mahapatra, T.R. and Panda, S.K. (2016a), "Nonlinear flexural analysis of single/doubly curved smart composite shell panels integrated with PFRC actuator", Eur. J. Mech. A-Solid., 60, 300-314. https://doi.org/10.1016/j.euromechsol.2016.08.006. DOI |
41 | Kumar, S. and Renji, K. (2019), "Estimation of strains in composite honeycomb sandwich panels subjected to low frequency diffused acoustic field", J. Sound. Vib., 449, 84-97. https://doi.org/10.1016/j.jsv.2019.02.013. DOI |
42 | Hu, S., Shen, S., Shuling, H.U. and Shengping, S. (2010), "Variational principles and governing equations in nanodielectrics with the flexoelectric effect", Sci. China Phys. Mech. Astronomy, 53(8), 1497-1504. https://doi.org/10.1007/s11433-010-4039-5. DOI |
43 | Jafari Mehrabadi, S., Sobhani Aragh, B., Khoshkhahesh, V. and Taherpour, A. (2012), "Mechanical buckling of nanocomposite rectangular plate reinforced by aligned and straight single-walled carbon nanotubes", Compos. Part B: Eng., 43(4), 2031- 2040. https://doi.org/10.1016/j.compositesb.2012.01.067. DOI |
44 | Katariya, P.V. and Panda, S.K. (2019a), "Frequency and deflection responses of shear deformable skew sandwich curved shell panel: A finite element approach", Arab. J. Sci. Eng., 44(2), 1631-1648. https://doi.org/10.1007/s13369-018-3633-0. DOI |
45 | Katariya, P.V. and Panda, S.K. (2019b), "Numerical frequency analysis of skew sandwich layered composite shell structures under thermal environment including shear deformation effects", Struct. Eng. Mech., Int. J., 71(6), 657-668. https://doi.org/10.12989/sem.2019.71.6.657 DOI |
46 | Khorasani, M., Eyvazian, A., Karbon, M., Tounsi, A., Lampani, L., Sebaey, T.A. and Sebaey, T.A. (2020), "Magneto-electro-elastic vibration analysis of modified couple stress-based three-layered micro rectangular plates exposed to multi-physical fields considering the flexoelectricity effects", Smart Struct. Syst., Int. J., 26(3), 331-343. https://doi.org/10.12989/SSS.2020.26.3.331. DOI |
47 | Liu, J., Cheng Y.S. and Lirf, A.U. (2010), "A semi-analyrical method for bending, buckling, and free vibration analyses of sandwich panels with square-honeycomb cores", Int. J. Struct. Stab. Dyn., 10(01), 127-151. https://doi.org/10.1142/S0219455410003361. DOI |
48 | Arshid, E., Amir, S. and Loghman, A. (2020b), "Bending and buckling behaviors of heterogeneous temperature-dependent micro annular/circular porous sandwich plates integrated by FGPEM nano-Composite layers", J. Sandw. Struct. Mater., 109963622095502. https://doi.org/10.1177/1099636220955027. DOI |
49 | Arshid, E., Arshid, H., Amir, S. and Mousavi, S.B. (2021), "Free vibration and buckling analyses of FG porous sandwich curved microbeams in thermal environment under magnetic field based on modified couple stress theory", Arch. Civil. Mech. Eng., 21(1), 6. https://doi.org/10.1007/s43452-020-00150-x. DOI |
50 | Li, Y. and Jin, Z. (2008), "Free flexural vibration analysis of symmetric rectangular honeycomb panels with SCSC edge supports", Compos. Struct., 83(2), 154-158. https://doi.org/10.1016/j.compstruct.2007.04.004. DOI |
51 | Ma, W. and Cross, L.E. (2001), "Observation of the flexoelectric effect in relaxor Pb(Mg1/3Nb2/3)O3 ceramics", Appl. Phys. Lett., 78(19), 2920-2921. https://doi.org/10.1063/1.1356444. DOI |
52 | Ebrahimi, F., Dabbagh, A., Tornabene, F. and Civalek, O. (2019a), "Hygro-thermal effects on wave dispersion responses of magnetostrictive sandwich nanoplates", Adv. Nano. Res., Int. J., 7(3), 157-167. https://doi.org/10.12989/anr.2019.7.3.157. DOI |
