The development of laminated composite plate theories: a review

This study investigates and reviews approaches to modelling laminated composite plates. It explores theories that have been proposed and developed and assesses their suitability and functionality. The particular focus in this study has been on normal stresses and the through-thickness distributions of transverse shear. These are important for composite plates as stress-induced failures can occur in three different ways. Therefore, it is essential to understand and calculate transverse shear and normal stress through the thickness of the plate accurately. In this study, previous laminated composite plate theories are categorised and reviewed in a general sense, i.e. not problem specific, and the advantages and disadvantages of each model are discussed. This research mainly focuses on how accurate and efficient the models can predict the transverse shear. It starts with displacement-based theories from very basic models such as Classical laminate plate theory to more complicated and higher-order shear deformation theory. Models are furthermore categorised by how the models consider the overall laminate. In this article, the theories are divided into two parts: Single layer theory, where the whole plate is considered as one layer; and Layerwise theory, where each layer is treated separately. The models based on zig-zag and Discrete Theories are then reviewed, and finally the mixed (hybrid) plate theories are studied.

[1]  G. Kirchhoff,et al.  Über das Gleichgewicht und die Bewegung einer elastischen Scheibe. , 1850 .

[2]  A. Love,et al.  The Mathematical Theory of Elasticity. , 1928 .

[3]  S. Timoshenko,et al.  Theory of elasticity , 1975 .

[4]  E. Reissner.  ON THE THEORY OF BENDING OF ELASTIC PLATES , 1944 .

[5]  E. Reissner,et al.  Bending and Stretching of Certain Types of Heterogeneous Aeolotropic Elastic Plates , 1961 .

[6]  Y. Stavsky,et al.  Bending and Stretching of Laminated Aeolotropic Plates , 1962 .

[7]  E. Dill,et al.  Theory of Elasticity of an Anisotropic Elastic Body , 1964 .

[8]  Y. Stavsky,et al.  Elastic wave propagation in heterogeneous plates , 1966 .

[9]  A. W. Leissa,et al.  Analysis of Heterogeneous Anisotropic Plates , 1969 .

[10]  N. J. Pagano,et al.  Elastic Behavior of Multilayered Bidirectional Composites , 1972 .

[11]  C. Sun,et al.  Theories for the Dynamic Response of Laminated Plates , 1973 .

[12]  A. K. Noor,et al.  Free vibrations of multilayered composite plates. , 1973 .

[13]  S. Srinivas,et al.  A refined analysis of composite laminates , 1973 .

[14]  C. Sun,et al.  A higher order theory for extensional motion of laminated composites , 1973 .

[15]  N. J. Pagano,et al.  Interlaminar Stresses in Composite Laminates—An Approximate Elasticity Solution , 1974 .

[16]  N. J. Pagano,et al.  On the Calculation of Interlaminar Normal Stress in Composite Laminate , 1974 .

[17]  Ahmed K. Noor,et al.  Stability of multilayered composite plates , 1975 .

[18]  S. Tang,et al.  A Boundary Layer Theory - Part I: Laminated Composites in Plane Stress , 1975 .

[19]  A. Levy,et al.  A Boundary Layer Theory - Part II: Extension of Laminated Finite Strip , 1975 .

[20]  Carl T. Herakovich,et al.  Edge Effects in Angle-Ply Composite Laminates* , 1977 .

[21]  N. J. Pagano,et al.  Stress fields in composite laminates , 1978 .

[22]  N. J. Pagano,et al.  Free Edge Stress Fields in Composite Laminates. , 1978 .

[23]  Tsu-Wei Chou,et al.  The effect of transverse shear on the longitudinal compressive strength of fibre composites , 1980 .

[24]  M. Levinson,et al.  An accurate, simple theory of the statics and dynamics of elastic plates , 1980 .

[25]  Paul Seide,et al.  An improved approximate theory for the bending of laminated plates , 1980 .

[26]  S. G. Lekhnit︠s︡kiĭ.  Theory of elasticity of an anisotropic body , 1981 .

