Analysis of cracked plate using higher-order shear deformation theory: Asymptotic crack-tip fields and XIGA implementation

Abstract In this work, an extended isogeometric analysis (XIGA) is used for the analysis of through-thickness crack in a homogeneous and isotropic plate. In isogeometric analysis (IGA), non-uniform rational B-splines (NURBS) are used as a basis function. The plate kinematics is modelled by Reddy’s higher-order shear deformation theory (HSDT). The C 1 continuity requirement of HSDT can be easily fulfilled by the NURBS basis functions. In order to obtain the plate fracture parameters (moment intensity factors), the expressions of crack-tip fields (auxiliary fields) are derived using separation of variables and Eigen-function approach. A new expression for moment intensity factors is developed using auxiliary fields solution (crack-tip fields) and interaction integral approach. Several cracked plate problems are solved by XIGA using HSDT. The results obtained by HSDT based XIGA (HSDT-XIGA) are compared with the FSDT based XIGA (FSDT-XIGA) and literature solutions.

[1]  Fuh-Gwo Yuan,et al.  Asymptotic crack-tip fields in an anisotropic plate subjected to bending, twisting moments and transverse shear loads , 2000 .

[2]  Said Rechak,et al.  Vibration analysis of cracked plates using the extended finite element method , 2009 .

[3]  Sharif Rahman,et al.  An interaction integral method for analysis of cracks in orthotropic functionally graded materials , 2003 .

[4]  Ted Belytschko,et al.  Modeling fracture in Mindlin–Reissner plates with the extended finite element method , 2000 .

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

[6]  A. S. Shedbale,et al.  Elasto-plastic fatigue crack growth analysis of plane problems in the presence of flaws using XFEM , 2015 .

[7]  B. K. Mishra,et al.  Fatigue crack growth analysis of an interfacial crack in heterogeneous materials using homogenized XIGA , 2016 .

[8]  James K. Knowles,et al.  On the Bending of an Elastic Plate Containing a Crack , 1960 .

[9]  H. Nguyen-Xuan,et al.  An extended isogeometric thin shell analysis based on Kirchhoff-Love theory , 2015 .

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

[11]  D. Rooke,et al.  The dual boundary element method: Effective implementation for crack problems , 1992 .

[12]  Bijay K. Mishra,et al.  Stochastic fatigue crack growth simulation of interfacial crack in bi-layered FGMs using XIGA , 2015 .

[13]  Timon Rabczuk,et al.  Linear buckling analysis of cracked plates by SFEM and XFEM , 2011 .

[14]  Michel Salaün,et al.  Stress intensity factors computation for bending plates with extended finite element method , 2012 .

[15]  Bijay K. Mishra,et al.  The numerical simulation of fatigue crack growth using extended finite element method , 2012 .

[16]  Ted Belytschko,et al.  Elastic crack growth in finite elements with minimal remeshing , 1999 .

[17]  Kuang-Chong Wu.  Stress intensity factors and energy release rate for anisotropic plates based on the classical plate theory , 2016 .

[18]  Sohichi Hirose,et al.  A stabilized discrete shear gap extended finite element for the analysis of cracked Reissner–Mindlin plate vibration problems involving distorted meshes , 2016 .

[19]  A. S. Shedbale,et al.  Evaluation of mechanical properties using spherical ball indentation and coupled finite element–element-free galerkin approach , 2016 .

[20]  Sohichi Hirose,et al.  Buckling failure analysis of cracked functionally graded plates by a stabilized discrete shear gap extended 3-node triangular plate element , 2015 .

[21]  Reaz A. Chaudhuri,et al.  A novel eigenfunction expansion solution for three-dimensional crack problems , 2000 .

[22]  T. Belytschko,et al.  X‐FEM in isogeometric analysis for linear fracture mechanics , 2011 .

[23]  Guangyu Shi,et al.  A new simple third-order shear deformation theory of plates , 2007 .

[24]  T. Q. Bui,et al.  Buckling and vibration extended isogeometric analysis of imperfect graded Reissner-Mindlin plates with internal defects using NURBS and level sets , 2016 .

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

[26]  Tinh Quoc Bui,et al.  Analysis of cracked shear deformable plates by an effective meshfree plate formulation , 2015 .

[27]  Michel Salaün,et al.  eXtended finite element methods for thin cracked plates with Kirchhoff–Love theory , 2010 .

[28]  Bijay K. Mishra,et al.  Numerical simulations of cracked plate using XIGA under different loads and boundary conditions , 2016 .

[29]  Yao Dai,et al.  Higher-order crack tip fields for functionally graded material plate with transverse shear deformation , 2016 .

[30]  C. S. Huang,et al.  Geometrically Induced Stress Singularities of a Thick FGM Plate Based on the Third-Order Shear Deformation Theory , 2009 .

[31]  P Kerfriden,et al.  Natural frequencies of cracked functionally graded material plates by the extended finite element method , 2011 .

[32]  M. Kuna,et al.  Eigenfunctions of crack problems in the Mindlin plate theory , 2015 .

[33]  T. Belytschko,et al.  Crack propagation by element-free Galerkin methods , 1995 .

