Partial-mixed special finite element for the analysis of multilayer composites and FGM
DOI:
https://doi.org/10.13052/17797179.2012.702433Keywords:
Hamiltonian approach, semi-analytical solution, partial-mixed finite element, state space method, composite, functionally graded material, plateAbstract
A partial-mixed special finite element (FE) is proposed for the static analysis of multilayer composite and functionally graded material plates. Using the Hamiltonian formalism, the three-dimensional elasticity equations are first reformulated so that a partial-mixed variational formulation, retaining as primary variables the translational displacements augmented with the transverse stresses only, is obtained; this allows, in particular, a straightforward fulfilment of the multilayer interfaces continuity conditions. After an only in-plane FE discretisation, the static problem is then reduced, for a single layer, to a Hamiltonian eigenvalue problem that is solved analytically, through the layer thickness, using the symplectic formalism; the multilayer solution is finally reached via the state-space method and the propagator matrix concept. The performance, in convergence and accuracy, of the proposed approach, applied to representative examples, is shown to be very satisfactory.
Downloads
References
Andrianarison, O., & Benjeddou, A. (2012). Hamiltonian partial mixed finite element – state space symplectic
semi-analytical approach for the piezoelectric composites and FGM analysis. Acta Mechanica.
doi:10.1007/s00707-012-0646-8.
Auricchio, F., & Sacco, E. (1999). A mixed-enhanced finite element for the analysis of laminated composite
plates. International Journal for Numerical Methods in Engineering, 44, 1481–1504.
Bahar, L.Y. (1977). A mixed method in elasticity. Journal of Applied Mechanics, 44, 790–791.
Benjeddou, A., & Andrianarison, O. (2006). A heat mixed variational theorem for thermoelastic multilayered
composites. Computers and Structures, 84(19–20), 1247–1255.
Benner, P., Mehrmann, V., & Xu, H. (1998). A numerically stable, structure preserving method for computing
the eigenvalues of real Hamiltonian or symplectic pencils. Numerische Matematik, 78, 3229–
Birman, V., & Byrd, L.W. (2007). Modeling and analysis of functionally graded materials and structures.
Applied Mechanics Reviews, 60(5), 195–216.
Brezzi, F., & Fortin, M. (1991). Mixed and hybrid finite element method. Springer-Verlag.
Courant, R., & Hilbert, D. (1989). Methods of mathematical physics I, II. USA: John Wiley.
Desai, Y.M., Ramtekkar, G.S., & Shah, A.H. (2003). A novel 3D mixed finite element model for statics
of angle-ply laminates. International Journal for Numerical Methods in Engineering, 57, 1695–
Goldstein, H., Poole, C., & Safko, J. (2002). Classical mechanics. London: Addison Wesley.
Kreja, I. (2011). A literature review on computational models for laminated composite and sandwich
panels. Central European Journal of Engineering, 1(1), 59–80.
Li, R., Zhong, Y., & Tian, B. (2011). On new symplectic superposition method for exact solutions of
rectangular cantilever thin plates. Mechanics Research Communications, 38, 111–116.
Pagano, N.J. (1970). Exact solutions for rectangular bidirectional composites and sandwich plates. Journal
of Composite Materials, 4, 20–34.
Reissner, E. (1984). On a certain mixed variational theory and proposed applications. International Journal
of Numerical Methods in Engineering, 20, 1366–1368.
Robaldo, A., Benjeddou, A., & Carrera, E. (2005). Unified formulation for FE thermoelastic analysis of
multilayered anisotropic composite plates. Journal of Thermal Stresses, 28(10), 1031–1065.
Sheng, H.Y., & Ye, J.Q. (2002). A state space finite element for laminated composite plates. Computer
Methods in Applied Mechanics and Engineering, 191, 4259–4276.
Sheng, H.Y., & Ye, J.Q. (2005). A state space finite element for axisymmetric bending of angle-ply laminated
cylinder with clamped edges. Composite Structures, 68, 119–128.
Souriau J.M. (1997). Structures of dynamical systems: A symplectic view of physics. Progress in Mathematics
Paris: Lavoisier.
Steele, C.R., & Kim, Y.Y. (1992). Modified mixed variational principle and the state-vector equation for
elastic bodies and shells of revolution. Journal of Applied Mechanics, 59, 587–595.
Van Loan, C. (1984). A symplectic method for approximating all the eigenvalues of a Hamiltonian
matrix. Linear Algebra Application, 61, 233–251.
Washizu, K. (1968). Variational method in elasticity and plasticity. New York, NY: Pergamon Press.
Wu, C.P., & Li, H.Y. (2010a). The RMVT and PVD based finite layer methods for the threedimensional
analysis of multilayered composite and FGM plates. Composite Structures, 92, 2476–
Wu, C.P., & Li, H.Y. (2010b). An RMVT-based third order shear deformation theory for multilayered
functionally graded material plate. Composite Structures, 92, 2591–2595.
Zhang, Y.X., & Yang, C.H. (2009). Recent developments in finite element analysis for laminated composite
plates. Composite Structures, 88, 147–157.
Zou, G., & Tang, L. (1995). A semi-analytical solution for laminated composite plates in Hamilltonian
system. Computer Methods in Applied Mechanics and Engineering, 128, 395–404.