[27]  Shuodao Wang,et al.  Boundary-Layer Effects in Composite Laminates: Part 2—Free-Edge Stress Solutions and Basic Characteristics , 1982 .

[28]  Shuodao Wang,et al.  The Interface Crack Behavior in Dissimilar Anisotropic Composites Under Mixed-Mode Loading , 1983 .

[29]  Shuodao Wang,et al.  The interface crack between dissimilar anisotropic composite materials , 1983 .

[30]  J. Reddy.  Energy and variational methods in applied mechanics : with an introduction to the finite element method , 1984 .

[31]  E. Reissner,et al.  Reflections on the Theory of Elastic Plates , 1985 .

[32]  J. N. Reddy,et al.  Analysis of laminated composite plates using a higher‐order shear deformation theory , 1985 .

[33]  C. Kassapoglou,et al.  An Efficient Method for the Calculation of Interlaminar Stresses in Composite Materials , 1986 .

[34]  M. Stein,et al.  Nonlinear theory for plates and shells including the effects of transverse shearing , 1986 .

[35]  J. Ren.  Bending theory of laminated plate , 1986 .

[36]  Reaz A. Chaudhuri,et al.  An equilibrium method for prediction of transverse shear stresses in a thick laminated plate , 1986 .

[37]  I. H. Marshall,et al.  Proceedings of the American Society for Composites First Technical Conference: October 1986, Dayton, Ohio, USA, Technomic Publishing Co., Lancaster, PA, USA , 1987 .

[38]  Christos Kassapoglou,et al.  Closed Form Solutions for the Interlaminar Stress Field in Angle-Ply and Cross-Ply Laminates , 1987 .

[39]  Dawn C. Jegley,et al.  EFFECTS OF TRANSVERSE SHEARING ON CYLINDRICAL BENDING, VIBRATION, AND BUCKLING OF LAMINATED PLATES , 1987 .

[40]  M. Di Sciuva,et al.  An Improved Shear-Deformation Theory for Moderately Thick Multilayered Anisotropic Shells and Plates , 1987 .

[41]  Tarun Kant,et al.  Flexural analysis of laminated composites using refined higher-order C ° plate bending elements , 1988 .

[42]  Ahmed K. Noor,et al.  Assessment of Shear Deformation Theories for Multilayered Composite Plates , 1989 .

[43]  J. N. Reddy,et al.  A plate bending element based on a generalized laminate plate theory , 1988 .

[44]  Christos Kassapoglou,et al.  Determination of Interlaminar Stresses in Composite Laminates under Combined Loads , 1990 .

[45]  K. H. Lee,et al.  An improved zig-zag model for the bending of laminated composite shells , 1990 .

[46]  J. N. Reddy,et al.  A GENERAL NON-LINEAR THIRD-ORDER THEORY OF PLATES WITH MODERATE THICKNESS , 1990 .

[47]  Charles W. Bert,et al.  Free vibrations of laminated rectangular plates analyzed by higher order individual-layer theory , 1991 .

[48]  M. Touratier,et al.  An efficient standard plate theory , 1991 .

[49]  Maenghyo Cho,et al.  An efficient higher-order plate theory for laminated composites , 1992 .

[50]  Maurice Touratier,et al.  A generalization of shear deformation theories for axisymmetric multilayered shells , 1992 .

[51]  H. Kabir,et al.  A double fourier series approach to the solution of a moderately thick simply supported plate with antisymmetric angle-ply laminations , 1992 .

[52]  J. N. Reddy,et al.  A Variational Approach to Three-Dimensional Elasticity Solutions of Laminated Composite Plates , 1992 .

[53]  Chih‐Ping Wu,et al.  Interlaminar stresses analysis for laminated composite plates based on a local high order lamination theory , 1992 .

[54]  K. Rohwer,et al.  Application of higher order theories to the bending analysis of layered composite plates , 1992 .

[55]  Ji-Fan He.  Cylindrical bending of arbitrary cross-ply laminated plates , 1992 .

