A review on plate and shell theories for laminated and sandwich structures highlighting the finite element method

Abstract In engineering, the amount of complex geometrical problems, which needs to be solved by using plates and shells theories, is remarkable. This is the reason why there are so many theories, which attempt to simplify three-dimensional problems into simpler ones. Additionally, the current increasing use of laminated and sandwich structures demands a minimum of accuracy from two-dimensional formulations. In the literature, one can find a variety of bi-dimensional theories and solution methods to solve these problems. Laminated and sandwich structure formulations are mainly classified according to the treatment of the variables in the normal direction of the plate/shell surface: equivalent single layer, Zig-Zag and layer-wise theories. The contribution of this paper is to set the stage for new theories and solution methods for laminated and sandwich structures by reviewing over 100 papers. To show the importance of the coupling between plate/shell theories and the respective solution method, a detailed review on theories and the respective solution methods is firstly given to update the current state of art. After that, solution methods based on the Finite Element Method are explained to exhibit how particular and/or complex an approach can get. In fact, this review gives a clearer picture on plate/shell theories.

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

[2]  L. Demasi.  An Invariant Model for any Composite Plate Theory and FEM applications: the Generalized Unifled Formulation , 2009 .

[3]  Marco Di Sciuva,et al.  Development of an anisotropic, multilayered, shear-deformable rectangular plate element , 1985 .

[4]  A. Noor,et al.  Assessment of computational models for sandwich panels and shells , 1995 .

[5]  Maria Radwańska,et al.  A survey of finite element models for the analysis of moderately thick shells , 1991 .

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

[7]  Erasmo Carrera,et al.  Analysis of laminated shells by a sinusoidal shear deformation theory and radial basis functions collocation, accounting for through-the-thickness deformations , 2011 .

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

[9]  Mohamad S. Qatu,et al.  A triangular conforming element for laminated shells , 1995 .

[10]  Tarun Kant,et al.  Higher-order shear deformable theories for flexure of sandwich plates—Finite element evaluations , 1988 .

[11]  R. P. Shimpi,et al.  A Review of Refined Shear Deformation Theories for Isotropic and Anisotropic Laminated Beams , 2001 .

[12]  Paulo A.F. Martins,et al.  A finite element model for the analysis of viscoelastic sandwich structures , 2011 .

[13]  David A. Dillard,et al.  On the problem of shear-locking in finite elements based on shear deformable plate theory , 1997 .

[14]  E. Carrera.  A study of transverse normal stress effect on vibration of multilayered plates and shells , 1999 .

[15]  K. Bathe,et al.  A four‐node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation , 1985 .

[16]  Salim Belouettar,et al.  Review and assessment of various theories for modeling sandwich composites , 2008 .

[17]  David R. Owen,et al.  A refined analysis of laminated plates by finite element displacement methods—I. Fundamentals and static analysis , 1987 .

[18]  L. Demasi.  Partially Zig-Zag Advanced Higher Order Shear Deformation Theories Based on the Generalized Unified Formulation , 2012 .

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

[20]  Erasmo Carrera,et al.  Radial basis functions collocation and a unified formulation for bending, vibration and buckling analysis of laminated plates, according to a variation of Murakami’s zig-zag theory , 2011 .

[21]  E. Carrera,et al.  A quasi-3D sinusoidal shear deformation theory for the static and free vibration analysis of functionally graded plates , 2012 .

[22]  Dahsin Liu,et al.  GENERALIZED LAMINATE THEORIES BASED ON DOUBLE SUPERPOSITION HYPOTHESIS , 1997 .

[23]  M. Shariyat,et al.  A finite element based global–local theory for static analysis of rectangular sandwich and laminated composite plates , 2014 .

[24]  R. A. Shenoi,et al.  Dynamic analysis of composite sandwich plates with damping modelled using high-order shear deformation theory , 2001 .

[25]  O. Rabinovitch,et al.  A high-order finite element for dynamic analysis of soft-core sandwich plates , 2012 .

[26]  Hidenori Murakami,et al.  A high-order laminated plate theory with improved in-plane responses☆ , 1987 .

[27]  Ren Xiaohui,et al.  An accurate higher-order theory and C0 finite element for free vibration analysis of laminated composite and sandwich plates , 2010 .

[28]  Isaac Fried,et al.  Minimal-degree thin triangular plate and shell bending finite elements of order two and four , 1986 .

[29]  Maurice Touratier,et al.  A refined theory for thick composite plates , 1988 .

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

[31]  Thomas Wallmersperger,et al.  Considerations on higher-order finite elements for multilayered plates based on a Unified Formulation , 2006 .

[32]  J. Reddy.  An introduction to the finite element method , 1989 .

[33]  Dewey H. Hodges,et al.  Asymptotic generalization of Reissner–Mindlin theory: accurate three-dimensional recovery for composite shells , 2002 .

