GDQFEM Numerical Simulations of Continuous Mediawith Cracks and Discontinuities

In the present paper the Generalized Differential Quadrature Finite Element Method (GDQFEM) is applied to deal with the static analysis of plane state structures with generic through the thickness material discontinuities and holes of various shapes. The GDQFEM numerical technique is an extension of the Generalized Differential Quadrature (GDQ) method and is based on the idea of conventional integral quadrature. In particular, the GDQFEM results in terms of stresses and displacements for classical and advanced plane stress problems with discontinuities are compared to the ones by the Cell Method (CM) and Finite Element Method (FEM). The multi-domain technique is implemented in a MATLAB code for solving irregular domains with holes and defects. In order to demonstrate the accuracy of the proposed methodology, several numerical examples of stress and displacement distributions are graphically shown and discussed.

[1]  Charles W. Bert,et al.  The differential quadrature method for irregular domains and application to plate vibration , 1996 .

[2]  Erasmo Viola,et al.  Analytical and numerical results for vibration analysis of multi-stepped and multi-damaged circular arches , 2007 .

[3]  Alessandro Marzani,et al.  Nonconservative stability problems via generalized differential quadrature method , 2008 .

[4]  Arthur W. Leissa,et al.  Accurate vibration analysis of thick, cracked rectangular plates , 2011 .

[5]  E. Viola,et al.  Static analysis of completely doubly-curved laminated shells and panels using general higher-order shear deformation theories , 2013 .

[6]  Alessandro Marzani,et al.  Crack effect on dynamic stability of beams under conservative and nonconservative forces , 2004 .

[7]  Yaojun Ge,et al.  Generation of inflow turbulence using the local differential quadrature method , 2013 .

[8]  Francesco Tornabene.  Meccanica delle Strutture a Guscio in Materiale Composito. Il metodo Generalizzato di Quadratura Differenziale , 2012 .

[9]  Erasmo Viola,et al.  Stress intensity factors for cracked T-sections and dynamic behaviour of T-beams , 2006 .

[10]  Q Li,et al.  APPLICATION OF MESHLESS LOCAL PETROV-GALERKIN (MLPG) TO PROBLEMS WITH SINGULARITIES, AND MATERIAL DISCONTINUITIES IN 3D ELASTICITY , 2003 .

[11]  E. Ferretti,et al.  Masonry Walls under Shear Test: a CM Modeling , 2008 .

[12]  Xinwei Wang,et al.  Static and free vibrational analysis of rectangular plates by the differential quadrature element method , 1998 .

[13]  Hongzhi Zhong,et al.  Solution of poisson and laplace equations by quadrilateral quadrature element , 1998 .

[14]  Satya N. Atluri,et al.  The Applications of Meshless Local Petrov-Galerkin (MLPG) Approaches in High-Speed Impact, Penetration and Perforation Problems , 2006 .

[15]  Erasmo Carrera,et al.  Analysis of thick isotropic and cross-ply laminated plates by generalized differential quadrature method and a Unified Formulation , 2014 .

[16]  Satya N. Atluri,et al.  Meshless Local Petrov-Galerkin Method for Heat Conduction Problem in an Anisotropic Medium , 2004 .

[17]  C. Shu,et al.  Free Vibration Analysis of Curvilinear Quadrilateral Plates by the Differential Quadrature Method , 2000 .

[18]  K. K. Gupta.  Development of a finite dynamic element for free vibration analysis of two-dimensional structures , 1978 .

[19]  Satya N. Atluri,et al.  Topology-optimization of Structures Based on the MLPG Mixed Collocation Method , 2008 .

[20]  Nicholas Fantuzzi,et al.  On the Stress Intensity Factors of Cracked Beams for Structural Analysis , 2011 .

[21]  Elena Ferretti.  A Cell Method Stress Analysis in Thin Floor TilesSubjected to Temperature Variation , 2013 .

[22]  E. Ferretti.  A discrete nonlocal formulation using local constitutive laws , 2004 .

