The paper presents a numerical assessment of the performance of the Refined Zigzag Theory to the analysis of bending (deflection and stress distributions) and free vibration of functionally graded material plates, monolayer and sandwich, under a set of different boundary conditions. The numerical assessment is performed comparing results from Refined Zigzag Theory using Ritz method with those from 3D, quasi-3D, and 2D theories and finite element method. In the framework of 2D theories, equivalent single-layer theories of different orders (sinusoidal, hyperbolic, inverse-hyperbolic, third-order shear deformation theory, first-order shear deformation theory, and classical plate theory) have been used to investigate deformation, stresses, and free vibration and compared with results from the Refined Zigzag Theory. After validating the convergence characteristics and the numerical accuracy of the developed approach using orthogonal admissible functions, a detailed parametric numerical investigation is carried out. Bending under transverse pressure and free vibration of functionally graded material square and rectangular plates of a different aspect ratio under various combinations of geometry (core-to-face sheet thickness ratio and plate to thickness ratio), boundary conditions and law of variation of volume fraction constituent in the thickness direction (power-law functionally graded material, exponential law functionally graded material, and sigmoidal-law functionally graded material) is studied. Monolayer and sandwich plates with homogeneous core and functionally graded face sheets are considered for the assessment. It is concluded that the Refined Zigzag Theory generally predicts the global (deflection and frequencies) and local (displacement and stress distributions) response of functionally graded material sandwich plates, more accurately than first-order shear deformation theory and third-order shear deformation theory, while retaining its simplicity.
ReddyJNMiraveteA.Practical analysis of composite laminates.
Boca Raton:
CRC Press, Inc. Epub ahead of print 1995. DOI: 10.1201/9780203742594.
2.
KoizumiMNiinoM.Overview of FGM research in Japan. MRS Bull1995;
20: 19–21.
3.
KoizumiM.FGM activities in Japan. Compos Part B: Eng1997;
28: 1–4.
4.
BohidarSKSharmaRMishraPR.Functionally graded materials: a critical review. Int J Res2014;
1: 13.
5.
UdupaGRaoSSGangadharanKV.Functionally graded composite materials: an overview. Proced Mater Sci2014;
5: 1291–1299.
6.
SwaminathanKSangeethaDM.Thermal analysis of FGM plates – a critical review of various modeling techniques and solution methods. Compos Struct2017;
160: 43–60.
7.
MahamoodRMAkinlabiET.Types of functionally graded materials and their areas of application. In: RMMahamoodETAkinlabi (eds) Functionally graded materials.
Cham:
Springer International Publishing, 2017, pp. 9–21.
8.
BirmanVByrdLW.Modeling and analysis of functionally graded materials and structures. Appl Mech Rev2007;
60: 195–216.
9.
ZuikerJR.Functionally graded materials: choice of micromechanics model and limitations in property variation. Compos Eng1995;
5: 807–819.
10.
YinHMSunLZPaulinoGH.Micromechanics-based elastic model for functionally graded materials with particle interactions. Acta Materialia2004;
52: 3535–3543.
11.
KlusemannBBöhmHJSvendsenB.Homogenization methods for multi-phase elastic composites with non-elliptical reinforcements: comparisons and benchmarks. Eur J Mech – A/Solids2012;
34: 21–37.
12.
LojaMARBarbosaJIMota SoaresCM.A study on the modeling of sandwich functionally graded particulate composites. Compos Struct2012;
94: 2209–2217.
13.
ShenH-SWangZ-X.Assessment of Voigt and Mori–Tanaka models for vibration analysis of functionally graded plates. Compos Struct2012;
94: 2197–2208.
14.
GuptaATalhaM.Recent development in modeling and analysis of functionally graded materials and structures. Prog Aerosp Sci2015;
79: 1–14.
15.
AshwinkumarAK.Review on functionally graded materials and various theories. Int Res J Eng Technol2017; 4: 04890–04893.
16.
SrividhyaSBasantKGuptaRK, et al.
Influence of the homogenization scheme on the bending response of functionally graded plates. Acta Mech2018;
229: 4071–4089.
17.