[56]  Dahsin Liu,et al.  Interlayer shear slip theory for cross-ply laminates with nonrigid interfaces , 1992 .

[57]  Dahsin Liu,et al.  An interlaminar stress continuity theory for laminated composite analysis , 1992 .

[58]  Marco Di Sciuva,et al.  Multilayered anisotropic plate models with continuous interlaminar stresses , 1992 .

[59]  Chien-Chang Lin,et al.  Method for Calculating the Interlaminar Stresses in Symmetric Laminates Containing a Circular Hole , 1991 .

[60]  X. Wang,et al.  Analytic solution for interlaminar stresses in a multilaminated cylindrical shell under thermal and mechanical loads , 1992 .

[61]  Kostas P. Soldatos,et al.  A transverse shear deformation theory for homogeneous monoclinic plates , 1992 .

[62]  S. K. Morton,et al.  An analytical solution for the thermal stresses at the free edges of laminated plates , 1993 .

[63]  Maenghyo Cho,et al.  Efficient higher order composite plate theory for general lamination configurations , 1993 .

[64]  Rakesh K. Kapania,et al.  Free vibration analysis of laminated plates using a layerwise theory , 1993 .

[65]  Kostas P. Soldatos,et al.  A unified formulation of laminated composite, shear deformable, five-degrees-of-freedom cylindrical shell theories , 1993 .

[66]  J. N. Reddy,et al.  Modelling of thick composites using a layerwise laminate theory , 1993 .

[67]  T. K. Varadan,et al.  Interlaminar Stresses In Composite Cylindrical Shells Under Transient Loads , 1993 .

[68]  Chien-Chang Lin,et al.  Interlaminar stresses around a hole in symmetric cross-ply laminates under bending/torsion , 1993 .

[69]  Jianhua Han,et al.  A three‐dimensional multilayer composite finite element for stress analysis of composite laminates , 1993 .

[70]  M. Di Sciuva,et al.  A general quadrilateral multilayered plate element with continuous interlaminar stresses , 1993 .

[71]  Wilfried Becker,et al.  Closed-form solution for the free-edge effect in cross-ply laminates , 1993 .

[72]  Maenghyo Cho,et al.  Finite element for composite plate bending based on efficient higher order theory , 1994 .

[73]  Chih‐Ping Wu,et al.  Vibration And Stability Of Laminated Plates Based On A Local High Order Plate Theory , 1994 .

[74]  Humayun R.H. Kabir,et al.  Analysis of a simply supported plate with symmetric angle-ply laminations , 1994 .

[75]  Matsunaga Hiroyuki.  Free vibration and stability of thick elastic plates subjected to in-plane forces , 1994 .

[76]  Linghui He,et al.  A linear theory of laminated shells accounting for continuity of displacements and transverse shear stresses at layer interfaces , 1994 .

[77]  J.-F. He,et al.  Vibration Analysis Of Laminated Plates Using A Refined Shear Deformation Theory , 1994 .

[78]  Ugo Icardi,et al.  Analysis of thick multilayered anisotropic plates by a higher order plate element , 1995 .

[79]  N. Ganesan,et al.  INTERLAMINAR STRESSES IN SHELLS OF REVOLUTION , 1996 .

[80]  Ji-Fan He,et al.  Bending analysis of antisymmetric angle-ply laminated plates including transverse shear effects , 1996 .

[81]  H. Kabir,et al.  BENDING OF A SIMPLY SUPPORTED RECTANGULAR PLATE WITH ARBITRARY LAMINATION , 1996 .

[82]  K. Soldatos,et al.  Accurate Stress Analysis of Laminated Plates Combining a Two-Dimensional Theory with the Exact Three-Dimensional Solution for Simply Supported Edges , 1997 .

[83]  Kostas P. Soldatos,et al.  A method for improving the stress analysis performance of one- and two-dimensional theories for laminated composites , 1997 .

[84]  Moussa Karama,et al.  Comparison of various laminated plate theories , 1997 .