[23]  E. Viola,et al.  Size effect investigation of a central interface crack between two bonded dissimilar materials , 2013 .

[24]  T. Anderson.  Fracture Mechanics: Fundamentals and Applications, Third Edition , 1994 .

[25]  Erasmo Viola,et al.  Free vibration analysis of axially loaded cracked Timoshenko beam structures using the dynamic stiffness method , 2007 .

[26]  Charles W. Bert,et al.  Differential quadrature: a powerful new technique for analysis of composite structures , 1997 .

[27]  W. E. Lorensen,et al.  The collapsed cubic isoparametric element as a ingular element for crack probblems , 1978 .

[28]  Francesco Tornabene,et al.  2-D GDQ solution for free vibrations of anisotropic doubly-curved shells and panels of revolution , 2011 .

[29]  Erasmo Viola,et al.  Free vibrations of three parameter functionally graded parabolic panels of revolution , 2009 .

[30]  Nicholas Fantuzzi,et al.  Generalized differential quadrature finite element method for cracked composite structures of arbitrary shape , 2013 .

[31]  Francesco Tornabene,et al.  Free vibrations of laminated composite doubly-curved shells and panels of revolution via the GDQ method , 2011 .

[32]  Alfredo Liverani,et al.  Laminated composite rectangular and annular plates: A GDQ solution for static analysis with a posteriori shear and normal stress recovery , 2012 .

[33]  Elena Ferretti.  Crack Propagation Modeling by Remeshing Using the Cell Method (CM) , 2003 .

[34]  Guang Liu,et al.  A Differential Quadrature Finite Element Method , 2010, A Differential Quadrature Hierarchical Finite Element Method.

[35]  E. Ferretti.  Shape-Effect in the Effective Laws of Plain and Rubberized Concrete , 2012 .

[36]  Bo Liu,et al.  High‐accuracy differential quadrature finite element method and its application to free vibrations of thin plate with curvilinear domain , 2009 .

[37]  Nasser Talebbeydokhti,et al.  Numerical modelling of the mild slope equation using localised differential quadrature method , 2012 .

[38]  Xinwei Wang,et al.  Static and free vibration analyses of rectangular plates by the new version of the differential quadrature element method , 2004 .

[39]  Chang-New Chen,et al.  The development of irregular elements for differential quadrature element method steady-state heat conduction analysis , 1999 .

[40]  Phillip L. Gould,et al.  A differential quadrature method solution for shear-deformable shells of revolution , 2005 .

[41]  Erasmo Viola,et al.  On the inclined crack problem in an orthotropic medium under biaxial loading , 2004 .

[42]  E. Ferretti.  The Cell Method: an Enriched Description of PhysicsStarting from the Algebraic Formulation , 2013 .

[43]  Alessandro Marzani,et al.  Critical Flow Speeds of Pipes Conveying Fluid Using the Generalized Differential Quadrature Method , 2010 .

[44]  E. Viola,et al.  General higher-order equivalent single layer theory for free vibrations of doubly-curved laminated composite shells and panels , 2013 .

[45]  Enzo Tonti,et al.  A Direct Discrete Formulation of Field Laws: The Cell Method , 2001 .

[46]  Alfredo Liverani,et al.  General anisotropic doubly-curved shell theory: A differential quadrature solution for free vibrations of shells and panels of revolution with a free-form meridian , 2012 .

[47]  S. Atluri,et al.  Fracture & Fatigue Analyses: SGBEM-FEM or XFEM?Part 2: 3D Solids , 2013 .

[48]  A. Ceruti,et al.  Mixed Static and Dynamic Optimization of Four-Parameter Functionally Graded Completely Doubly Curved and Degenerate Shells and Panels Using GDQ Method , 2013 .

[49]  E. Viola,et al.  General higher-order shear deformation theories for the free vibration analysis of completely doubly-curved laminated shells and panels , 2013 .