BiloueiBSKolahchiRBidgoliMR.Buckling of concrete columns retrofitted with Nano-Fiber Reinforced Polymer (NFRP). Comput Concr2016;
18: 1053–1063.
18.
ZamanianMKolahchiRBidgoliMR.Agglomeration effects on the buckling behaviour of embedded concrete columns reinforced with SiO2 nano-particles. Wind Struct2017;
24: 43–57.
19.
AmniehHBZamzamMSKolahchiR.Dynamic analysis of non-homogeneous concrete blocks mixed by SiO2 nanoparticles subjected to blast load experimentally and theoretically. Constr Build Mater2018;
174: 633–644.
20.
HosseiniHKolahchiR.Seismic response of functionally graded-carbon nanotubes-reinforced submerged viscoelastic cylindrical shell in hygrothermal environment. Phys E: Low-Dimens Syst Nanostruct2018;
102: 101–109.
21.
FakharAKolahchiR.Dynamic buckling of magnetorheological fluid integrated by visco-piezo-GPL reinforced plates. Int J Mech Sci2018;
144: 788–799.
22.
HajmohammadMHKolahchiRZareiMS, et al.
Earthquake induced dynamic deflection of submerged viscoelastic cylindrical shell reinforced by agglomerated CNTs considering thermal and moisture effects. Compos Struct2018;
187: 498–508.
23.
HajmohammadMHMalekiMKolahchiR.Seismic response of underwater concrete pipes conveying fluid covered with nano-fiber reinforced polymer layer. Soil Dyn Earthq Eng2018;
110: 18–27.
24.
GolabchiHKolahchiRBidgoliMR.Vibration and instability analysis of pipes reinforced by SiO2 nanoparticles considering agglomeration effects. Comput Concr2018;
21: 431–440.
25.
ZareiMSKolahchiRHajmohammadMH, et al.
Seismic response of underwater fluid-conveying concrete pipes reinforced with SiO2 nanoparticles and fiber reinforced polymer (FRP) layer. Soil Dyn Earthq Eng2017;
103: 76–85.
26.
HajmohammadMHFarrokhianAKolahchiR.Smart control and vibration of viscoelastic actuator-multiphase nanocomposite conical shells-sensor considering hygrothermal load based on layerwise theory. Aerosp Sci Technol2018;
78: 260–270.
27.
AraniAJKolahchiR.Buckling analysis of embedded concrete columns armed with carbon nanotubes. Comput Concr2016;
17: 567–578.
28.
HajmohammadMHZareiMSNouriA, et al.
Dynamic buckling of sensor/functionally graded-carbon nanotube-reinforced laminated plates/actuator based on sinusoidal-visco-piezoelasticity theories. J Sandw Struct Mater2017; 14: 1–33. 1099636217720373.
29.
KolahchiRZareiMSHajmohammadMH, et al.
Wave propagation of embedded viscoelastic FG-CNT-reinforced sandwich plates integrated with sensor and actuator based on Refined Zigzag Theory. Int J Mech Sci2017;
130: 534–545.
30.
MadaniHHosseiniHShokraviM.Differential cubature method for vibration analysis of embedded FG-CNT-reinforced piezoelectric cylindrical shells subjected to uniform and non-uniform temperature distributions. Steel Compos Struct2016;
22: 889–913.
31.
KolahchiRSafariMEsmailpourM.Dynamic stability analysis of temperature-dependent functionally graded CNT-reinforced visco-plates resting on orthotropic elastomeric medium. Compos Struct2016;
150: 255–265.
32.
ShakeriMMirzaeifarR.Static and dynamic analysis of thick functionally graded plates with piezoelectric layers using layerwise finite element model. Mech Adv Mater Struct2009;
16: 561–575.
33.
Vo-DuyTHo-HuuVNguyen-ThoiT.Free vibration analysis of laminated FG-CNT reinforced composite beams using finite element method. Front Struct Civ Eng2018; 13: 324–336.
34.
PandeySPradyumnaS.Analysis of functionally graded sandwich plates using a higher-order layerwise theory. Compos Part B: Eng2018;
153: 325–336.
35.
