Split by PDF SplitterMODELING DISCONTINUITIES USING XFEM

1.19

Modeling discontinuities using XFEM

Crack propagation of a single-edge notch simulated using XFEM, Section 1.19.1 Crack propagation in a plate with a hole simulated using XFEM, Section 1.19.2

1.191

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Split by PDF SplitterXFEM: SINGLE-EDGE NOTCH

1.19.1

CRACK PROPAGATION OF A SINGLE-EDGE NOTCH SIMULATED USING XFEM

Product: Abaqus/Standard Problem description

This example veries and illustrates the use of the extended nite element method (XFEM) in Abaqus/Standard to predict crack initiation and propagation of a single-edge notch in a specimen along an arbitrary path by modeling the crack as an enriched feature. Both two- and three-dimensional models are studied. The specimen is subjected to loadings ranging from pure Mode I to pure Mode II to mixed-mode. The results presented are compared to the available analytical solutions and those obtained using cohesive elements.Geometry and model

Two single-edge notch specimens are studied. The rst specimen is shown in Figure 1.19.11 and has a length of 3 m, a thickness of 1 m, a width of 3 m, and an initial crack length of 0.3 m, loaded under pure Mode I loading. Equal and opposite displacements are applied at both ends in the longitudinal direction. The maximum displacement value is set equal to 0.001 m. The second specimen has a length of 6 m, a thickness of 1 m, a width of 3 m, and an initial crack length of 1.5 m, loaded under pure Mode II or mixed-mode loading. Equal and opposite displacements are applied at both ends in the width direction under pure Mode II loading, while equal and opposite displacements are applied at both ends in both the longitudinal and width directions under mixed-mode loading. The maximum displacement value is set equal to 0.0035 m.Material

The material data for the bulk material properties in the enriched elements are GPa and . The response of cohesive behavior in the enriched elements in the model is specied. The maximum principal stress failure criterion is selected for damage initiation; and a mixed-mode, energy-based damage evolution law based on a power law criterion is selected for damage propagation. The relevant material data are as follows: MPa, 103 N/m, 103 N/m, 3 42.2 10 N/m, and .Results and discussion

Figure 1.19.12 shows plots of the prescribed displacement versus the corresponding reaction force under the pure Mode I loading compared with the results obtained using cohesive elements. The results displayed are from the two-dimensional plane strain analyses. The results obtained using the XFEM method agree well with those obtained using cohesive elements. The results from the equivalent threedimensional models show similar agreement.

1.19.11

Split by PDF SplitterXFEM: SINGLE-EDGE NOTCH

Under the pure Mode II or mixed-mode loading, the crack will no longer propagate along a straight path and will instead propagate along a path based on the maximum tangential stress criterion according to Erdogan and Sih (1963). The direction of crack propagation is given by

where the crack propagation angle, , is measured with respect to the crack plane. represents the crack propagation in the straight-ahead direction. if , while if . Under pure Mode II loading, the above equation predicts that the crack will propagate at an angle of 70 while the crack propagation angle predicted using XFEM is 66.5.Input files

crackprop_modeI_xfem_cpe4r.inp crackprop_modeI_xfem_cpe4.inp crackprop_modeI_xfem_cps4r.inp crackprop_modeI_xfem_cps4.inp crackprop_modeI_xfem_c3d4.inp crackprop_modeI_xfem_c3d8r.inp crackprop_modeI_xfem_c3d8.inp crackprop_modeII_xfem_cpe4r.inp crackprop_modeII_xfem_cpe4.inp crackprop_modeII_xfem_cps4r.inp crackprop_modeII_xfem_cps4.inp crackprop_modeII_xfem_c3d4.inp crackprop_modeII_xfem_c3d8r.inp crackprop_modeII_xfem_c3d8.inp

Abaqus/Standard two-dimensional plane strain model with reduced integration under pure Mode I loading. Abaqus/Standard two-dimensional plane strain model under pure Mode I loading. Abaqus/Standard two-dimensional plane stress model with reduced integration under pure Mode I loading. Abaqus/Standard two-dimensional plane stress model under pure Mode I loading. Abaqus/Standard three-dimensional tetrahedron model under pure Mode I loading. Abaqus/Standard three-dimensional brick model with reduced integration under pure Mode I loading. Abaqus/Standard three-dimensional brick model under pure Mode I loading. Abaqus/Standard two-dimensional plane strain model with reduced integration under pure Mode II loading. Abaqus/Standard two-dimensional plane strain model under pure Mode II loading. Abaqus/Standard two-dimensional plane stress model with reduced integration under pure Mode II loading. Abaqus/Standard two-dimensional plane stress model under pure Mode II loading. Abaqus/Standard three-dimensional tetrahedron model under pure Mode II loading. Abaqus/Standard three-dimensional brick model with reduced integration under pure Mode II loading. Abaqus/Standard three-dimensional brick model under pure Mode II loading.

1.19.12

Split by PDF SplitterXFEM: SINGLE-EDGE NOTCH

crackprop_mixmode_xfem_cpe4r.inp crackprop_mixmode_xfem_cpe4.inp crackprop_mixmode_xfem_cps4r.inp crackprop_mixmode_xfem_cps4.inp crackprop_mixmode_xfem_c3d4.inp crackprop_mixmode_xfem_c3d8r.inp crackprop_mixmode_xfem_c3d8.inp

Abaqus/Standard two-dimensional plane strain model with reduced integration under mixed-mode loading. Abaqus/Standard two-dimensional plane strain model under mixed-mode loading. Abaqus/Standard two-dimensional plane stress model with reduced integration under mixed-mode loading. Abaqus/Standard two-dimensional plane stress model under mixed-mode loading. Abaqus/Standard three-dimensional tetrahedron model under mixed-mode loading. Abaqus/Standard three-dimensional brick model with reduced integration under mixed-mode loading. Abaqus/Standard three-dimensional brick model under mixed-mode loading.

