###Plate with semi-elliptical surface crack under uniaxial stress###

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

Problem description

A plate with a horizontal semi-elliptical surface crack is subjected to uniaxial stress as indicated in Figure 1 below. We want to calculate the Mode I stress intensity factor (KI) and compare the results to the reference solution given by Raju & Newman [1].

Figure 1 Plate with semi-elliptical surface crack

Files

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

• semi-elliptic_crack.jou (CUBIT journal)

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

Geometry

Considering the figure above, the dimensions of the plate relative to the plate thickness, t, are:

H/t = 16, W/t = 8, a/t = 0.6, c/t = 2.5, and a/c = 0.24

where H is the half-height, W is the half-width, a is the mid-plane crack depth and c is the surface crack half-length.

Therefore, if t = 1.6667 then H = 26.667, W = 13.333, a = 1, c = 4.167.

Material properties

Young's modulus E = 3E+7

Poisson's ratio v = 0.3

Loads

The stress applied at the top and bottom of the plate is equal to 1.

Finite Element Model

The finite element model consists of a symmetry model of 6,127 of 4-node tetrahedral elements as shown in Figure 2. The mesh density is high at the crack front in order to capture adequately the semi-elliptical shape of the crack in the mesh. The mesh becomes relatively coarser away from the crack and towards the loading boundaries.

In addition to the loads shown above and displacement supports perpendicular to the symmetry plane, we apply displacement restraints at three vertices of the plate to prevent rigid body motions (see semi-elliptical_crack.jou).

We use automatically different orders of approximation to improve the solution. We do not use automatic mesh refinement for this problem since the mesh is already fine at the crack front.

The crack is defined geometrically in CUBIT by a single surface. It is worthwhile to note that the visible spherical 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 (see the User Manual).

Figure 2. Geometry (left) and mesh (right)

Analysis procedure

The analysis can be executed using:

./convergence_study.sh -l 1e-7 -o 1,2,3,4,5 -r 0 -p 4

where is the stress intensity factor and G is the Griffith force. The stress intensity factor is calculated along the crack, with the current position along the crack curve calculated in degrees using:

where x is the horizontal coordinate of the current point and x0 is the horizontal coordinate of the point furthest along the curve (-6.667).

Results

Figure 3 X

References

1. Raju & Newman (1979), Stress-Intensity Factors for a Wide Range of Semi-Elliptical Surface Cracks in Finite-Thickness Plates, Eng. Fract. Mech.
2. Abaqus Benchmarks Guide, Fracture Mechanics, Contour integral evaluation: three-dimensional case, https://www.sharcnet.ca/Software/Abaqus/6.13.3/books/bmk/default.htm?startat=ch01s16ach122.html