MohammadimehrMMostafavifarM.Free vibration analysis of sandwich plate with a transversely flexible core and FG-CNTs reinforced nanocomposite face sheets subjected to magnetic field and temperature-dependent material properties using SGT. Compos Part B: Eng2016;
94: 253–270.
36.
LeeW-HHanS-CParkW-T.A refined higher order shear and normal deformation theory for E-, P-, and S-FGM plates on Pasternak elastic foundation. Compos Struct2015;
122: 330–342.
37.
MezianeMAAAbdelazizHHTounsiA.An efficient and simple refined theory for buckling and free vibration of exponentially graded sandwich plates under various boundary conditions. J Sandw Struct Mater2014;
16: 293–318.
38.
ChiS-HChungY-L.Mechanical behavior of functionally graded material plates under transverse load—part I: analysis. Int J Solids Struct2006;
43: 3657–3674.
39.
ChiS-HChungY-L.Mechanical behavior of functionally graded material plates under transverse load—part II: numerical results. Int J Solids Struct2006;
43: 3675–3691.
40.
JhaDKKantTSinghRK.A critical review of recent research on functionally graded plates. Compos Struct2013;
96: 833–849.
41.
SobhyM.Buckling and free vibration of exponentially graded sandwich plates resting on elastic foundations under various boundary conditions. Compos Struct2013;
99: 76–87.
42.
AnjuMSubhaK.A review on functionally graded plate. Int Res J Eng Technol2018;
5: 75–81.
43.
SinghSJHarshaSP.Exact solution for free vibration and buckling of sandwich S-FGM plates on pasternak elastic foundation with various boundary conditions. Int J Str Stab Dyn2018; 19: 1950028.
44.
BirmanVKardomateasGA.Review of current trends in research and applications of sandwich structures. Compos Part B: Eng2018;
142: 221–240.
45.
ThaiH-TKimS-E.A review of theories for the modeling and analysis of functionally graded plates and shells. Compos Struct2015;
128: 70–86.
46.
SayyadASGhugalYM.Modeling and analysis of functionally graded sandwich beams: a review. Mech Adv Mater Struct2018;
34: 1–20.
47.
ReddyJN.Analysis of functionally graded plates. Int J Numer Meth Eng2000;
47: 663–684.
PanEHanF.Exact solution for functionally graded and layered magneto-electro-elastic plates. Int J Eng Sci2005;
43: 321–339.
50.
LiQIuVPKouKP.Three-dimensional vibration analysis of functionally graded material sandwich plates. J Sound Vib2008;
311: 498–515.
51.
KashtalyanM.Three-dimensional elasticity solution for bending of functionally graded rectangular plates. Eur J Mech – A/Solids2004;
23: 853–864.
52.
JinGSuZShiS, et al.
Three-dimensional exact solution for the free vibration of arbitrarily thick functionally graded rectangular plates with general boundary conditions. Compos Struct2014;
108: 565–577.
53.
NevesAMAFerreiraAJMCarreraE, et al.
A quasi-3D sinusoidal shear deformation theory for the static and free vibration analysis of functionally graded plates. Compos Part B: Eng2012;
43: 711–725.
54.
NevesAMAFerreiraAJMCarreraE, et al.
A quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates. Compos Struct2012;
94: 1814–1825.
55.
NevesAMAFerreiraAJMCarreraE, et al.
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: Eng2013;
44: 657–674.
56.
HebaliHTounsiAHouariMSA, et al.
New quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates. J Eng Mech2014;
140: 374–383.
57.
ThaiH-TVoTPBuiTQ, et al.
A quasi-3D hyperbolic shear deformation theory for functionally graded plates. Acta Mech2014;
225: 951–964.
58.
Farzam-RadSAHassaniBKaramodinA.Isogeometric analysis of functionally graded plates using a new quasi-3D shear deformation theory based on physical neutral surface. Compos Part B: Eng2017;
108: 174–189.
59.
AbrateSDi SciuvaM.Equivalent single layer theories for composite and sandwich structures: a review. Compos Struct2017;
179: 482–494.
60.
AbrateSDi SciuvaM. Multilayer models for composite and sandwich structures. In: Beaumont PWR, Zweben CH (eds) Comprehensive composite materials II. Amsterdam: Elsevier, 2017, pp. 399–425.
