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Page 1: Modeling Discon XFEM

MODELING 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

1.19–1

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XFEM: SINGLE-EDGE NOTCH

1.19.1 CRACK PROPAGATION OF A SINGLE-EDGE NOTCH SIMULATED USING XFEM

Product: Abaqus/Standard

Problem description

This example verifies and illustrates the use of the extended finite element method (XFEM) inAbaqus/Standard to predict crack initiation and propagation of a single-edge notch in a specimenalong an arbitrary path by modeling the crack as an enriched feature. Both two- and three-dimensionalmodels are studied. The specimen is subjected to loadings ranging from pure Mode I to pure Mode IIto mixed-mode. The results presented are compared to the available analytical solutions and thoseobtained using cohesive elements.

Geometry and model

Two single-edge notch specimens are studied. The first specimen is shown in Figure 1.19.1–1 and has alength of 3 m, a thickness of 1 m, a width of 3 m, and an initial crack length of 0.3 m, loaded under pureMode 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 ormixed-mode loading. Equal and opposite displacements are applied at both ends in the width directionunder pure Mode II loading, while equal and opposite displacements are applied at both ends in both thelongitudinal and width directions under mixed-mode loading. The maximum displacement value is setequal 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 specified. The maximumprincipal stress failure criterion is selected for damage initiation; and a mixed-mode, energy-baseddamage evolution law based on a power law criterion is selected for damage propagation. The relevantmaterial data are as follows: MPa, × 103 N/m, × 103 N/m,

42.2× 103 N/m, and .

Results and discussion

Figure 1.19.1–2 shows plots of the prescribed displacement versus the corresponding reaction forceunder the pure Mode I loading compared with the results obtained using cohesive elements. The resultsdisplayed are from the two-dimensional plane strain analyses. The results obtained using the XFEMmethod agree well with those obtained using cohesive elements. The results from the equivalent three-dimensional models show similar agreement.

1.19.1–1

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XFEM: SINGLE-EDGE NOTCH

Under the pure Mode II or mixed-mode loading, the crack will no longer propagate along a straightpath and will instead propagate along a path based on the maximum tangential stress criterion accordingto 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 thecrack propagation in the “straight-ahead” direction. if , while if . Underpure Mode II loading, the above equation predicts that the crack will propagate at an angle of 70° whilethe crack propagation angle predicted using XFEM is 66.5°.

Input files

crackprop_modeI_xfem_cpe4r.inp Abaqus/Standard two-dimensional plane strain modelwith reduced integration under pure Mode I loading.

crackprop_modeI_xfem_cpe4.inp Abaqus/Standard two-dimensional plane strain modelunder pure Mode I loading.

crackprop_modeI_xfem_cps4r.inp Abaqus/Standard two-dimensional plane stress modelwith reduced integration under pure Mode I loading.

crackprop_modeI_xfem_cps4.inp Abaqus/Standard two-dimensional plane stress modelunder pure Mode I loading.

crackprop_modeI_xfem_c3d4.inp Abaqus/Standard three-dimensional tetrahedron modelunder pure Mode I loading.

crackprop_modeI_xfem_c3d8r.inp Abaqus/Standard three-dimensional brick model withreduced integration under pure Mode I loading.

crackprop_modeI_xfem_c3d8.inp Abaqus/Standard three-dimensional brick model underpure Mode I loading.

crackprop_modeII_xfem_cpe4r.inp Abaqus/Standard two-dimensional plane strain modelwith reduced integration under pure Mode II loading.

crackprop_modeII_xfem_cpe4.inp Abaqus/Standard two-dimensional plane strain modelunder pure Mode II loading.

crackprop_modeII_xfem_cps4r.inp Abaqus/Standard two-dimensional plane stress modelwith reduced integration under pure Mode II loading.

crackprop_modeII_xfem_cps4.inp Abaqus/Standard two-dimensional plane stress modelunder pure Mode II loading.

crackprop_modeII_xfem_c3d4.inp Abaqus/Standard three-dimensional tetrahedron modelunder pure Mode II loading.

crackprop_modeII_xfem_c3d8r.inp Abaqus/Standard three-dimensional brick model withreduced integration under pure Mode II loading.

crackprop_modeII_xfem_c3d8.inp Abaqus/Standard three-dimensional brick model underpure Mode II loading.

1.19.1–2

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XFEM: SINGLE-EDGE NOTCH

crackprop_mixmode_xfem_cpe4r.inp Abaqus/Standard two-dimensional plane strain modelwith reduced integration under mixed-mode loading.

crackprop_mixmode_xfem_cpe4.inp Abaqus/Standard two-dimensional plane strain modelunder mixed-mode loading.

crackprop_mixmode_xfem_cps4r.inp Abaqus/Standard two-dimensional plane stress modelwith reduced integration under mixed-mode loading.

crackprop_mixmode_xfem_cps4.inp Abaqus/Standard two-dimensional plane stress modelunder mixed-mode loading.

crackprop_mixmode_xfem_c3d4.inp Abaqus/Standard three-dimensional tetrahedron modelunder mixed-mode loading.

crackprop_mixmode_xfem_c3d8r.inp Abaqus/Standard three-dimensional brick model withreduced integration under mixed-mode loading.

crackprop_mixmode_xfem_c3d8.inp Abaqus/Standard three-dimensional brick model undermixed-mode loading.

