Stress analysis in anisotropic functionally graded materials by the MLPG method

Abstract A meshless method based on the local Petrov–Galerkin approach is proposed for stress analysis in two-dimensional (2D), anisotropic and linear elastic/viscoelastic solids with continuously varying material properties. The correspondence principle is applied for non-homogeneous, anisotropic and linear viscoelastic solids where the relaxation moduli are separable in space and time. The inertial dynamic term in the governing equations is considered too. A unit step function is used as the test functions in the local weak-form. It leads to local boundary integral equations (LBIEs). The analyzed domain is divided into small subdomains with a circular shape. The moving least squares (MLS) method is adopted for approximating the physical quantities in the LBIEs. For time-dependent problems, the Laplace-transform technique is utilized. Several numerical examples are given to verify the accuracy and the efficiency of the proposed method.

[1]  E. L. Albuquerque,et al.  The boundary element method applied to time dependent problems in anisotropic materials , 2002 .

[2]  S. G. Lekhnit︠s︡kiĭ.  Theory of elasticity of an anisotropic body , 1981 .

[3]  P. Sollero,et al.  Anisotropic analysis of cracks in composite laminates using the dual boundary element method , 1995 .

[4]  Bernard Amadei,et al.  2-D BEM analysis of anisotropic half-plane problems—application to rock mechanics , 1998 .

[5]  M. Aliabadi,et al.  A Galerkin boundary element formulation with dual reciprocity for elastodynamics , 2000 .

[6]  V. Sladek,et al.  Local boundary integral equation (LBIE) method for solving problems of elasticity with nonhomogeneous material properties , 2000 .

[7]  Ernian Pan,et al.  Fracture mechanics analysis of cracked 2-D anisotropic media with a new formulation of the boundary element method , 1996 .

[8]  S. Atluri,et al.  The meshless local Petrov-Galerkin (MLPG) method , 2002 .

[9]  Satya N. Atluri,et al.  Meshless Local Petrov-Galerkin Method in Anisotropic Elasticity , 2004 .

[10]  N. A. Schclar.  Anisotropic Analysis Using Boundary Elements , 1994 .

[11]  R. Christensen,et al.  Theory of Viscoelasticity , 1971 .

[12]  C. Brebbia,et al.  Boundary Element Techniques , 1984 .

[13]  Alok Sutradhar,et al.  Transient heat conduction in homogeneous and non-homogeneous materials by the Laplace transform Galerkin boundary element method , 2002 .

[14]  J. D. Eshelby,et al.  Anisotropic elasticity with applications to dislocation theory , 1953 .

[15]  Mark A Fleming,et al.  Meshless methods: An overview and recent developments , 1996 .

[16]  G. Paulino,et al.  A viscoelastic functionally graded strip containing a crack subjected to in-plane loading , 2002 .

[17]  J. W. Eischen,et al.  Fracture of nonhomogeneous materials , 1987, International Journal of Fracture.

[18]  L. Gaul,et al.  A 3-D Boundary Element Method for Dynamic Analysis of Anisotropic Elastic Solids , 2000 .

[19]  Fazil Erdogan.  Fracture mechanics of functionally graded materials , 1995 .

[20]  E. Pan.  A BEM analysis of fracture mechanics in 2D anisotropic piezoelectric solids , 1999 .

[21]  S. Atluri,et al.  A local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach , 1998 .

[22]  T. Cruse,et al.  Interactive Program for Analysis and Design Problems in Advanced Composites Technology , 1971 .

[23]  M. H. Aliabadi,et al.  Fracture mechanics analysis of anisotropic plates by the boundary element method , 1993 .

[24]  Glaucio H. Paulino,et al.  Green's function for a two–dimensional exponentially graded elastic medium , 2004, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[25]  F. Afagh,et al.  TREATMENT OF BODY-FORCE VOLUME INTEGRALS IN BEM BY EXACT TRANSFORMATION FOR 2-D ANISOTROPIC ELASTICITY , 1997 .

[26]  Satya N. Atluri,et al.  The local boundary integral equation (LBIE) and it's meshless implementation for linear elasticity , 2000 .

[27]  Michael H. Santare,et al.  Numerical Calculation of Stress Intensity Factors in Functionally Graded Materials , 2000 .

[28]  D. Beskos,et al.  Boundary Element Methods in Elastodynamics , 1988 .

[29]  Robert J. Asaro,et al.  A Simplified Method for Calculating the Crack-Tip Field of Functionally Graded Materials Using the Domain Integral , 1999 .

[30]  J. Sládek,et al.  Numerical Analysis of Cracked Functionally Graded Materials , 2003 .

[31]  G. Paulino,et al.  Correspondence Principle in Viscoelastic Functionally Graded Materials , 2001 .

[32]  J. Sládek,et al.  An advanced numerical method for computing elastodynamic fracture parameters in functionally graded materials , 2005 .

[33]  G. Manolis,et al.  A Green's Function for Variable Density Elastodynamics under Plane Strain Conditions by Hormander's Method , 2002 .

[34]  T. Cruse,et al.  Boundary-integral equation analysis of cracked anisotropic plates , 1975 .

[35]  J. Sládek,et al.  Meshless local boundary integral equation method for 2D elastodynamic problems , 2003 .

[36]  Sergey E. Mikhailov,et al.  Localized boundary-domain integral formulations for problems with variable coefficients , 2002 .

[37]  W. Ang,et al.  A numerical Green's function for multiple cracks in anisotropic bodies , 2004 .

[38]  J. Achenbach,et al.  Two-dimensional time domain BEM for scattering of elastic waves in solids of general anisotropy , 1996 .

[39]  Sharif Rahman,et al.  Mesh-free analysis of cracks in isotropic functionally graded materials , 2003 .

[40]  D. Clements,et al.  A boundary integral equation method for a class of crack problems in anisotropic elasticity , 1983 .

[41]  G. Paulino,et al.  Finite element evaluation of mixed mode stress intensity factors in functionally graded materials , 2002 .

[42]  J. Sládek,et al.  APPLICATION OF MESHLESS LOCAL PETROV-GALERKIN (MLPG) METHOD TO ELASTO-DYNAMIC PROBLEMS IN CONTINUOUSLY NONHOMOGENEOUS SOLIDS , 2003 .

[43]  Bhushan Lal Karihaloo,et al.  Comprehensive structural integrity , 2003 .

[44]  G. Paulino,et al.  Viscoelastic Functionally Graded Materials Subjected to Antiplane Shear Fracture , 2001 .

[45]  P. C. Dumir,et al.  Boundary element solution for elastic orthotropic half-plane problems , 1987 .

[46]  J. Domínguez.  Boundary elements in dynamics , 1993 .

[47]  Leonard J. Gray,et al.  On Green's function for a three–dimensional exponentially graded elastic solid , 2002, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[48]  D. Haojiang,et al.  The united point force solution for both isotropic and transversely isotropic media , 1997 .

[49]  S. Suresh,et al.  Fundamentals of functionally graded materials , 1998 .

[50]  Vladimir Sladek,et al.  Stress analysis by boundary element methods , 1989 .

[51]  Ch. Zhang,et al.  Effects of material gradients on transient dynamic mode-III stress intensity factors in a FGM , 2003 .

[52]  H. Antes,et al.  Application of ‘Operational Quadrature Methods’ in Time Domain Boundary Element Methods , 1997 .