61.
FerreiraAJMBatraRCRoqueCMC, et al.
Static analysis of functionally graded plates using third-order shear deformation theory and a meshless method. Compos Struct2005;
69: 449–457.
62.
ZenkourAM.A comprehensive analysis of functionally graded sandwich plates: part 1—deflection and stresses. Int J Solids Struct2005;
42: 5224–5242.
63.
ZenkourAM.A comprehensive analysis of functionally graded sandwich plates: part 2—buckling and free vibration. Int J Solids Struct2005;
42: 5243–5258.
BessaimAHouariMSTounsiA, et al.
A new higher-order shear and normal deformation theory for the static and free vibration analysis of sandwich plates with functionally graded isotropic face sheets. J Sandw Struct Mater2013;
15: 671–703.
66.
PradhanKKChakravertyS.Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh–Ritz method. Compos Part B: Eng2013;
51: 175–184.
67.
TajMGChakrabartiATalhaM.Bending analysis of functionally graded skew sandwich plates with through-the thickness displacement variations. J Sandw Struct Mater2014;
16: 210–248.
68.
TounsiAHouariMSABenyoucefS, et al.
A refined trigonometric shear deformation theory for thermoelastic bending of functionally graded sandwich plates. Aerosp Sci Technol2013;
24: 209–220.
69.
XiangSKangGYangM, et al.
Natural frequencies of sandwich plate with functionally graded face and homogeneous core. Compos Struct2013;
96: 226–231.
70.
ZenkourAM.Bending analysis of functionally graded sandwich plates using a simple four-unknown shear and normal deformations theory. J Sandw Struct Mater2013;
15: 629–656.
71.
NguyenV-HNguyenT-KThaiH-T, et al.
A new inverse trigonometric shear deformation theory for isotropic and functionally graded sandwich plates. Compos Part B: Eng2014;
66: 233–246.
72.
RedddyBSKumarJSReddyCE, et al.
Free vibration behaviour of functionally graded plates using higher-order shear deformation theory. J Appl Sci Eng2014;
17: 231–241.
73.
RedddyBSKumarJSReddyCE, et al.
Static bending behavior of functionally graded plates subjected to mechanical loading. Jordan J Mech Ind Eng2014;
8: 14.
74.
MahiAAdda BediaEATounsiA.A new hyperbolic shear deformation theory for bending and free vibration analysis of isotropic, functionally graded, sandwich and laminated composite plates. Appl Math Model2015;
39: 2489–2508.
75.
MantariJLGranadosEV.A refined FSDT for the static analysis of functionally graded sandwich plates. Thin-Walled Struct2015;
90: 150–158.
76.
NguyenKTThaiTHVoTP.A refined higher-order shear deformation theory for bending, vibration and buckling analysis of functionally graded sandwich plates. Steel Compos Struct2015;
18: 91–120.
77.
BennounMHouariMSATounsiA.A novel five-variable refined plate theory for vibration analysis of functionally graded sandwich plates. Mech Adv Mater Struct2016;
23: 423–431.
78.
TounsiAHouariMSABessaimA.A new 3-unknowns non-polynomial plate theory for buckling and vibration of functionally graded sandwich plate. Struct Eng Mech2016;
60: 547–565.
KolahchiRKeshtegarBFakharMH.Optimization of dynamic buckling for sandwich nanocomposite plates with sensor and actuator layer based on sinusoidal-visco-piezoelasticity theories using Grey Wolf algorithm. J Sandw Struct Mater2017; 1–25.
81.
KolahchiR.A comparative study on the bending, vibration and buckling of viscoelastic sandwich nano-plates based on different nonlocal theories using DC, HDQ and DQ methods. Aerosp Sci Technol2017;
66: 235–248.
82.
PandeySPradyumnaS.Free vibration of functionally graded sandwich plates in thermal environment using a layerwise theory. Eur J Mech – A/Solids2015;
51: 55–66.
83.
NatarajanSGanapathiM.Bending and vibration of functionally graded material sandwich plates using an accurate theory. Finite Elem Anal Des2012;
57: 32–42.
