###Brick Slice with Slot###

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

Problem description

A graphite brick slice with a slot in the place of a loose keyway is placed in a test rig as indicated in Figure 1 below with a force applied to the loading plate assembly. The applied loading creates a moment on the brick slice which generates internal hoop stresses. We want to calculate the propagation of the crack through the brick slice and to examine the load-displacement response.

Figure 1. Brick slice with slot dimensions

Files

The CUBIT journal file required to generate the models can be downloaded from here:

• brick_slice_with_slot.jou (CUBIT journal)

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

Geometry

Considering Figure 1 above, the brick slice is 25 mm thick, has an inner radius of 263.25 mm and an outer radius of 459.8 mm. The width of the interstatial keyway is 32.95 mm and the width of the loose keyway is 20.25 mm with all keyways 37 mm deep. The slot is in the place of a loose keyway and the slot angle is 10˚. The left-hand loading assembly consists of two steel plates, one either side of the brick slice, 191.43 mm high, 40 mm wide and 20 mm thick. The right-hand loading assembly also consists of two steel plates, one either side of the brick slice, 243 mm high, 50 mm high and 11.43 mm thick.

Material properties

Graphite Young's modulus E = 1.09 N/mm²

Graphite Poisson's ratio v = 0.2

Graphite Griffith energy g = 2.3e-5 N/mm

Steel Young's modulus E = 21 N/mm²

Steel Poisson's ratio v = 0.3

For convenience, the values of E and g above have been scaled by e-4. This is to achieve a Young's modulus of approximately 1 N/mm² which will simplify the computational performance.

Loads

The reference loads, which are scaled, applied to the both plates of the right hand loading assembly are 4.33e-1 N/mm in tension and 4.3e-1 N/mm in compression. This generates a moment on the brick slice, generating internal hoop stresses.

Finite Element Model

The finite element model comprises of the graphite brick slice and loading assembly only. The model consists of a relatively coarse uniform mesh of 4-node tetrahedral elements as shown in Figure 2. We will use automatically different orders of approximation and levels of mesh refinement to improve the solution. 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 2. 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 1st (linear) to 4th (quartic) order 4-node elements. For each order, analyses with 1 level 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 brick_slice_with_slot.cub -l -1 -g 2.3e-5 -o 1,2,3 -r 1 -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 1st (linear) order with 1 level of automatic mesh refinement, for example, the script can be executed using:

$ ./arc_length.sh -f $DIRNAME/out_spatial_1_1.h5m -a 5 -n 500 -g 2.3e-5 -t 1e-8 -p $NPROC

$ ./face_split_arc_length.sh -f $DIRNAME/out_spatial_1_1.h5m -a 5 -n 500 -g 2.3e-5 -t 1e-8 -p $NPROC

Results

Figure 3 shows the evolution of the crack through the graphite brick for 2nd order approximation using smooth crack propagation with flip with no automatic mesh refinement (MoFEM v0.2).

Figure 3. Propagation of the crack through the graphite brick slice with a slot for 2nd order approximation with no automatic mesh refinement using smooth crack propagation with flip analysis (MoFEM v0.1) (green).

Figure 4 shows the evolution of the crack through the graphite brick for 2nd order approximation using face splitting with one level of automatic mesh refinement (MoFEM v0.1).

Figure 4. Propagation of the crack through the graphite brick slice with a slot for 2nd order approximation with one level of automatic mesh refinement using face splitting analysis (MoFEM v0.1) (red).

Notice the difference in crack propagation between the two analysis types. The script file required to extract the desired data (from log files) and plot graphs can be downloaded from here:

• plot_graphs.sh (shell script)

The files are also located under /benchmarks/brick_slice_with_slot in the MoFEM source and build directories. The graphs can be plotted using:

$ ./plot_graphs.sh -o 1,2,3 -r 1 -ent 813,6172 -dim 0,1,2 -s 1e4*1*1355.519433

where -s is a scaling factor which is the product of the applied load (see CUBIT journal file), the area over which the load is applied and the inverse of the scaling factor applied to E and g. This scaling is highly important to ensure correct results. Log files must be placed into a directory named log_files and log files must follow the naming convention log_ORDER_REF-LEVEL, with the plot file located one level above log_files. For more information on plotting, see Plotting information.

Figure 5. Load versus crack area for smooth crack propagation and crack propagation with face splitting; note points have been removed for steps that did not converge

Figure 6. Load versus crack mouth opening displacement for smooth crack propagation and crack propagation with face splitting; note points have been removed for steps that did not converge

Figure 7. Load versus crack mouth sliding displacement for smooth crack propagation and crack propagation with face splitting; note points have been removed for steps that did not converge

Discussion and Further Results

• Linear elements are not usually recommended due to the occurence of shear locking.
• The crack propagation with face splitting is a lower bound solution and underestimates the strength of the brick slice.
• The solutions improve as the level of approximation increases.

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