Restricted accessResearch articleFirst published online 2014-6
Reissner’s mixed variational theorem-based finite cylindrical layer methods for the three-dimensional free vibration analysis of sandwich circular hollow cylinders with an embedded functionally graded material layer
Based on Reissner’s mixed variational theorem (RMVT), finite cylindrical layer methods (FCLMs) were developed for the three-dimensional (3D) free vibration analysis of simply supported, functionally graded material (FGM) sandwich circular hollow cylinders. The FGM sandwich cylinder consists of a thick and soft FGM core bounded with two thin and stiff homogeneous material face sheets, in which the material properties of the FGM core are assumed to obey an exponent-law varying exponentially with the thickness coordinate. In this formulation, the FGM sandwich cylinder is divided into a number of equal-thickness cylindrical layers, where the trigonometric functions and Lagrange polynomials are used to interpolate the in- and out-of-surface variations of the field variables of each individual layer, respectively. An h-refinement process instead of a p-refinement one is adopted to yield the convergent solutions in this study, and the layerwise linear, quadratic or cubic function distribution through the thickness coordinate is thus assumed for the related field variables. The accuracy and convergence of the RMVT-based FCLMs developed in this article are assessed by comparing their solutions with the exact 3D solutions available in the literature.
AkhrasGLiWC (2007) Three-dimensional static, vibration and stability analysis of piezoelectric composite plates using a finite layer method. Smart Materials and Structures16: 561–569.
2.
AkhrasGLiWC (2008) Three-dimensional thermal buckling analysis of piezoelectric composite plates using the finite layer method. Smart Materials and Structures17: 1–8.
3.
AkhrasGLiWC (2010) Three-dimensional stability analysis of piezoelectric antisymmetric angle-ply laminates using finite layer method. Journal of Intelligent Material Systems and Structures21: 719–727.
4.
AlibeiglooANouriV (2010) Static analysis of functionally graded cylindrical shell with piezoelectric layers using differential quadrature method. Composite Structures92: 1775–1785.
5.
CarreraE (2001) Developments, ideas, and evaluations based upon Reissner’s Mixed Variational Theorem in the modeling of multilayered plates and shells. Applied Mechanics Reviews54: 301–329.
6.
CarreraE (2003a) Historical review of zig-zag theories for multilayered plates and shells. Applied Mechanics Reviews56: 287–308.
7.
CarreraE (2003b) Theories and finite elements for multilayered plates and shells: A unified compact formulation with numerical assessment and benchmarks. Archives of Computational Methods in Engineering10: 215–296.
8.
CarreraEBrischettoS (2009) A survey with numerical assessment of classical and refined theories for the analysis of sandwich plates. Applied Mechanics Reviews62: 1–17.
9.
CarreraECiuffredaA (2005) A unified formulation to assess theories of multilayered plates for various bending problems. Composite Structures69: 271–293.
10.
CarreraEPetroloM (2010) Guidelines and recommendations to construct theories for metallic and composite plates. AIAA Journal48: 2852–2866.
11.
CarreraEBrischettoSRobaldoA (2008) Variable kinematic model for the analysis of functionally graded material plates. AIAA Journal46: 194–203.
12.
ChenWQLvCFBianZG (2003) Elasticity solution for free vibration of laminated beams. Composite Structures62: 75–82.
13.
ChenWQBianZGDingHJ (2004a) Three-dimensional vibration analysis of fluid-filled orthotropic FGM cylindrical shells. International Journal of Mechanical Sciences46: 159–171.
14.
ChenWQLvCFBianZG (2004b) Free vibration analysis of generally laminated beams via state-space-based differential quadrature. Composite Structures63: 417–425.
15.
CheungYKJiangCP (2001) Finite layer method in analysis of piezoelectric composite laminates. Computer Methods in Applied Mechanics and Engineering191: 879–901.
16.
CinefraMBelouettarSSoaveMCarreraE (2010) Variable kinematic models applied to free-vibration analysis of functionally graded material shells. European Journal of Mechanics A/Solids29: 1078–1087.
17.
KangJHLeissaAW (2005) Free vibrations of thick, complete conical shells of revolution from a three-dimensional theory. Journal of Applied Mechanics72: 797–800.
18.
LeissaAWJinyoungS (1995) Three-dimensional vibrations of truncated hollow cones. Journal of Vibration and Control1: 145–158.
19.
LiewKMLimCWKitipornchaiS (1997) Vibration of shallow shells: a review with bibliography. Applied Mechanics Reviews50: 431–444.
20.
LiewKMBergmanLANgTYLamKY (2000) Three-dimensional vibration of cylindrical shell panels-solution by continuum and discrete approaches. Computational Mechanics26: 208–221.
21.
LiewKMPengLXNgTY (2002) Three-dimensional vibration analysis of spherical shell panels subjected to different boundary conditions. International Journal of Mechanical Sciences44: 2103–2117.
22.
LimCWKitipornchaiSLiewKM (1997) Comparative accuracy of shallow and deep shell theories for vibration of cylindrical shells. Journal of Vibration and Control3: 119–143.
23.
LüCFChenWQShaoJW (2008) Semi-analytical three-dimensional elasticity solutions for generally laminated composite plates. European Journal of Mechanics A/Solids27: 899–917.
24.
NieGJZhongZ (2007) Semi-analytical solution for three-dimensional vibration of functionally graded circular plates. Computer Methods in Applied Mechanics and Engineering196: 4901–4910.
