###Finite plate with horizontal crack under uniaxial stress###

Keywords: Fracture, Stress Intensity Factor, High-Order Approximation, Automatic Mesh Refinement, Benchmark

Problem description

A finite plate with a horizontal through-thickness crack is subjected to uniaxial stress as indicated in Figure 1 below. We want to calculate the Mode I stress intensity factor () and compare the results to the analytical solution given by Rooke, D.P. and Cartwright, D.J. [2].

Figure 1. Finite plate with horizontal crack

Equation 1. Analytical solution from Rooke and Cartwright [2].

where σ is the applied stress (see Loads section below).

Files

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

• horizontal_crack.jou (CUBIT journal)

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

Geometry

Considering the figure above:

a = 1, b = 2.5, h = 5.

We assume that the plate thickness is equal to 1.

Material properties

Young's modulus E = 1000

Poisson's ratio v = 0.3

Loads

The stress (σ) at the top and bottom of the plate is equal to 1.

Finite Element Model

The finite element 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 loads shown above, we apply displacement constraints at three vertices of the plate to prevent rigid body motions (see horizontal_crack.jou).

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 automatically using a script which specifies analyses for 1st (linear) to 5th (high-order) 4-node elements. For each order, analyses with 0 (no refinement) to 4 levels of automatic refinement are also run. All analyses run on 4 processors with the FGMRES (Flexible Generalized Minimal Residual Method) Krylov solver for systems of linear equations and the LU preconditioner.

The analysis can be executed using:

./convergence_study.sh -f horizontal_crack.cub -l 1 -o 1,2,3,4,5 -r 0,1,2,3,4 -p 4

where is the stress intensity factor and G is the Griffith force.

Results

Figure 3 shows the deformed shape of the plate for the 5th order solution. The results from all analyses are illustrated in Figure 5. This shows the error percentage of against the number of degrees of freedom for each analysis. The error is defined as:

where is the reference solution from the analytical formula, which is calculated to be 1.9499.

Figure 3. Deformed shape and displacement contours (left); Deformed shape, automatically refined mesh around crack and vectors of material resistance force (right)

Figure 4. Levels of mesh refinement for fifth order approximation

The script file and gnuplot plot file required to extract the desired data (from the log_griffith_forces file) and plot the graph can be downloaded from here:

• create_graph_horiontal_crack.sh (shell script)
• plot_file_horizontal_crack.in (gnuplot script)

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

./create_graph_horizontal_crack.sh >/dev/null -o 1,2,3,4,5

Figure 5. KI Error (%) versus no. of DOF (log10)

It is evident from Figure 5 that, for the same coarse mesh and number of nodes, the solution improves from -22.65% error for 1st-order elements to -1.12% for 5th-order elements. The 5th-order solution improves further to -0.54% error with four levels of automatic local mesh refinement. It is worthwhile to note the well known pathological nature of the 1st order solution with mesh refinement, which does not achieve convergence due to shear locking.

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. Rooke, D.P. and Cartwright, D.J. (1976), Compendium of stress intensity factors, HMSO Ministry of Defence, Procurement Executive

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