1. Reference problem#
1.1. Geometry of the problem#
It is a square with side \(L=10m\). This bar has an interface-type discontinuity (a non-meshed interface that is introduced into the model via level-sets using the DEFI_FISS_XFEM operator). The square is thus entirely crossed by the discontinuity (in terms of approximating the displacement field, only Heaviside enrichment is taken into account). The discontinuity is circular with center \(O(\mathrm{0,}-2)\) and radius \(R=9m\).
The geometry of the problem is represented in the figure.
Figure 1.1-a : 2D Problem Geometry
1.2. Material properties#
The parameters given in the Table correspond to the parameters used for the 4 models. The behavior is elastic (“ELAS”).
Elastic parameters |
Young’s modulus \(E(\mathit{en}\mathit{MPa})\) Poisson’s Ratio \(\nu\) Thermal expansion coefficient \(\alpha (\mathit{en}{K}^{\text{-1}})\) |
\(0\) \(0\) |
Table 1.2-1 : Material Properties
1.3. Boundary conditions and loading#
The following Dirichlet conditions apply:
on the bottom side of the square, movements are blocked in all directions (\({u}_{\text{x}}=0\) and \({u}_{\text{y}}=0\)),
on the upper side of the square, the movements following \(x\) are blocked \({u}_{\text{x}}=0\) and the next square \(y\), \({u}_{\text{y}}={u}_{\text{y, impo}}=-{1.E}^{-6}\) is imposed.
« Contact pressure and shear » are applied to the crack. That is to say, we apply the forces that would be there at the crack if the square was not cracked or if there was perfect adherent contact. The Neuman conditions are thus as follows:
On each of the lips of the interface, a distributed pressure \(p(\theta )={\sigma }_{\text{yy}}\ast \mathit{cos²}(\theta )\) is imposed using AFFE_CHAR_MECA and the keyword FISSUREdePRES_REP.
On each of the lips of the interface, a distributed shear \(t(\theta )=-{\sigma }_{\text{yy}}\ast \mathrm{cos}(\theta )\mathrm{sin}(\theta )\) is imposed by means of AFFE_CHAR_MECA and the keyword FISSUREdeCISA_2D.