25.
NoorAKRarigPL (1974) Three-dimensional solutions of laminated cylinders. Computer Methods in Applied Mechanics and Engineering3: 319–334.
26.
NoorAKBurtonWSPetersJM (1991) Assessment of Computational models for multilayered composite cylinders. International Journal of Solids and Structures27: 1269–1286.
27.
NoorAKBurtonWSBertCW (1996) Computational model for sandwich panels and shells. Applied Mechanics Reviews49: 155–199.
28.
RamirezFHeyligerPRPanE (2006a) Static analysis of functionally graded elastic anisotropic plates using a discrete layer approach. Composites Part B: Engineering37: 10–20.
29.
RamirezFHeyligerPRPanE (2006b) Discrete layer solution to free vibrations of functionally graded magneto-electro-elastic plates. Mechanics of Advanced Materials and Structures13: 249–2466.
30.
ReissnerE (1984) On a certain mixed variational theory and a proposed application. International Journal for Numerical Methods in Engineering20: 1366–1368.
31.
Reissner E (1985) On mixed variational formulations in finite elasticity. Acta Mechanica 56: 117–125.
32.
ReissnerE (1986) On a mixed variational theorem and on a shear deformable plate theory. International Journal for Numerical Methods in Engineering23: 193–198.
33.
QatuM (2002a) Recent research advances in the dynamic behavior of shells: 1989-2000, Part 1: Laminated composite shells. Applied Mechanics Reviews55: 325–349.
34.
QatuM (2002b) Recent research advances in the dynamic behavior of shells: 1989-2000, Part 2: Homogeneous shells. Applied Mechanics Reviews55: 415–434.
35.
QatuMSullivanRWWangW (2009) Recent research advances on the dynamic analysis of composite shells: 2000_2009. Composite Structures93: 14–31.
36.
SaravanosDAHeyligerPR (1999) Mechanics and computational models for laminated piezoelectric beams, plates, and shells. Applied Mechanics Reviews52: 305–320.
37.
ShengHYYeJQ (2003) A three-dimensional state space finite element solution for laminated composite cylindrical shells. Computer Methods in Applied Mechanics and Engineering192: 2441–2459.
38.
TajeddiniVOhadiA (2012) Three-dimensional vibration analysis of functionally graded thick, annular plates with variable thickness via polynomial-Ritz method. Journal of Vibration and Control. 18: 1698--1707.
39.
Wu CP and Chang YT (2012) A unified formulation of RMVT-based finite cylindrical layer methods for sandwich circular hollow cylinders with an embedded FGM layer. Composites: Part B: Engineering 43: 3318--3333.
40.
WuCPChenWY (1994) Vibration and stability of laminated plates based on a local high order plate theory. Journal of Sound and Vibration177: 503–520.
41.
WuCPChiuKH (2011) RMVT-based meshless collocation and element-free Galerkin methods for the quasi-3D free vibration analysis of multilayered composite and FGM plates. Composite Structures93: 1433–1448.
42.
WuCPHungYC (1999) Asymptotic theory of laminated circular conical shells. International Journal of Engineering Science37: 977–1005.
43.
Wu CP and Kuo CH (2012) A unified formulation of PVD-based finite cylindrical layer methods for functionally graded material sandwich cylinders. Applied Mathematical Modeling 37: 916–938.
44.
WuCPKuoHC (1992) Interlaminar stresses analysis for laminated composite plates based on a local high order lamination theory. Composite Structures20: 237–247.
45.
WuCPLiHY (2010a) The RMVT- and PVD-based finite layer methods for the three-dimensional analysis of multilayered composite and FGM plates. Composite Structures92: 2476–2496.
46.
WuCPLiHY (2010b) RMVT- and PVD-based finite layer methods for the quasi-3D free vibration analysis of multilayered composite and FGM plates. CMC-Computers, Materials, & Continua19: 155–198.
47.
WuCPLiYC (2012) Exact solutions of free vibration of rotating multilayered FGM cylinders. Smart Structures and Systems9: 105–125.
48.
WuCPLinCC (1993) Analysis of sandwich plates using a mixed finite element. Composite Structures25: 397–405.
49.
Wu CP and Lin CC (1994) Stress and displacement of thick doubly curved laminated shells. Journal of Engineering Mechanics 120: 1403–1428.
50.
WuCPLiuCC (1995) Mixed finite element analysis of thick doubly curved laminated shells. Journal of Aerospace Engineering8: 43–53.
51.
WuCPTsaiYH (2004) Asymptotic DQ solutions of functionally graded annular spherical shells. European Journal of Mechanics A/Solids23: 283–299.
WuCPHungYCLoJY (2002) A refined asymptotic theory of laminated circular conical shells. European Journal of Mechanics A/Solids21: 281–300.
54.
WuCPChiuKHWangYM (2008) A review of the three-dimensional analytical approaches of multilayered and functionally graded piezoelectric plates and shells. CMC-Computers, Materials, & Continua8: 93–132.
55.
WuCPChiuKHWangYM (2011) RMVT-based meshless collocation and element-free Galerkin methods for the analysis of quasi-3D analysis of multilayered composite and FGM plates. Composite Structures93: 923–943.
56.
YeJQShengHYQinQH (2004) A state space finite element for laminated composites with free edges and subjected to transverse and in-plane loads. Computers and Structures82: 1131–1141.