53 | Hirwani, C.K. and Panda, S.K. (2018), "Numerical nonlinear frequency analysis of pre-damaged curved layered composite structure using higher-order finite element method", Int. J. Nonlinear Mech., 102, 14-24. https://doi.org/10.1016/j.ijnonlinmec.2018.03.005. DOI |
54 | Shariati, A., Ebrahimi, F., Karimiasl, M., Vinyas, M. and Toghroli, A. (2020), "On transient hygrothermal vibration of embedded viscoelastic flexoelectric/piezoelectric nanobeams under magnetic loading", Adv. Nano. Res., Int. J., 8(1), 49-58. https://doi.org/10.12989/anr.2020.8.1.049. DOI |
55 | Singh, V.K. and Panda, S.K. (2015), "Large amplitude free vibration analysis of laminated composite spherical shells embedded with piezoelectric layers", Smart Struct. Syst., Int. J., 16(5), 853-872. http://doi.org/10.12989/sss.2015.16.5.853. DOI |
56 | Sobhy, M. (2020), "Differential quadrature method for magneto-hygrothermal bending of functionally graded graphene/Al sandwich-curved beams with honeycomb core via a new higher-order theory", J. Sandw. Struct. Mater., 1099636219900668. https://doi.org/10.1177/1099636219900668. DOI |
57 | Katariya, P.V., Panda, S.K. and Mahapatra, T.R. (2018), "Bending and vibration analysis of skew sandwich plate", Aircr. Eng. Aerosp. Tech., 90(6), 885-895. https://doi.org/10.1108/AEAT-05-2016-0087. DOI |
58 | Yang, W., Liang, X. and Shen, S. (2015), "Electromechanical responses of piezoelectric nanoplates with flexoelectricity", Acta Mechanica, 226(9), 3097-3110. https://doi.org/10.1007/s00707-015-1373-8. DOI |
59 | Yudin, P.V. and Tagantsev, A.K. (2013), "Fundamentals of flexoelectricity in solids", Nanotechnology, 24(43), 432001. https://doi.org/10.1088/0957-4484/24/43/432001. DOI |
60 | Arshid, E., Khorshidvand, A.R. and Khorsandijou, S.M. (2019a), "The effect of porosity on free vibration of SPFG circular plates resting on visco-Pasternak elastic foundation based on CPT, FSDT and TSDT", Struct. Eng. Mech., Int. J., 70(1), 97-112. https://doi.org/10.1177/1464420719832626. DOI |
61 | Tagantsev, A.K. (1986), "Piezoelectricity and flexoelectricity in crystalline dielectrics", Phys. Rev. B, 34(8), 5883-5889. https://doi.org/10.1103/PhysRevB.34.5883. DOI |
62 | Tagantsev, A.K. and Gerra, G. (2006), "Interface-induced phenomena in polarization response of ferroelectric thin films", J. Appl. Phys., 100(5), 051607. https://doi.org/10.1063/1.2337009. DOI |
63 | Amir, S., Khorasani, M. and BabaAkbar-Zarei, H. (2020a), "Buckling analysis of nanocomposite sandwich plates with piezoelectric face sheets based on flexoelectricity and first-order shear deformation theory", J. Sandw. Struct. Mater., 22(7), 2186-2209. https://doi.org/10.1177/1099636218795385. DOI |
64 | Abdulrazzaq, M.A., Fenjan, R.M., Ahmed, R.A. and Faleh, N.M. (2020), "Thermal buckling of nonlocal clamped exponentially graded plate according to a secant function based refined theory", Steel. Compos. Struct., Int. J., 35(1), 147-157. https://doi.org/10.12989/scs.2020.35.1.149. DOI |
65 | Akbas, S.D. (2018), "Forced vibration analysis of cracked functionally graded microbeams", Adv. Nano. Res., Int. J., 6(1), 39-55. https://doi.org/10.12989/anr.2018.6.1.039. DOI |
66 | Al-Maliki, A.F.H., Ahmed, R.A., Moustafa, N.M. and Faleh, N.M. (2020), "Finite element based modeling and thermal dynamic analysis of functionally graded graphene reinforced beams", Adv. Computat. Des., Int. J., 5(2), 177-193. https://doi.org/10.12989/acd.2020.5.2.177. DOI |
67 | Zhang, Z., Yan, Z. and Jiang, L. (2014), "Flexoelectric effect on the electroelastic responses and vibrational behaviors of a piezoelectric nanoplate", J. Appl. Phys., 116(1), 014307. https://doi.org/10.1063/1.4886315. DOI |
