1.15.1 Contour integral evaluation: two-dimensional case

Product: ABAQUS/Standard  

The J-integral, stress intensity factors, and T-stress are widely used in fracture mechanics; and their accurate estimation for postulated flaws under given load conditions is an important aspect of the use of fracture mechanics in design. The domain integral method of Shih et al. (1986) provides a useful method for numerically evaluating contour integrals for the J-integral, stress intensity factors, and T-stress. This method provides high accuracy with rather coarse models in two–dimensions; in three-dimensions coarse meshes still give reasonably accurate values. It adds only a small increment to the cost of the stress analysis and can be specified easily. Contour integral evaluation is available in ABAQUS for any loading (including thermal loading: see Single-edged notched specimen under a thermal load, Section 1.15.8) and for elastic, elastic-plastic, and viscoplastic (creep) behaviors, the latter two cases being based on the equivalent hypoelastic material concept. The evaluation of the contour integral in three-dimensional cases is also often of interest: see Contour integral evaluation: three-dimensional case, Section 1.15.2.

Problem description

Geometry and model

The geometry of the first example is shown in Figure 1.15.1–1. The plane strain structure is a section of a plate with symmetric edge cracks at its center line, leaving an uncracked ligament of half the plate's width. The specimen is loaded in Mode I by uniform tension applied to its top and bottom surfaces. Symmetry about x = 0 and about y = 0 can be used to model only the top right-hand quadrant of the plate. The mesh used for the quarter model is shown in Figure 1.15.1–2. Second-order elements (8-node quadrilaterals, 20-node bricks) are used. In the case of the full-model seam crack, left and right contour integrals are defined in ABAQUS/CAE, as shown in Figure 1.15.1–3. Either the normal to the crack extension or the q vectors can be used to define the crack extension direction.

One advantage of using second-order elements is that they can be used to model the desired singularity at the crack tip. To obtain a singularity term, the following conditions must be met:

  1. The elements around the crack tip must be focused on the crack tip. One edge of each element must be collapsed to zero length (as shown in Figure 1.15.1–2) so that the nodes of this zero length edge are located at the crack tip.

  2. The “midside” nodes of the edges radiating out from the crack tip of each of the elements attached to the crack tip must be placed at one-quarter of the distance from the crack tip to the other node of the edge.

The region around the crack tip can be partitioned as shown in Figure 1.15.1–4 and swept meshed with elements having a quad-dominated shape. The node connectivity table is adjusted internally by ABAQUS/CAE to create the degenerate quadrilateral elements.

If the coincident nodes at the crack tip are constrained to displace together, the only singularity term in strain is ; if the crack-tip nodes are free to displace independently, the crack-tip singularity in strain includes a term in addition to the term. These methods of creating singularities in standard isoparametric elements are explained in detail by Barsoum (1976).

The material type determines the singularity at the crack tip. A linear elastic material exhibits a singularity in strain at a sharp crack tip, whereas a perfectly plastic material exhibits a singularity in strain. The singularity in strain for a plastic hardening material lies somewhere between and .

Frequently the need for meshing simplicity with a preprocessor is greater than the need for extreme accuracy of contour integral results. The contour integral results often will be adequate as long as some singularity is included. For example, a singularity is introduced if the element edges are collapsed, the nodes at the crack tip are free to displace independently, and the midside nodes are not moved to the quarter points (they remain at the midside points). This singularity is often quite adequate for elastic-plastic problems.

The model in this example problem uses a linear elastic material and, thus, should be modeled with only a singularity term.

For the quarter model of the double-edged notch specimen, symmetry is used for calculating the contour integral results. Thus, the results for the contour integrals are multiplied by two before being output. The three-dimensional model uses the same mesh with one layer of 20-node bricks, as shown in Figure 1.15.1–5. The loading applied is either a uniform edge load in two dimensions or a uniform surface pressure (of negative magnitude) in three dimensions.

