###Mixed Mode###

Keywords: Crack Propagation, Fracture, High-Order Approximation, Automatic Mesh Refinement

Problem description

A graphite beam is loaded in a manner so as to induce mixed mode cracking, as indicated in Figure 1 below. The ratio of the applied loads will be changed to examine the effect this has on the propagation of the crack.

Figure 1. Graphite beam in mixed mode setup showing loading arrangement and boundary conditions

Files

The CUBIT mesh file can be downloaded from here:

• mixed mode 25 (CUBIT mesh)
• mixed mode 34 (CUBIT mesh)
• mixed mode 49 (CUBIT mesh)
• mixed mode 59 (CUBIT mesh)
• mixed mode 66 (CUBIT mesh)

The files are also located under /mofem_install/mofem/benchmarks/mixed_mode in the MoFEM source and build directories.

Geometry

Considering figure 1 above, the beam is 270 mm long, 100 mm deep and 100 mm high. One entire end of the beam is restricted from moving. Two pressures are applied to the beam, P1 to an area on the top surface of the beam and P2 to the free end of the beam. P1 is kept constant but P2 is gradually increased to change the direction of propagation of the crack.

Material properties

Young's modulus E = 109 N/m²

Poisson's ratio v = 0.2

Griffith energy g = 2.3e-6 N/m

For convenience, the values of E and g above have been scaled by e-8. This is to achieve a Young's modulus of approximately 100 N/m² which will simplify the computational performance.

Loads

There are two reference loads applied in this problem. P1 is applied to a small area (0.00125m²) on the top surface of the beam and is 0.1N/mm² in magnitude. P2 is appied to the free end of the beam (0.01m²) and has the value of either 0.03 N/mm², 0.04 N/mm², 0.05N/mm², 0.055 N/mm² and 0.058 N/mm².

Finite Element Model

The finite element model comprises of the graphite beam only. The model consists of a relatively coarse uniform mesh of 4-node tetrahedral elements as shown in Figure 2. In addition to the load shown above, we apply displacement constraints to prevent rigid body motions (see CUBIT journal files).

The crack is defined geometrically in CUBIT by a single surface. It is worthwhile to note that the visible cylindrical cutout around the crack is only used to construct the crack geometry and optionally give better mesh control around the crack. The crack tips and crack surface are specified by custom sidesets.

Figure 3. Geometry (left) and mesh (right)

Analysis procedure

The analysis is run in two stages. The first stage uses a script to calculate the Griffith forces for 2nd (quadratic) order 4-node elements. For this order, analyses with no levels of automatic refinement are run. All analyses were run on the ARCHIE_WeSt high performance computer using 12 processors with the FGMRES (Flexible Generalized Minimal Residual Method) Krylov solver for systems of linear equations and the LU preconditioner. The first script can be executed using:

$ ./convergence_study.sh -f cas3_mode_mixte_25deg.cub -l -1 -g 2.3e-6 -o 2 -r 0 -p $NPROC

If desired, the Mode I stress intensity factor () can be calculated directly from the output Griffith forces, see the Analysis procedure section of the plate with horizontal crack benchmark.

The second stage uses a script to calculate, over a prescribed number of steps, the propagation of the crack for a specified approximation order and level of mesh refinement, using the resultant Griffith forces from the first stage. All analyses were run on the ARCHIE_WeSt high performance computer using 12 processors with the FGMRES (Flexible Generalized Minimal Residual Method) Krylov solver for systems of linear equations and the LU preconditioner. To calculate the propagation of the crack for the 2nd (quadratic) order with no levels of automatic mesh refinement, for example, the script can be executed using (MoFEM v0.2):

$ ./arc_length.sh -f $DIRNAME/out_spatial_2_0.h5m -a 1e-4 -n 500 -g 2.3e-6 -t 1e-8 -m 1e-7 -i 12 -p $NPROC

Results

Figure 4 shows resultant crack paths for each set of loads for a 2nd order solution with no levels of mesh refinement using the smooth crack propagation with flip analysis MOFEM v0.2. The deformation in each case is illustrated in Figures 5 to 9.

Figure 4. Resultant crack paths for 2nd order approximation with no levels of automatic mesh refinement using smooth crack propagation with flip analysis in MoFEM v0.2

Figure 4. Deformed shape for 25° crack path with 2nd order approximation with no levels of automatic mesh refinement using smooth crack propagation with flip analysis in MoFEM v0.2.

Figure 4. Deformed shape for 34° crack path with 2nd order approximation with no levels of automatic mesh refinement using smooth crack propagation with flip analysis in MoFEM v0.2.

Figure 4. Deformed shape for 49° crack path with 2nd order approximation with no levels of automatic mesh refinement using smooth crack propagation with flip analysis in MoFEM v0.2.

Figure 4. Deformed shape for 59° crack path with 2nd order approximation with no levels of automatic mesh refinement using smooth crack propagation with flip analysis in MoFEM v0.2.

Figure 4. Deformed shape for 66° crack path with 2nd order approximation with no levels of automatic mesh refinement using smooth crack propagation with flip analysis in MoFEM v0.2.

Acknowledgements

Results were obtained using the EPSRC funded ARCHIE-WeSt High Performance Computer (www.archie-west.ac.uk). EPSRC grant no. EP/K000586/1

References

1. Three‐dimensional brittle fracture: configurational‐force‐driven crack propagation, Ł Kaczmarczyk, MM Nezhad, C Pearce, International Journal for Numerical Methods in Engineering 97 (7) 531-550

• Created by Ross Mackenzie
• any difficulties or suggestions email cmatgu@googlegroups.com