[50]  Erasmo Viola,et al.  Vibration analysis of spherical structural elements using the GDQ method , 2007, Comput. Math. Appl..

[51]  Chang-New Chen.  The differential quadrature element method irregular element torsion analysis model , 1999 .

[52]  F. Liu.  Differential quadrature element method for static analysis of shear deformable cross‐ply laminates , 1999 .

[53]  E. Viola,et al.  An alternative complex variable formulation for an inclined crack in an orthotropic medium , 2003 .

[54]  Nicholas Fantuzzi,et al.  Generalized Differential Quadrature Finite Element Method applied to Advanced Structural Mechanics , 2013 .

[55]  Nicholas Fantuzzi,et al.  Numerical investigation of functionally graded cylindrical shells and panels using the generalized unconstrained third order theory coupled with the stress recovery , 2012 .

[56]  Mohamed Nassar,et al.  Vibration analysis of structural elements using differential quadrature method , 2012, Journal of advanced research.

[57]  Zekeriya Girgin,et al.  Buckling Analyses of Axially Functionally Graded Nonuniform Columns with Elastic Restraint Using a Localized Differential Quadrature Method , 2013 .

[58]  Erasmo Viola,et al.  Vibration Analysis of Damaged Circular Arches with Varying Cross-section , 2005 .

[59]  Faruk Civan,et al.  Differential quadrature for multi-dimensional problems , 1984 .

[60]  Faruk Civan.  Application of Differential Quadrature to Solution of Pool Boiling Cavities , 1985 .

[61]  Francesco Tornabene,et al.  Free vibration analysis of functionally graded conical, cylindrical shell and annular plate structures with a four-parameter power-law distribution , 2009 .

[62]  Jian-an Sun,et al.  Upwind local differential quadrature method for solving incompressible viscous flow , 2000 .

[63]  K. Y. Lam,et al.  A numerical study of wave propagation in a poroelastic medium by use of localized differential quadrature method , 2004 .

[64]  Erasmo Viola,et al.  FREE VIBRATIONS OF FOUR-PARAMETER FUNCTIONALLY GRADED PARABOLIC PANELS AND SHELLS OF REVOLUTION , 2009 .

[65]  C. Shu.  Differential Quadrature and Its Application in Engineering , 2000 .

[66]  E. Ferretti.  A Cell Method (CM) Code for Modeling the Pullout Test Step-wise , 2004 .

[67]  Arthur W. Leissa,et al.  Vibration analysis of rectangular plates with side cracks via the Ritz method , 2009 .

[68]  Elena Ferretti.  The Cell Method: A Purely Algebraic Computational Method in Physics and Engineering , 2014 .

[70]  A. Ceruti,et al.  Free-Form Laminated Doubly-Curved Shells and Panels of Revolution Resting on Winkler-Pasternak Elastic Foundations: A 2-D GDQ Solution for Static and Free Vibration Analysis , 2013 .

[71]  J. N. Reddy,et al.  FGM and Laminated Doubly-Curved and Degenerate Shells Resting on Nonlinear Elastic Foundations: A GDQ Solution for Static Analysis with a Posteriori Stress and Strain Recovery , 2013 .

[72]  Hongzhi Zhong,et al.  A weak form quadrature element method for plane elasticity problems , 2009 .

[73]  Alfredo Liverani,et al.  FGM and laminated doubly curved shells and panels of revolution with a free-form meridian: A 2-D GDQ solution for free vibrations , 2011 .

[74]  Satya N. Atluri,et al.  Fracture & Fatigue Analyses: SGBEM-FEM or XFEM? Part 1: 2D Structures , 2013 .

[75]  Satya N. Atluri,et al.  The MLPG Mixed Collocation Method for Material Orientation and Topology Optimization of Anisotropic Solids and Structures , 2008 .

[76]  Grant P. Steven,et al.  ASYMPTOTIC SOLUTIONS FOR PREDICTED NATURAL FREQUENCIES OF TWO-DIMENSIONAL ELASTIC SOLID VIBRATION PROBLEMS IN FINITE ELEMENT ANALYSIS , 1996 .