68 | Zeng, S., Wang, B.L. and Wang, K.F. (2019), "Nonlinear vibration of piezoelectric sandwich nanoplates with functionally graded porous core with consideration of flexoelectric effect", Compos. Struct., 207, 340-351. https://doi.org/10.1016/j.compstruct.2018.09.040. DOI |
69 | Zenkour, A.M. (2016), "Buckling of a single-layered graphene sheet embedded in visco-Pasternak", Adv. Nano. Res., Int. J., 4(4), 309-326. https://doi.org/10.12989/anr.2016.4.4.309. DOI |
70 | Zhang, Z. and Jiang, L. (2014), "Size effects on electromechanical coupling fields of a bending piezoelectric nanoplate due to surface effects and flexoelectricity", J. Appl. Phys., 116, 134308. https://doi.org/10.1063/1.4897367. DOI |
71 | Zhao, M., Qian, C., Lee, S.W.R., Tong, P., Suemasu, H. and Zhang, T.-Y. (2007), "Electro-elastic analysis of piezoelectric laminated plates", Adv. Compos. Mater., 16(1), 63-81. https://doi.org/10.1163/156855107779755273. DOI |
72 | Zubko, P., Catalan, G. and Tagantsev, A.K. (2013), "Flexoelectric Effect in Solids", Annual Rev. Mater. Res., 43(1), 387-421. https://doi.org/10.1146/annurev-matsci-071312-121634. DOI |
73 | Liu, C., Hu, S. and Shen, S. (2012), "Effect of flexoelectricity on electrostatic potential in a bent piezoelectric nanowire", Smart Mater. Struct., 21(11), 115024. https://doi.org/10.1088/0964-1726/21/11/115024. DOI |
74 | Mehar, K., Mishra, P.K. and Panda, S.K. (2020a), "Numerical investigation of thermal frequency responses of graded hybrid smart nanocomposite (CNT-SMA-Epoxy) structure", Mech. Adv. Mater. Struct., 1-13. https://doi.org/10.1080/15376494.2020.1725193. DOI |
75 | Mindlin, R.D. (1968), "Polarization gradient in elastic dielectrics", Int. J. Solids. Struct., 4(6), 637-642. https://doi.org/10.1016/0020-7683(68)90079-6. DOI |
76 | Barati, M.R. (2017a), "Coupled effects of electrical polarizationstrain gradient on vibration behavior of double-layered flexoelectric nanoplates", Smart Struct. Syst., Int. J., 20(5), 573-581. https://doi.org/10.12989/sss.2017.20.5.573. DOI |
77 | Barati, M.R. (2017b), "Nonlocal-strain gradient forced vibration analysis of metal foam nanoplates with uniform and graded", Adv. Nano. Res., Int. J., 5(4), 393-414. https://doi.org/10.12989/anr.2017.5.4.393. DOI |
78 | Amir, S., BabaAkbar-Zarei, H. and Khorasani, M. (2020b), "Flexoelectric vibration analysis of nanocomposite sandwich plates", Mech. Based Des. Struct. Mach., 48(2), 146-163. https://doi.org/10.1080/15397734.2019.1624175. DOI |
79 | Amir, S., Arshid, E., Khoddami Maraghi, Z., Loghman, A. and Ghorbanpour Arani, A. (2020c), "Vibration analysis of magnetorheological fluid circular sandwich plates with magnetostrictive facesheets exposed to monotonic magnetic field located on visco-Pasternak substrate", J. Vib. Control, 26(17-18), 1523-1537. https://doi.org/10.1177/1077546319899203. DOI |
80 | Amir, S., Soleimani-Javid, Z. and Arshid, E. (2019), "Size-dependent free vibration of sandwich micro beam with porous core subjected to thermal load based on SSDBT", ZAMM - J. Appl. Math. Mech. / Zeitschrift Fur Angewandte Mathematik Und Mechanik, 99(9). https://doi.org/10.1002/zamm.201800334. DOI |
81 | Thai, H.T. and Choi, D.H. (2013), "Size-dependent functionally graded Kirchhoff and Mindlin plate models based on a modified couple stress theory", Compos. Struct., 95, 142-153. https://doi.org/10.1016/j.compstruct.2012.08.023. DOI |
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