The axisymmetric model corresponds to a penny-shaped crack in a round bar. The model is shown in Figure 1.15.1–6 and is loaded in Mode I by uniform tension applied to the top and bottom surfaces. Symmetry about r = 0 and z = 0 allows you to model only the top right-hand quadrant with a mesh, as shown in Figure 1.15.1–7. Second-order elements (CAX8 and CAX8R) are used.

The last two examples are both single-edged notch specimens under Mode I tension, as shown in Figure 1.15.1–8. One specimen contains a crack in the symmetry plane in a homogeneous linear elastic material, while the other specimen contains a crack lying along the interface between two dissimilar elastic materials. Due to symmetry only the top half-plane is modeled for the specimen with a homogeneous linear elastic material. The complete body is modeled for the specimen with an interface crack. Second-order elements CPE8 and CPS8 are used for these examples.

The J-integral, stress intensity factors, and T-stress should be path independent, and ABAQUS provides for its evaluation on as many contours as you request. The first contour is normally at the crack tip, and subsequent contours are generated automatically as contours passing through the nearest neighboring elements, moving out from the crack tip. The mesh used in this case has several rings of elements surrounding the crack tip and as many contours as the number of rings that can be requested. The contour integral should be path independent, so the variation of values between contours can be taken as an indicator of the quality of the mesh for determining the fracture parameters. Path independence of the contour integral values is sufficient to indicate mesh convergence for stress, strain, or displacements.

Results and discussion

Submodeling around the crack tip

The submodeling technique is capable of providing a more accurate analysis of the stresses around the crack tip. The global model has a coarse mesh, while the submodel has a refined mesh. For the double-edged notch specimen and the single-edged notch specimen, the submodel region is a semicircular region of radius 127 mm (5 inches). Thus, the submodel boundary is the same as the partition around the crack tip in the global model. The submodel uses a focused mesh with six rows of elements around the crack tip. For the axisymmetric penny crack specimen, the submodel region is a semicircular region of radius 114.3 mm (4.5 inches) and coincides with the outer partition around the crack tip in the global model. The global mesh in all three problems gives satisfactory J-integral results; hence, we assume that the displacements at the submodel boundary are sufficiently accurate to drive the deformation in the submodel. No attempt has been made to study the effect of making the submodel region larger or smaller. The meshed global model with the boundary of the submodel (in dashed lines) is shown on the left, and on the right an enlarged view of the submodel is shown in Figure 1.15.1–9 and Figure 1.15.1–10 for the double-edged notch specimen and the axisymmetric penny crack specimen, respectively.

Contours of the vertical displacement field in the submodel and the global model are shown for a double-edged notch specimen in Figure 1.15.1–11. The continuity of the contour lines verifies that the proper displacement values are prescribed on the submodel boundary. Contour integral values are calculated using five contours. The results of the J-integral are listed in Table 1.15.1–7. The J-integral results obtained with the global mesh are quite accurate; hence, only minor improvements in J-integral values are expected. The same trend also prevails for the calculated stress intensity factor and the T-stress. The agreement with Bowie's approximate solution is indeed slightly better, and a somewhat better path independence can be observed as well. A submodel analysis is also carried out for the axisymmetric model. The calculated J-integral values for the submodel analysis of the axisymmetric penny crack are listed in Table 1.15.1–8.

Python scripts

Input files

References

Tables

Table 1.15.1–1 J-integral values: two-dimensional symmetric double-edged notch specimen modeled using plane strain elements. Bowie's approximate solution: J = 2.245 N/m (0.0128 lb/in).

ContourFull integrationReduced integration
N/mlb/inN/mlb/in
12.2840.013032.2850.01304
22.2820.013022.2800.01301
32.2820.013022.2820.01302
42.2820.013022.2820.01302
52.2820.013022.2820.01302

Table 1.15.1–2 J-integral values: three-dimensional symmetric double-edged notch specimen modeled using continuum elements.