[34]  H. B. Coda,et al.  A geometrically nonlinear FEM formulation for the analysis of fiber reinforced laminated plates and shells , 2015 .

[35]  E. Carrera.  Theories and Finite Elements for Multilayered Plates and Shells:A Unified compact formulation with numerical assessment and benchmarking , 2003 .

[36]  Erasmo Carrera,et al.  Reissner’s mixed variational theorem toward MITC finite elements for multilayered plates , 2013 .

[37]  Maria Cinefra,et al.  A variable kinematic doubly-curved MITC9 shell element for the analysis of laminated composites , 2016 .

[38]  Mohamad S. Qatu,et al.  Static and vibration analyses of thick deep laminated cylindrical shells using 3D and various shear deformation theories , 2012 .

[39]  On the validity of nonlinear shear deformation theories for laminated plates and shells , 1994 .

[40]  E. Kansa.  Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates , 1990 .

[41]  R. A. Shenoi,et al.  Transient response of composite sandwich plates , 2004 .

[42]  Maurice Touratier,et al.  A refined theory of laminated shallow shells , 1992 .

[43]  J. Reddy.  Mechanics of laminated composite plates and shells : theory and analysis , 1996 .

[44]  Ugo Icardi,et al.  Co plate element for global/local analysis of multilayered composites, based on a 3D zig-zag model and strain energy updating , 2005 .

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

[46]  A. Rao,et al.  Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates , 1970 .

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

[48]  Volnei Tita,et al.  A finite element formulation for smart piezoelectric composite shells: Mathematical formulation, computational analysis and experimental evaluation , 2015 .

[49]  E. Carrera.  On the use of the Murakami's zig-zag function in the modeling of layered plates and shells , 2004 .

[50]  R. Christensen,et al.  A High-Order Theory of Plate Deformation—Part 2: Laminated Plates , 1977 .

[51]  Ole Thybo Thomsen,et al.  Thermo-mechanical load interactions in foam cored axi-symmetric sandwich circular plates – high-order and FE models , 2011 .

[52]  E. Kansa.  MULTIQUADRICS--A SCATTERED DATA APPROXIMATION SCHEME WITH APPLICATIONS TO COMPUTATIONAL FLUID-DYNAMICS-- II SOLUTIONS TO PARABOLIC, HYPERBOLIC AND ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS , 1990 .

[53]  C. M. Mota Soares,et al.  Static analysis of functionally graded sandwich plates according to a hyperbolic theory considering Zig-Zag and warping effects , 2012, Adv. Eng. Softw..

[54]  Ahmed K. Noor,et al.  Shear-Flexible Finite-Element Models of Laminated Composite Plates and Shells. , 1975 .

[55]  Erasmo Carrera,et al.  Accuracy of refined finite elements for laminated plate analysis , 2011 .

[56]  Tarun Kant,et al.  A Simple Finite Element Formulation of a Higher-order Theory for Unsymmetrically Laminated Composite Plates , 1988 .

[57]  E. Reissner.  On a mixed variational theorem and on shear deformable plate theory , 1986 .

[58]  Ole Thybo Thomsen,et al.  Sandwich Materials for Wind Turbine Blades — Present and Future , 2009 .

[59]  Carlo Sansour,et al.  The Cosserat surface as a shell model, theory and finite-element formulation , 1995 .

[60]  Liviu Librescu,et al.  Recent developments in the modeling and behavior of advanced sandwich constructions: a survey , 2000 .

[61]  Demetres Briassoulis.  The performance of a reformulated four-node plate bending element in moderately thick to very thin plate applications , 1993 .

[62]  J. Altenbach,et al.  On generalized Cosserat-type theories of plates and shells: a short review and bibliography , 2010 .

[63]  Gaetano Giunta,et al.  Hierarchical modelling of doubly curved laminated composite shells under distributed and localised loadings , 2011 .

[64]  Luciano Demasi,et al.  ∞3 Hierarchy plate theories for thick and thin composite plates: The generalized unified formulation , 2008 .

[65]  E. Carrera,et al.  Variable Kinematic Shell Elements for the Analysis of Electro-Mechanical Problems , 2015 .

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

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

[68]  Raimund Rolfes,et al.  A three-layered sandwich element with improved transverse shear stiffness and stresses based on FSDT , 2006 .

[69]  David R. Owen,et al.  A refined analysis of laminated plates by finite element displacement methods—II. Vibration and stability , 1987 .

[70]  Spyros G. Voutsinas,et al.  STATE OF THE ART IN WIND TURBINE AERODYNAMICS AND AEROELASTICITY , 2006 .

[71]  K. Bathe,et al.  A continuum mechanics based four‐node shell element for general non‐linear analysis , 1984 .

[72]  Eugenio Oñate,et al.  A layer-wise triangle for analysis of laminated composite plates and shells , 1999 .

