A generalized unconstrained theory and isogeometric finite element analysis based on Bézier extraction for laminated composite plates

This work presents an isogeometric finite element formulation based on Bézier extraction of the non-uniform rational B-splines (NURBS) in combination with a generalized unconstrained higher-order shear deformation theory (UHSDT) for laminated composite plates. The proposed approach relaxes zero-shear stresses at the top and bottom surfaces of the plates and no shear correction factors are required. A weak form of static, free vibration and transient response analyses for laminated composite plates is then established and is numerically solved using isogeometric Bézier finite elements. NURBS can be written in terms of Bernstein polynomials and the Bézier extraction operator. IGA is implemented with the presence of C°-continuous Bézier elements which allow to easily incorporate into existing finite element codes without adding many changes as the former IGA. As a result, all computations can be performed based on the basis functions defined previously as the same way in finite element method (FEM). Numerical results performed over static, vibration and transient analysis show high efficiency of the present method.

[1]  A. Leung.  An unconstrained third-order plate theory , 1991 .

[2]  Rakesh K. Kapania,et al.  Geometrically nonlinear NURBS isogeometric finite element analysis of laminated composite plates , 2012 .

[3]  J. N. Reddy,et al.  Exact solutions for the transient response of symmetric cross-ply laminates using a higher-order plate theory , 1989 .

[4]  Santosh Kapuria,et al.  Levy-type piezothermoelastic solution for hybrid plate by using first-order shear deformation theory , 1997 .

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

[6]  H. Nguyen-Xuan,et al.  A node-based smoothed finite element method with stabilized discrete shear gap technique for analysis of Reissner–Mindlin plates , 2010 .

[7]  H. Parisch.  A continuum‐based shell theory for non‐linear applications , 1995 .

[8]  K. M. Liew,et al.  Vibration analysis of symmetrically laminated plates based on FSDT using the moving least squares differential quadrature method , 2003 .

[9]  Thomas J. R. Hughes,et al.  Isogeometric Analysis: Toward Integration of CAD and FEA , 2009 .

[10]  J. Reddy.  A Simple Higher-Order Theory for Laminated Composite Plates , 1984 .

[11]  Stéphane Bordas,et al.  Isogeometric analysis of laminated composite and sandwich plates using a new inverse trigonometric shear deformation theory , 2014 .

[12]  Alessandro Reali,et al.  Studies of Refinement and Continuity in Isogeometric Structural Analysis (Preprint) , 2007 .

[13]  E. Ramm,et al.  On the physical significance of higher order kinematic and static variables in a three-dimensional shell formulation , 2000 .

[14]  S. A. Ambartsumian,et al.  On the theory of bending of anisotropic plates and shallow shells , 1960 .

[15]  John A. Evans,et al.  Isogeometric finite element data structures based on Bézier extraction of NURBS , 2011 .

[16]  Hung Nguyen-Xuan,et al.  Isogeometric Analysis of Laminated Composite Plates Using the Higher-Order Shear Deformation Theory , 2015 .

[17]  E. Ramm,et al.  Three‐dimensional extension of non‐linear shell formulation based on the enhanced assumed strain concept , 1994 .

[18]  J. N. Reddy,et al.  Analysis of composite plates using various plate theories -Part 1: Formulation and analytical solutions , 1998 .

[19]  Hung Nguyen-Xuan,et al.  Isogeometric finite element analysis of composite sandwich plates using a higher order shear deformation theory , 2013 .

[20]  T. Hughes,et al.  Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement , 2005 .

[21]  Hung Nguyen-Xuan,et al.  Isogeometric analysis of laminated composite and sandwich plates using a layerwise deformation theory , 2013 .

[22]  António J.M. Ferreira,et al.  A formulation of the multiquadric radial basis function method for the analysis of laminated composite plates , 2003 .

[23]  Thomas J. R. Hughes,et al.  Isogeometric shell analysis: The Reissner-Mindlin shell , 2010 .

[24]  Thomas J. R. Hughes,et al.  A large deformation, rotation-free, isogeometric shell , 2011 .

[25]  Yuri Bazilevs,et al.  The bending strip method for isogeometric analysis of Kirchhoff–Love shell structures comprised of multiple patches , 2010 .

[26]  H. Nguyen-Xuan,et al.  Isogeometric analysis using polynomial splines over hierarchical T-meshes for two-dimensional elastic solids , 2011 .

[27]  S. Sahraee,et al.  Axisymmetric bending and buckling analysis of thick functionally graded circular plates using unconstrained third-order shear deformation plate theory , 2009 .

[28]  Thomas J. R. Hughes,et al.  Nonlinear Isogeometric Analysis , 2009 .

[29]  H. Rothert,et al.  Solution of a laminated cylindrical shell using an unconstrained third-order theory , 1998 .

[30]  K. Y. Lam,et al.  A strip element method for the transient analysis of symmetric laminated plates , 2001 .

[31]  Tianxu Zhang,et al.  Three-dimensional linear analysis for composite axially symmetrical circular plates , 2004 .

[32]  Loc V. Tran,et al.  Isogeometric analysis of functionally graded plates using higher-order shear deformation theory , 2013 .

[33]  Silvia Bertoluzza,et al.  A high order collocation method for the static and vibration analysis of composite plates using a first-order theory , 2009 .

[34]  Liping Liu.  THEORY OF ELASTICITY , 2012 .

[35]  Liviu Librescu,et al.  Analysis of symmetric cross-ply laminated elastic plates using a higher-order theory. II - Buckling and free vibration , 1988 .

[36]  C. Lim,et al.  A new unconstrained third-order plate theory for Navier solutions of symmetrically laminated plates , 2003 .

[37]  R. M. Natal Jorge,et al.  Static and dynamic analysis of laminated plates based on an unconstrained third order theory and using a radial point interpolator meshless method , 2011 .

[38]  K. M. Liew,et al.  Response of Plates of Arbitrary Shape Subject to Static Loading , 1992 .

[39]  Alessandro Reali,et al.  Isogeometric Analysis of Structural Vibrations , 2006 .

[40]  Cv Clemens Verhoosel,et al.  An isogeometric solid‐like shell element for nonlinear analysis , 2013 .

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

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

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

[44]  Hung Nguyen-Xuan,et al.  Static, free vibration, and buckling analysis of laminated composite Reissner–Mindlin plates using NURBS‐based isogeometric approach , 2012 .

[45]  M. Lezgy-Nazargah,et al.  NURBS-based isogeometric analysis of laminated composite beams using refined sinus model , 2015 .

[46]  Chen Wanji,et al.  Free vibration of laminated composite and sandwich plates using global–local higher-order theory , 2006 .

[47]  N. Pagano,et al.  Exact Solutions for Rectangular Bidirectional Composites and Sandwich Plates , 1970 .

[48]  K. M. Liew,et al.  Vibration of clamped circular symmetric laminates , 1994 .

[49]  Ekkehard Ramm,et al.  Nonlinear shell formulations for complete three-dimensional constitutive laws including composites and laminates , 1994 .

[50]  Timon Rabczuk,et al.  Transient analysis of laminated composite plates using isogeometric analysis , 2012 .

[51]  John A. Evans,et al.  Bézier projection: A unified approach for local projection and quadrature-free refinement and coarsening of NURBS and T-splines with particular application to isogeometric design and analysis , 2014, 1404.7155.

[52]  Roland Wüchner,et al.  Isogeometric shell analysis with Kirchhoff–Love elements , 2009 .