[85]  H. Matsunaga,et al.  Buckling instabilities of thick elastic plates subjected to in-plane stresses , 1997 .

[86]  E. Carrera.  Layer-Wise Mixed Models for Accurate Vibrations Analysis of Multilayered Plates , 1998 .

[87]  M. Karama,et al.  Bending, buckling and free vibration of laminated composite with a transverse shear stress continuity model , 1998 .

[88]  Hiroyuki Matsunaga,et al.  Vibration and stability of cross-ply laminated composite plates according to a global higher-order plate theory , 2000 .

[89]  Transverse shear and normal stresses in nonlinear shell theory , 2000 .

[90]  Djordje Vuksanović,et al.  Linear analysis of laminated composite plates using single layer higher-order discrete models , 2000 .

[91]  Tarun Kant,et al.  Estimation of transverse/interlaminar stresses in laminated composites – a selective review and survey of current developments , 2000 .

[92]  Hiroyuki Matsunaga,et al.  Vibration and stability of angle-ply laminated composite plates subjected to in-plane stresses , 2001 .

[93]  V. V..  1903 An Improved Transverse Shear Deformation Theory for Laminated Anisotropic Plates , 2001 .

[94]  Hiroyuki Matsunaga,et al.  Assessment of a global higher-order deformation theory for laminated composite and sandwich plates , 2002 .

[95]  T. Tran-Cong,et al.  Finite strip elements for laminated composite plates with transverse shear strain discontinuities , 2002 .

[96]  Erasmo Carrera,et al.  Classical and advanced multilayered plate elements based upon PVD and RMVT. Part 1: Derivation of finite element matrices , 2002 .

[97]  Erasmo Carrera,et al.  Classical and advanced multilayered plate elements based upon PVD and RMVT. Part 2: Numerical implementations , 2002 .

[98]  Song Cen,et al.  A new hybrid-enhanced displacement-based element for the analysis of laminated composite plates , 2002 .

[99]  Sébastien Mistou,et al.  Mechanical behaviour of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity , 2003 .

[100]  Abdul Hamid Sheikh,et al.  A new plate bending element based on higher-order shear deformation theory for the analysis of composite plates , 2003 .

[101]  Worsak Kanok-Nukulchai,et al.  An element-based 9-node resultant shell element for large deformation analysis of laminated composite plates and shells , 2004 .

[102]  Jinho Oh,et al.  A finite element based on cubic zig-zag plate theory for the prediction of thermo-electric-mechanical behaviors , 2004 .

[103]  Kostas P. Soldatos,et al.  Complex potential formalisms for bending of inhomogeneous monoclinic plates including transverse shear deformation , 2004 .

[104]  Erasmo Carrera,et al.  A unified formulation to assess theories of multilayered plates for various bending problems , 2005 .

[105]  K. S. Kim,et al.  Linear and Geometrically nonlinear analysis of plates and shells by a new refined non-conforming triangular plate/shell element , 2005 .

[106]  Zhen Wu,et al.  Refined laminated composite plate element based on global–local higher-order shear deformation theory , 2005 .

[107]  Hiroyuki Matsunaga,et al.  Thermal buckling of angle-ply laminated composite and sandwich plates according to a global higher-order deformation theory , 2006 .

[108]  Rüdiger Schmidt,et al.  Large rotations in first-order shear deformation FE analysis of laminated shells , 2006 .

[109]  K. Soldatos.  General Solutions for the Statics of Anisotropic, Transversely Inhomogeneous Elastic Plates in Terms of Complex Functions , 2006 .

[110]  Metin Aydogdu,et al.  Comparison of Various Shear Deformation Theories for Bending, Buckling, and Vibration of Rectangular Symmetric Cross-ply Plate with Simply Supported Edges , 2006 .

[111]  H. Arya,et al.  New First-Order Shear Deformation Plate Theories , 2007 .

[112]  M. Touratier,et al.  An efficient and simple refined model for temperature analysis in thin laminated composites , 2007 .

