1. Reference problem#

1.1. Geometry#

Structure \(\mathrm{2D}\) is a unitary square plate (\(\mathit{LX}\mathrm{=}\mathrm{0,1}\), \(\mathit{LY}\mathrm{=}\mathrm{0,1}\)). A « straight » interface divides the domain into 2 sub-domains.

The interface branches off in the vicinity of the center of square \((\mathrm{0,}0.5)\) with an angle \(\theta \mathrm{=}\mathrm{67,5}°\) from the horizontal.

_images/10000201000000EE000000E060973FFD7F104D23.png

Figure 1.1-1 : Geometry of the cracked plate

The interface is defined analytically by the union of 2 line segments, whose respective equations are:

  • \(X=0\) for the vertical segment.

  • \(Y\mathrm{=}2x+0.55\) for the oblique segment.

The intersection point of the 2 line segments is calculated to be located on the edge where the interface forks:

  • \((\mathrm{0,}0.55)\) for \(A\) and \(B\) models, see

  • \((\mathrm{0,}0.6)\) for \(C\) and \(D\) models, see

_images/1000000000000360000002098B3E85B805699CF1.png

Figure 1.1-2: « zoom » in the vicinity of the bifurcation point

_images/10000201000003600000020958070A9972EF1609.png

Figure 1.1-3: « zoom » in the vicinity of the bifurcation point

1.2. Material properties#

Young’s module: \(E=210{10}^{9}\mathrm{Pa}\)

Poisson’s ratio: \(\nu =0\)

1.3. Boundary conditions and loads#

Modeling \(A\):

The interface divides the domain into 2 sub-domains. The solution is imposed on the move on each sub-domain, thanks to Dirichlet conditions at the edges:

  • on edge \(\mathit{LIG2}\): \(\mathit{DX}=-1\) and \(\mathit{DY}=0\)

  • on edge \(\mathit{LIG4}\): \(\mathit{DX}=\text{+}1\) and \(\mathit{DY}=0\)

There is no contact at the interface level. We are just interested in the movements of the 2 blocks, which can interpenetrate each other.

The imposed Dirichlet conditions also block rigid body movements in the 2 sub-domains.

Models \(B\), \(C\) and \(D\):

Same load as in modeling \(A\).

1.4. Benchmark solution#

Modeling \(A\):

By construction, the field of movement is uniform on each sub-block.

  • On the sub-domain on the right, we have: \(\mathit{DX}=-1\) and \(\mathit{DY}=0\)

  • On the sub-domain on the left, we have: \(\mathit{DX}=\text{+}1\) and \(\mathit{DY}=0\)

Modeling \(B\):

Same analytical solution as modeling \(A\)

Modeling \(C\):

Same analytical solution as modeling \(A\)

Modeling \(D\):

Same analytical solution as modeling \(A\)

1.5. Bibliographical references#

  1. GENIAUT S., MASSIN P.: eXtended Finite Element Method, Code_Aster Reference Manual, [R7.02.12]