Isogeometric Analysis of Laminated Composite Plates Using the Higher-Order Shear Deformation Theory

Isogeometric analysis (IGA) aims at simplifying the computer aided design (CAD) and computer aided engineering (CAE) pipeline by using the same functions to describe the geometry (CAD) and the unknown fields (Analysis). IGA can be based on a variety of CAD descriptions, the most widely used today being non-uniform rational B-splines (NURBS). In this article, the suitability of NURBS-based isogeometric analysis within a third-order shear deformation theory for the simulation of the static, dynamic, and buckling response of laminated composite plates is investigated. The method employs NURBS basis functions to both represent the geometry (exactly) and the unknown field variables. One of the main advantages of the present method is directly inherited from IGA, that is to easily increase the approximation order. To avoid using a shear correction factor, a third-order shear deformation theory (TSDT) is introduced. It requires C1-continuity of generalized displacements and the NURBS basis functions are well suited for this requirement. Several numerical examples are used to demonstrate the performance of the present method compared with other published ones.

[1]  N. Valizadeh,et al.  Extended isogeometric analysis for simulation of stationary and propagating cracks , 2012 .

[2]  Alessandro Reali,et al.  Duality and unified analysis of discrete approximations in structural dynamics and wave propagation : Comparison of p-method finite elements with k-method NURBS , 2008 .

[3]  Hung Nguyen-Xuan,et al.  An alternative alpha finite element method with discrete shear gap technique for analysis of laminated composite plates , 2011, Appl. Math. Comput..

[4]  John A. Evans,et al.  Robustness of isogeometric structural discretizations under severe mesh distortion , 2010 .

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

[6]  D. J. Gorman,et al.  THE VIBRATION OF MINDLIN PLATES , 1999 .

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

[8]  Sandra Maurer,et al.  Matlab Codes For Finite Element Analysis Solids And Structures , 2016 .

[9]  Hung Nguyen-Xuan,et al.  An edge-based smoothed finite element method (ES-FEM) with stabilized discrete shear gap technique for analysis of Reissner–Mindlin plates , 2010 .

[10]  K. M. Liew,et al.  Buckling and vibration analysis of isotropic and laminated plates by radial basis functions , 2011 .

[11]  Renato Natal Jorge,et al.  Natural frequencies of functionally graded plates by a meshless method , 2006 .

[12]  Jorge Belinha,et al.  Analysis of plates and laminates using the element-free Galerkin method , 2006 .

[13]  J. Reddy,et al.  Stability and vibration of isotropic, orthotropic and laminated plates according to a higher-order shear deformation theory , 1985 .

[14]  Paul Steinmann,et al.  Isogeometric analysis of 2D gradient elasticity , 2011 .

[15]  H. Nguyen-Xuan,et al.  A smoothed finite element method for plate analysis , 2008 .

[16]  Renato Natal Jorge,et al.  Free vibration analysis of symmetric laminated composite plates by FSDT and radial basis functions , 2005 .

[17]  Hyun-Jung Kim,et al.  Isogeometric analysis for trimmed CAD surfaces , 2009 .

[18]  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 .

[19]  Ashraf M. Zenkour,et al.  Buckling and free vibration of non-homogeneous composite cross-ply laminated plates with various plate theories , 1999 .

[20]  S. Xiang,et al.  Free vibration analysis of symmetric laminated composite plates by trigonometric shear deformation theory and inverse multiquadric RBF , 2009 .

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

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

[23]  Abdul Hamid Sheikh,et al.  Buckling of Laminated Composite Plates by a New Element Based on Higher Order Shear Deformation Theory , 2003 .

[24]  R. Shimpi,et al.  REFINED PLATE THEORY AND ITS VARIANTS , 2002 .

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

[26]  Qiusheng Li,et al.  Bending and buckling analysis of antisymmetric laminates using the moving least square differential quadrature method , 2004 .

[27]  J. Whitney,et al.  The Effect of Transverse Shear Deformation on the Bending of Laminated Plates , 1969 .

[28]  Victor M. Calo,et al.  The role of continuity in residual-based variational multiscale modeling of turbulence , 2007 .

[29]  T. Rabczuk,et al.  A Meshfree Thin Shell for Arbitrary Evolving Cracks Based on An Extrinsic Basis , 2006 .

[30]  K. M. Liew,et al.  Vibration of Mindlin plates. Programming the p‐version Ritz method. (Liew, K. M., Wang, C. M., Xiang, Y., Kitipornchai, S.) , 1999 .

[31]  António J.M. Ferreira,et al.  A local radial basis functions—Finite differences technique for the analysis of composite plates , 2011 .

[32]  Timon Rabczuk,et al.  Coupling of mesh‐free methods with finite elements: basic concepts and test results , 2006 .

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

[34]  John A. Evans,et al.  Isogeometric analysis using T-splines , 2010 .

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

[36]  S. Vel,et al.  Analytical Solution for Rectangular Thick Laminated Plates Subjected to Arbitrary Boundary Conditions , 1999 .

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

[38]  K. M. Liew,et al.  Free vibration and buckling analyses of shear-deformable plates based on FSDT meshfree method , 2004 .

[39]  P. Mohan,et al.  Consistent and variationally correct finite elements for higher-order laminated plate theory , 1994 .

[40]  G. Sangalli,et al.  Linear independence of the T-spline blending functions associated with some particular T-meshes , 2010 .