[73]  Chen Wanji,et al.  A QUADRILATERAL ELEMENT BASED ON REFINED GLOBAL-LOCAL HIGHER-ORDER THEORY FOR COUPLING BENDING AND EXTENSION THERMO-ELASTIC MULTILAYERED PLATES , 2007 .

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

[75]  M. S. Qatu,et al.  Bending analysis of laminated plates and shells by different methods , 1994 .

[76]  J. Reddy.  Theory and Analysis of Elastic Plates and Shells , 2006 .

[77]  Luciano Demasi,et al.  Three-dimensional closed form solutions and exact thin plate theories for isotropic plates , 2007 .

[78]  Ulrich Gabbert,et al.  Numerically Efficient Finite Element Formulation for Modeling Active Composite Laminates , 2006 .

[79]  E. Carrera,et al.  A finite element model using a unified formulation for the analysis of viscoelastic sandwich laminates , 2013 .

[80]  Olivier Polit,et al.  A multilayered/sandwich triangular finite element applied to linear and non-linear analyses , 2002 .

[81]  Kaïss Bouayed,et al.  Finite element analysis of the dynamic behavior of a laminated windscreen with frequency dependent viscoelastic core. , 2012, The Journal of the Acoustical Society of America.

[82]  E. Carrera.  Theories and finite elements for multilayered, anisotropic, composite plates and shells , 2002 .

[83]  Erasmo Carrera,et al.  Guidelines and Recommendations on the Use of Higher Order Finite Elements for Bending Analysis of Plates , 2011 .

[84]  Ole Thybo Thomsen,et al.  Non-linear thermal response of sandwich panels with a flexible core and temperature dependent mechanical properties , 2008 .

[85]  E. Carrera.  Historical review of Zig-Zag theories for multilayered plates and shells , 2003 .

[86]  Ahmed K. Noor,et al.  Predictor-corrector procedures for stress and free vibration analysis of multilayered composite plates and shells , 1990 .

[87]  Luciano Demasi,et al.  2D, Quasi 3D and 3D Exact Solutions for Bending of Thick and Thin Sandwich Plates , 2008 .

[88]  E. Carrera,et al.  Shell finite elements with different through‐the‐thickness kinematics for the linear analysis of cylindrical multilayered structures , 2013 .

[89]  Jasbir S. Arora,et al.  Structural design sensitivity analysis with general boundary conditions: Dynamic problem , 1984 .

[90]  J. Argyris,et al.  An efficient and locking-free flat anisotropic plate and shell triangular element , 1994 .

[91]  Erasmo Carrera,et al.  Two higher order Zig-Zag theories for the accurate analysis of bending, vibration and buckling response of laminated plates by radial basis functions collocation and a unified formulation , 2011 .

[92]  N. J. Pagano,et al.  Dynamic characteristics of composite laminates , 1972 .

[93]  L. Demasi.  Treatment of stress variables in advanced multilayered plate elements based upon Reissner’s mixed variational theorem , 2006 .

[94]  R. Jorge,et al.  Analysis of composite plates by trigonometric shear deformation theory and multiquadrics , 2005 .

[95]  M. D. Sciuva.  Evaluation of some multilayered, shear-deformable plate elements , 1985 .

[96]  C. Soares,et al.  A new trigonometric layerwise shear deformation theory for the finite element analysis of laminated composite and sandwich plates , 2012 .

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

[98]  J. Reddy,et al.  Transient analysis of composite and sandwich plates by radial basis functions* , 2011 .

[99]  John Argyris,et al.  Linear and geometrically nonlinear bending of isotropic and multilayered composite plates by the natural mode method , 1994 .

[100]  Hidenori Murakami,et al.  Laminated Composite Plate Theory With Improved In-Plane Responses , 1986 .

[101]  Noureddine Atalla,et al.  Modeling thick composite laminate and sandwich structures with linear viscoelastic damping , 2011 .

[102]  E. Dvorkin.  Nonlinear analysis of shells using the MITC formulation , 1995 .

[103]  J. Argyris,et al.  A practicable and locking-free laminated shallow shell triangular element of varying and adaptable curvature , 1994 .

[104]  Erasmo Carrera,et al.  MITC technique extended to variable kinematic multilayered plate elements , 2010 .

[105]  Rakesh K. Kapania,et al.  A survey of recent shell finite elements , 2000 .

[106]  R. A. S. Moreira,et al.  A generalized layerwise finite element for multi-layer damping treatments , 2006 .

[107]  Yeoshua Frostig,et al.  Hygothermal (environmental) effects in high-order bending of sandwich beams with a flexible core and a discontinuous skin , 1997 .

[108]  Chen Wanji,et al.  A C0-type higher-order theory for bending analysis of laminated composite and sandwich plates , 2010 .

[109]  C. Soares,et al.  A new higher order shear deformation theory for sandwich and composite laminated plates , 2012 .