###Graphite Torsion Test###

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

Problem description

A graphite beam with a deep notch is placed in a test rig and is examined in a torsion test as indicated in Figure 1 below. We want to calculate the propagation of the crack from the notch through the graphite beam. This is the same test as the concrete torsion test, but with the change from concrete to graphite.

Figure 1. Graphite beam with deep notch in torsion test

Files

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

For the Gilscocarbon grade of graphite (GCMB):

• torsion_GCMB_450L_coarse.jou (CUBIT journal)
• torsion_GCMB_270L_coarse.jou (CUBIT journal)
• torsion_GCMB_450L_medium.jou (CUBIT journal)
• torsion_GCMB_270L_medium.jou (CUBIT journal)

For the commercial grade of graphite (MCCA):

• torsion_MCCA_450L_coarse.jou (CUBIT journal)
• torsion_MCCA_270L_coarse.jou (CUBIT journal)
• torsion_MCCA_450L_medium.jou (CUBIT journal)
• torsion_MCCA_270L_medium.jou (CUBIT journal)

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

Geometry

Considering the figure above, the graphite beam is examined for lengths of 270 mm and 450 mm, and has a square cross-section of 100 mm by 100 mm. The notch measures 5 mm wide and 50 mm deep machined at 45˚ to the length direction centre-line of the specimen. The specimen is restrained at each end in a two-part loading collar which features lugs which mount on hardened steel, spherical ended, load applicators positioned at each corner of the specimen. At two diagonally opposite corners the specimen is restrained on the underside of the loading lugs whilst on the remaining two corners the specimen is restrained on the upper side of the loading lugs. One of the load applicators mounted on the upper side of a loading lug is used to apply load to the specimen. The distance between load applicators across the specimen width is 250 mm, whilst the corresponding distance along the length of the specimen is 245 mm or 425 mm, depending on the beam examined.

Material properties

Two different grades of graphite are examined in this test, Gilsocarbon graphite (GCMB) and a commercially available graphite (MCCA):

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

Loads

The reference load, which is scaled, applied to one point on one lug only is equal to 1.

Finite Element Model

Two finite element models are created, one consists of a relatively coarse uniform mesh of 4-node tetrahedral elements as shown in Figure 2, and the other consists of a medium 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 at two edges of the beam 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. Top row: Geometry (left), coarse mesh (middle) and medium mesh (right) for the 450 mm long beam Geometry (left); Bottom row: Geometry (left), coarse mesh (middle) and medium mesh (right) for the 270 mm long beam.

Analysis procedure

The analysis is run in two stages. The first stage uses a script to calculate the Griffith forces for 1st (linear) to 3rd (cubic) order 4-node elements. For each order, analyses with 1 and 2 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 stage can be executed using, for example:

$ ./convergence_study.sh -f torsion_GCMB_450L_coarse.cub -l -1 -g 2.3e-5 -o 1,2,3,4 -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 2nd (quadratic) order with 1 level of automatic mesh refinement with the GCMB graphite grade, for example, the script can be executed using:

$ ./arc_length.sh -f $DIRNAME/out_spatial_2_1.h5m -a 1 -n 500 -g 2.3e-5 -t 1e-8 -i 12 -p $NPROC

$ ./face_split_arc_length.sh -f $DIRNAME/out_spatial_2_1.h5m -a 1 -n 500 -g 2.3e-5 -t 1e-8 -i 12 -p $NPROC

Results

Part 1: 450 mm Gilsocarbon Graphite Beam

Firstly, we assess the 450 mm gilsocarbon beam for both the coarse and medium meshes. Figure 3 shows the evolution of the crack for the 3rd order solution with one level of automatic mesh refinement.

Figure 3. Propagation of the crack for the 450 mm gilsocarbon graphite (GCMB) beam for 3rd order approximation with one level of automatic mesh refinement of a medium mesh using smooth crack propagation analysis (green) and crack propagation with face splitting analysis (red).

Notice the difference in crack propagation between the two different analysis approaches. Also, it is important to note that in the smooth crack propagation analysis, the results are discounted after the point when the crack front approaches the top side of the beam as MoFEM is currently unable to capture this accurately. Please refer to the Discussion section.

The script file required to extract the desired data (from log files) and plot the graphs can be downloaded from here:

• plot_graphs.sh (shell script)

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

$ ./plot_graphs.sh -o 1,2,3,4 -r 1 -ent 1,429 -dim 0,1,2 -s 1e4*1*1

for the coarse mesh, or:

$ ./plot_graphs.sh -o 1,2,3,4 -r 1 -ent 1,2156 -dim 0,1,2 -s 1e4*1*1

for the medium mesh. The entities are different because the meshes are different. -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.

