1. 3D reference problem#
1.1. Geometry#
The structure is a cubic domain, separated in two by an oblique interface XFEM.
The length of each cube edge is: \(L\mathrm{=}4\). The cube is centered in \((0,0)\).
The interface position is: \(x+y+z+\mathit{cte}=0\)
The constant is adjusted so that the interface is flush with one row of nodes in the mesh,
for models \(A\) and \(C\), the constant is equal to: \(\mathit{cte}=0.01\),
for modeling \(B\), the constant is equal to: \(\mathit{cte}=0.1\),
for models \(D\) and \(E\), the constant is equal to: \(\mathit{cte}=0.011\)
1.2. Material properties#
Poisson’s ratio: \(\nu \mathrm{=}0.3\)
Young’s module: \(E\mathrm{=}{10}^{9}N\mathrm{/}{m}^{2}\)
1.3. Boundary conditions and loads#
On each sub-domain (on both sides of the interface XFEM), a loading in compression is imposed. In fact, the uniform pressure \(p\mathrm{=}\mathrm{-}10\mathit{MPa}\) is applied to each face of the cube and to the interface XFEM.
Then Dirichlet limit conditions are imposed, to fix the analytical solution in motion and to block the 12 rigid body movements.
Reminder :
When the XFEM interface cuts the entire structure and the two blocks are not in contact, everything happens as if there were two decoupled mechanical problems, in the presence of two solids. This therefore leads to a total of 12 rigid modes in the equivalent problem XFEM.
We therefore choose a few nodes on either side of the interface, to fix the solution on the go:
3 nodes are locked above the interface \(\text{{}{\text{NS}}_{1},{\text{NS}}_{2},{\text{NS}}_{4}\text{}}\)
\(\begin{array}{c}{\text{NS}}_{1}=\left(\text{+}\mathrm{2,}\text{+}\mathrm{2,}\text{+}2\right)\\ {\text{NS}}_{2}=\left(-\mathrm{2,}\text{+}\mathrm{2,}\text{+}2\right)\\ {\text{NS}}_{4}=\left(-\mathrm{2,}-\mathrm{2,}\text{+}2\right)\end{array}\)
3 nodes are locked below the interface \(\text{{}{\text{NI}}_{1},{\text{NI}}_{2},{\text{NI}}_{4}\text{}}\)
\(\begin{array}{c}{\text{NI}}_{1}\mathrm{=}(\mathrm{-}\mathrm{2,}\mathrm{-}\mathrm{2,}\mathrm{-}2)\\ {\text{NI}}_{2}\mathrm{=}(+\mathrm{2,}\mathrm{-}\mathrm{2,}\mathrm{-}2)\\ {\text{NI}}_{4}\mathrm{=}(+\mathrm{2,}+\mathrm{2,}\mathrm{-}2)\end{array}\)
Knowing that, the analytical solution on the go, is valid at every point in the field:
\({U}_{x}(x,y,z)\mathrm{=}\mathrm{\{}\begin{array}{c}k\mathrm{\times }x+2\text{si}x+y+z+\mathit{cte}>0\\ k\mathrm{\times }x\mathrm{-}2\text{si}x+y+z+\mathit{cte}\mathrm{\le }0\end{array}\)
\({U}_{y}(x,y,z)\mathrm{=}\mathrm{\{}\begin{array}{c}k\mathrm{\times }y+2\text{si}x+y+z+\mathit{cte}>0\\ k\mathrm{\times }y\mathrm{-}2\text{si}x+y+z+\mathit{cte}\mathrm{\le }0\end{array}\)
\({U}_{z}(x,y,z)\mathrm{=}\mathrm{\{}\begin{array}{c}k\mathrm{\times }z+2\text{si}x+y+z+\mathit{cte}>0\\ k\mathrm{\times }z\mathrm{-}2\text{si}x+y+z+\mathit{cte}\mathrm{\le }0\end{array}\)
where, \(k\mathrm{=}\mathrm{-}\frac{p(1\mathrm{-}2\nu )}{E}\mathrm{=}{4.10}^{\mathrm{-}3}\).
1.4. Bibliographical references#
Bechet, H. Minnebo, N. Moes, B. Burgardt, B. Burgardt, Improved implementation and robustness study of the x-fem method for stress analysis around cracks, International Journal for Numerical Methods in Engineering, 64:1033—1056, 2005.
Siavelis, M.L.E. Guiton, P. Massin, P. Massin, N. Moës, Large sliding contact along branched discontinuities with X- FEM, Computational Mechanics, 52-1, 52-1:201—219, 2013.
Extended Finite Element Method, Code_Aster documentation, R7.02.12