Robust \(C^{0}\) high-order plate finite element for thin to very thick structures: mechanical and thermo-mechanical analysis. (English) Zbl 1242.74150
Summary: This paper presents a new \(C^{0}\) eight-node quadrilateral finite element (FE) for geometrically linear elastic plates. This finite element aims at modeling both thin and thick plates without any pathologies of the classical plate finite elements (shear and Poisson or thickness locking, spurious modes, etc). A \(C^{1}\) FE was previously developed by the first author based on the kinematics proposed by Touratier. This new FE can be viewed as an evolution towards three directions: (1) use of only \(C^{0}\) FE approximations; (2) modeling of thick to thin structures; and (3) capability in multifield problems. The transverse normal stress is included allowing use of the three-dimensional constitutive law. The element performances are evaluated on some standard plate tests, and comparisons are given with exact three-dimensional solutions for plates under mechanical and thermal loads. Comparisons are made with other plate models using \(C^{1}\) and semi-\(C^{1}\) FE approximations as well as with an eight node \(C^{0}\) FE based on the Reissner-Mindlin model. All results indicate that the present element is highly insensitive to mesh distortion, has very fast convergence properties and gives accurate results for displacements and stresses.
Keywords:
sinus model; normal stresses; thick plates; \(C^{0}\) finite element; numerical locking; mechanical load; thermo-mechanical loadReferences:
| [1] | Mackerle, Finite element linear and nonlinear, static and dynamic analysis of structural elements: a bibliography (1992-1995), Engineering Computations 14 (4) pp 347– (1997) · Zbl 0983.74500 · doi:10.1108/02644409710178494 |
| [2] | Mackerle, Finite- and boundary-element linear and nonlinear analysis of shells and shell-like structures : a bibliography (1999-2001), Finite Elements in Analysis and Design 38 pp 765– (2002) · Zbl 1008.74001 · doi:10.1016/S0168-874X(01)00103-2 |
| [3] | Hughes, Finite elements based upon Mindlin plate theory with particular reference to the four-node bilinear isoparametric element, ASME - Journal of Applied Mechanics 48 pp 587– (1981) · Zbl 0459.73069 · doi:10.1115/1.3157679 |
| [4] | Bathe, A four node plate bending element based on Mindlin-Reissner plate theory and a mixed interpolation, International Journal for Numerical Methods in Engineering 35 pp 503– (1985) · Zbl 0551.73072 |
| [5] | Somashekar, A field consistent four-node laminated anisotropic plate/shell element, Computers & Structures 25 pp 345– (1987) · Zbl 0599.73064 · doi:10.1016/0045-7949(87)90127-1 |
| [6] | Park, Analytical and Computational Models of Shells pp 359– (1989) |
| [7] | Batoz, A discrete shear triangular nine DOF element for the analysis of thick to very thin plates, International Journal for Numerical Methods in Engineering 28 pp 533– (1989) · Zbl 0675.73042 · doi:10.1002/nme.1620280305 |
| [8] | Simo, On stress resultant geometrically exact shell model. Part IV : variable thickness shells with through the thickness stretching, Computer Methods in Applied Mechanics and Engineering 81 pp 53– (1990) · Zbl 0746.73016 · doi:10.1016/0045-7825(90)90143-A |
| [9] | Bletzinger, A unified approach for shear-locking-free triangular and rectangular shell finite elements, Computers & Structures 75 pp 321– (2000) · doi:10.1016/S0045-7949(99)00140-6 |
| [10] | Reddy, Mechanics of laminated composite plates and shells (2004) · Zbl 1075.74001 |
