1. Reference problem#

1.1. Geometry#

The geometry, dimensions and materials are taken to be identical to those of Bittencourt et al. [1] and Ventura et al. [2].

Structure \(\mathrm{2D}\) is a rectangular plate (\(20\mathrm{mm}\times 8\mathrm{mm}\)) with 3 holes, including a through crack (). The length of the initial crack is \(a=\mathrm{1,5}\mathrm{mm}\).

The nodes marked \(\mathrm{P1}\), \(\mathrm{P2}\), and \(\mathrm{P3}\) on the serve to impose boundary conditions, which are explained in paragraph [§ 1.3].

_images/100000000000037B000001D1FEB09A073CEAB243.jpg

Figure 1.1-1: geometry of the cracked plate

1.2. Material properties#

Young’s module: \(E=205000\mathrm{MPa}\)

Poisson’s ratio: \(\nu \mathrm{=}\mathrm{0,3}\)

1.3. Boundary conditions and loads#

In order to block rigid modes, we block the movements of nodes \(\mathrm{P1}\) and \(\mathrm{P2}\) as follows:

      • \({\mathrm{DY}}^{\mathrm{P1}}={\mathrm{DY}}^{\mathrm{P2}}=0\);

      • \({\mathrm{DX}}^{\mathrm{P1}}=0.\)

In order to simulate fatigue propagation, a unit nodal force is applied in \(\mathrm{P3}\): \(\mathit{FY}\mathrm{=}\mathrm{-}1\). A load cycle will correspond to: zero load → max load → zero load. 35 propagation steps are simulated. At each propagation step, the crack advances by an imposed length equal to 0.1 m.

1.4. Benchmark solution#

Given the lack of precision of the diagrams in article [1], it is not possible to derive/to establish precise numerical values. All we have to do is check that the crack paths have the same appearance (see § 2.3).

For the test, the values of the stress intensity factors \({K}_{I}\) and \({K}_{\mathrm{II}}\) calculated by modeling A at the end of propagation are used as a reference:

\(\begin{array}{c}{K}_{I}^{\mathit{ref}}\mathrm{=}\mathrm{1,142045}{\mathit{MPa.mm}}^{\mathrm{0,5}}\\ {K}_{\mathit{II}}^{\mathit{ref}}\mathrm{=}\mathrm{-}\mathrm{0,057097}{\mathit{MPa.mm}}^{\mathrm{0,5}}\end{array}\)

1.5. Bibliographical references#

  1. T.N. Bittencourt, P.A. Wawrzynek, A.R. Ingraffea, A.R. Ingraffea, J.L. Sousa, Quasi-automatic simulation of crack propagation for 2D LEFM problems, Engineering Fracture Mechanics, vol. 55, vol. 55, pp. 321—334, 1996

    1. Ventura, J.X. Xu, T. Belytschko, T. Belytschko, A vector level set method and new discontinuity approximations for crack growth by EFG, International Journal for Numerical Methods in Engineering, vol. 54, vol. 54, pp. 923—944, 2002