The results from all analyses are illustrated in Figures 4 to 9. These show the crack mouth opening displacement (CMOD) against load, crack mouth sliding displacement (CMSD) against load, and crack area against load for each analysis. The figures below showing CMOD and CMSD compare the results from MoFEM with those from experimental data [2], Experiment 2 and Experiment 3, where the failure load was measured as 3.02 kN and 3.17 kN respectively. It can be seen that the failure load of the beam is over-estimated by approximately 20%.

Figure 4. Crack mouth opening displacement versus load for smooth crack propagation (SC) and for crack propagation with face splitting (FS) of the 450 mm gilsocarbon graphite beam using a coarse mesh; note points have been removed for steps that did not converge.

Figure 5. Crack mouth sliding displacement versus load for smooth crack propagation (SC) and for crack propagation with face splitting (FS) of the 450 mm gilsocarbon graphite beam using a coarse mesh; note points have been removed for steps that did not converge.

Figure 6. Crack area versus load for smooth crack propagation (SC) and for crack propagation with face splitting (FS) of the 450 mm gilsocarbon graphite beam using a coarse mesh; note points have been removed for steps that did not converge.

Figure 7. Crack mouth opening displacement versus load for smooth crack propagation (SC) and crack propagation with face splitting (FS) of the 450 mm gilsocarbon graphite beam using a medium mesh; note points have been removed for steps that did not converge.

Figure 8. Crack mouth sliding displacement versus load for smooth crack propagation (SC) and crack propagation with face splitting (FS) of the 450 mm gilsocarbon graphite beam using a medium mesh; note points have been removed for steps that did not converge.

Figure 9. Crack area versus load for smooth crack propagation (SC) and crack propagation with face splitting (FS) of the 450 mm gilsocarbon graphite beam using a medium mesh; note points have been removed for steps that did not converge.

Part 2: 450 mm MCCA Graphite Beam

Next, we assess the 450 mm MCCA beam for both the coarse and medium meshes. Figure 10 shows the evolution of the crack for the 3rd order solution with one level of automatic mesh refinement.

Figure 10. Propagation of the crack for the 450 mm MCCA graphite beam for 3rd order approximation with one level of automatic mesh refinement of a medium mesh using smooth crack propagation analysis (green) and crack propagation with face splitting analysis (red).

Notice the difference in crack propagation between the two different analysis approaches. Also, it is important to note that in the smooth crack propagation analysis, the results are discounted after the point when the crack front approaches the top side of the beam as MoFEM is currently unable to capture this accurately. Please refer to the Discussion section.

The results from all analyses are illustrated in Figures 11 to 16. These show the crack mouth opening displacement (CMOD) against load, crack mouth sliding displacement (CMSD) against load, and crack area against load for each analysis. The figures below showing CMOD and CMSD compare the results from MoFEM with those from experimental data [2], Experiment 1, where the failure load was measured as 2.28 kN. It can be seen that the failure load of the beam is over-estimated by approximately 20%.

Figure 11. Crack mouth opening displacement versus load for smooth crack propagation (SC) and for crack propagation with face splitting (FS) of the 450 mm MCCA graphite beam using a coarse mesh; note points have been removed for steps that did not converge.

Figure 12. Crack mouth sliding displacement versus load for smooth crack propagation (SC) and for crack propagation with face splitting (FS) of the 450 mm MCCA graphite beam using a coarse mesh; note points have been removed for steps that did not converge.

Figure 13. Crack area versus load for smooth crack propagation (SC) and for crack propagation with face splitting (FS) of the 450 mm MCCA graphite beam using a coarse mesh; note points have been removed for steps that did not converge.

Figure 14. Crack mouth opening displacement versus load for smooth crack propagation (SC) and crack propagation with face splitting (FS) of the 450 mm MCCA graphite beam using a medium mesh; note points have been removed for steps that did not converge.

Figure 15. Crack mouth sliding displacement versus load for smooth crack propagation (SC) and crack propagation with face splitting (FS) of the 450 mm MCCA graphite beam using a medium mesh; note points have been removed for steps that did not converge.

Figure 16. Crack area versus load for smooth crack propagation (SC) and crack propagation with face splitting (FS) of the 450 mm MCCA graphite beam using a medium mesh; note points have been removed for steps that did not converge.

Part 3: 270 mm Gilsocarbon Graphite Beam

Now we assess the 270 mm gilsocarbon graphite beam for both coarse and medium meshes. Figure 17 shows the evolution of the crack for the 3rd order solution with one level of automatic mesh refinement.

Figure 17. Propagation of the crack for the 270 mm gilsocarbon graphite beam for 3rd order approximation with one level of automatic mesh refinement of a medium mesh using smooth crack propagation (green) and crack propagation with face splitting analysis (red).

