Reference problem
=====================


Geometry and loading
-----------------------

Consider a hollow cylinder with a length :math:`L`, an inner radius :math:`{R}_{f}` and an outer radius :math:`R`. Let's say a rigid frame with a circular cross section of radius :math:`{R}_{f}` embedded in its center. The inner and outer surfaces of the hollow cylinder are denoted :math:`{\Gamma }_{i}` and :math:`{\Gamma }_{e}` (see [:ref:`Figure 1.1-a <Figure 1.1-a>`]). The loading consists in applying, on the rigid frame, a displacement :math:`{U}^{i}{e}_{z}` (:math:`{U}^{i}>0`) as well as a zero displacement on the outer edge :math:`{\Gamma }_{e}`.


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Figure 1.1-a: Domain diagram and load

We hypothesize an axisymmetric solution which allows us to restrict our study to a 2D rectangular domain :math:`\Omega`. The dimensions of the domain are as follows:

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. The load on the rigid frame will be taken into account by applying the imposed displacement

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all over the side

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of the domain :math:`\mathrm{2D}` as well as zero displacement to the side

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to take into account the embedment of the cylinder. Finally, a zero radial displacement is imposed on the lower and upper faces of the domain in order to avoid a singularity linked to a change in condition at the limits at the points.

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and :math:`A\text{'}` (see [:ref:`Figure 1.1-a <Figure 1.1-a>`]). These boundary conditions will lead to an anti-plane solution (independent of

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) which makes it easier to obtain an analytical solution.