1. Reference problem

1.1. Geometry

The crack is circular (penny shaped crack) with radius \(a\), in the \(\mathrm{Oxy}\) plane. For the medium to be considered infinite, the characteristic quantities of the massif are of the order of 5 times greater than radius \(a\).

1.2. Material properties

Young’s module: \(E={2.10}^{5}\mathrm{MPa}\)

Poisson’s ratio: \(\nu =0.3\)

Density: \(\rho =7850\mathrm{kg}/\mathrm{m³}\)

1.3. Boundary conditions and loads

Underside	: uniform tensile stress \({\sigma }_{z}\mathrm{=}1.\mathit{MPa}\)
Upper face	: uniform tensile stress \({\sigma }_{z}=1.\mathrm{MPa}\)

According to the modeling, we also have conditions at the limits of symmetry and the blocking of rigid body movements.

In modeling D, where only a quarter of the parallelepiped is represented, antisymmetry boundary conditions are used for torsional loading: they amount to imposing zero tangential displacements on one face. Torsional loading is introduced in the form of a tangential surface force (distributed shear) applied to the lips of the crack.

• Upper lip: \({F}_{X}\mathrm{=}\mathrm{-}\tau \frac{Y}{a}\) and \({F}_{Y}\mathrm{=}+\tau \frac{X}{a}\)

• Lower lip: \({F}_{X}\mathrm{=}+\tau \frac{Y}{a}\) and \({F}_{Y}\mathrm{=}\mathrm{-}\tau \frac{X}{a}\)