The results from all analyses are illustrated in Figures 18 to 23. These show the crack mouth opening displacement (CMOD) against load, crack mouth sliding displacement (CMSD) against load, and crack area against load for each analysis. The figures below showing CMOD and CMSD compare the results from MoFEM with those from experimental data [2], Experiment 6, where the failure load was measured as 3.02 kN. It can be seen that the failure load of the beam is over-estimated by approximately 20%.

Figure 18. Crack mouth opening displacement versus load for smooth crack propagation (SC) and for crack propagation with face splitting (FS) of the 270 mm gilsocarbon graphite beam using a coarse mesh; note points have been removed for steps that did not converge.

Figure 19. Crack mouth sliding displacement versus load for smooth crack propagation (SC) and for crack propagation with face splitting (FS) of the 270 mm gilsocarbon graphite beam using a coarse mesh; note points have been removed for steps that did not converge.

Figure 20. Crack area versus load for smooth crack propagation (SC) and crack propagation with face splitting (FS) of the 270 mm gilsocarbon graphite beam using a coarse mesh; note points have been removed for steps that did not converge.

Figure 21. Crack mouth opening displacement versus load for smooth crack propagation (SC) and crack propagation with face splitting (FS) of the 270 mm gilsocarbon graphite beam using a medium mesh; note points have been removed for steps that did not converge.

Figure 22. Crack mouth sliding displacement versus load for smooth crack propagation (SC) and crack propagation with face splitting (FS) of the 270 mm gilsocarbon graphite beam using a medium mesh; note points have been removed for steps that did not converge.

Figure 23. Crack area versus load for smooth crack propagation (SC) and crack propagation with face splitting (FS) of the 270 mm gilsocarbon graphite beam using a medium mesh; note points have been removed for steps that did not converge.

Part 4: 270 mm MCCA Graphite Beam

Finaly, we assess the 270 mm MCCA graphite beam for both a coarse and medium mesh. Figure 24 shows the evolution of the crack for the 3rd order solution with one level of automatic mesh refinement.

Figure 24. Propagation of the crack for the 270 mm MCCA graphite beam for 3rd order approximation with one level of automatic mesh refinement of a medium mesh using smooth crack propagation (green) and crack propagation with face splitting analysis (red).

The results from all analyses are illustrated in Figures 25 to 30. These show the crack mouth opening displacement (CMOD) against load, crack mouth sliding displacement (CMSD) against load, and crack area against load for each analysis. The figures below showing CMOD and CMSD compare the results from MoFEM with those from experimental data [2], Experiment 5, where the failure load was measured as 2.20 kN. It can be seen that the failure load of the beam is over-estimated by approximately 20%.

Figure 25. Crack mouth opening displacement versus load for smooth crack propagation (SC) and for crack propagation with face splitting (FS) of the 270 mm MCCA graphite beam using a coarse mesh; note points have been removed for steps that did not converge.

Figure 26. Crack mouth sliding displacement versus load for smooth crack propagation (SC) and for crack propagation with face splitting (FS) of the 270 mm MCCA graphite beam using a coarse mesh; note points have been removed for steps that did not converge.

Figure 27. Crack area versus load for smooth crack propagation (SC) and for crack propagation with face splitting (FS) of the 270 mm MCCA graphite beam using a coarse mesh; note points have been removed for steps that did not converge.

Figure 28. Crack mouth opening displacement versus load for smooth crack propagation (SC) and crack propagation with face splitting (FS) of the 270 mm MCCA graphite beam using a medium mesh; note points have been removed for steps that did not converge.

Figure 29. Crack mouth sliding displacement versus load for smooth crack propagation (SC) and crack propagation with face splitting (FS) of the 270 mm MCCA graphite beam using a medium mesh; note points have been removed for steps that did not converge.

Figure 30. Crack area versus load for smooth crack propagation (SC) and crack propagation with face splitting (FS) of the 270 mm MCCA graphite beam using a medium mesh; note points have been removed for steps that did not converge.

Discussion

In general:

• Linear elements are not recommended due to the occurrence of shear locking.
• The crack propagation with face splitting (FS) is a lower bound solution and under estimates the strength of both the long and short beams.
• The solutions improve as both the level of approximation and the level of automatic mesh refinement increase.

In more detail:

• For this problem relative size of microstructure is small and the predicted failure load is much closer to the actual failure load than the concrete beam assessed previously. However, despite the good qualitative predictions, the numerical analyses still over predict the experimental ultimate load by approximately 20%.
• When using smooth crack propagation, the results after the point at which the crack front approaches the top side of the beam are discounted as MoFEM is currently unable to capture this accurately. However, this capability will be added at a later date.

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
2. Graphite Core PLEX: Measurement of Dynamic Crack Front Profile, Commissioning of Torsion Test Rig, P. Hutchison AMEC, AMEC ref 103282-TR-0003 Issue 1, 17th January 2014
3. Size Effect on Structural Strength: A Review, Z.P. Bažant, Archive of Applied Mechanics 69 (1999) 703-725

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