An enriched meshless method for non‐linear fracture mechanics

Abstract

SUMMARYThis paper presents an enriched meshless method for fracture analysis of cracks in homogeneous, isotropic, non-linear-elastic, two-dimensional solids, subject to mode-I loading conditions. The method involves an element-free Galerkin formulation and two new enriched basis functions (Types I and II) to capture the Hutchinson-Rice-Rosengren singularity ÿeld in non-linear fracture mechanics. The Type I enriched basis function can be viewed as a generalized enriched basis function, which degenerates to the linear-elastic basis function when the material hardening exponent is unity. The Type II enriched basis function entails further improvements of the Type I basis function by adding trigonometric functions. Four numerical examples are presented to illustrate the proposed method. The boundary layer analysis indicates that the crack-tip ÿeld predicted by using the proposed basis functions matches with the theoretical solution very well in the whole region considered, whether for the near-tip asymptotic ÿeld or for the far-tip elastic ÿeld. Numerical analyses of standard fracture specimens by the proposed meshless method also yield accurate estimates of the J -integral for the applied load intensities and material properties considered. Also, the crack-mouth opening displacement evaluated by the proposed meshless method is in good agreement with ÿnite element results. Furthermore, the meshless results show excellent agreement with the experimental measurements, indicating that the new basis functions are also capable of capturing elastic-plastic deformations at a stress concentration e ectively.

An explicit dynamics extended finite element method. Part 1: Mass lumping for arbitrary enrichment functions

“…For elastic-plastic materials with power-law hardening, enrichment functions derived from a Fourier analysis of the Hutchinson-Rice-Rosengren fields were proposed in Rao and Rahman [31] and Elguedj et al [10]:…”

“…(31). As presented in the introduction, these functions have two important roles: introduce more physics in the solution and localize the crack tip inside the element.…”

Multiple crack weight for solution of multiple interacting cracks by meshless numerical methods

“…Although X-FEM has recently received greater attention than meshless methods, they remain an efficient and accurate approach to solve fracture mechanics problems. Recent developments of meshless methods for the solution of different classes of problems such as multiple interacting cracks [11], 3D cracks [12], and cracks in elastic-plastic materials [13] improve these numerical methods, make them more attractive to the user.…”

J-Integral modeling and validation for GTS reservoirs.

“…if ( ($max_min eq "max" && $curr_elem_val > $elem_val && $max_dir eq $dir_str) || ($max_min eq "min" && $curr_elem_val < $elem_val && $max_dir eq $dir_str) ) { $result = $i; [2],$node_ids [6],$node_ids [5]); return @facenodes; } if ($fid == 3) { @facenodes=($node_ids [2],$node_ids [3],$node_ids [7],$node_ids [6]); return @facenodes; } if ($fid == 4) { @facenodes=($node_ids[0],$node_ids [4],$node_ids [7],$node_ids [3] [4],$node_ids [5],$node_ids [6],$node_ids [7] [7]]); $diags [2] = point_to_point_SQdistance($nodal_coords[$nod_ids [2]],$nodal_coords[$nod_ids [4]]); $diags [3] = point_to_point_SQdistance($nodal_coords[$nod_ids [3]],$nodal_coords[$nod_ids [5] [2]; } ##print "Element Coordinates:\n"; ##print "x = ".join(', ',@x); ##print "\n"; ##print "y = ".join(', ',@y); ##print "\n"; ##print "z = ".join(', ',@z); ##print "\n"; my @alpha = (-1,1,1,-1,-1,1,1,-1); my @beta = (-1,-1,1,1,-1,-1,1,1); my @gamma = (-1,-1,-1,-1,1,1,1,1 } ##print "Jacobian Elements = "; ##print "$j1,$j2,$j3,$j4,$j5,$j6,$j7,$j8,$j9\n"; ### Jacobian Determinant my $jdet = -$j3*$j5*$j7+$j2*$j6*$j7+$j3*$j4*$j8-$j1*$j6*$j8-$j2*$j4*$j9+$j1*$j5*$j9; ##print "Jacobian Determinant = $jdet\n"; ### Inverse Jacobian my ($jI1,$jI2,$jI3,$jI4,$jI5,$jI6,$jI7,$jI8,$jI9) = (0,0,0,0,0,0,0,0,0); $jI1 = (1/$jdet)*(-$j6*$j8+$j5*$j9); $jI2 = (1/$jdet)*($j3*$j8-$j2*$j9); $jI3 = (1/$jdet)*(-$j3*$j5+$j2*$j6); $jI4 = (1/$jdet)*($j6*$j7-$j4*$j9); $jI5 = (1/$jdet)*(-$j3*$j7+$j1*$j9); $jI6 = (1/$jdet)*($j3*$j4-$j1*$j6); $jI7 = (1/$jdet)*(-$j5*$j7+$j4*$j8); $jI8 = (1/$jdet)*($j2*$j7-$j1*$j8); $jI9 = (1/$jdet)*(-$j2*$j4+$j1*$j5); ##print "Inverse Jacobian Elements = "; ##print "$jI1,$jI2,$jI3,$jI4,$jI5,$jI6,$jI7,$jI8,$jI9\n"; ### Adjust @nc0 …”

“…It also contains a separate module, j3d_beam_subs.pm, for performing some of the lower level searches, and two additional modules, tims_general_subs.pm and tims_netcdf_subs_4_9_06.pm, for accessing information from Exodus II files. [2]; my $next_plane_dir = $ARGV [3]; my $num_paths = $ARGV [4]; my $num_planes = $ARGV [5]; my $offset = 1; if (@ARGV > 6) { $offset = $ARGV [6]; } ## the output file which will be used as input for j3d my $out_file = "output.dat"; ## do some input error checking die "ERROR: The plane_offset must not be < 0, you specified $offset\n" if ( $offset < 0 ); die "ERROR: The number of paths must be > 0, you specified $num_paths\n" if ( $num_paths < 1 ); die "ERROR: The number of planes must be > 0, you specified $num_planes\n" if ( …”