1. Reference problem#

1.1. Geometry#

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The sphere has an inner radius \({R}_{i}=1m\) and an outer radius \({R}_{e}=10m\).

A circular XFEM interface with radius \({R}_{f}=\mathrm{5,5}m\) is introduced.

1.2. Material properties#

Material characteristics are \(E=1E+5\mathit{Pa}\) and \(\nu =0.3\). The law of behavior is elastic linear.

1.3. Boundary conditions and loading#

The following conditions are imposed:

  • An imposed pressure of \(300\mathit{Pa}\) on the inner face AB of radius \({R}_{i}\) of the sphere \(\sigma \mathrm{.}{e}_{r}(r={R}_{i})={P}_{i}{e}_{r}\);

  • An imposed pressure of \(150\mathit{Pa}\) on the external face CD with radius \({R}_{e}\) of the sphere \(\sigma \mathrm{.}{e}_{r}(r={R}_{e})=-{P}_{e}{e}_{r}\);

  • Symmetry conditions: \(\mathit{DY}=0\) on BD, \(\mathit{DX}=0\) on AC.

The X- FEM interface is subject to contact friction so that the solution of the problem with interface is the solution of the continuous problem (i.e. the hollow sphere without an interface). The reference solution calculated in the following paragraph is the solution of the problem of a hollow sphere subjected to internal pressure and external pressure. Here, frictionless contact is applied.