| [11] | Kapania, A review on the analysis of laminated shells, ASME - Journal of Pressure Vessel Technology 111 (2) pp 88– (1989) · doi:10.1115/1.3265662 |
| [12] | Noor, Assessment of computational models for multilayered composite shells, Applied Mechanics Reviews 43 (4) pp 67– (1989) · doi:10.1115/1.3119162 |
| [13] | Reddy, Theories and computational models for composite laminates, Applied Mechanics Reviews 47 (6) pp 147– (1994) · doi:10.1115/1.3111076 |
| [14] | Kant, Estimation of transverse/interlaminar stresses in laminated composites-a selective review and survey of current developments, Composite Structures 49 pp 65– (2000) · doi:10.1016/S0263-8223(99)00126-9 |
| [15] | Carrera, Theories and finite elements for multilayered, anisotropic, composite plates and shells, Archives of Computational Methods in Engineering 9 (2) pp 87– (2002) · Zbl 1062.74048 · doi:10.1007/BF02736649 |
| [16] | Carrera, Historical review of ZIGZAG theories for multilayered plates and shells, Applied Mechanics Reviews 56 pp 287– (2003) · doi:10.1115/1.1557614 |
| [17] | Zhang, Recent developments in finite elements analysis for laminated composite plates, Composite Structures 88 pp 147– (2009) · doi:10.1016/j.compstruct.2008.02.014 |
| [18] | Yu, Asympotically accurate 3D recovery from Reissner-like composite plate finite elements, Computers & Structures 81 pp 439– (2003) · doi:10.1016/S0045-7949(03)00011-7 |
| [19] | Yu, Mathematical construction of a Reissner-Mindlin plate theory for composite laminates, International Journal of Solids and Structures 42 pp 6680– (2005) · Zbl 1119.74470 · doi:10.1016/j.ijsolstr.2005.02.049 |
| [20] | Polit, A multilayered/sandwich triangular finite element applied to linear and nonlinear analysis, Composite Structures 58 (1) pp 121– (2002) · doi:10.1016/S0263-8223(02)00033-8 |
| [21] | Dau, C1 plate and shell finite elements for geometrically non linear analysis of multilayered structures, Computers & Structures 84 pp 1264– (2006) · doi:10.1016/j.compstruc.2006.01.031 |
| [22] | Vidal, A thermomechanical finite element for the analysis of rectangular sandwich beams, Finite Elements in Analysis and Design 42 pp 868– (2006) · doi:10.1016/j.finel.2006.01.005 |
| [23] | Vidal, A family of sinus finite elements for the analysis of rectangular laminated beams, Composite Structures 84 pp 56– (2008) · doi:10.1016/j.compstruct.2007.06.009 |
| [24] | Touratier, An efficient standard plate theory, International Journal of Engineering Science 29 pp 901– (1991) · Zbl 0825.73299 · doi:10.1016/0020-7225(91)90165-Y |
| [25] | Touratier, A refined theory of laminated shallow shells, International Journal of Solids and Structures 29 (11) pp 1401– (1992) · Zbl 0759.73041 · doi:10.1016/0020-7683(92)90086-9 |
| [26] | Ganapathi, A c1 finite element including transverse shear and torsion warping for rectangular sandwich beams, International Journal for Numerical Methods in Engineering 45 (1) pp 47– (1999) · Zbl 0958.74064 · doi:10.1002/(SICI)1097-0207(19990510)45:1<47::AID-NME578>3.0.CO;2-B |
| [27] | Polit, High order triangular sandwich plate finite element for linear and nonlinear analyses, Computer Methods in Applied Mechanics and Engineering 185 pp 305– (2000) · Zbl 1064.74670 · doi:10.1016/S0045-7825(99)00264-9 |
| [28] | D’Ottavio, Sensitivity analysis of thickness assumptions for piezoelectric plate models, Journal of Intelligent Material Systems and Structures 20 pp 1815– (2009) · doi:10.1177/1045389X09343023 |
| [29] | Vidal, A refined sine-based finite element with transverse normal deformation for the analysis of laminated under thermomechanical loads, Journal of Mechanics of Materials and Structures 4 pp 1127– (2009) · doi:10.2140/jomms.2009.4.1127 |