Python scripts

crackprop_mixmode_xfem_cpe4.py crackprop_mixmode_xfem_c3d8.py

Script to generate the two-dimensional plane strain model under mixed-mode loading in Abaqus/CAE. Script to generate the three-dimensional brick model under mixed-mode loading in Abaqus/CAE.

Reference

Erdogan, F., and G. C. Sih, On the Crack Extension in Plates under Plane Loading and Transverse Shear, Journal of Basic Engineering, vol. 85, p. 519527, 1963.

1.19.13

Split by PDF SplitterXFEM: SINGLE-EDGE NOTCH

Figure 1.19.11

Model geometry for crack propagation in a single-edge notch specimen.

Cohesive elements XFEM [x1.E9] 0.20

0.15

Force (N)

0.10

0.05

0.00 0.0

0.2

0.4

0.6

0.8

1.0 [x1.E3]

Displacement (m)Figure 1.19.12 Reaction force versus prescribed displacement: XFEM and cohesive element results.

1.19.14

Split by PDF SplitterXFEM: PLATE WITH HOLE

1.19.2

CRACK PROPAGATION IN A PLATE WITH A HOLE SIMULATED USING XFEM

Product: Abaqus/Standard Problem description

This example veries and illustrates the use of the extended nite element method (XFEM) in Abaqus/Standard to predict crack initiation and propagation due to stress concentration in a plate with a hole. The specimen is subjected to pure Mode I loading. The results presented are compared to the available analytical solution.Geometry and model

A plate with a circular hole is studied. The specimen, shown in Figure 1.19.21, has a length of 0.34 m, a thickness of 0.02 m, a width of 0.2 m, and a hole radius of 0.02 m, under pure Mode I loading. Figure 1.19.21 denes the dimensions used to calculate the variation of crack length, : a is the crack length, b is half the specimen width, and c is the hole radius. Equal and opposite displacements are applied at both ends in the longitudinal direction. The maximum displacement value is set equal to 0.00055 m. To examine the mesh sensitivity, three different mesh discretizations of the same geometry are studied. Symmetry conditions reduce the specimen to a half model. The original mesh, as depicted in Figure 1.19.22, has 2060 plane strain elements. The second mesh has four times as many elements as the original one, while the third mesh has sixteen times as many elements as the original one.Material

The material data for the bulk material properties in the enriched elements are GPa and = 0.3. The response of cohesive behavior in the enriched elements in the model is specied. The maximum principal stress failure criterion is selected for damage initiation, and an energy-based damage evolution law based on a BK law criterion is selected for damage propagation. The relevant material data are as follows: MPa, 103 N/m, 103 N/m, and .Results and discussion

Figure 1.19.23 shows plots of the prescribed displacement versus the corresponding reaction force with different mesh discretizations. The gure clearly illustrates the convergence of the response to the same solution with mesh renement. A plot of the applied stress versus the variation of crack length is presented in Figure 1.19.24 and compared with the analytical solution of Tada et al. (1985). The agreement is better than 10% except when the crack length is small, in which case the stress singularity ahead of the crack is not considered by the XFEM approach. However, as indicated in this gure, the crack initiates (i.e., ) when the applied stress, , reaches a level of 8.37 MPa, giving a ratio of equal to 2.63. This value is in close agreement with the stress concentration factor of 2.52 obtained analytically for the same geometry.

1.19.21

Split by PDF SplitterXFEM: PLATE WITH HOLE

Input file

crackprop_hole_xfem_cpe4.inp

Abaqus/Standard two-dimensional plane strain model with a hole under pure Mode I loading.

Python script

crackprop_hole_xfem_cpe4.py

Script to generate the two-dimensional plane strain model with a hole under pure Mode I loading in Abaqus/CAE.

Reference

Tada, H., P. C. Paris, and G. R. Irwin, The Stress Analysis of Cracks Handbook, 2nd Edition, Paris Productions Incorporated, 226 Woodbourne Drive, St. Louis, Missouri, 63105, 1985.

0.02 m

+

0.34 m

c

a

b

0.20 m

Figure 1.19.21

Model geometry of the plate with a hole specimen.

1.19.22

Split by PDF SplitterXFEM: PLATE WITH HOLE

Figure 1.19.22

Original mesh of the half model for crack propagation in a plate with a hole.

1.19.23

Split by PDF SplitterXFEM: PLATE WITH HOLE

Response for the original mesh Response for 4 x the original mesh Response for 16 x the original mesh [x1.E3] 20.

15.

Force (N)

10.

5.

0. 0.00

0.10

0.20

0.30

0.40

0.50

[x1.E3]

Displacement (m)Figure 1.19.23 Reaction force versus prescribed displacement with XFEM with different mesh discretizations.

Analytical solutions XFEM [x1.E6]

12.

8. 4. 0.20 0.30 0.40 0.50 0.60 0.70 0.80

(a+c)/bFigure 1.19.24 Applied stress versus variation of crack length: XFEM and analytical solution.

1.19.24

Split by PDF SplitterELEMENT TESTS

2.

Element TestsContinuum elements, Section 2.1 Innite elements, Section 2.2 Structural elements, Section 2.3 Acoustic elements, Section 2.4 Fluid elements, Section 2.5 Connector elements, Section 2.6 Special-purpose elements, Section 2.7