22. T modeling

Validation of the implementation of law CZM_LIN_MIX in formulation X- FEM. We are testing the opening mode. This modeling is an adaptation to X- FEM of the \(G\) modeling.

22.1. Characteristics of modeling

The line of discontinuity is modelled by an X- FEM interface, which is introduced into the model by the DEFI_FISS_XFEM operator, with TYPE_DISCONTINUITE =” INTERFACE “. This line crosses the block from one end to the other. It is located at a distance of \(\mathrm{0,4}\mathit{mm}\) from the left edge and crosses the elements (see fig.2). The interface is said to be non-compliant.

The contact elements are introduced by the discretization CONTACT =” MORTAR “in the operator MODI_MODELE_XFEM.

The cohesive law is then defined in the operator DEFI_CONTACT, by the keywords ALGO_CONT =” CZM “and RELATION =” CZM_LIN_MIX”.

The surface elements are of type C_ PLAN.

22.2. Characteristics of the mesh

The square is discretized at the rate of \(4\) elements per side. Therefore:

Number of elements, type HEXA4: 16

Number of knots: 65.

22.3. Tested sizes and results

The remark on control mentioned for modeling A also applies to this modeling. Strictly speaking, no longer having any interface elements in the model, we replace the tests on SIGN and SIGTX by tests on the contact multipliers X- FEM LAGS_C and LAGS_F1 respectively.

Fashion \(I\)

Size tested	No time	Reference	Tolerance ( \(\text{\%}\) )
DX on node 2	4	1.71003596	0.10
ETA_PILO	4	7.898168291D-01	0.10
SIXXsur mesh 32	2	6.599934D-01	0.10
LAGS_Csur node 9	9	1.099989D-01	0.10
LAGS_F1sur node 9	8	0.D+00	0.10

These values are obtained according to the explanations in part 2.2.

The move jump at time \(t\) is: \(⟦u⟧=\frac{t}{\text{COEF\_MULT}}\frac{2{G}_{c}}{{\sigma }_{c}}\). So, \(⟦u⟧=0.163636t\) (in \(\mathit{mm}\)).

We can then obtain the displacement standard applied by \(U=⟦u⟧+L\frac{{\sigma }_{c}}{E}\left(1-⟦u⟧\frac{{\sigma }_{c}}{2{G}_{c}}\right)\). For \(t=4\), we get \(U=1.97\mathit{mm}\), from where \(\text{DX}=U\mathrm{cos}(\frac{\pi }{6})=1.71\mathit{mm}\) and \(\text{ETA\_PILO}=\frac{U}{{U}_{0}}=0.78\mathit{mm}\)

Starting with \(⟦u⟧\), we get the constraint by \(\sigma ={\sigma }_{c}\left(1-⟦u⟧\frac{{\sigma }_{c}}{2{G}_{c}}\right)\). For \(t=9\), \(\sigma =0.11\mathit{Mpa}\), which, considering the loading in pure I mode, can be tested by the lagrange multiplier LAGS_C. For \(t=2\), we find \(\sigma =0.88\mathit{Mpa}\), and we have \({\sigma }_{\mathit{xx}}=\sigma {\mathrm{cos}}^{2}(\frac{\pi }{6})=0.66\mathit{MPa}\), which is the tested value.