Full integration
ContourFront faceMiddle surfaceBack face
N/mlb/inN/mlb/inN/mlb/in
12.2120.012622.3060.013162.2120.01262
22.2770.012992.2770.012992.2770.01299
32.2800.013012.2800.013012.2800.01301
Reduced integration
ContourFront faceMiddle surfaceBack face
N/mlb/inN/mlb/inN/mlb/in
12.2730.012972.2840.013032.2730.01297
22.2770.012992.2770.012992.2770.01299
32.2800.013012.2800.013012.2800.01301

Table 1.15.1–3 J-integral values: axisymmetric penny-shaped crack specimen. Tada et al. approximate solution: J = 0.7635 N/m (0.00436 lb/in).

ContourFull integrationReduced integration
N/mlb/inN/mlb/in
10.78700.004490.78530.00448
20.78180.004460.78530.00448
30.78350.004470.78700.00449
40.78350.004470.78700.00449
50.78350.004470.78700.00449

Table 1.15.1–4 Nondimensional stress intensity factor for two-dimensional symmetric single-edged notch specimen. Tada et al. approximate solution: 2.826.

ContourCPE8CPS8
12.82502.8249
22.82302.8231
32.82372.8238
42.82382.8239
52.82382.8239

Table 1.15.1–5 Nondimensional T-stress for single-edged notch specimen. Nakamura and Parks approximate solution: –0.43.

ContourCPE8CPS8
1–0.4307–0.4298
2–0.4204–0.4202
3–0.4226–0.4224
4–0.4226–0.4224
5–0.4225–0.4223

Table 1.15.1–6 Nondimensional , , and values of an interface crack.

ContourJ-integral value estimated by the stress intensity factorsJ-integral value estimated directly
from
12.82450.012117.3617.30
22.82260.012717.3317.30
32.82320.012717.3417.30
42.82330.012717.3417.30
52.82320.012717.3417.30

Table 1.15.1–7 J-integral values: two-dimensional submodel analysis of a double-edged notch specimen using plane strain elements.

ContourFull integrationReduced integration
N/mlb/inN/mlb/in
12.2820.013022.2850.01304
22.2780.0132.2800.01301
32.2800.013012.2820.01302
42.2800.013012.2820.01302
52.2800.013012.2820.01302

Table 1.15.1–8 J-integral values: submodel analysis of an axisymmetric penny-shaped crack specimen.

ContourFull integrationReduced integration
N/mlb/inN/mlb/in
10.77650.004430.78530.00448
20.77480.004420.78350.00447
30.77650.004430.78350.00447
40.77650.004430.78350.00447
50.77650.004430.78350.00447


Figures

Figure 1.15.1–1 Double-edged notch example.

Figure 1.15.1–2 Symmetric finite element model of double-edged notch specimen.

Figure 1.15.1–3 Model showing the seam cracks defined in bold and the q vectors defined at the left and right crack tips.

Figure 1.15.1–4 Two-dimensional double-edged notch specimen showing the partitions created around the crack tip. The circular partitioned region is meshed using the sweep meshing technique with quad-dominated elements.

Figure 1.15.1–5 Finite element model of a three-dimensional quarter model of the double-edged notch specimen meshed with one layer of C3D20 elements.

Figure 1.15.1–6 Penny-shaped crack in round bar.

Figure 1.15.1–7 Axisymmetric finite element model of penny-shaped crack in round bar.

Figure 1.15.1–8 Single-edged notch specimen.

Figure 1.15.1–9 Left: meshed global model of two-dimensional double-edged notch specimen with boundary of submodel shown with dashed lines, only two elements in submodel region. Right: enlarged view of submodel, refined mesh with six rows of elements.

Figure 1.15.1–10 Left: meshed global model of axisymmetric penny-shaped crack specimen with boundary of submodel shown with dashed lines, only two elements in submodel region. Right: enlarged view of submodel, refined mesh with six rows of elements.

Figure 1.15.1–11 Displacement contours of the global model and the submodel for a two-dimensional double-edged notch specimen.