Python scripts

crackprop_mixmode_xfem_cpe4.py Script to generate the two-dimensional plane strain modelunder mixed-mode loading in Abaqus/CAE.

crackprop_mixmode_xfem_c3d8.py Script to generate the three-dimensional brick modelunder mixed-mode loading in Abaqus/CAE.

Reference

• Erdogan, F., and G. C. Sih, “On the Crack Extension in Plates under Plane Loading and TransverseShear,” Journal of Basic Engineering, vol. 85, p. 519–527, 1963.

1.19.1–3

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XFEM: SINGLE-EDGE NOTCH

Figure 1.19.1–1 Model geometry for crack propagation in a single-edge notch specimen.

Displacement (m)0.0 0.2 0.4 0.6 0.8 1.0 [x1.E−3]

For

ce (

N)

0.00

0.05

0.10

0.15

0.20[x1.E9]

Cohesive elementsXFEM

Figure 1.19.1–2 Reaction force versus prescribed displacement: XFEM and cohesive element results.

1.19.1–4

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XFEM: PLATE WITH HOLE

1.19.2 CRACK PROPAGATION IN A PLATE WITH A HOLE SIMULATED USING XFEM

Product: Abaqus/Standard

Problem description

This example verifies and illustrates the use of the extended finite element method (XFEM) inAbaqus/Standard to predict crack initiation and propagation due to stress concentration in a plate witha hole. The specimen is subjected to pure Mode I loading. The results presented are compared to theavailable analytical solution.

Geometry and model

A plate with a circular hole is studied. The specimen, shown in Figure 1.19.2–1, 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 1.19.2–1 defines the dimensions used to calculate the variation of crack length, : a is thecrack length, b is half the specimen width, and c is the hole radius. Equal and opposite displacementsare applied at both ends in the longitudinal direction. The maximum displacement value is set equal to0.00055 m. To examine the mesh sensitivity, three different mesh discretizations of the same geometryare studied. Symmetry conditions reduce the specimen to a half model. The original mesh, as depictedin Figure 1.19.2–2, has 2060 plane strain elements. The second mesh has four times as many elementsas 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 specified. The maximum

principal stress failure criterion is selected for damage initiation, and an energy-based damage evolutionlaw based on a BK law criterion is selected for damage propagation. The relevant material data are asfollows: MPa, × 103 N/m, × 103 N/m, and .

Results and discussion

Figure 1.19.2–3 shows plots of the prescribed displacement versus the corresponding reaction forcewith different mesh discretizations. The figure clearly illustrates the convergence of the response tothe same solution with mesh refinement. A plot of the applied stress versus the variation of crack lengthis presented in Figure 1.19.2–4 and compared with the analytical solution of Tada et al. (1985). Theagreement is better than 10% except when the crack length is small, in which case the stress singularityahead of the crack is not considered by the XFEM approach. However, as indicated in this figure, thecrack 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 obtainedanalytically for the same geometry.

1.19.2–1

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XFEM: PLATE WITH HOLE

Input file

crackprop_hole_xfem_cpe4.inp Abaqus/Standard two-dimensional plane strain modelwith a hole under pure Mode I loading.

Python script

crackprop_hole_xfem_cpe4.py Script to generate the two-dimensional plane strain modelwith 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

a

0.20 m

b

c

Figure 1.19.2–1 Model geometry of the plate with a hole specimen.

1.19.2–2

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XFEM: PLATE WITH HOLE

Figure 1.19.2–2 Original mesh of the half model for crack propagation in a plate with a hole.

1.19.2–3

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XFEM: PLATE WITH HOLE

Displacement (m)0.00 0.10 0.20 0.30 0.40 0.50 [x1.E−3]

For

ce (

N)

0.

5.

10.

15.

20.[x1.E3]

Response for the original mesh

Response for 4 x the original mesh

Response for 16 x the original mesh

Figure 1.19.2–3 Reaction force versus prescribed displacement with XFEMwith different mesh discretizations.

(a+c)/b0.20 0.30 0.40 0.50 0.60 0.70 0.80

σ

4.

8.

12.

[x1.E6]

Analytical solutionsXFEM

Figure 1.19.2–4 Applied stress versus variation of crack length: XFEM and analytical solution.

1.19.2–4

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ELEMENT TESTS

2. Element Tests

• “Continuum elements,” Section 2.1• “Infinite 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

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