[113]  Erasmo Carrera.  On the use of transverse shear stress homogeneous and non-homogeneous conditions in third-order orthotropic plate theory , 2007 .

[114]  Hiroyuki Matsunaga,et al.  Free vibration and stability of angle-ply laminated composite and sandwich plates under thermal loading , 2007 .

[115]  C. M. Mota Soares,et al.  Mixed least-squares finite element model for the static analysis of laminated composite plates , 2008 .

[116]  K. K. Ang,et al.  Computation of stress resultants in plate bending problems using higher-order triangular elements , 2008 .

[117]  Francesco Ubertini,et al.  A hybrid stress approach for laminated composite plates within the First-order Shear Deformation Theory , 2008 .

[118]  Ala Tabiei,et al.  Geometrically nonlinear analysis of laminated composite thin shells using a modified first-order shear deformable element-based Lagrangian shell element , 2008 .

[119]  M.Kh. Elmarghany,et al.  A refined zigzag nonlinear first-order shear deformation theory of composite laminated plates , 2008 .

[120]  Yiming Fu,et al.  Analysis of Inter-Laminar Stresses for Composite Laminated Plate with Interfacial Damage , 2008 .

[121]  K. Swaminathan,et al.  Analytical solutions using a higher order refined computational model with 12 degrees of freedom for the free vibration analysis of antisymmetric angle-ply plates , 2008 .

[122]  L. Demasi.  ∞6 Mixed plate theories based on the Generalized Unified Formulation.: Part II: Layerwise theories , 2009 .

[123]  L. Demasi.  ∞6 Mixed plate theories based on the Generalized Unified Formulation. Part IV: Zig-zag theories , 2009 .

[124]  Metin Aydogdu,et al.  A new shear deformation theory for laminated composite plates , 2009 .

[125]  Luciano Demasi,et al.  ∞6 Mixed plate theories based on the Generalized Unified Formulation. Part I: Governing equations , 2009 .

[126]  Dimitris A. Saravanos,et al.  Higher-order layerwise laminate theory for the prediction of interlaminar shear stresses in thick composite and sandwich composite plates , 2009 .

[127]  Marco Gherlone,et al.  Refined Zigzag Theory for Laminated Composite and Sandwich Plates , 2015 .

[128]  Mixed least-squares finite element models for static and free vibration analysis of laminated composite plates , 2009 .

[129]  L. Demasi.  ∞6 Mixed plate theories based on the Generalized Unified Formulation. Part III: Advanced mixed high order shear deformation theories , 2009 .

[130]  Chien Ming Wang,et al.  A higher-order plate element for accurate prediction of interlaminar stresses in laminated composite plates , 2009 .

[131]  L. Demasi.  ∞6 Mixed plate theories based on the Generalized Unified Formulation. Part V: Results , 2009 .

[132]  Fabrizio Vestroni,et al.  A generalized higher-order theory for buckling of thick multi-layered composite plates with normal and transverse shear strains , 2010 .

[133]  Chih‐Ping Wu,et al.  An RMVT-based third-order shear deformation theory of multilayered functionally graded material plates , 2010 .

[134]  S. Noroozi,et al.  Optimum Design of Fibre Orientation Angles in Composite Laminate Plates for Minimum Thickness , 2010 .

[135]  Paul M. Weaver,et al.  An enhanced single-layer variational formulation for the effect of transverse shear on laminated orthotropic plates , 2010 .

[136]  C. M. Mota Soares,et al.  Layerwise mixed least-squares finite element models for static and free vibration analysis of multilayered composite plates , 2010 .

[137]  Song Xiang,et al.  A nth-order meshless generalization of Reddy’s third-order shear deformation theory for the free vibration on laminated composite plates , 2011 .

[138]  K. Liew,et al.  A review of meshless methods for laminated and functionally graded plates and shells , 2011 .

[139]  S. K. Satsangi,et al.  Accurate transverse stress evaluation in composite/sandwich thick laminates using a C0 HSDT and a novel post-processing technique , 2011 .