[41]  T. Belytschko,et al.  A three dimensional large deformation meshfree method for arbitrary evolving cracks , 2007 .

[42]  C.M.C. Roque,et al.  Analysis of composite plates using higher-order shear deformation theory and a finite point formulation based on the multiquadric radial basis function method , 2003 .

[43]  B. Simeon,et al.  Adaptive isogeometric analysis by local h-refinement with T-splines , 2010 .

[44]  K. M. Liew,et al.  SOLVING THE VIBRATION OF THICK SYMMETRIC LAMINATES BY REISSNER/MINDLIN PLATE THEORY AND THEp-RITZ METHOD , 1996 .

[45]  Manfred Bischoff,et al.  Numerical efficiency, locking and unlocking of NURBS finite elements , 2010 .

[46]  J. Reddy.  Mechanics of laminated composite plates : theory and analysis , 1997 .

[47]  Nathan Ida,et al.  Introduction to the Finite Element Method , 1997 .

[48]  N. J. Pagano,et al.  Global-local laminate variational model , 1983 .

[49]  Mark A. Bradford,et al.  On the use of bubble functions in the local buckling analysis of plate structures by the spline finite strip method , 2000 .

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

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

[52]  Yuri Bazilevs,et al.  Rotation free isogeometric thin shell analysis using PHT-splines , 2011 .

[53]  P. Marcal,et al.  Introduction to the Finite-Element Method , 1973 .

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

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

[56]  T. Hughes,et al.  Isogeometric fluid-structure interaction: theory, algorithms, and computations , 2008 .

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

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

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

[60]  H. Nguyen-Xuan,et al.  A cell — based smoothed finite element method for free vibration and buckling analysis of shells , 2011 .

[61]  M. A. McCarthy,et al.  Analysis of thick composite laminates using a higher-order shear and normal deformable plate theory (HOSNDPT) and a meshless method , 2008 .

[62]  S. Srinivas,et al.  An exact analysis for vibration of simply-supported homogeneous and laminated thick rectangular plates , 1970 .

[63]  J. N. Reddy,et al.  A refined nonlinear theory of plates with transverse shear deformation , 1984 .

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

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

[66]  J. Ren,et al.  A new theory of laminated plate , 1986 .

[67]  Gregory E. Fasshauer,et al.  Computation of natural frequencies of shear deformable beams and plates by an RBF-pseudospectral method , 2006 .

[68]  Liviu Librescu,et al.  On the theory of anisotropic elastic shells and plates , 1967 .

[69]  E. Reissner,et al.  A Consistent Treatment of Transverse Shear Deformations in Laminated Anisotropic Plates , 1972 .

[70]  Ted Belytschko,et al.  Cracking particles: a simplified meshfree method for arbitrary evolving cracks , 2004 .

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

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

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

[74]  William Portilho de Paiva,et al.  Modal analysis of anisotropic plates using the boundary element method , 2011 .

[75]  K. M. Liew,et al.  Analysis of rectangular laminated composite plates via FSDT meshless method , 2002 .

[76]  Chien H. Thai,et al.  Analysis of laminated composite plates using higher-order shear deformation plate theory and node-based smoothed discrete shear gap method , 2012 .

[77]  L. K. Stevens,et al.  A Higher Order Theory for Free Vibration of Orthotropic, Homogeneous, and Laminated Rectangular Plates , 1984 .

[78]  Dhanjoo N. Ghista,et al.  Mesh-free radial basis function method for static, free vibration and buckling analysis of shear deformable composite laminates , 2007 .

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

[80]  T. Rabczuk,et al.  Discontinuous modelling of shear bands using adaptive meshfree methods , 2008 .

[81]  T. Rabczuk,et al.  Coupling of meshfree methods with finite elements : Basic concepts and test results , 2006 .

[82]  Warna Karunasena,et al.  Buckling and vibration analysis of laminated composite plate/shell structures via a smoothed quadrilateral flat shell element with in-plane rotations , 2011 .

[83]  N. E. Meiche,et al.  A new hyperbolic shear deformation theory for buckling and vibration of functionally graded sandwich plate , 2011 .

[84]  T. Rabczuk,et al.  A meshfree thin shell method for non‐linear dynamic fracture , 2007 .

[85]  Mo Shing Cheung,et al.  Finite Strip Analysis of Anisotropic Laminated Composite Plates Using Higher-Order Shear Deformation Theory" , 1994 .

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

[87]  W. Wall,et al.  Isogeometric structural shape optimization , 2008 .

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

[89]  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 .

[90]  T. Rabczuk,et al.  A two-dimensional Isogeometric Boundary Element Method for elastostatic analysis , 2012 .

[91]  J. N. Reddy,et al.  Buckling and vibration of laminated composite plates using various plate theories , 1989 .

[92]  R. Shimpi,et al.  A two variable refined plate theory for orthotropic plate analysis , 2006 .

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

[94]  António J.M. Ferreira,et al.  Analysis of Composite Plates Using a Layerwise Theory and Multiquadrics Discretization , 2005 .

[95]  Ernest Hinton,et al.  Numerical methods and software for dynamic analysis of plates and shells , 1988 .

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

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

[98]  A. Reis,et al.  Computation of moments and stresses in laminated composite plates by the boundary element method , 2011 .

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