To obtain accurate results with a single analysis, a refined three-dimensional model must be used, requiring considerable computer time. Alternatively, the submodeling capability can be used to obtain accurate results by running two, much smaller, models: first a global model to obtain the displacement solution with moderate accuracy away from the crack tip and then a submodel to obtain a more accurate solution and, hence, more accurate -integrals along the crack front. These models are much smaller than the full three-dimensional model, allowing them to be run on a smaller computer than that required to run the full model.

The geometry analyzed is a conical crack in a block, as shown in Figure 1.4.2–1. The block dimensions (300 × 300 × 300 units) are chosen to be large enough so that the results for the crack are not affected. The crack intersects the free surface at 45° and extends 15 units into the block. Pressure loading is applied on the region of the block surface circumscribed by the crack. The full three-dimensional and axisymmetric meshes are shown in Figure 1.4.2–2 and Figure 1.4.2–3, respectively. The full three-dimensional model represents one-quarter of the problem, using symmetry about the – and – planes. The model is partitioned such that a swept mesh can be created around the crack line (see Figure 1.4.2–4). A smaller partition around the crack line enables the creation of wedge elements with collapsed edges using the swept meshing technique (see Figure 1.4.2–5). The outer ring is meshed with hexahedral elements using the structured meshing technique. The entire model is meshed using reduced-integration elements (C3D20R). A refined mesh is used along the crack line, and the element size is increased as you move away from the area. The crack properties are defined for the conical crack. The slanted face of the cone is identified as a seam crack face as shown in Figure 1.4.2–6; and the crack line and the crack front are defined.

The vectors are specified to define the crack extension direction using a start and end point. Currently, the ABAQUS/CAE capability for defining vectors is limited to a crack line with the same curvature at all points on the crack line. However, in this case the curvature is different at all points on the crack line. Thus, the vectors need to be defined at several points along the crack line. This can be done easily for an orphan mesh, since there are multiple nodes along the crack line. Thus, after defining a single vector on the ABAQUS/CAE model and using a singularity of 0.27, an input file is written. A new orphan mesh model is created by reading the input file back into ABAQUS/CAE, and the vectors are edited as shown in Figure 1.4.2–7.

Since the mesh around the crack line is highly refined, a quarter-point singularity is not used since the symmetric model may produce negative volume errors. Negative volume errors are usually produced at exactly the quarter point, because the Jacobian goes singular. These errors can be avoided either by refining the mesh further or by increasing the size of the elements. A refined mesh will help to keep the edges curved, but mesh refinement is associated with a higher computational cost for the model. By increasing the element size around the crack line, the focus may shift away from the crack line, thus leading to an inaccuracy in the results. Thus, a singularity of 0.27 is used for this model, which keeps the computational cost of the model low and still accurately captures the results around the crack line. For the axisymmetric model a singularity of 0.25 could be used with the existing mesh. However, for the purpose of comparison a singularity of 0.27 is used. The desired 1/ strain singularity improves the modeling of the strain field near the crack tip by moving the midside nodes, which results in more accurate contour integral values. Collapsed elements are necessary at the crack tip to provide the desired singularity.

Figure 1.4.2–8 shows the displaced shape of the mesh near the crack for the three-dimensional case.

The submodeling technique is used to focus on a small region of the entire problem. The mesh in this region is refined. The global model, which has a coarser mesh, is first analyzed and then used to drive the submodel. Only 2 rings of elements are used in the focused part of the global model mesh, compared to 13 rings in the full model. For the three-dimensional global model only 18 elements are used along the crack line, whereas in the submodel 38 elements are used along the crack line. Figure 1.4.2–9 shows the meshes for the three-dimensional global model and submodel. The axisymmetric global model and the submodel meshes are shown in Figure 1.4.2–10. The axisymmetric submodel has a refined mesh around the crack tip with 12 rings of elements, whereas the axisymmetric global model has only 2 rings. Quarter-symmetry boundary conditions are applied to the three-dimensional submodel as well as to the global model. It is assumed that the global model's coarse mesh is sufficiently accurate to drive the submodel: if the global model's displacement field far from the crack tip is accurate, the submodel can obtain accurate contour integral results.

The output database (.odb) file from the global model is used to drive the submodel for both the full model and the axisymmetric model. In the axisymmetric submodel, submodeling boundary conditions are applied on two edges; in the full model, submodeling boundary conditions are applied on three faces.