84.
ThaiH-TVoTP.A new sinusoidal shear deformation theory for bending, buckling, and vibration of functionally graded plates. Appl Math Model2013;
37: 3269–3281.
85.
ThaiH-TNguyenT-KVoTP, et al.
Analysis of functionally graded sandwich plates using a new first-order shear deformation theory. Eur J Mech – A/Solids2014;
45: 211–225.
86.
IurlaroLGherloneMDi SciuvaM.Bending and free vibration analysis of functionally graded sandwich plates using the Refined Zigzag Theory. J Sandw Struct Mater2014;
16: 669–699.
87.
ElishakoffIGentiliniC.Three-dimensional flexure of rectangular plates made of functionally graded materials. J Appl Mech2005;
72: 788–791.
88.
KumarY.Free vibration of two-directional functionally graded annular plates using Chebyshev collocation technique and differential quadrature method. Int J Str Stab Dyn2014;
15: 1450086.
89.
DasMBarutAMadenciE, et al.
A triangular plate element for thermo-elastic analysis of sandwich panels with a functionally graded core. Int J Numer Meth Eng2006;
68: 940–966.
90.
XiangSJinYBiZ, et al.
A n-order shear deformation theory for free vibration of functionally graded and composite sandwich plates. Compos Struct2011;
93: 2826–2832.
91.
XiangSKangG.Static analysis of functionally graded plates by the various shear deformation theory. Compos Struct2013;
99: 224–230.
92.
ZhaoXLeeYYLiewKM.Free vibration analysis of functionally graded plates using the element-free kp-Ritz method. J Sound Vib2009;
319: 918–939.
93.
ZuoHYangZChenX, et al.
Bending, free vibration and buckling analysis of functionally graded plates via wavelet finite element method. Comput Mater Continua2014;
44: 167–204.
94.
GuptaATalhaM.Influence of porosity on the flexural and vibration response of gradient plate using nonpolynomial higher-order shear and normal deformation theory. Int J Mech Mater Des2018;
14: 277–296.
95.
GuptaATalhaM.Influence of porosity on the flexural and free vibration responses of functionally graded plates in thermal environment. Int J Str Stab Dyn2017;
18: 1850013.
96.
GuptaATalhaM.Influence of micro-structural defects on post-buckling and large-amplitude vibration of geometrically imperfect gradient plate. Nonlinear Dyn2018;
94: 39–56.
97.
KianiYSadighiMSalamiSJ, et al.
Low velocity impact response of thick FGM beams with general boundary conditions in thermal field. Compos Struct2013;
104: 293–303.
98.
YahiaSAAtmaneHAHouariMSA, et al.
Wave propagation in functionally graded plates with porosities using various higher-order shear deformation plate theories. Struct Eng Mech2015;
53: 1143–1165.
99.
BousahlaAABenyoucefSTounsiA, et al.
On thermal stability of plates with functionally graded coefficient of thermal expansion. Struct Eng Mech2016;
60: 313–335.
100.
ChauhanPKKhanIA.Review on analysis of functionally graded material beam type structure. Int J Adv Mech Eng2014;
4: 299–306.
101.
DaiH-LRaoY-NDaiT.A review of recent researches on FGM cylindrical structures under coupled physical interactions, 2000–2015. Compos Struct2016;
152: 199–225.
102.
KumarY.The Rayleigh–Ritz method for linear dynamic, static and buckling behavior of beams, shells and plates: a literature review. J Vib Control2018;
24: 1205–1227.
103.
Moreno-GarcíaPdos SantosJVALopesH.A review and study on Ritz method admissible functions with emphasis on buckling and free vibration of isotropic and anisotropic beams and plates. Arch Comput Methods Eng2018;
25: 785–815.
104.
TesslerADi SciuvaMGherloneM.Refinement of Timoshenko beam theory for composite and sandwich beams using zigzag kinematics. Nasa/TP-2007-2150862007: 1–45.
105.
TesslerADi SciuvaMGherloneM.A refined zigzag beam theory for composite and sandwich beams. J Compos Mater2009;
43: 1051–1081.
106.
