1. Reference problem#

1.1. Geometry#

We consider a cube with side \(\mathrm{2L}\) and a lens-shaped crack with radius \(R\) such as \(\frac{L}{R}\mathrm{=}5\) and central angle \(\alpha \mathrm{=}\frac{\pi }{4}\) (see).

The equation characteristic of the shape of the crack surface is:

\({x}^{2}+{y}^{2}+{(z\mathrm{-}R)}^{2}\mathrm{=}{R}^{2}\) with \(0\mathrm{\le }z\mathrm{\le }(1\mathrm{-}\mathrm{cos}\alpha )R\).

We pose \(a\mathrm{=}R\mathrm{sin}\alpha\).

1.2. The characteristic equation of the crack background is:#

\({x}^{2}+{y}^{2}\mathrm{=}{a}^{2}\) with \(z\mathrm{=}(1\mathrm{-}\mathrm{cos}\alpha )R\)

_images/1000000000000445000002EBAB849099C5E163DF.png

Figure 1: geometry of the cracked cube

1.3. Material properties#

The material is elastic isotropic in properties:

\(E\mathrm{=}210000\mathit{MPa}\)

\(\mathrm{\nu }=\mathrm{0,22}\)

1.4. Boundary conditions and loads#

The cube is subjected to hydrostatic tension \(\sigma\).