[34]  Tatacipta Dirgantara,et al.  Crack Growth analysis of plates Loaded by bending and tension using dual boundary element method , 2000 .

[35]  Shuodao Wang,et al.  A Mixed-Mode Crack Analysis of Isotropic Solids Using Conservation Laws of Elasticity , 1980 .

[36]  B. K. Mishra,et al.  A new criterion for modeling multiple discontinuities passing through an element using XIGA , 2015 .

[37]  Saeed Shojaee,et al.  Free vibration analysis of thin plates by using a NURBS-based isogeometric approach , 2012 .

[38]  Timon Rabczuk,et al.  Phase-field analysis of finite-strain plates and shells including element subdivision , 2016 .

[39]  Glaucio H. Paulino,et al.  Mixed-mode J-integral formulation and implementation using graded elements for fracture analysis of nonhomogeneous orthotropic materials , 2003 .

[40]  Timon Rabczuk,et al.  Finite strain fracture of plates and shells with configurational forces and edge rotations , 2013 .

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

[42]  Bijay K. Mishra,et al.  A coupled finite element and element-free Galerkin approach for the simulation of stable crack growth in ductile materials , 2014 .

[43]  B. K. Mishra,et al.  Numerical simulation of functionally graded cracked plates using NURBS based XIGA under different loads and boundary conditions , 2015 .

[44]  Ted Belytschko,et al.  A finite element method for crack growth without remeshing , 1999 .

[45]  Kun Zhou,et al.  Extended isogeometric analysis based on Bézier extraction for an FGM plate by using the two-variable refined plate theory , 2017 .

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

[47]  G. Sangalli,et al.  A fully ''locking-free'' isogeometric approach for plane linear elasticity problems: A stream function formulation , 2007 .

[48]  On the Singularity Induced by Boundary Conditions in a Third-Order Thick Plate Theory , 2002 .

[49]  D. Choi,et al.  Finite element formulation of various four unknown shear deformation theories for functionally graded plates , 2013 .

[50]  T. Belytschko,et al.  Fracture and crack growth by element free Galerkin methods , 1994 .

[51]  Tinh Quoc Bui,et al.  On the thermal buckling analysis of functionally graded plates with internal defects using extended isogeometric analysis , 2016 .

[52]  C. Hui,et al.  A theory for the fracture of thin plates subjected to bending and twisting moments , 1993 .

[53]  B. K. Mishra,et al.  A simple, efficient and accurate Bézier extraction based T-spline XIGA for crack simulations , 2017 .

[54]  M. Williams.  The Bending Stress Distribution at the Base of a Stationary Crack , 1961 .

[55]  Loc V. Tran,et al.  Vibration analysis of cracked FGM plates using higher-order shear deformation theory and extended isogeometric approach , 2015 .

[56]  E. Ramm,et al.  A unified approach for shear-locking-free triangular and rectangular shell finite elements , 2000 .

[57]  G. C. Sih,et al.  Effect of Plate Thickness on the Bending Stress Distribution Around Through Cracks , 1968 .

[58]  Alan T. Zehnder,et al.  Crack tip stress fields for thin, cracked plates in bending, shear and twisting: A comparison of plate theory and three-dimensional elasticity theory solutions , 2000 .

[59]  Les A. Piegl,et al.  The NURBS Book , 1995, Monographs in Visual Communication.

[60]  Soheil Mohammadi,et al.  XFEM buckling analysis of cracked composite plates , 2015 .

[61]  A. Leung,et al.  Mixed mode cracks in Reissner plates , 2001 .

[62]  J. Rice.  A path-independent integral and the approximate analysis of strain , 1968 .

[63]  J. W. Eischen,et al.  Computation of stress intensity factors for plate bending via a path-independent integral , 1986 .

[64]  J. Reddy.  Analysis of functionally graded plates , 2000 .

[65]  T. Q. Bui.  Extended isogeometric dynamic and static fracture analysis for cracks in piezoelectric materials using NURBS , 2015 .

[66]  A. S. Shedbale,et al.  Ductile failure modeling and simulations using coupled FE–EFG approach , 2016, International Journal of Fracture.

[67]  Long Yu-qiu,et al.  Calculation of stress intensity factors of cracked reissner plates by the sub-region mixed finite element method , 1988 .

[68]  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 .

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

[70]  Tinh Quoc Bui,et al.  Accurate evaluation of mixed-mode intensity factors of cracked shear-deformable plates by an enriched meshfree Galerkin formulation , 2017 .

[71]  Stéphane Bordas,et al.  Isogeometric analysis of functionally graded plates using a refined plate theory , 2014 .

[72]  A. S. Shedbale,et al.  A coupled FE–EFG approach for modelling crack growth in ductile materials , 2016 .

[73]  T. Rabczuk,et al.  Phase-field modeling of fracture in linear thin shells , 2014 .

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

[75]  B. K. Mishra,et al.  Fatigue crack growth in functionally graded material using homogenized XIGA , 2015 .

[76]  Huu-Tai Thai,et al.  A new sinusoidal shear deformation theory for bending, buckling, and vibration of functionally graded plates , 2013 .

[77]  Jeeoot Singh,et al.  Nonlinear flexural analysis of laminated composite plates using RBF based meshless method , 2012 .