| [30] | Polit, A new eight-node quadrilateral shear-bending plate finite element, International Journal for Numerical Methods in Engineering 37 pp 387– (1994) · Zbl 0788.73072 · doi:10.1002/nme.1620370303 |
| [31] | Touratier, A generalization of shear deformation theories for axisymmetric multilayered shells, International Journal of Solids and Structures 29 pp 1379– (1992) · Zbl 0759.73040 · doi:10.1016/0020-7683(92)90085-8 |
| [32] | Büchter, Three-dimensional extension of non-linear shell formulation based on the enhanced assumed strain concept, International Journal for Numerical Methods in Engineering 37 pp 2551– (1994) · Zbl 0808.73046 · doi:10.1002/nme.1620371504 |
| [33] | Parisch, A continuum-based shell theory for non-linear applications, International Journal for Numerical Methods in Engineering 38 pp 1855– (1995) · Zbl 0826.73041 · doi:10.1002/nme.1620381105 |
| [34] | Tessler, An improved plate theory of 1,2-order for thick composite laminates, International Journal of Solids and Structures 30 pp 981– (1993) · Zbl 0804.73027 · doi:10.1016/0020-7683(93)90022-Y |
| [35] | Barut, Analysis of thick sandwich construction by a 3,2-order theory, International Journal of Solids and Structures 38 pp 6063– (2001) · Zbl 1075.74518 · doi:10.1016/S0020-7683(00)00367-X |
| [36] | Carrera, Analysis of thickness locking in classical, refined and mixed plate theories, Composite Structures 82 (4) pp 549– (2008) · doi:10.1016/j.compstruct.2007.02.002 |
| [37] | Cheng, Elasticity theory of plates and a refined theory, ASME - Journal of Applied Mechanics 46 pp 644– (1979) · Zbl 0409.73047 · doi:10.1115/1.3424620 |
| [38] | Ganapathi, A c0 eight-node membrane-shear-bending element for geometrically non-linear (static and dynamic) analysis of laminates, International Journal for Numerical Methods in Engineering 39 pp 3453– (1996) · Zbl 0884.73066 · doi:10.1002/(SICI)1097-0207(19961030)39:20<3453::AID-NME9>3.0.CO;2-7 |
| [39] | Pagano, Exact solutions for rectangular bidirectional composites and sandwich plates, Journal of Composite Materials 4 pp 20– (1970) · doi:10.1177/002199837000400102 |
| [40] | Bhaskar, Thermoelastic solutions for orthotropic and anisotropic composite laminates, Composites Part B: Engineering 27B pp 415– (1996) · doi:10.1016/1359-8368(96)00005-4 |
| [41] | Carrera, Developments, ideas, and evaluations based upon Reissner’s mixed variational theorem in the modelling of multilayered plates and shells, Applied Mechanics Reviews 54 pp 301– (2001) · doi:10.1115/1.1385512 |
| [42] | Carrera, Theories and finite elements for multilayered plates and shells: a unified compact formulation with numerical assessment and benchmarking, Archives of Computational Methods in Engineering 10 (3) pp 216– (2003) · Zbl 1140.74549 · doi:10.1007/BF02736224 |
| [43] | Babuska, On locking and robustness in the finite element method, SIAM Journal Numerical Analysis 51 pp 221– (1992) |
| [44] | Bischoff, On the physical significance of higher order kinematic and static variables in a three-dimensional shell formulation, International Journal of Solids and Structures 37 pp 6933– (2000) · Zbl 1003.74045 · doi:10.1016/S0020-7683(99)00321-2 |
| [45] | Klinkel, A continuum based three-dimensional shell element for laminated structures, Computers & Structures 71 pp 43– (1999) · Zbl 0942.74649 · doi:10.1016/S0045-7949(98)00222-3 |
| [46] | Demasi, 3 plate theories for thick and thin plates: the generalized unified formulation, Composite Structures 84 pp 256– (2008) · doi:10.1016/j.compstruct.2007.08.004 |
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.