[77]  N. Fantuzzi,et al.  On mixed mode crack initiation and direction in shafts: Strain energy density factor and maximum tangential stress criteria , 2013 .

[78]  D. L. Young,et al.  Local Differential Quadrature Method for 2-D Flow and Forced-Convection Problems in Irregular Domains , 2009 .

[79]  Zhi Zong,et al.  A localized differential quadrature (LDQ) method and its application to the 2D wave equation , 2002 .

[80]  Erasmo Carrera,et al.  Static analysis of doubly-curved anisotropic shells and panels using CUF approach, differential geometry and differential quadrature method , 2014 .

[81]  Alessandro Marzani,et al.  Cracks Interaction Effect on the Dynamic Stability of Beams under Conservative and Nonconservative Forces , 2011 .

[82]  J. N. Reddy,et al.  Winkler–Pasternak foundation effect on the static and dynamic analyses of laminated doubly-curved and degenerate shells and panels , 2014 .

[83]  S.S.E. Lam,et al.  Application of the differential quadrature method to two-dimensional problems with arbitrary geometry , 1993 .

[84]  Arthur W. Leissa,et al.  Vibration analysis of rectangular plates with edge V-notches , 2008 .

[85]  Erasmo Viola,et al.  Analytical and differential quadrature results for vibration analysis of damaged circular arches , 2005 .

[86]  Lucio Nobile,et al.  Formulation of Cracked Beam Element for Structural Analysis , 2002 .

[87]  A. Leissa,et al.  Vibrations of rectangular plates with internal cracks or slits , 2011 .

[88]  Song Cen,et al.  Quadrilateral membrane elements with analytical element stiffness matrices formulated by the new quadrilateral area coordinate method (QACM‐II) , 2009 .

[89]  F. Ubertini,et al.  A consistent approach for mixed stress finite element formulations in linear elastodynamics , 2008 .

[90]  Erasmo Viola,et al.  Vibration analysis of conical shell structures using GDQ Method , 2006 .

[91]  Francesco Tornabene,et al.  Free vibrations of anisotropic doubly-curved shells and panels of revolution with a free-form meridian resting on Winkler–Pasternak elastic foundations , 2011 .

[92]  Erasmo Viola,et al.  Static analysis of functionally graded doubly-curved shells and panels of revolution , 2013 .

[93]  Chang-New Chen.  DQEM and DQFDM for the analysis of composite two-dimensional elasticity problems , 2003 .

[94]  Elena Ferretti,et al.  Cell Method Analysis of Crack Propagation in Tensioned Concrete Plates , 2009 .

[95]  Erasmo Viola,et al.  Free vibration analysis of functionally graded panels and shells of revolution , 2009 .

[96]  S. Atluri,et al.  Development of 3D T-Trefftz Voronoi Cell Finite Elements with/without Spherical Voids &/or Elastic/Rigid Inclusions for Micromechanical Modeling of Heterogeneous Materials , 2012 .

[97]  E. Ferretti.  A Local Strictly Nondecreasing Material Law for Modeling Softening and Size-Effect: a Discrete Approach , 2005 .

[98]  Robert D. Cook,et al.  Error estimation and adaptive meshing for vibration problems , 1992 .

[99]  Erasmo Viola,et al.  2-D solution for free vibrations of parabolic shells using generalized differential quadrature method , 2008 .

[100]  D. L. Young,et al.  The localized differential quadrature method for two-dimensional stream function formulation of Navier–Stokes equations , 2011 .

[101]  Alfredo Liverani,et al.  Static analysis of laminated composite curved shells and panels of revolution with a posteriori shear and normal stress recovery using generalized differential quadrature method , 2012 .

[102]  Ashraf M. Zenkour,et al.  Analysis of Sandwich Plates by Generalized Differential Quadrature Method , 2013 .