4. Implementation: with contact on an interface#

This part explains step by step the different steps of a calculation with X- FEM where the contact is defined on the lips of the crack (here, an interface). It is recommended that you have read the previous chapter. We can refer to test cases SSNV182, SSNV186,, SSNV201,, SSNV209, SSNP503 or SSNP504.

For this formulation, the creation of the healthy model is done exactly as in the non-contact case.

4.1. Interface definition#

An interface is defined by a single level set: the normal level-set. The type of discontinuity must be specified in DEFI_FISS_XFEM: TYPE_DISCONTINUITE =” INTERFACE “.

Unlike the case of a real crack, it is not necessary to define the true distance function. Any level-set is sufficient, as long as the iso-zero of the level-set coincides with the interface. For example, the real distance function signed to the circle with center \(C\) and radius \(R\) is:

\(\mathrm{LSN}=\sqrt{({(X-{X}_{C})}^{2}+{(Y-{Y}_{C})}^{2})}-R\)

but a level-set function whose iso-zero coincides with the circle is for example:

\(\mathrm{LSN}={(X-{X}_{C})}^{2}+{(Y-{Y}_{C})}^{2}-{R}^{2}\)

4.2. Creating the rich model#

The particularity of the case where the contact is defined on the interface or crack X- FEM is that it must be specified when creating the enriched model, by the keyword CONTACT =” STANDARD “of the command MODI_MODELE_XFEM. In the context of cohesive approaches, however, the keyword CONTACT =” MORTAR “will be used. The restriction on the positioning of multiple cracks is the same as explained in § 3.3.

4.3. Bi-material#

Defining a different material on either side of the interface is not easy because normally, a material is defined by groups of meshes, so on a mesh, we necessarily have the same material. However, in the case of an interface crossing a mesh, it may be desirable to have different materials on both sides of the interface. However, this is possible if:

  • the normal level-set is an explicit function of space,

  • the law of behavior is the same across the board.

In this case, we can define the material characteristics (like \(E\) and \(\nu\)) according to a control variable \(\mathit{VC}\), which is in fact the normal level-set \(\mathit{LSN}\). We therefore proceed as follows:

\((X,Y)\to \mathrm{LSN}(X,Y)\to \mathrm{VC}=\mathrm{LSN}(X,Y)\to (E,\nu )=f(\mathrm{VC})\)

We use a control variable because the material cannot depend directly on the coordinates of the space. Function \(f\) is a function of the type:

\(\mathrm{if}\mathit{VC}>0\mathrm{then}(E,\nu )\mathrm{=}({E}^{\mathrm{1,}}{\nu }^{1})\mathrm{else}(E,\nu )\mathrm{=}({E}^{\mathrm{2,}}{\nu }^{2})\)

where \(({E}^{\mathrm{1,}}{\nu }^{1})\) and \(({E}^{\mathrm{2,}}{\nu }^{2})\) are the properties of the two materials in question.

4.4. Contact load#

When creating the contact load, you must specify METHODE = “XFEM”, and give the name of the crack under the keyword FISS_MAIT. The contact method activated is the continuous method.

It may be useful for the convergence of the algorithm to specify the integration diagram by INTEGRATION =” NOEUD “or INTEGRATION =” GAUSS” or INTEGRATION =” SIMPSON “.

4.5. Visualization post-processing#

Everything explained in § 3.5 remains valid, but the healthy mesh to fill in is the initial linear mesh (in POST_MAIL_XFEM and POST_CHAM_XFEM). In addition, the visualization model is a copy of the healthy model, but with the classic “3D”, “CPLAN”, “DPLAN” and “AXIS” models.

4.6. Post-treatment in fracture mechanics#

It is entirely possible to calculate \(G\) and \(K\) on a fully or partially closed crack. The calculation is done with the operators POST_K1_K2_K3 or CALC_G. For operator POST_K1_K2_K3, the mesh to be filled in is the initial linear mesh.

If the crack bottom is closed, the theoretical value of \(G\) or \(K\) is 0, but it cannot be approached with the command CALC_Gqu “with very small meshes.

As for a non-meshed crack, it is possible to perform G calculations in the presence of an initial stress field.

4.7. Contact post-treatment#

As with a classic contact calculation, field CONT_NOEU, which gives information on the values of certain quantities related to the contact, is calculated and can be post-processed. However, if the contact terms are integrated numerically by a Gauss method, the quantities defined at the Gauss points of the contact facets cannot be displayed.