Di SciuvaMGherloneMTesslerA.A robust and consistent first-order zigzag theory for multilayered beams. In: RGilatLBanks-Sills (eds) Advances in mathematical modeling and experimental methods for materials and structures: the Jacob Aboudi volume.
Dordrecht:
Springer, 2009, pp. 255–268.
107.
TesslerASciuvaMDGherloneM.Refined Zigzag Theory for laminated composite and sandwich plates. Nasa/TP-2009-2155612009;
52: 1–53.
108.
TesslerADi SciuvaMGherloneM.A consistent refinement of first-order shear deformation theory for laminated composite and sandwich plates using improved zigzag kinematics. J Mech Mater Struct2010;
5: 341–367.
109.
VersinoD.Refined theories and discontinuous Galerkin methods for the analysis of multilayered composite structures. PhD Thesis, Politecnico di Torino, Italy, 2012.
110.
IurlaroLGherloneMDi SciuvaM, et al.
Assessment of the Refined Zigzag Theory for bending, vibration, and buckling of sandwich plates: a comparative study of different theories. Compos Struct2013;
106: 777–792.
111.
GherloneM.On the use of zigzag functions in equivalent single layer theories for laminated composite and sandwich beams: a comparative study and some observations on external weak layers. J Appl Mech2013;
80: 061004–061019.
112.
IurlaroL.Development of refined models for multilayered composite and sandwich structures: analytical formulation, FEM implementation and experimental assessment. PhD Thesis, Politecnico di Torino, Italy, 2015.
113.
GherloneMTesslerADi SciuvaM.C0 beam elements based on the Refined Zigzag Theory for multilayered composite and sandwich laminates. Compos Struct2011;
93: 2882–2894.
114.
VersinoDGherloneMMattoneMC, et al.
C0 triangular elements based on the Refined Zigzag Theory for multilayer composite and sandwich plates. Compos Part B: Eng2013;
44B: 218–230.
115.
VersinoDGherloneMDi SciuvaM.Four-node shell element for doubly curved multilayered composites based on the Refined Zigzag Theory. Compos Struct2014;
118: 392–402.
116.
TrevisoAMundoDTournourM.A C0-continuous RZT beam element for the damped response of laminated structures. Compos Struct2015;
131: 987–994.
117.
TrevisoAMundoDTournourM.Dynamic response of laminated structures using a Refined Zigzag Theory shell element. Compos Struct2017;
159: 197–205.
118.
OñateEEijoAOllerS.Simple and accurate two-noded beam element for composite laminated beams using a Refined Zigzag Theory. Comput Methods Appl Mech Eng2012;
213–216: 362–382.
119.
OñateE.Structural analysis with the finite element method. Linear statics: volume 2: beams, plates and shells.
Netherlands:
Springer, https://www.springer.com/la/book/9781402087424 (2013, accessed 14 February 2019).
120.
Di SciuvaMGherloneMIurlaroL, et al.
A class of higher-order C0 composite and sandwich beam elements based on the Refined Zigzag Theory. Compos Struct2015;
132: 784–803.
121.
GrohRMWeaverPMTesslerA.Application of the Refined Zigzag Theory to the modeling of delaminations in laminated composites. NASA/TM-2015-2188082015;
22: 1–22.
122.
KefalATesslerAOterkusE.An efficient inverse finite element method for shape and stress sensing of laminated composite and sandwich plates and shells. Report, NASA, https://strathprints.strath.ac.uk/65515/ (31 July 2018, accessed 14 February 2019).
123.
PaganoNJ.Exact solutions for rectangular bidirectional composites and sandwich plates. J Compos Mater1970;
4: 20–34.
124.
ReddyJN.A simple higher-order theory for laminated composite plates. J Appl Mech1984;
51: 745–752.
125.
Madabhusi-RamanPDavalosJF.Static shear correction factor for laminated rectangular beams. Compos Part B: Eng1996;
27: 285–293.
126.
ZhangYFanQDongL, et al.
Are “higher-order” and “layer-wise zig-zag” plate & shell theories necessary for functionally graded materials and structures?Computers Modeling in Engineering and Sciences2016;
112: 1–32.
