Mixed Elastic Modeling of Multilayer Composite Plates by Using Dimension Reduction Approach

Document Type : Original Research Paper


1 Mechanical Engineering Department, Arak University, Arak, Iran.

2 Mechanical Engineering Department, Bu-Ali Sina University, Hamadan, Iran


In this paper, a mixed modeling approach for orthotropic laminated plates is developed. By adopting Hellinger-Reissner functional and dimension reduction method along the thickness, the governing equations were derived. By considering other theories i.e. classical plate theory, first order shear deformation theory and elasticity theory, the advantages of the current work are illustrated with some numerical results. Excellent agreements were observed by comparing the obtained results with three-dimensional elasticity theory for laminated thick plates. In the presented method, shear correction factor was not required for considering shear strain components. Furthermore, finite element simulation was implemented in Abaqus software by using two-dimensional shell elements and compared with obtained results. It is seen that although finite element model predicts good results for displacement field but it cannot provide any suitable results in thickness direction.


[1] R. Khandan, S. Noroozi, P. Sewell, J. Vinney, The development of laminated composite plate theories: a review, J. Mater. Sci., 47(16) (2012) 5901-5910.
[2] J.N. Reddy, J. Kim, A nonlinear modified couple stress-based third-order theory of functionally graded plates, Compos. Struct., 94(3) (2012) 1128-1143.
[3] N.J. Pagano, Exact solutions for rectangular bidirectional composites and sandwich plates, J. Compos. Mater., 4(1) (1970) 20-34.
[4] E. Reissner, On a mixed variational theorem and on shear deformable plate theory, Int. J. Numer. Meth. Eng., 23(2) (1986) 193-198.
[5] S.M. Alessandrini, D.N. Arnold, R.S. Falk, A.L. Madureira, Derivation and justification of plate models by variational methods, Plates and Shells, (1999) 1-21.
[6] W. Chih-Ping, L. Hao-Yuan, The RMVT-and PVD-based finite layer methods for the threedimensional analysis of multilayered composite and FGM plates, Compos. Struct., 92(10) (2010) 2476-2496.
[7] C.P. Wu, H.Y. Li, An RMVT-based third-order shear deformation theory of multilayered functionally graded material plates, Compos. Struct., 92(10) (2010) 2591-2605.
[8] E. Carrera, An assessment of mixed and classical theories on global and local response of multilayered orthotropic plates, Compos. Struct., 50(2) (2000) 183-198.
[9] L. Demasi, 6 mixed plate theories based on the generalized unified formulation, Part I: Governing equations, Compos. Struct., 87(1) (2009) 1-11.
[10] L. Demasi, 6 Mixed plate theories based on the generalized unified formulation, Part II: Layerwise theories, Compos. Struct., 87(1) (2009): 12-22.
[11] L. Demasi, 6 Mixed plate theories based on the generalized unified formulation, Part III: Advanced mixed high order shear deformation theories, Compos. Struct., 87(3) (2009) 183-194.
[12] L. Demasi, 6 Mixed plate theories based on the generalized unified formulation, Part IV: Zig-zag theories, Compos. Struct., 87(3) (2009) 195-205.
[13] L. Demasi, 6 Mixed plate theories based on the Generalized Unified Formulation, Part V: Results. Compos. Struct., 88(1) (2009) 1-16.
[14] M. Vogelius, I. Babuška, On a dimensional reduction method, I. The optimal selection of basis functions, Math. Comput., 37(155) (1981) 31-46.
[15] K.M. Liu, Dimensional reduction for the plate in elasticity on an unbounded domain, Math. Comput. Modell., 30(5-6) (1999) 1-22.
[16] F. Auricchio, E. Sacco, A mixed enhanced finite‐element for the analysis of laminated composite plates, Int. J. Numer. Meth. Eng., 44(10) (1999) 1481-1504.
[17] F. Daghia, S. De Miranda, F. Ubertini, E. Viola, A hybrid stress approach for laminated composite plates within the first-order shear deformation theory, Int. J. Solids Struct., 45(6) (2008) 1766-1787.
[18] F. Moleiro, C.M. Mota Soares, C.A. Mota Soares, J.N. Reddy, A layerwise mixed least-squares finite element model for static analysis of multilayered composite plates, Comput. Struct., 89(19-20) (2011) 1730-1742.
[19] F. Auricchio, G. Balduzzi, C. Lovadina, A new modeling approach for planar beams: finiteelement solutions based on mixed variational derivations, Int. J. Mater. Struct., 5(5) (2010) 771-794.
[20] F. Auricchio, B. Giuseppe, C. Lovadina, The dimensional reduction modelling approach for 3D beams: Differential equations and finite-element solutions based on Hellinger–Reissner principle, Int. J. Solids Struct., 50(25-26) (2013) 4184-4196.
[21] M. D’Ottavio, A Sublaminate Generalized Unified Formulation for the analysis of composite structures, Compos. Struct., 142 (2016) 187-199.
[22] M. Arefi, M. Kiani, A.M. Zenkour, Size-dependent free vibration analysis of a three-layered exponentially graded nano-/micro-plate with piezomagnetic face sheets resting on Pasternak’s foundation via MCST, J. Sandwich Struct. Mater., (2017) 1099636217734279.
[23] M. Arefi, A.M. Zenkour, Size-dependent electroelastic analysis of a sandwich microbeam based on higher-order sinusoidal shear deformation theory and strain gradient theory, Int. J. Solids Struct., 29(7) (2018) 1394-1406.
[24] M.L. Ribeiro, G.F.O. Ferreira, R. De Medeiros, A.J.M. Ferreira, V. Tita, Experimental and numerical dynamic analysis of laminate plates via carrera unified formulation, Compos. Struct., 202 (2018) 1176-1185.
[25] J.N. Reddy, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, CRC press, (2004).